Research ArticleCourse Control of Underactuated Ship Based on NonlinearRobust Neural Network Backstepping Method

Junjia Yuan Hao Meng Qidan Zhu and Jiajia Zhou

College of Automation Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Hao Meng menghaohrbeueducn

Received 25 August 2015 Revised 25 November 2015 Accepted 31 January 2016

Academic Editor Chaomin Luo

Copyright copy 2016 Junjia Yuan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of course control for underactuated surface ship is addressed in this paper Firstly neural networks are adopted todetermine the parameters of the unknown part of ideal virtual backstepping control even the weight values of neural networkare updated by adaptive technique Then uniform stability for the convergence of course tracking errors has been proven throughLyapunov stability theory Finally simulation experiments are carried out to illustrate the effectiveness of proposed control method

1 Introduction

Tracking control performance for surface vessel along thepredefined route has been an essential control problemfor marine autopilot system design and it has receivedconsiderable attractions from control community In 1922proportional-integral-derivative (PID) autopilot for shipsteering was presented by Nicholas Minosky [1] PID con-troller greatly improved the performance of autopilots Untilthe 1980s almost all makes of autopilots were based on thesecontrollers One challenge for tracking control of surfacevessel based on above method is that the systems are oftenunderactuated by the swaymotion due to weight complexityand efficiency considerations and exhibit nonholonomicconstraints which meets Brocketrsquos theorem that there is nocontinuous or even smooth time-invariant state feedbacklaw that can stabilize the system to the origin [2] Anotherchallenge is that the vessel model itself exhibits severenonlinear characteristic and model uncertainties induced bythe ocean environment [3 4]

For the ship with nonlinear maneuvering characteristicsand without uncertainties a state feedback linearizationcontrol law was designed [5] while feedback linearizationwith saturation and slew rate limiting actuators was dis-cussed [6] Later combined with a genetic algorithm thebackstepping method was employed to develop a nonlinearship course controller by Witkowska and Smierzchalski [7]

where the ship course parameters were automatically tunedto the optimal values with the aid of a genetic algorithmEven considering the ship steering model with both con-stant parametric uncertainties and input disturbance withunknown bound a robust adaptive nonlinear control lawwas presented based on projection approach and Lyapunovstability theory [8] Recently many papers have tackled theseproblems based on Lyapunov theory [9ndash12] In [13ndash15] aglobal tracking controller for underactuated ship is addressedwith nonzero off-diagonal terms the reference trajectoryis generated by using a virtual target guidance algorithmand the controller designed is facilitated by an introductionof changing the ship outputs several coordinate trans-formations and backstepping method And the controllerdesign is heavily depending on accurate dynamic model therobustness against disturbance has not been addressed Amethod using backstepping adaptive dynamical slidingmodecontrol is presented for path following control of USV in[16] the control system takes account of the modeling errorsand disturbances and simplified tracking error dynamicsare obtained by assuming that the sway velocity is smallwhich can be neglected in the controller design and only forstraight line path tracking can be achieved The LOS basedguidance law is also used in the controller design whichcauses the complexity of computing high-order derivative ofvirtual control In [17] a transformation of vessel kinematicsto the Serret-Frenet frame is introduced by exploring an extra

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2016 Article ID 3013280 11 pageshttpdxdoiorg10115520163013280

2 Computational Intelligence and Neuroscience

degree of freedom by controlling explicitly the progressionrate of the virtual target along the path and overcomes themajor singular problem approach angle is introduced forcontroller design via backstepping method Neural networksare introduced to enhance system stability and transientperformance which can handle the known dynamics anduncertainties of systems well [18ndash20] Particularly in [12]a single hidden layer neural network (SHLNN) is adoptedto obtain the adaptive signal online but the choice of thesingle hidden layer neural network is limited by the numberof hidden layer node selections that will affect the onlinelearning speed and accuracy and cannot produce a betterestimation effect on the fast changing disturbances

Therefore a solution to the course control of underac-tuated surface vessel is addressed in this paper In view ofthe characteristics of the underactuated performance thebackstepping control method is used to deal with aboveproblem The direct adaptive neural network is adopted todesign control law by using the RBF neural network toovercome the problem that the ideal virtual control cannot beused directly in practice The weights of the neural networkare updated by adaptive technique to guarantee the stabilityof the closed-loop system through Lyapunov stability theorySimulation results are illustrated to verify the performance ofthe proposed adaptive neural network controller with goodprecision

2 Adaptive Robust Neural NetworkController Design

21 Problem Description Consider the following nonlinearsystems

119894= 119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894 1 le 119894 le 119899 minus 1

119899= 119891

119899(119909

119899) + 119892

119899(

119899) 119906 + 119889

119899 119899 ge 2

119910 = 119909

1

(1)

where 119909119894= [119909

1 119909

2 119909

119894] is system state 119906 is control input

and 119910 is system output The control objective is to design anadaptive neural network controller and make 119910 track 119910

119889 119910119889

meets the smooth bounded reference model as follows

119889119894= 119891

119889119894(119909

119889) 1 le 119894 le 119898

119910

119889= 119909

1198891 119898 ge 119899

(2)

where 119909119889= [119909

1198891 119909

1198892 119909

119889119898]

119879isin 119877

119898 is state constant 119910119889isin

119877 represents system output and119891119889119894(sdot) 119894 = 1 2 119898 denote

nonlinear function assuming that the reference model foreach state is bounded as 119909

119889isin Ω

119889 forall119905 ge 0

Assumption 1 There is an unknown constant 119901lowast119894to meet

forall(119909

119899 119905) isin 119877

119899times119877

+ |119889119894(119909

119899 119905)| le 119901

lowast

119894120588

119894(119909

119894) and120588

119894(119909

119894) is a known

positive smooth function

22 Direct Adaptive Neural Network Controller Design Inview of the problems and solutions described in the lastsection the direct adaptive neural network controller for

nonlinear systems with RBF neural network is chosenDetailed design steps will be described in the following

Step 1 Let 1199111= 119909

1minus 119909

1198891 1199112= 119909

2minus 120572

1 and then

1= 119891

1(119909

1) + 119892

1(119909

1) 119909

2+ 119889

1minus

1198891 (3)

Consider the following Lyapunov function

119881

1=

1

2119892

1(119909

1)

119911

2

1+

1

2

119882

119879

1Γ

minus1

1

119882

1 (4)

where 1198821=

119882

1minus119882

lowast

1119882lowast1represents the ideal weight vector

of neural network 1198821represents the estimated value of the

neural network weight vector 1198821represents the estimation

error of weight vector Γ1= Γ

119879

1gt 0 is the adaptive gainmatrix

and the derivation of 1198811can be computed as

119881

1=

119911

1

1

119892

1(119909

1)

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

=

119911

1

119892

1(119909

1)

(119891

1(119909

1) + 119892

1(119909

1) 119909

2+ 119889

1minus

1198891)

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

= 119911

1(119911

2+ 120572

1+

119891

1(119909

1) minus

1198891

119892

1(119909

1)

) +

119911

1119889

1

119892

1(119909

1)

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

(5)

According to Assumption 1 we can get

119881

1le 119911

1(119911

2+ 120572

1+

119891

1(119909

1) minus

1198891

119892

1(119909

1)

) +

119911

2

1120588

2

1

2119892

2

1(119909

1)

+

119875

lowast2

1

2

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

= 119911

1(119911

2+ 120572

1+

119891

1(119909

1) minus

1198891

119892

1(119909

1)

+

119911

1120588

2

1

2119892

2

1(119909

1)

) +

119875

lowast2

1

2

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

(6)

There is an ideal virtual feedback control law

120572

lowast

1= minus119888

1119911

1minus [

119891

1(119909

1) minus

1198891

119892

1(119909

1)

+

119911

1120588

2

1

2119892

2

1(119909

1)

] (7)

where 1198881gt 0 is designed controller parameter

Because of the unknown smooth functions 1198911(119909

1) and

119892

1(119909

1) we cannot actually get the ideal feedback control law

120572

lowast

1 from (7) we can see that the unknown part (119891

1(119909

1) minus

1198891)119892

1(119909

1) is smooth function of 119909

1and

1198891 so that

ℎ

1(119885

1) ≜

119891

1(119909

1) minus

1198891

119892

1(119909

1)

+

119911

1120588

2

1

2119892

2

1(119909

1)

119885

1≜ [119909

1

1198891]

119879sub 119877

2

(8)

Computational Intelligence and Neuroscience 3

RBF neural network1198821198791119878

1(119885

1) is used to approximate the

unknown function ℎ1(119885

1) and 120572lowast

1can be expressed as

120572

lowast

1= minus119888

1119911

1minus119882

lowast119879

1119878

1(119885

1) minus 119890

1 (9)

where |1198901| le 119890

lowast

1is estimated error and meets 119890lowast

1gt 0

Because 119882

lowast

1is unknown the virtual control law is

selected as follows

120572

1= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1)

(10)

and then

119881

1le 119911

1119911

2minus 119888

1119911

2

1+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+ 119911

1119890

1+

119875

lowast2

1

2

minus

119882

119879

1119878

1119911

1+

119882

119879

1Γ

minus1

1

119882

1

(11)

Adaptive law can be chosen as follows

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(12)

where 1205901gt 0 and then

119881

1le 119911

1119911

2minus 119888

1119911

2

1+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+ 119911

1119890

1+

119875

lowast2

1

2

minus 120590

1

119882

119879

1

119882

1

(13)

Let 1198881= 119888

10+ 119888

11 where 119888

10gt 0 and 119888

11gt 0 and then the

upper equation becomes

119881

1le 119911

1119911

2minus (119888

10+

1

2119892

2

1

)119911

2

1minus 119888

11119911

2

1+ 119911

1119890

1+

119875

lowast2

1

2

minus 120590

1

119882

119879

1

119882

1

(14)

According to the complete square formula

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

minus119888

11119911

2

1+ 119911

1119890

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

119890

1

1003816

1003816

1003816

1003816

le

119890

2

1

4119888

11

le

119890

lowast2

1

4119888

11

(15)

Because minus(11988810+ (

12119892

2

1))119911

2

1le minus(119888

10minus (119892

11198892119892

2

1119898))119911

2

1

we can make (119888lowast10≜ 119888

10minus (119892

11198892119892

2

1119898)) gt 0 by choosing the

appropriate 11988810and obtain the following inequality

119881

1le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

1

4119888

11

+

119875

lowast2

1

2

(16)

The cross coupling 1199111119911

2in (16) will be eliminated in the

next step

Step 2 Let 1199112= 119909

2minus 120572

1 then

2= 119891

2(119909

2) + 119892

2(119909

2) 119909

3+ 119889

2minus

1 (17)

From (10) we can see that 1205721is a function of 119909

1 119909119889 and

119882

1 and

1can be written as

1=

120597120572

1

120597119909

1

1+

120597120572

1

120597119909

119889

119889+

120597120572

1

120597

119882

1

119882

1

=

120597120572

1

120597119909

1

(119892

1(119909

1) 119909

2+ 119891

1(119909

1)) + 120593

1

(18)

where 1206011= (120597120572

1120597119909

119889)

119889+ (120597120572

1120597

119882

1)[Γ

1(119878

1(119885

1)119911

1minus 120590

1

119882

1)]

can be calculatedConsider the following Lyapunov function

119881

2= 119881

1+

1

2119892

2(119909

2)

119911

2

2+

1

2

119882

119879

2Γ

minus1

2

119882

2 (19)

where Γ2= Γ

119879

2gt 0 is an adaptive gain matrix

Then the derivation of 1198812can be calculated as

119881

2=

119881

1+

119911

2

2

119892

2(119909

2)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+

119911

2

119892

2(119909

2)

(119891

2(119909

2) + 119892

2(119909

2) 119909

3+ 119889

2minus

1)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

) +

119911

2119889

2

119892

2(119909

2)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

(20)

According to Assumption 1 we can get

119881

2le

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

) +

119911

2

2120588

2

2

2119892

2

2(119909

2)

+

119875

lowast2

2

2

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

)

+

119875

lowast2

2

2

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

(21)

4 Computational Intelligence and Neuroscience

There is an ideal feedback control law

120572

lowast

2= minus119911

1minus 119888

2119911

2minus [

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

] (22)

where 1198882gt 0 is a designed controller parameter

Because of the unknown smooth functions 1198912(119909

2) and

119892

2(119909

2) we cannot actually get the ideal feedback control law

120572

lowast

2 from (22) we can see that the unknown part is a smooth

function of 1199092and

1 let

ℎ

2(119885

2) ≜

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

(23)

where 1198852≜ [119909

119879

2 (120597120572

1120597119909

1) 120601

1]

119879sub 119877

4 RBF neural network119882

119879

2119878

2(119885

2) is used to approximate the unknown function

ℎ

2(119885

2) and 120572lowast

2can be expressed as

120572

lowast

2= minus119911

1minus 119888

2119911

2minus119882

lowast119879

2119878

2(119885

2) minus 119890

2 (24)

where119882lowast2is expressed as the ideal constant weight vector and

|119890

2| le 119890

lowast

2is the estimated error and meets 119890lowast

2gt 0

Because 119882lowast2

is unknown select the following virtualcontrol law

120572

2= minus119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(25)

where 1198822is the estimated value of119882lowast

2 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus

119882

119879

2119878

2119911

2+

119882

119879

2Γ

minus1

2

119882

2

(26)

where 1198822=

119882

2minus119882

lowast

2

Adaptive law can be chosen as

119882

2=

119882

2= Γ

2[119878

2(119885

2) 119911

2minus 120590

2

119882

2]

(27)

where 1205902gt 0 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(28)

Let 1198882= 119888

20+ 119888

21 11988820 119888

21gt 0 then the upper equation

becomes

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus (119888

20+

2(119909

2)

2119892

2

2(119909

2)

) 119911

2

2minus 119888

21119911

2

2

+ 119911

2119890

2+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(29)

According to the complete square formula

minus120590

2

119882

119879

2

119882

2= minus120590

2

119882

119879

2(

119882

2+119882

lowast

2)

le minus120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

+ 120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

le minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

minus119888

21119911

2

2+ 119911

2119890

2le minus119888

21119911

2

2+ 119911

2

1003816

1003816

1003816

1003816

119890

2

1003816

1003816

1003816

1003816

le

119890

2

2

4119888

21

le

119890

lowast2

2

4119888

21

(30)

Because minus(11988820+(

22119892

2

2))119911

2

2le minus(119888

20minus(119892

21198892119892

2

2119898))119911

2

2 then

we can make (119888lowast20≜ 119888

20minus (119892

21198892119892

2

2119898)) gt 0 by selecting the

proper 11988820 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

lowast

20119911

2

2minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

2

4119888

21

+

119875

lowast2

2

2

le 119911

2119911

3minus

2

sum

119896=1

119888

lowast

1198960119911

2

119896minus

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

119890

lowast2

119896

4119888

1198961

(31)

The cross coupling 1199112119911

3in (31) will be eliminated in the

next step

Step 119894 (3 le 119894 le 119899 minus 1) The derivative of 119911119894= 119909

119894minus 120572

119894minus1can be

calculated as

119894= 119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1minus

119894minus1 (32)

where

119894minus1=

119894minus1

sum

119896=1

120597120572

119894minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120593

119894minus1

120601

119894minus1=

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597119909

119889

)

119889

+

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(33)

Consider the following Lyapunov function

119881

119894= 119881

119894minus1+

1

2119892

119894(119909

119894)

119911

2

119894+

1

2

119882

119879

119894Γ

minus1

119894

119882

119894 (34)

where Γ119894= Γ

119879

119894gt 0 is an adaptive gain matrix

Computational Intelligence and Neuroscience 5

Then the derivation of 119881119894can be calculated as

119881

119894=

119881

119894minus1+

119911

119894

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+

119911

119894

119892

119894(119909

119894)

(119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894minus

119894minus1)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

) +

119911

119894119889

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(35)

According to Assumption 1 we can get

119881

119894le

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1

+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(36)

There is an ideal feedback control law as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (37)

where 119888119894gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119894(119909

119894) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

120572

lowast

119894 from (37) we can see that the unknown part is a smooth

function of 119909119894and

119894minus1 and let

ℎ

119894(119885

119894) ≜

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

(38)

where

119885

119894≜ [119909

119879

119894

120597120572

119894minus1

120597119909

1

120597120572

119894minus1

120597119909

119894minus1

120593

119894minus1]

119879

sub 119877

2119894

(39)

By introducing the direct variable (120597120572

119894minus1120597119909

1)

(120597120572

119894minus1120597119909

119894minus1) 120593119894minus1

we can make the number of neuralnetworks minimized RBF neural network119882119879

119894119878

119894(119885

119894) is used

to approximate the unknown function ℎ119894(119885

119894) and 120572lowast

119894can be

expressed as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus119882

lowast119879

119894119878

119894(119885

119894) minus 119890

119894 (40)

where |119890119894| le 119890

lowast

119894is estimated error and meets 119890lowast

119894gt 0

Because 119882lowast119894

is unknown select the following virtualcontrol law

120572

119894= minus119911

119894minus1minus 119888

119894119911

119894minus

119882

119879

119894119878

119894(119885

119894)

(41)

where119882lowast119894is the estimated value of 119882

119894 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus

119882

119879

119894119878

119894119911

119894+

119882

119879

119894Γ

minus1

119894

119882

119894

(42)

where 119882119894=

119882

119894minus119882

lowast

119894

The following adaptive law can be selected as

119882

119894=

119882

119894= Γ

119894[119878

119894(119885

119894) 119911

119894minus 120590

119894

119882

119894]

(43)

where 120590119894gt 0 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(44)

Let 119888119894= 119888

1198940+ 119888

1198941 1198881198940 119888

1198941gt 0 then (44) can be rewritten as

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus (119888

1198940+

119894(119909

119894)

2119892

2

119894(119909

119894)

) 119911

2

119894

minus 119888

1198941119911

2

119894+ 119911

119894119890

119894+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(45)

According to the complete square formula

minus120590

119894

119882

119879

119894

119882

119894= minus120590

119894

119882

119879

119894(

119882

119894+119882

lowast

119894)

le minus120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

le minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

minus119888

1198941119911

2

119894+ 119911

119894119890

119894le minus119888

1198941119911

2

119894+ 119911

119894

1003816

1003816

1003816

1003816

119890

119894

1003816

1003816

1003816

1003816

le

119890

2

119894

4119888

1198941

le

119890

lowast2

119894

4119888

1198941

(46)

6 Computational Intelligence and Neuroscience

Because minus(1198881198940+ (

1198942119892

2

119894))119911

2

119894le minus(119888

1198940minus (119892

1198941198892119892

2

119894119898))119911

2

119894 then

we can make (119888lowast1198940≜ 119888

1198940minus (119892

1198941198892119892

2

119894119898)) gt 0 by selecting the

proper 1198881198940 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

lowast

1198940119911

2

119894minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

119894

4119888

1198941

+

119875

lowast2

119894

2

le 119911

119894119911

119894+1minus

119894

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119894

sum

119896=1

119875

lowast2

119896

2

(47)

The cross coupling 119911119894119911

119894+1in (47) will be eliminated in the

next step

Step 119899 The derivative of 119911119899= 119909

119899minus 120572

119899minus1can be calculated as

119899= 119891

119899(119909

119899) + 119892

119899(119909

119899minus1) 119906 minus

119899minus1 (48)

where

119899minus1=

119899minus1

sum

119896=1

120597120572

119899minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120601

119899minus1 (49)

where

120601

119899minus1=

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597119909

119889

)

119889

+

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(50)

Consider the following Lyapunov function

119881

119899= 119881

119899minus1+

1

2119892

119899(119909

119899)

119911

2

119899+

1

2

119882

119879

119899Γ

minus1

119899

119882

119899 (51)

where Γ119899= Γ

119879

119899gt 0 is an adaptive gain matrix Then the

derivation of 119881119899can be calculated as

119881

119899=

119881

119899minus1+

119911

119899

119899

119892

119894(119909

119894)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+

119911

119899

119892

119899(119909

119899)

(119891

119899(119909

119899) + 119892

119899(119909

119899) 119906 + 119889

119899minus

119899minus1)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

)

+

119911

119899119889

119899

119892

119899(119909

119899)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(52)

According to Assumption 1 we can get

119881

119899le

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119894(119909

119894) minus

119899minus1

119892

119894(119909

119894)

)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(53)

There is an ideal feedback control law as

119906

lowast= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (54)

where 119888119899gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119899(119909

119899) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

119906

lowast from (54) we can see the unknown part is a smoothfunction of 119909

119899and

119899minus1 and let

ℎ

119899(119885

119894) ≜

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

119899120588

2

119899

2119892

2

119899(119909

119899)

(55)

where 119885119899≜ [119909

119879

119899 120597120572

119899minus1120597119909

1 120597120572

119899minus1120597119909

119899minus1 120601

119899minus1]

119879sub 119877

2119899RBF neural network119882119879

119899119878

119899(119885

119899) is used to approximate the

unknown function ℎ119899(119885

119899) and 119906lowast can be expressed as

119906

lowast= minus119911

119899minus1minus 119888

119899119911

119899minus119882

lowast119879

119899119878

119899(119885

119899) minus 119890

119899 (56)

where |119890119899| le 119890

lowast

119899is estimated error and meets 119890lowast

119899gt 0

Because 119882lowast119899

is unknown select the following virtualcontrol law

119906 = minus119911

119899minus1minus 119888

119899119911

119899minus

119882

119879

119899119878

119899(119885

119899)

(57)

where 119882119894is the estimated value of119882lowast

119894 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus

119882

119879

119899119878

119899119911

119899+

119882

119879

119899Γ

minus1

119899

119882

119899

(58)

where 119882119899=

119882

119899minus119882

lowast

119899

The following adaptive law can be selected as

119882

119899=

119882

119899= Γ

119899[119878

119899(119885

119899) 119911

119899minus 120590

119899

119882

119899]

(59)

where 120590119899gt 0 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(60)

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

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2 Computational Intelligence and Neuroscience

degree of freedom by controlling explicitly the progressionrate of the virtual target along the path and overcomes themajor singular problem approach angle is introduced forcontroller design via backstepping method Neural networksare introduced to enhance system stability and transientperformance which can handle the known dynamics anduncertainties of systems well [18ndash20] Particularly in [12]a single hidden layer neural network (SHLNN) is adoptedto obtain the adaptive signal online but the choice of thesingle hidden layer neural network is limited by the numberof hidden layer node selections that will affect the onlinelearning speed and accuracy and cannot produce a betterestimation effect on the fast changing disturbances

Therefore a solution to the course control of underac-tuated surface vessel is addressed in this paper In view ofthe characteristics of the underactuated performance thebackstepping control method is used to deal with aboveproblem The direct adaptive neural network is adopted todesign control law by using the RBF neural network toovercome the problem that the ideal virtual control cannot beused directly in practice The weights of the neural networkare updated by adaptive technique to guarantee the stabilityof the closed-loop system through Lyapunov stability theorySimulation results are illustrated to verify the performance ofthe proposed adaptive neural network controller with goodprecision

2 Adaptive Robust Neural NetworkController Design

21 Problem Description Consider the following nonlinearsystems

119894= 119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894 1 le 119894 le 119899 minus 1

119899= 119891

119899(119909

119899) + 119892

119899(

119899) 119906 + 119889

119899 119899 ge 2

119910 = 119909

1

(1)

where 119909119894= [119909

1 119909

2 119909

119894] is system state 119906 is control input

and 119910 is system output The control objective is to design anadaptive neural network controller and make 119910 track 119910

119889 119910119889

meets the smooth bounded reference model as follows

119889119894= 119891

119889119894(119909

119889) 1 le 119894 le 119898

119910

119889= 119909

1198891 119898 ge 119899

(2)

where 119909119889= [119909

1198891 119909

1198892 119909

119889119898]

119879isin 119877

119898 is state constant 119910119889isin

119877 represents system output and119891119889119894(sdot) 119894 = 1 2 119898 denote

nonlinear function assuming that the reference model foreach state is bounded as 119909

119889isin Ω

119889 forall119905 ge 0

Assumption 1 There is an unknown constant 119901lowast119894to meet

forall(119909

119899 119905) isin 119877

119899times119877

+ |119889119894(119909

119899 119905)| le 119901

lowast

119894120588

119894(119909

119894) and120588

119894(119909

119894) is a known

positive smooth function

22 Direct Adaptive Neural Network Controller Design Inview of the problems and solutions described in the lastsection the direct adaptive neural network controller for

nonlinear systems with RBF neural network is chosenDetailed design steps will be described in the following

Step 1 Let 1199111= 119909

1minus 119909

1198891 1199112= 119909

2minus 120572

1 and then

1= 119891

1(119909

1) + 119892

1(119909

1) 119909

2+ 119889

1minus

1198891 (3)

Consider the following Lyapunov function

119881

1=

1

2119892

1(119909

1)

119911

2

1+

1

2

119882

119879

1Γ

minus1

1

119882

1 (4)

where 1198821=

119882

1minus119882

lowast

1119882lowast1represents the ideal weight vector

of neural network 1198821represents the estimated value of the

neural network weight vector 1198821represents the estimation

error of weight vector Γ1= Γ

119879

1gt 0 is the adaptive gainmatrix

and the derivation of 1198811can be computed as

119881

1=

119911

1

1

119892

1(119909

1)

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

=

119911

1

119892

1(119909

1)

(119891

1(119909

1) + 119892

1(119909

1) 119909

2+ 119889

1minus

1198891)

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

= 119911

1(119911

2+ 120572

1+

119891

1(119909

1) minus

1198891

119892

1(119909

1)

) +

119911

1119889

1

119892

1(119909

1)

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

(5)

According to Assumption 1 we can get

119881

1le 119911

1(119911

2+ 120572

1+

119891

1(119909

1) minus

1198891

119892

1(119909

1)

) +

119911

2

1120588

2

1

2119892

2

1(119909

1)

+

119875

lowast2

1

2

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

= 119911

1(119911

2+ 120572

1+

119891

1(119909

1) minus

1198891

119892

1(119909

1)

+

119911

1120588

2

1

2119892

2

1(119909

1)

) +

119875

lowast2

1

2

+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+

119882

119879

1Γ

minus1

1

119882

1

(6)

There is an ideal virtual feedback control law

120572

lowast

1= minus119888

1119911

1minus [

119891

1(119909

1) minus

1198891

119892

1(119909

1)

+

119911

1120588

2

1

2119892

2

1(119909

1)

] (7)

where 1198881gt 0 is designed controller parameter

Because of the unknown smooth functions 1198911(119909

1) and

119892

1(119909

1) we cannot actually get the ideal feedback control law

120572

lowast

1 from (7) we can see that the unknown part (119891

1(119909

1) minus

1198891)119892

1(119909

1) is smooth function of 119909

1and

1198891 so that

ℎ

1(119885

1) ≜

119891

1(119909

1) minus

1198891

119892

1(119909

1)

+

119911

1120588

2

1

2119892

2

1(119909

1)

119885

1≜ [119909

1

1198891]

119879sub 119877

2

(8)

Computational Intelligence and Neuroscience 3

RBF neural network1198821198791119878

1(119885

1) is used to approximate the

unknown function ℎ1(119885

1) and 120572lowast

1can be expressed as

120572

lowast

1= minus119888

1119911

1minus119882

lowast119879

1119878

1(119885

1) minus 119890

1 (9)

where |1198901| le 119890

lowast

1is estimated error and meets 119890lowast

1gt 0

Because 119882

lowast

1is unknown the virtual control law is

selected as follows

120572

1= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1)

(10)

and then

119881

1le 119911

1119911

2minus 119888

1119911

2

1+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+ 119911

1119890

1+

119875

lowast2

1

2

minus

119882

119879

1119878

1119911

1+

119882

119879

1Γ

minus1

1

119882

1

(11)

Adaptive law can be chosen as follows

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(12)

where 1205901gt 0 and then

119881

1le 119911

1119911

2minus 119888

1119911

2

1+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+ 119911

1119890

1+

119875

lowast2

1

2

minus 120590

1

119882

119879

1

119882

1

(13)

Let 1198881= 119888

10+ 119888

11 where 119888

10gt 0 and 119888

11gt 0 and then the

upper equation becomes

119881

1le 119911

1119911

2minus (119888

10+

1

2119892

2

1

)119911

2

1minus 119888

11119911

2

1+ 119911

1119890

1+

119875

lowast2

1

2

minus 120590

1

119882

119879

1

119882

1

(14)

According to the complete square formula

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

minus119888

11119911

2

1+ 119911

1119890

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

119890

1

1003816

1003816

1003816

1003816

le

119890

2

1

4119888

11

le

119890

lowast2

1

4119888

11

(15)

Because minus(11988810+ (

12119892

2

1))119911

2

1le minus(119888

10minus (119892

11198892119892

2

1119898))119911

2

1

we can make (119888lowast10≜ 119888

10minus (119892

11198892119892

2

1119898)) gt 0 by choosing the

appropriate 11988810and obtain the following inequality

119881

1le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

1

4119888

11

+

119875

lowast2

1

2

(16)

The cross coupling 1199111119911

2in (16) will be eliminated in the

next step

Step 2 Let 1199112= 119909

2minus 120572

1 then

2= 119891

2(119909

2) + 119892

2(119909

2) 119909

3+ 119889

2minus

1 (17)

From (10) we can see that 1205721is a function of 119909

1 119909119889 and

119882

1 and

1can be written as

1=

120597120572

1

120597119909

1

1+

120597120572

1

120597119909

119889

119889+

120597120572

1

120597

119882

1

119882

1

=

120597120572

1

120597119909

1

(119892

1(119909

1) 119909

2+ 119891

1(119909

1)) + 120593

1

(18)

where 1206011= (120597120572

1120597119909

119889)

119889+ (120597120572

1120597

119882

1)[Γ

1(119878

1(119885

1)119911

1minus 120590

1

119882

1)]

can be calculatedConsider the following Lyapunov function

119881

2= 119881

1+

1

2119892

2(119909

2)

119911

2

2+

1

2

119882

119879

2Γ

minus1

2

119882

2 (19)

where Γ2= Γ

119879

2gt 0 is an adaptive gain matrix

Then the derivation of 1198812can be calculated as

119881

2=

119881

1+

119911

2

2

119892

2(119909

2)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+

119911

2

119892

2(119909

2)

(119891

2(119909

2) + 119892

2(119909

2) 119909

3+ 119889

2minus

1)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

) +

119911

2119889

2

119892

2(119909

2)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

(20)

According to Assumption 1 we can get

119881

2le

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

) +

119911

2

2120588

2

2

2119892

2

2(119909

2)

+

119875

lowast2

2

2

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

)

+

119875

lowast2

2

2

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

(21)

4 Computational Intelligence and Neuroscience

There is an ideal feedback control law

120572

lowast

2= minus119911

1minus 119888

2119911

2minus [

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

] (22)

where 1198882gt 0 is a designed controller parameter

Because of the unknown smooth functions 1198912(119909

2) and

119892

2(119909

2) we cannot actually get the ideal feedback control law

120572

lowast

2 from (22) we can see that the unknown part is a smooth

function of 1199092and

1 let

ℎ

2(119885

2) ≜

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

(23)

where 1198852≜ [119909

119879

2 (120597120572

1120597119909

1) 120601

1]

119879sub 119877

4 RBF neural network119882

119879

2119878

2(119885

2) is used to approximate the unknown function

ℎ

2(119885

2) and 120572lowast

2can be expressed as

120572

lowast

2= minus119911

1minus 119888

2119911

2minus119882

lowast119879

2119878

2(119885

2) minus 119890

2 (24)

where119882lowast2is expressed as the ideal constant weight vector and

|119890

2| le 119890

lowast

2is the estimated error and meets 119890lowast

2gt 0

Because 119882lowast2

is unknown select the following virtualcontrol law

120572

2= minus119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(25)

where 1198822is the estimated value of119882lowast

2 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus

119882

119879

2119878

2119911

2+

119882

119879

2Γ

minus1

2

119882

2

(26)

where 1198822=

119882

2minus119882

lowast

2

Adaptive law can be chosen as

119882

2=

119882

2= Γ

2[119878

2(119885

2) 119911

2minus 120590

2

119882

2]

(27)

where 1205902gt 0 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(28)

Let 1198882= 119888

20+ 119888

21 11988820 119888

21gt 0 then the upper equation

becomes

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus (119888

20+

2(119909

2)

2119892

2

2(119909

2)

) 119911

2

2minus 119888

21119911

2

2

+ 119911

2119890

2+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(29)

According to the complete square formula

minus120590

2

119882

119879

2

119882

2= minus120590

2

119882

119879

2(

119882

2+119882

lowast

2)

le minus120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

+ 120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

le minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

minus119888

21119911

2

2+ 119911

2119890

2le minus119888

21119911

2

2+ 119911

2

1003816

1003816

1003816

1003816

119890

2

1003816

1003816

1003816

1003816

le

119890

2

2

4119888

21

le

119890

lowast2

2

4119888

21

(30)

Because minus(11988820+(

22119892

2

2))119911

2

2le minus(119888

20minus(119892

21198892119892

2

2119898))119911

2

2 then

we can make (119888lowast20≜ 119888

20minus (119892

21198892119892

2

2119898)) gt 0 by selecting the

proper 11988820 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

lowast

20119911

2

2minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

2

4119888

21

+

119875

lowast2

2

2

le 119911

2119911

3minus

2

sum

119896=1

119888

lowast

1198960119911

2

119896minus

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

119890

lowast2

119896

4119888

1198961

(31)

The cross coupling 1199112119911

3in (31) will be eliminated in the

next step

Step 119894 (3 le 119894 le 119899 minus 1) The derivative of 119911119894= 119909

119894minus 120572

119894minus1can be

calculated as

119894= 119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1minus

119894minus1 (32)

where

119894minus1=

119894minus1

sum

119896=1

120597120572

119894minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120593

119894minus1

120601

119894minus1=

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597119909

119889

)

119889

+

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(33)

Consider the following Lyapunov function

119881

119894= 119881

119894minus1+

1

2119892

119894(119909

119894)

119911

2

119894+

1

2

119882

119879

119894Γ

minus1

119894

119882

119894 (34)

where Γ119894= Γ

119879

119894gt 0 is an adaptive gain matrix

Computational Intelligence and Neuroscience 5

Then the derivation of 119881119894can be calculated as

119881

119894=

119881

119894minus1+

119911

119894

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+

119911

119894

119892

119894(119909

119894)

(119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894minus

119894minus1)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

) +

119911

119894119889

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(35)

According to Assumption 1 we can get

119881

119894le

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1

+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(36)

There is an ideal feedback control law as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (37)

where 119888119894gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119894(119909

119894) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

120572

lowast

119894 from (37) we can see that the unknown part is a smooth

function of 119909119894and

119894minus1 and let

ℎ

119894(119885

119894) ≜

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

(38)

where

119885

119894≜ [119909

119879

119894

120597120572

119894minus1

120597119909

1

120597120572

119894minus1

120597119909

119894minus1

120593

119894minus1]

119879

sub 119877

2119894

(39)

By introducing the direct variable (120597120572

119894minus1120597119909

1)

(120597120572

119894minus1120597119909

119894minus1) 120593119894minus1

we can make the number of neuralnetworks minimized RBF neural network119882119879

119894119878

119894(119885

119894) is used

to approximate the unknown function ℎ119894(119885

119894) and 120572lowast

119894can be

expressed as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus119882

lowast119879

119894119878

119894(119885

119894) minus 119890

119894 (40)

where |119890119894| le 119890

lowast

119894is estimated error and meets 119890lowast

119894gt 0

Because 119882lowast119894

is unknown select the following virtualcontrol law

120572

119894= minus119911

119894minus1minus 119888

119894119911

119894minus

119882

119879

119894119878

119894(119885

119894)

(41)

where119882lowast119894is the estimated value of 119882

119894 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus

119882

119879

119894119878

119894119911

119894+

119882

119879

119894Γ

minus1

119894

119882

119894

(42)

where 119882119894=

119882

119894minus119882

lowast

119894

The following adaptive law can be selected as

119882

119894=

119882

119894= Γ

119894[119878

119894(119885

119894) 119911

119894minus 120590

119894

119882

119894]

(43)

where 120590119894gt 0 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(44)

Let 119888119894= 119888

1198940+ 119888

1198941 1198881198940 119888

1198941gt 0 then (44) can be rewritten as

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus (119888

1198940+

119894(119909

119894)

2119892

2

119894(119909

119894)

) 119911

2

119894

minus 119888

1198941119911

2

119894+ 119911

119894119890

119894+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(45)

According to the complete square formula

minus120590

119894

119882

119879

119894

119882

119894= minus120590

119894

119882

119879

119894(

119882

119894+119882

lowast

119894)

le minus120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

le minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

minus119888

1198941119911

2

119894+ 119911

119894119890

119894le minus119888

1198941119911

2

119894+ 119911

119894

1003816

1003816

1003816

1003816

119890

119894

1003816

1003816

1003816

1003816

le

119890

2

119894

4119888

1198941

le

119890

lowast2

119894

4119888

1198941

(46)

6 Computational Intelligence and Neuroscience

Because minus(1198881198940+ (

1198942119892

2

119894))119911

2

119894le minus(119888

1198940minus (119892

1198941198892119892

2

119894119898))119911

2

119894 then

we can make (119888lowast1198940≜ 119888

1198940minus (119892

1198941198892119892

2

119894119898)) gt 0 by selecting the

proper 1198881198940 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

lowast

1198940119911

2

119894minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

119894

4119888

1198941

+

119875

lowast2

119894

2

le 119911

119894119911

119894+1minus

119894

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119894

sum

119896=1

119875

lowast2

119896

2

(47)

The cross coupling 119911119894119911

119894+1in (47) will be eliminated in the

next step

Step 119899 The derivative of 119911119899= 119909

119899minus 120572

119899minus1can be calculated as

119899= 119891

119899(119909

119899) + 119892

119899(119909

119899minus1) 119906 minus

119899minus1 (48)

where

119899minus1=

119899minus1

sum

119896=1

120597120572

119899minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120601

119899minus1 (49)

where

120601

119899minus1=

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597119909

119889

)

119889

+

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(50)

Consider the following Lyapunov function

119881

119899= 119881

119899minus1+

1

2119892

119899(119909

119899)

119911

2

119899+

1

2

119882

119879

119899Γ

minus1

119899

119882

119899 (51)

where Γ119899= Γ

119879

119899gt 0 is an adaptive gain matrix Then the

derivation of 119881119899can be calculated as

119881

119899=

119881

119899minus1+

119911

119899

119899

119892

119894(119909

119894)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+

119911

119899

119892

119899(119909

119899)

(119891

119899(119909

119899) + 119892

119899(119909

119899) 119906 + 119889

119899minus

119899minus1)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

)

+

119911

119899119889

119899

119892

119899(119909

119899)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(52)

According to Assumption 1 we can get

119881

119899le

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119894(119909

119894) minus

119899minus1

119892

119894(119909

119894)

)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(53)

There is an ideal feedback control law as

119906

lowast= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (54)

where 119888119899gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119899(119909

119899) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

119906

lowast from (54) we can see the unknown part is a smoothfunction of 119909

119899and

119899minus1 and let

ℎ

119899(119885

119894) ≜

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

119899120588

2

119899

2119892

2

119899(119909

119899)

(55)

where 119885119899≜ [119909

119879

119899 120597120572

119899minus1120597119909

1 120597120572

119899minus1120597119909

119899minus1 120601

119899minus1]

119879sub 119877

2119899RBF neural network119882119879

119899119878

119899(119885

119899) is used to approximate the

unknown function ℎ119899(119885

119899) and 119906lowast can be expressed as

119906

lowast= minus119911

119899minus1minus 119888

119899119911

119899minus119882

lowast119879

119899119878

119899(119885

119899) minus 119890

119899 (56)

where |119890119899| le 119890

lowast

119899is estimated error and meets 119890lowast

119899gt 0

Because 119882lowast119899

is unknown select the following virtualcontrol law

119906 = minus119911

119899minus1minus 119888

119899119911

119899minus

119882

119879

119899119878

119899(119885

119899)

(57)

where 119882119894is the estimated value of119882lowast

119894 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus

119882

119879

119899119878

119899119911

119899+

119882

119879

119899Γ

minus1

119899

119882

119899

(58)

where 119882119899=

119882

119899minus119882

lowast

119899

The following adaptive law can be selected as

119882

119899=

119882

119899= Γ

119899[119878

119899(119885

119899) 119911

119899minus 120590

119899

119882

119899]

(59)

where 120590119899gt 0 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(60)

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

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Computational Intelligence and Neuroscience 3

RBF neural network1198821198791119878

1(119885

1) is used to approximate the

unknown function ℎ1(119885

1) and 120572lowast

1can be expressed as

120572

lowast

1= minus119888

1119911

1minus119882

lowast119879

1119878

1(119885

1) minus 119890

1 (9)

where |1198901| le 119890

lowast

1is estimated error and meets 119890lowast

1gt 0

Because 119882

lowast

1is unknown the virtual control law is

selected as follows

120572

1= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1)

(10)

and then

119881

1le 119911

1119911

2minus 119888

1119911

2

1+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+ 119911

1119890

1+

119875

lowast2

1

2

minus

119882

119879

1119878

1119911

1+

119882

119879

1Γ

minus1

1

119882

1

(11)

Adaptive law can be chosen as follows

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(12)

where 1205901gt 0 and then

119881

1le 119911

1119911

2minus 119888

1119911

2

1+

1(119909

1) 119911

2

1

2119892

2

1(119909

1)

+ 119911

1119890

1+

119875

lowast2

1

2

minus 120590

1

119882

119879

1

119882

1

(13)

Let 1198881= 119888

10+ 119888

11 where 119888

10gt 0 and 119888

11gt 0 and then the

upper equation becomes

119881

1le 119911

1119911

2minus (119888

10+

1

2119892

2

1

)119911

2

1minus 119888

11119911

2

1+ 119911

1119890

1+

119875

lowast2

1

2

minus 120590

1

119882

119879

1

119882

1

(14)

According to the complete square formula

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

minus119888

11119911

2

1+ 119911

1119890

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

119890

1

1003816

1003816

1003816

1003816

le

119890

2

1

4119888

11

le

119890

lowast2

1

4119888

11

(15)

Because minus(11988810+ (

12119892

2

1))119911

2

1le minus(119888

10minus (119892

11198892119892

2

1119898))119911

2

1

we can make (119888lowast10≜ 119888

10minus (119892

11198892119892

2

1119898)) gt 0 by choosing the

appropriate 11988810and obtain the following inequality

119881

1le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

1

4119888

11

+

119875

lowast2

1

2

(16)

The cross coupling 1199111119911

2in (16) will be eliminated in the

next step

Step 2 Let 1199112= 119909

2minus 120572

1 then

2= 119891

2(119909

2) + 119892

2(119909

2) 119909

3+ 119889

2minus

1 (17)

From (10) we can see that 1205721is a function of 119909

1 119909119889 and

119882

1 and

1can be written as

1=

120597120572

1

120597119909

1

1+

120597120572

1

120597119909

119889

119889+

120597120572

1

120597

119882

1

119882

1

=

120597120572

1

120597119909

1

(119892

1(119909

1) 119909

2+ 119891

1(119909

1)) + 120593

1

(18)

where 1206011= (120597120572

1120597119909

119889)

119889+ (120597120572

1120597

119882

1)[Γ

1(119878

1(119885

1)119911

1minus 120590

1

119882

1)]

can be calculatedConsider the following Lyapunov function

119881

2= 119881

1+

1

2119892

2(119909

2)

119911

2

2+

1

2

119882

119879

2Γ

minus1

2

119882

2 (19)

where Γ2= Γ

119879

2gt 0 is an adaptive gain matrix

Then the derivation of 1198812can be calculated as

119881

2=

119881

1+

119911

2

2

119892

2(119909

2)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+

119911

2

119892

2(119909

2)

(119891

2(119909

2) + 119892

2(119909

2) 119909

3+ 119889

2minus

1)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

) +

119911

2119889

2

119892

2(119909

2)

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

(20)

According to Assumption 1 we can get

119881

2le

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

) +

119911

2

2120588

2

2

2119892

2

2(119909

2)

+

119875

lowast2

2

2

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

=

119881

1+ 119911

2(119911

3+ 120572

2+

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

)

+

119875

lowast2

2

2

+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+

119882

119879

2Γ

minus1

2

119882

2

(21)

4 Computational Intelligence and Neuroscience

There is an ideal feedback control law

120572

lowast

2= minus119911

1minus 119888

2119911

2minus [

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

] (22)

where 1198882gt 0 is a designed controller parameter

Because of the unknown smooth functions 1198912(119909

2) and

119892

2(119909

2) we cannot actually get the ideal feedback control law

120572

lowast

2 from (22) we can see that the unknown part is a smooth

function of 1199092and

1 let

ℎ

2(119885

2) ≜

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

(23)

where 1198852≜ [119909

119879

2 (120597120572

1120597119909

1) 120601

1]

119879sub 119877

4 RBF neural network119882

119879

2119878

2(119885

2) is used to approximate the unknown function

ℎ

2(119885

2) and 120572lowast

2can be expressed as

120572

lowast

2= minus119911

1minus 119888

2119911

2minus119882

lowast119879

2119878

2(119885

2) minus 119890

2 (24)

where119882lowast2is expressed as the ideal constant weight vector and

|119890

2| le 119890

lowast

2is the estimated error and meets 119890lowast

2gt 0

Because 119882lowast2

is unknown select the following virtualcontrol law

120572

2= minus119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(25)

where 1198822is the estimated value of119882lowast

2 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus

119882

119879

2119878

2119911

2+

119882

119879

2Γ

minus1

2

119882

2

(26)

where 1198822=

119882

2minus119882

lowast

2

Adaptive law can be chosen as

119882

2=

119882

2= Γ

2[119878

2(119885

2) 119911

2minus 120590

2

119882

2]

(27)

where 1205902gt 0 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(28)

Let 1198882= 119888

20+ 119888

21 11988820 119888

21gt 0 then the upper equation

becomes

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus (119888

20+

2(119909

2)

2119892

2

2(119909

2)

) 119911

2

2minus 119888

21119911

2

2

+ 119911

2119890

2+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(29)

According to the complete square formula

minus120590

2

119882

119879

2

119882

2= minus120590

2

119882

119879

2(

119882

2+119882

lowast

2)

le minus120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

+ 120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

le minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

minus119888

21119911

2

2+ 119911

2119890

2le minus119888

21119911

2

2+ 119911

2

1003816

1003816

1003816

1003816

119890

2

1003816

1003816

1003816

1003816

le

119890

2

2

4119888

21

le

119890

lowast2

2

4119888

21

(30)

Because minus(11988820+(

22119892

2

2))119911

2

2le minus(119888

20minus(119892

21198892119892

2

2119898))119911

2

2 then

we can make (119888lowast20≜ 119888

20minus (119892

21198892119892

2

2119898)) gt 0 by selecting the

proper 11988820 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

lowast

20119911

2

2minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

2

4119888

21

+

119875

lowast2

2

2

le 119911

2119911

3minus

2

sum

119896=1

119888

lowast

1198960119911

2

119896minus

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

119890

lowast2

119896

4119888

1198961

(31)

The cross coupling 1199112119911

3in (31) will be eliminated in the

next step

Step 119894 (3 le 119894 le 119899 minus 1) The derivative of 119911119894= 119909

119894minus 120572

119894minus1can be

calculated as

119894= 119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1minus

119894minus1 (32)

where

119894minus1=

119894minus1

sum

119896=1

120597120572

119894minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120593

119894minus1

120601

119894minus1=

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597119909

119889

)

119889

+

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(33)

Consider the following Lyapunov function

119881

119894= 119881

119894minus1+

1

2119892

119894(119909

119894)

119911

2

119894+

1

2

119882

119879

119894Γ

minus1

119894

119882

119894 (34)

where Γ119894= Γ

119879

119894gt 0 is an adaptive gain matrix

Computational Intelligence and Neuroscience 5

Then the derivation of 119881119894can be calculated as

119881

119894=

119881

119894minus1+

119911

119894

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+

119911

119894

119892

119894(119909

119894)

(119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894minus

119894minus1)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

) +

119911

119894119889

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(35)

According to Assumption 1 we can get

119881

119894le

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1

+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(36)

There is an ideal feedback control law as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (37)

where 119888119894gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119894(119909

119894) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

120572

lowast

119894 from (37) we can see that the unknown part is a smooth

function of 119909119894and

119894minus1 and let

ℎ

119894(119885

119894) ≜

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

(38)

where

119885

119894≜ [119909

119879

119894

120597120572

119894minus1

120597119909

1

120597120572

119894minus1

120597119909

119894minus1

120593

119894minus1]

119879

sub 119877

2119894

(39)

By introducing the direct variable (120597120572

119894minus1120597119909

1)

(120597120572

119894minus1120597119909

119894minus1) 120593119894minus1

we can make the number of neuralnetworks minimized RBF neural network119882119879

119894119878

119894(119885

119894) is used

to approximate the unknown function ℎ119894(119885

119894) and 120572lowast

119894can be

expressed as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus119882

lowast119879

119894119878

119894(119885

119894) minus 119890

119894 (40)

where |119890119894| le 119890

lowast

119894is estimated error and meets 119890lowast

119894gt 0

Because 119882lowast119894

is unknown select the following virtualcontrol law

120572

119894= minus119911

119894minus1minus 119888

119894119911

119894minus

119882

119879

119894119878

119894(119885

119894)

(41)

where119882lowast119894is the estimated value of 119882

119894 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus

119882

119879

119894119878

119894119911

119894+

119882

119879

119894Γ

minus1

119894

119882

119894

(42)

where 119882119894=

119882

119894minus119882

lowast

119894

The following adaptive law can be selected as

119882

119894=

119882

119894= Γ

119894[119878

119894(119885

119894) 119911

119894minus 120590

119894

119882

119894]

(43)

where 120590119894gt 0 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(44)

Let 119888119894= 119888

1198940+ 119888

1198941 1198881198940 119888

1198941gt 0 then (44) can be rewritten as

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus (119888

1198940+

119894(119909

119894)

2119892

2

119894(119909

119894)

) 119911

2

119894

minus 119888

1198941119911

2

119894+ 119911

119894119890

119894+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(45)

According to the complete square formula

minus120590

119894

119882

119879

119894

119882

119894= minus120590

119894

119882

119879

119894(

119882

119894+119882

lowast

119894)

le minus120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

le minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

minus119888

1198941119911

2

119894+ 119911

119894119890

119894le minus119888

1198941119911

2

119894+ 119911

119894

1003816

1003816

1003816

1003816

119890

119894

1003816

1003816

1003816

1003816

le

119890

2

119894

4119888

1198941

le

119890

lowast2

119894

4119888

1198941

(46)

6 Computational Intelligence and Neuroscience

Because minus(1198881198940+ (

1198942119892

2

119894))119911

2

119894le minus(119888

1198940minus (119892

1198941198892119892

2

119894119898))119911

2

119894 then

we can make (119888lowast1198940≜ 119888

1198940minus (119892

1198941198892119892

2

119894119898)) gt 0 by selecting the

proper 1198881198940 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

lowast

1198940119911

2

119894minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

119894

4119888

1198941

+

119875

lowast2

119894

2

le 119911

119894119911

119894+1minus

119894

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119894

sum

119896=1

119875

lowast2

119896

2

(47)

The cross coupling 119911119894119911

119894+1in (47) will be eliminated in the

next step

Step 119899 The derivative of 119911119899= 119909

119899minus 120572

119899minus1can be calculated as

119899= 119891

119899(119909

119899) + 119892

119899(119909

119899minus1) 119906 minus

119899minus1 (48)

where

119899minus1=

119899minus1

sum

119896=1

120597120572

119899minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120601

119899minus1 (49)

where

120601

119899minus1=

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597119909

119889

)

119889

+

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(50)

Consider the following Lyapunov function

119881

119899= 119881

119899minus1+

1

2119892

119899(119909

119899)

119911

2

119899+

1

2

119882

119879

119899Γ

minus1

119899

119882

119899 (51)

where Γ119899= Γ

119879

119899gt 0 is an adaptive gain matrix Then the

derivation of 119881119899can be calculated as

119881

119899=

119881

119899minus1+

119911

119899

119899

119892

119894(119909

119894)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+

119911

119899

119892

119899(119909

119899)

(119891

119899(119909

119899) + 119892

119899(119909

119899) 119906 + 119889

119899minus

119899minus1)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

)

+

119911

119899119889

119899

119892

119899(119909

119899)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(52)

According to Assumption 1 we can get

119881

119899le

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119894(119909

119894) minus

119899minus1

119892

119894(119909

119894)

)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(53)

There is an ideal feedback control law as

119906

lowast= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (54)

where 119888119899gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119899(119909

119899) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

119906

lowast from (54) we can see the unknown part is a smoothfunction of 119909

119899and

119899minus1 and let

ℎ

119899(119885

119894) ≜

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

119899120588

2

119899

2119892

2

119899(119909

119899)

(55)

where 119885119899≜ [119909

119879

119899 120597120572

119899minus1120597119909

1 120597120572

119899minus1120597119909

119899minus1 120601

119899minus1]

119879sub 119877

2119899RBF neural network119882119879

119899119878

119899(119885

119899) is used to approximate the

unknown function ℎ119899(119885

119899) and 119906lowast can be expressed as

119906

lowast= minus119911

119899minus1minus 119888

119899119911

119899minus119882

lowast119879

119899119878

119899(119885

119899) minus 119890

119899 (56)

where |119890119899| le 119890

lowast

119899is estimated error and meets 119890lowast

119899gt 0

Because 119882lowast119899

is unknown select the following virtualcontrol law

119906 = minus119911

119899minus1minus 119888

119899119911

119899minus

119882

119879

119899119878

119899(119885

119899)

(57)

where 119882119894is the estimated value of119882lowast

119894 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus

119882

119879

119899119878

119899119911

119899+

119882

119879

119899Γ

minus1

119899

119882

119899

(58)

where 119882119899=

119882

119899minus119882

lowast

119899

The following adaptive law can be selected as

119882

119899=

119882

119899= Γ

119899[119878

119899(119885

119899) 119911

119899minus 120590

119899

119882

119899]

(59)

where 120590119899gt 0 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(60)

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Computational Intelligence and Neuroscience

There is an ideal feedback control law

120572

lowast

2= minus119911

1minus 119888

2119911

2minus [

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

] (22)

where 1198882gt 0 is a designed controller parameter

Because of the unknown smooth functions 1198912(119909

2) and

119892

2(119909

2) we cannot actually get the ideal feedback control law

120572

lowast

2 from (22) we can see that the unknown part is a smooth

function of 1199092and

1 let

ℎ

2(119885

2) ≜

119891

2(119909

2) minus

1

119892

2(119909

2)

+

119911

2120588

2

2

2119892

2

2(119909

2)

(23)

where 1198852≜ [119909

119879

2 (120597120572

1120597119909

1) 120601

1]

119879sub 119877

4 RBF neural network119882

119879

2119878

2(119885

2) is used to approximate the unknown function

ℎ

2(119885

2) and 120572lowast

2can be expressed as

120572

lowast

2= minus119911

1minus 119888

2119911

2minus119882

lowast119879

2119878

2(119885

2) minus 119890

2 (24)

where119882lowast2is expressed as the ideal constant weight vector and

|119890

2| le 119890

lowast

2is the estimated error and meets 119890lowast

2gt 0

Because 119882lowast2

is unknown select the following virtualcontrol law

120572

2= minus119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(25)

where 1198822is the estimated value of119882lowast

2 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus

119882

119879

2119878

2119911

2+

119882

119879

2Γ

minus1

2

119882

2

(26)

where 1198822=

119882

2minus119882

lowast

2

Adaptive law can be chosen as

119882

2=

119882

2= Γ

2[119878

2(119885

2) 119911

2minus 120590

2

119882

2]

(27)

where 1205902gt 0 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

2119911

2

2+

2(119909

2) 119911

2

2

2119892

2

2(119909

2)

+ 119911

2119890

2

+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(28)

Let 1198882= 119888

20+ 119888

21 11988820 119888

21gt 0 then the upper equation

becomes

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus (119888

20+

2(119909

2)

2119892

2

2(119909

2)

) 119911

2

2minus 119888

21119911

2

2

+ 119911

2119890

2+

119875

lowast2

2

2

minus 120590

2

119882

119879

2

119882

2

(29)

According to the complete square formula

minus120590

2

119882

119879

2

119882

2= minus120590

2

119882

119879

2(

119882

2+119882

lowast

2)

le minus120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

+ 120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

le minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

minus119888

21119911

2

2+ 119911

2119890

2le minus119888

21119911

2

2+ 119911

2

1003816

1003816

1003816

1003816

119890

2

1003816

1003816

1003816

1003816

le

119890

2

2

4119888

21

le

119890

lowast2

2

4119888

21

(30)

Because minus(11988820+(

22119892

2

2))119911

2

2le minus(119888

20minus(119892

21198892119892

2

2119898))119911

2

2 then

we can make (119888lowast20≜ 119888

20minus (119892

21198892119892

2

2119898)) gt 0 by selecting the

proper 11988820 then

119881

2le

119881

1minus 119911

1119911

2+ 119911

2119911

3minus 119888

lowast

20119911

2

2minus

120590

2

1003817

1003817

1003817

1003817

1003817

119882

2

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

2

1003817

1003817

1003817

1003817

119882

lowast

2

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

2

4119888

21

+

119875

lowast2

2

2

le 119911

2119911

3minus

2

sum

119896=1

119888

lowast

1198960119911

2

119896minus

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

2

sum

119896=1

119890

lowast2

119896

4119888

1198961

(31)

The cross coupling 1199112119911

3in (31) will be eliminated in the

next step

Step 119894 (3 le 119894 le 119899 minus 1) The derivative of 119911119894= 119909

119894minus 120572

119894minus1can be

calculated as

119894= 119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1minus

119894minus1 (32)

where

119894minus1=

119894minus1

sum

119896=1

120597120572

119894minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120593

119894minus1

120601

119894minus1=

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597119909

119889

)

119889

+

119894minus1

sum

119896=1

(

120597120572

119894minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(33)

Consider the following Lyapunov function

119881

119894= 119881

119894minus1+

1

2119892

119894(119909

119894)

119911

2

119894+

1

2

119882

119879

119894Γ

minus1

119894

119882

119894 (34)

where Γ119894= Γ

119879

119894gt 0 is an adaptive gain matrix

Computational Intelligence and Neuroscience 5

Then the derivation of 119881119894can be calculated as

119881

119894=

119881

119894minus1+

119911

119894

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+

119911

119894

119892

119894(119909

119894)

(119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894minus

119894minus1)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

) +

119911

119894119889

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(35)

According to Assumption 1 we can get

119881

119894le

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1

+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(36)

There is an ideal feedback control law as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (37)

where 119888119894gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119894(119909

119894) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

120572

lowast

119894 from (37) we can see that the unknown part is a smooth

function of 119909119894and

119894minus1 and let

ℎ

119894(119885

119894) ≜

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

(38)

where

119885

119894≜ [119909

119879

119894

120597120572

119894minus1

120597119909

1

120597120572

119894minus1

120597119909

119894minus1

120593

119894minus1]

119879

sub 119877

2119894

(39)

By introducing the direct variable (120597120572

119894minus1120597119909

1)

(120597120572

119894minus1120597119909

119894minus1) 120593119894minus1

we can make the number of neuralnetworks minimized RBF neural network119882119879

119894119878

119894(119885

119894) is used

to approximate the unknown function ℎ119894(119885

119894) and 120572lowast

119894can be

expressed as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus119882

lowast119879

119894119878

119894(119885

119894) minus 119890

119894 (40)

where |119890119894| le 119890

lowast

119894is estimated error and meets 119890lowast

119894gt 0

Because 119882lowast119894

is unknown select the following virtualcontrol law

120572

119894= minus119911

119894minus1minus 119888

119894119911

119894minus

119882

119879

119894119878

119894(119885

119894)

(41)

where119882lowast119894is the estimated value of 119882

119894 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus

119882

119879

119894119878

119894119911

119894+

119882

119879

119894Γ

minus1

119894

119882

119894

(42)

where 119882119894=

119882

119894minus119882

lowast

119894

The following adaptive law can be selected as

119882

119894=

119882

119894= Γ

119894[119878

119894(119885

119894) 119911

119894minus 120590

119894

119882

119894]

(43)

where 120590119894gt 0 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(44)

Let 119888119894= 119888

1198940+ 119888

1198941 1198881198940 119888

1198941gt 0 then (44) can be rewritten as

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus (119888

1198940+

119894(119909

119894)

2119892

2

119894(119909

119894)

) 119911

2

119894

minus 119888

1198941119911

2

119894+ 119911

119894119890

119894+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(45)

According to the complete square formula

minus120590

119894

119882

119879

119894

119882

119894= minus120590

119894

119882

119879

119894(

119882

119894+119882

lowast

119894)

le minus120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

le minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

minus119888

1198941119911

2

119894+ 119911

119894119890

119894le minus119888

1198941119911

2

119894+ 119911

119894

1003816

1003816

1003816

1003816

119890

119894

1003816

1003816

1003816

1003816

le

119890

2

119894

4119888

1198941

le

119890

lowast2

119894

4119888

1198941

(46)

6 Computational Intelligence and Neuroscience

Because minus(1198881198940+ (

1198942119892

2

119894))119911

2

119894le minus(119888

1198940minus (119892

1198941198892119892

2

119894119898))119911

2

119894 then

we can make (119888lowast1198940≜ 119888

1198940minus (119892

1198941198892119892

2

119894119898)) gt 0 by selecting the

proper 1198881198940 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

lowast

1198940119911

2

119894minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

119894

4119888

1198941

+

119875

lowast2

119894

2

le 119911

119894119911

119894+1minus

119894

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119894

sum

119896=1

119875

lowast2

119896

2

(47)

The cross coupling 119911119894119911

119894+1in (47) will be eliminated in the

next step

Step 119899 The derivative of 119911119899= 119909

119899minus 120572

119899minus1can be calculated as

119899= 119891

119899(119909

119899) + 119892

119899(119909

119899minus1) 119906 minus

119899minus1 (48)

where

119899minus1=

119899minus1

sum

119896=1

120597120572

119899minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120601

119899minus1 (49)

where

120601

119899minus1=

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597119909

119889

)

119889

+

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(50)

Consider the following Lyapunov function

119881

119899= 119881

119899minus1+

1

2119892

119899(119909

119899)

119911

2

119899+

1

2

119882

119879

119899Γ

minus1

119899

119882

119899 (51)

where Γ119899= Γ

119879

119899gt 0 is an adaptive gain matrix Then the

derivation of 119881119899can be calculated as

119881

119899=

119881

119899minus1+

119911

119899

119899

119892

119894(119909

119894)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+

119911

119899

119892

119899(119909

119899)

(119891

119899(119909

119899) + 119892

119899(119909

119899) 119906 + 119889

119899minus

119899minus1)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

)

+

119911

119899119889

119899

119892

119899(119909

119899)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(52)

According to Assumption 1 we can get

119881

119899le

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119894(119909

119894) minus

119899minus1

119892

119894(119909

119894)

)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(53)

There is an ideal feedback control law as

119906

lowast= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (54)

where 119888119899gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119899(119909

119899) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

119906

lowast from (54) we can see the unknown part is a smoothfunction of 119909

119899and

119899minus1 and let

ℎ

119899(119885

119894) ≜

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

119899120588

2

119899

2119892

2

119899(119909

119899)

(55)

where 119885119899≜ [119909

119879

119899 120597120572

119899minus1120597119909

1 120597120572

119899minus1120597119909

119899minus1 120601

119899minus1]

119879sub 119877

2119899RBF neural network119882119879

119899119878

119899(119885

119899) is used to approximate the

unknown function ℎ119899(119885

119899) and 119906lowast can be expressed as

119906

lowast= minus119911

119899minus1minus 119888

119899119911

119899minus119882

lowast119879

119899119878

119899(119885

119899) minus 119890

119899 (56)

where |119890119899| le 119890

lowast

119899is estimated error and meets 119890lowast

119899gt 0

Because 119882lowast119899

is unknown select the following virtualcontrol law

119906 = minus119911

119899minus1minus 119888

119899119911

119899minus

119882

119879

119899119878

119899(119885

119899)

(57)

where 119882119894is the estimated value of119882lowast

119894 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus

119882

119879

119899119878

119899119911

119899+

119882

119879

119899Γ

minus1

119899

119882

119899

(58)

where 119882119899=

119882

119899minus119882

lowast

119899

The following adaptive law can be selected as

119882

119899=

119882

119899= Γ

119899[119878

119899(119885

119899) 119911

119899minus 120590

119899

119882

119899]

(59)

where 120590119899gt 0 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(60)

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

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Computational Intelligence and Neuroscience 5

Then the derivation of 119881119894can be calculated as

119881

119894=

119881

119894minus1+

119911

119894

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+

119911

119894

119892

119894(119909

119894)

(119891

119894(119909

119894) + 119892

119894(119909

119894) 119909

119894+1+ 119889

119894minus

119894minus1)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

) +

119911

119894119889

119894

119892

119894(119909

119894)

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(35)

According to Assumption 1 we can get

119881

119894le

119881

119894minus1+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

=

119881

119894minus1

+ 119911

119894(119911

119894+1+ 120572

119894+

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

2

119894120588

2

119894

2119892

2

119894(119909

119894)

)

+

119875

lowast2

119894

2

+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+

119882

119879

119894Γ

minus1

119894

119882

119894

(36)

There is an ideal feedback control law as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (37)

where 119888119894gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119894(119909

119894) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

120572

lowast

119894 from (37) we can see that the unknown part is a smooth

function of 119909119894and

119894minus1 and let

ℎ

119894(119885

119894) ≜

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

(38)

where

119885

119894≜ [119909

119879

119894

120597120572

119894minus1

120597119909

1

120597120572

119894minus1

120597119909

119894minus1

120593

119894minus1]

119879

sub 119877

2119894

(39)

By introducing the direct variable (120597120572

119894minus1120597119909

1)

(120597120572

119894minus1120597119909

119894minus1) 120593119894minus1

we can make the number of neuralnetworks minimized RBF neural network119882119879

119894119878

119894(119885

119894) is used

to approximate the unknown function ℎ119894(119885

119894) and 120572lowast

119894can be

expressed as

120572

lowast

119894= minus119911

119894minus1minus 119888

119894119911

119894minus119882

lowast119879

119894119878

119894(119885

119894) minus 119890

119894 (40)

where |119890119894| le 119890

lowast

119894is estimated error and meets 119890lowast

119894gt 0

Because 119882lowast119894

is unknown select the following virtualcontrol law

120572

119894= minus119911

119894minus1minus 119888

119894119911

119894minus

119882

119879

119894119878

119894(119885

119894)

(41)

where119882lowast119894is the estimated value of 119882

119894 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus

119882

119879

119894119878

119894119911

119894+

119882

119879

119894Γ

minus1

119894

119882

119894

(42)

where 119882119894=

119882

119894minus119882

lowast

119894

The following adaptive law can be selected as

119882

119894=

119882

119894= Γ

119894[119878

119894(119885

119894) 119911

119894minus 120590

119894

119882

119894]

(43)

where 120590119894gt 0 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

119894119911

2

119894+

119894(119909

119894) 119911

2

119894

2119892

2

119894(119909

119894)

+ 119911

119894119890

119894

+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(44)

Let 119888119894= 119888

1198940+ 119888

1198941 1198881198940 119888

1198941gt 0 then (44) can be rewritten as

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus (119888

1198940+

119894(119909

119894)

2119892

2

119894(119909

119894)

) 119911

2

119894

minus 119888

1198941119911

2

119894+ 119911

119894119890

119894+

119875

lowast2

119894

2

minus 120590

119894

119882

119879

119894

119882

119894

(45)

According to the complete square formula

minus120590

119894

119882

119879

119894

119882

119894= minus120590

119894

119882

119879

119894(

119882

119894+119882

lowast

119894)

le minus120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

le minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

minus119888

1198941119911

2

119894+ 119911

119894119890

119894le minus119888

1198941119911

2

119894+ 119911

119894

1003816

1003816

1003816

1003816

119890

119894

1003816

1003816

1003816

1003816

le

119890

2

119894

4119888

1198941

le

119890

lowast2

119894

4119888

1198941

(46)

6 Computational Intelligence and Neuroscience

Because minus(1198881198940+ (

1198942119892

2

119894))119911

2

119894le minus(119888

1198940minus (119892

1198941198892119892

2

119894119898))119911

2

119894 then

we can make (119888lowast1198940≜ 119888

1198940minus (119892

1198941198892119892

2

119894119898)) gt 0 by selecting the

proper 1198881198940 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

lowast

1198940119911

2

119894minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

119894

4119888

1198941

+

119875

lowast2

119894

2

le 119911

119894119911

119894+1minus

119894

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119894

sum

119896=1

119875

lowast2

119896

2

(47)

The cross coupling 119911119894119911

119894+1in (47) will be eliminated in the

next step

Step 119899 The derivative of 119911119899= 119909

119899minus 120572

119899minus1can be calculated as

119899= 119891

119899(119909

119899) + 119892

119899(119909

119899minus1) 119906 minus

119899minus1 (48)

where

119899minus1=

119899minus1

sum

119896=1

120597120572

119899minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120601

119899minus1 (49)

where

120601

119899minus1=

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597119909

119889

)

119889

+

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(50)

Consider the following Lyapunov function

119881

119899= 119881

119899minus1+

1

2119892

119899(119909

119899)

119911

2

119899+

1

2

119882

119879

119899Γ

minus1

119899

119882

119899 (51)

where Γ119899= Γ

119879

119899gt 0 is an adaptive gain matrix Then the

derivation of 119881119899can be calculated as

119881

119899=

119881

119899minus1+

119911

119899

119899

119892

119894(119909

119894)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+

119911

119899

119892

119899(119909

119899)

(119891

119899(119909

119899) + 119892

119899(119909

119899) 119906 + 119889

119899minus

119899minus1)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

)

+

119911

119899119889

119899

119892

119899(119909

119899)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(52)

According to Assumption 1 we can get

119881

119899le

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119894(119909

119894) minus

119899minus1

119892

119894(119909

119894)

)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(53)

There is an ideal feedback control law as

119906

lowast= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (54)

where 119888119899gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119899(119909

119899) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

119906

lowast from (54) we can see the unknown part is a smoothfunction of 119909

119899and

119899minus1 and let

ℎ

119899(119885

119894) ≜

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

119899120588

2

119899

2119892

2

119899(119909

119899)

(55)

where 119885119899≜ [119909

119879

119899 120597120572

119899minus1120597119909

1 120597120572

119899minus1120597119909

119899minus1 120601

119899minus1]

119879sub 119877

2119899RBF neural network119882119879

119899119878

119899(119885

119899) is used to approximate the

unknown function ℎ119899(119885

119899) and 119906lowast can be expressed as

119906

lowast= minus119911

119899minus1minus 119888

119899119911

119899minus119882

lowast119879

119899119878

119899(119885

119899) minus 119890

119899 (56)

where |119890119899| le 119890

lowast

119899is estimated error and meets 119890lowast

119899gt 0

Because 119882lowast119899

is unknown select the following virtualcontrol law

119906 = minus119911

119899minus1minus 119888

119899119911

119899minus

119882

119879

119899119878

119899(119885

119899)

(57)

where 119882119894is the estimated value of119882lowast

119894 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus

119882

119879

119899119878

119899119911

119899+

119882

119879

119899Γ

minus1

119899

119882

119899

(58)

where 119882119899=

119882

119899minus119882

lowast

119899

The following adaptive law can be selected as

119882

119899=

119882

119899= Γ

119899[119878

119899(119885

119899) 119911

119899minus 120590

119899

119882

119899]

(59)

where 120590119899gt 0 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(60)

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

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Volume 2014

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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6 Computational Intelligence and Neuroscience

Because minus(1198881198940+ (

1198942119892

2

119894))119911

2

119894le minus(119888

1198940minus (119892

1198941198892119892

2

119894119898))119911

2

119894 then

we can make (119888lowast1198940≜ 119888

1198940minus (119892

1198941198892119892

2

119894119898)) gt 0 by selecting the

proper 1198881198940 then

119881

119894le

119881

119894minus1minus 119911

119894minus1119911

119894+ 119911

119894119911

119894+1minus 119888

lowast

1198940119911

2

119894minus

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

119890

lowast2

119894

4119888

1198941

+

119875

lowast2

119894

2

le 119911

119894119911

119894+1minus

119894

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119894

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119894

sum

119896=1

119875

lowast2

119896

2

(47)

The cross coupling 119911119894119911

119894+1in (47) will be eliminated in the

next step

Step 119899 The derivative of 119911119899= 119909

119899minus 120572

119899minus1can be calculated as

119899= 119891

119899(119909

119899) + 119892

119899(119909

119899minus1) 119906 minus

119899minus1 (48)

where

119899minus1=

119899minus1

sum

119896=1

120597120572

119899minus1

120597119909

119896

(119892

119896(119909

119896) 119909

119896+1+ 119891

119896(119909

119896)) + 120601

119899minus1 (49)

where

120601

119899minus1=

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597119909

119889

)

119889

+

119899minus1

sum

119896=1

(

120597120572

119899minus1

120597

119882

119896

) [Γ

119896(119878

119896(119885

119896) 119911

119896minus 120590

119896

119882

119896)]

(50)

Consider the following Lyapunov function

119881

119899= 119881

119899minus1+

1

2119892

119899(119909

119899)

119911

2

119899+

1

2

119882

119879

119899Γ

minus1

119899

119882

119899 (51)

where Γ119899= Γ

119879

119899gt 0 is an adaptive gain matrix Then the

derivation of 119881119899can be calculated as

119881

119899=

119881

119899minus1+

119911

119899

119899

119892

119894(119909

119894)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+

119911

119899

119892

119899(119909

119899)

(119891

119899(119909

119899) + 119892

119899(119909

119899) 119906 + 119889

119899minus

119899minus1)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

)

+

119911

119899119889

119899

119892

119899(119909

119899)

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(52)

According to Assumption 1 we can get

119881

119899le

119881

119899minus1+ 119911

119899(119911

119899+1+ 119906 +

119891

119894(119909

119894) minus

119899minus1

119892

119894(119909

119894)

)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

=

119881

119899minus1

+ 119911

119899(119911

119899+1+ 119906 +

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

2

119899120588

2

119899

2119892

2

119899(119909

119899)

)

+

119875

lowast2

119899

2

+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+

119882

119879

119899Γ

minus1

119899

119882

119899

(53)

There is an ideal feedback control law as

119906

lowast= minus119911

119894minus1minus 119888

119894119911

119894minus [

119891

119894(119909

119894) minus

119894minus1

119892

119894(119909

119894)

+

119911

119894120588

2

119894

2119892

2

119894(119909

119894)

] (54)

where 119888119899gt 0 is designed controller parameter

Because of the unknown smooth functions 119891119899(119909

119899) and

119892

119894(119909

119894) we cannot actually get the ideal feedback control law

119906

lowast from (54) we can see the unknown part is a smoothfunction of 119909

119899and

119899minus1 and let

ℎ

119899(119885

119894) ≜

119891

119899(119909

119899) minus

119899minus1

119892

119899(119909

119899)

+

119911

119899120588

2

119899

2119892

2

119899(119909

119899)

(55)

where 119885119899≜ [119909

119879

119899 120597120572

119899minus1120597119909

1 120597120572

119899minus1120597119909

119899minus1 120601

119899minus1]

119879sub 119877

2119899RBF neural network119882119879

119899119878

119899(119885

119899) is used to approximate the

unknown function ℎ119899(119885

119899) and 119906lowast can be expressed as

119906

lowast= minus119911

119899minus1minus 119888

119899119911

119899minus119882

lowast119879

119899119878

119899(119885

119899) minus 119890

119899 (56)

where |119890119899| le 119890

lowast

119899is estimated error and meets 119890lowast

119899gt 0

Because 119882lowast119899

is unknown select the following virtualcontrol law

119906 = minus119911

119899minus1minus 119888

119899119911

119899minus

119882

119879

119899119878

119899(119885

119899)

(57)

where 119882119894is the estimated value of119882lowast

119894 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus

119882

119879

119899119878

119899119911

119899+

119882

119879

119899Γ

minus1

119899

119882

119899

(58)

where 119882119899=

119882

119899minus119882

lowast

119899

The following adaptive law can be selected as

119882

119899=

119882

119899= Γ

119899[119878

119899(119885

119899) 119911

119899minus 120590

119899

119882

119899]

(59)

where 120590119899gt 0 then

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus 119888

119899119911

2

119899+

119899(119909

119899) 119911

2

119899

2119892

2

119899(119909

119899)

+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(60)

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 7

Let 119888119899= 119888

1198990+ 119888

1198991 1198881198990 119888

1198991gt 0 (60) can be rewritten as

119881

119899le

119881

119899minus1minus 119911

119899minus1119911

119899+ 119911

119899119911

119899+1minus (119888

1198990+

119899(119909

119899)

2119892

2

119899(119909

119899)

) 119911

2

119899

minus 119888

1198991119911

2

119899+ 119911

119899119890

119899+

119875

lowast2

119899

2

minus 120590

119899

119882

119879

119899

119882

119899

(61)

According to the complete square formula

minus120590

119899

119882

119879

119899

119882

119899= minus120590

119899

119882

119879

119899(

119882

119899+119882

lowast

119899)

le minus120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

+ 120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

le minus

120590

119899

1003817

1003817

1003817

1003817

1003817

119882

119899

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

119899

1003817

1003817

1003817

1003817

119882

lowast

119899

1003817

1003817

1003817

1003817

2

2

minus119888

1198991119911

2

119899+ 119911

119899119890

119899le minus119888

1198991119911

2

119899+ 119911

119899

1003816

1003816

1003816

1003816

119890

119899

1003816

1003816

1003816

1003816

le

119890

2

119899

4119888

1198991

le

119890

lowast2

119899

4119888

1198991

(62)

Becauseminus(1198881198990+(

1198992119892

2

119899))119911

2

119899le minus(119888

1198990minus(119892

1198991198892119892

2

119899119898))119911

2

119899 then

we can make (119888lowast1198990≜ 119888

1198990minus (119892

1198991198892119892

2

119899119898)) gt 0 by selecting the

proper 1198881198990 then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

119882

lowast

119896

1003817

1003817

1003817

1003817

2

2

+

119899

sum

119896=1

119890

lowast2

119896

4119888

1198961

+

119899

sum

119896=1

119875

lowast2

119896

2

(63)

Let 120575 ≜ sum

119899

119896=1(120590

119896119882

lowast

119896

22) + sum

119899

119896=1(119890

lowast2

1198964119888

1198961) +

sum

119899

119896=1(119901

lowast2

1198962) 119888lowast1198960ge (1205742119892

119896119898) 1198881198960gt (1205742119892

119896119898) + (119892

1198961198892119892

2

119896119898)

119896 = 1 2 119899 where 120574 gt 0 120590119896ge 120574120582maxΓ

minus1

119896 119896 = 1 2 119899

then

119881

119899le minus

119899

sum

119896=1

119888

lowast

1198960119911

2

119896minus

119899

sum

119896=1

120590

119896

1003817

1003817

1003817

1003817

1003817

119882

119896

1003817

1003817

1003817

1003817

1003817

2

2

+ 120575

le minus

119899

sum

119896=1

120574

2119892

119896119898

119911

2

119896minus

119899

sum

119896=1

120574

119882

119879

119896Γ

minus1

119896

119882

119896

2119892

119896119898

+ 120575

le minus120574

[

[

119899

sum

119896=1

1

2119892

119896

119911

2

119896+

119899

sum

119896=1

119882

119879

119896Γ

minus1

119896

119882

119896

2

]

]

+ 120575

le minus120574119881

119899+ 120575

(64)

The stability and control performance of the closed-loopadaptive system are demonstrated by the following theorem

Theorem 2 In the initial conditions by formula (1) referencemodel (2) control law (57) and neural network weight updaterate in (12) (27) (43) and (59) supposing that there is a largeenough set of closed sets Ω

119894isin 119877

2119894 119894 = 1 2 119899 for any givenmoment 119905 ge 0 making 119885

119894isin Ω

119894 the following conclusions can

be obtained as follows

(1) The signal of the whole closed-loop system is boundedand the state variable 119909

119899and the neural network

estimation errors 1198821198791

119882

119879

119899will eventually converge

to the closed set as follows

Ω

1199041≜ 119909

119899

119882

1

119882

119899| 119881 lt

120575

120574

119909

119889isin Ω

119889 (65)

(2) By choosing the proper control parameters the outputtracking error 119910(119905) minus119910

1198891(119905) is close to a small neighbor-

hood of zero [21]

3 Adaptive Robust Neural Network Controlfor Ship Course

31 Problem Formulation This section introduces a sim-plified dynamic model of an underactuated surface vehiclewith only one control input 120575 for heading control A surfaceship usually has three degrees of freedom for path followingcontrol in horizontal plane Assuming that the vessel hasthree planes of symmetry for most underactuated vesselshave portstarboard symmetry it can be neglected to simplifythe vessel model for controller design The detailed modelwhich considers the environment disturbances can be set asfollows

= 119880 sin120595

120595 = 119903

119903 = minus

1

119879

119903 minus

120572

119879

119903

3+

119870

119879

120575 + Δ

119910

1= 119910

119910

2= 120595

(66)

where 119910 denotes transverse displacement in the earth inertialcoordinates 119880 =

radic

119906

2+ V2 is resultant velocity of ship 120595

is course angle 119903 is yawing angular velocity 119870119879 representperformance index for ship steering 120572 is coefficient ofnonlinear term 120575 is control rudder angle 119910

1 119910

2represent

system outputThe control objective is to design the controller 120575 to make

the control output 119910 120595 achieve the setting value (119910119889 120595

119889)

Because the dimension of the system control input is less thanthe degree of freedom of the system it is an underactuatedsystem

32 Dynamic Controller Design Selection of coordinatetransformation is as follows

119908

119890= 120595 + arcsin(

119896119910

radic

1 + (119896119910)

2

) (67)

Theoriginal system can be transformed into a single inputsingle output system

1=

119896

1 + (119896119910)

2+ 119909

2

2= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ

(68)

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Computational Intelligence and Neuroscience

where 1198861= 1119879 119886

2= 120572119879 119887 = 119870119879 119909

1= 119908

119890 1199092= 119903 119906 = 120575

and the output of whole system is 1199091

For system model (67) and (68) the controller design iscarried out by using backstepping method

Step 1 Let 1199111= 119909

1 1199091198891= 0 then

1=

119896

1 + (119896119910)

2+ 119909

2 (69)

For the subsystem 119911

1 120572lowast1≜ 119909

2is chosen as virtual control

input Select the Lyapunov function 1198811199111= (12)119911

2

1 and there

is

119881

1199111= 119911

1

1= (

119896

1 + (119896119910)

2+ 119909

2)119911

1 (70)

Let 1199112= 119909

2minus 120572

1 then 119909

2= 119911

2+ 120572

1

119881

1199111= (

119896

1 + (119896119910)

2+ 119911

2+ 120572

1)119911

1 (71)

Select the following virtual control law

120572

lowast

1= minus119888

1119911

1minus

119896

1 + (119896119910)

2 (72)

119881

1199111= 119911

1119911

2minus 119888

1119911

2

1 because 119896(1 + (119896119910)2) is unknown

function ℎ1(119885

1) = 119896(1 + (119896119910)

2) and we will adopt RBF

NN to estimate ℎ1(119885

1) and get ℎ

1(119885

1) = 119882

lowast119879

1119878

1(119885

1) + 120576

1 But

the actual use of theNN for the system is ℎ1(119885

1) =

119882

119879

1119878

1(119885

1)

Actual virtual control input is 1205721= minus119888

1119911

1minus

119882

119879

1119878

1(119885

1) then

1=

119896

1 + (119896119910)

2+ 119911

2+ 120572

1

= (119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1)

(73)

where 1198821=

119882

1minus119882

lowast

1

Select Lyapunov function as

119881

1= 119881

1199111+

1

2

119882

119879

1Γ

minus1

119882

1 (74)

then

119881

1=

119881

1199111+

119882

1Γ

minus1

119882

1le 119911

1(119911

2+ 120572

1+ ℎ

1(119885

1))

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 119882

lowast

1119878

1(119885

1) + 120576

1]

+

119882

1Γ

minus1

119882

1

= 119911

1[119911

2minus 119888

1119911

1minus

119882

1119878

1(119885

1) + 120576

1] +

119882

1Γ

minus1

119882

1

(75)

The adaptive law of neural network can be designed as

119882

1=

119882

1= Γ

1[119878

1(119885

1) 119911

1minus 120590

1

119882

1]

(76)

where 1205901gt 0 Let 119888

1= 119888

10+ 119888

11 where 119888

10 119888

11gt 0

Furthermore

119881

1= 119911

1119911

2minus 119888

10119911

2

1minus 119888

11119911

2

1+ 119911

1120576

1minus 120590

1

119882

119879

1

119882

1

(77)

then

minus120590

1

119882

119879

1

119882

1= minus120590

1

119882

119879

1(

119882

1+119882

lowast

1)

le minus120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

+ 120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

le minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

(78)

because

minus119888

11119911

2

1+ 119911

1120576

1le minus119888

11119911

2

1+ 119911

1

1003816

1003816

1003816

1003816

120576

1

1003816

1003816

1003816

1003816

le

120576

2

1

4119888

11

le

120576

lowast2

1

4119888

11

(79)

Finally we can get

119881

1lt 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

(80)

Step 2 Let 1199112= 119909

2minus 120572

1 derivation of 119911

2can be calculated as

2= 119891

2(119909

2) + 119892

2(119909

2) 119906 + Δ minus

1

= minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1

(81)

Because 1198811199112= (12119887)119911

2

2 then

119881

1199112=

1

119887

119911

2

2=

1

119887

119911

2(minus119886

1119909

2minus 119886

2119909

2

3+ 119887119906 + Δ minus

1)

= 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1)] +

Δ

119887

119911

2

le 119911

2[119906 +

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

)]

+

119901

2

2

(82)

where Δ le 119901 sdot 120588(119909) 119901 is unknown parameter 120588(119909) is knownnonlinear function and then

119906

lowast= minus119911

1minus 119888

2119911

2minus

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (83)

Let

ℎ

2(119885

2) =

1

119887

(minus119886

1119909

2minus 119886

2119909

2

3minus

1+

120588

2119911

2

2119887

) (84)

Equation (83) can be rewritten as

119906

lowast= minus119911

1minus 119888

2119911

2minus ℎ

2(119885

2) (85)

In the same way we use RBF NN estimate ℎ2(119885

2)

ℎ

2(119885

2) = 119882

lowast

2

119879119878

2(119885

2) + 120576

2

(86)

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 9

The actual use of theNN for the system and controller canbe expressed as

ℎ

2(119885

2) =

119882

119879

2119878

2(119885

2)

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(87)

Select Lyapunov function as

119881

2= 119881

1+ 119881

1199112+

1

2

119882

119879

2Γ

minus1

119882

2 (88)

The derivation of 1198812can be calculated as

119881

2=

119881

1+

119881

1199112+

119882

1Γ

minus1

119882

1

le 119911

1119911

2minus 119888

lowast

10119911

2

1minus

120590

1

1003817

1003817

1003817

1003817

1003817

119882

1

1003817

1003817

1003817

1003817

1003817

2

2

+

120590

1

1003817

1003817

1003817

1003817

119882

lowast

1

1003817

1003817

1003817

1003817

2

2

+

120576

lowast2

1

4119888

11

+ 119911

2[minus119911

1minus 119888

2119911

2minus

119882

2119878

2(119885

2) + 119882

lowast

2119878

2(119885

2) + 120576

2]

+

119901

2

2

+

119882

1Γ

minus1

119882

1

= minus

2

sum

119894=1

119888

lowast

1198940119911

2

119894minus

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

1003817

119882

119894

1003817

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120590

119894

1003817

1003817

1003817

1003817

119882

lowast

119894

1003817

1003817

1003817

1003817

2

2

+

2

sum

119894=1

120576

lowast2

119894

4119888

11

+

119901

2

2

(89)

Therefore all signals in the close loop of course trackingsystem are stable and the tracking errors can be made arbi-trarily small by selecting appropriate controller parametersSo the final control law can be designed as

119906 = 119911

1minus 119888

2119911

2minus

119882

119879

2119878

2(119885

2)

(90)

4 Numerical Simulations and Analysis

The simulation experiment can be operated based on anexperimental shipThe nonlinearmathematicalmodel for theship has been presented in [22] which captures the funda-mental characteristics of dynamics and offers good maneu-verability in the open-loop test To illustrate the effectivenessof the theoretical results the proposed control scheme isimplemented and simulated with the above nonlinear modelwith tracking task

The characteristic parameters of the ship used in thesimulation are given as 119870 = 0478 119879 = 216 and 120572 = 30Neural network contains 25 neurons that is 119897

1= 25 the

center vector 120583119897(119897 = 1 2 119897

1) is uniformly distributed in

thewidth [minus2 2]times[minus2 2]times[minus2 2] Neural network1198821198792119878

2(119885

2)

contains 135 neurons that is 1198972= 125 the center vector

120583

119897(119897 = 1 2 119897

2) is uniformly distributed in the width

[minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times [minus4 4] times

[minus4 0] times [minus6 6] The controller design parameters are givenas follows which satisfy the condition mentioned in designprocedure 119896 = 01394 119888

1= 4 119888

2= 120 Γ

1= diag3

Γ

2= diag4 and 120590

1= 4 120590

2= 2 The initial linear and

0 20 40 60 80 100 120 140 160 180minus30

minus20

minus10

0

10

20

30

40

50

Desired trajectoryShip trajectory

Y(m

)

X (m)

Figure 1 Ship tracking performance of proposed control method

angular velocity of ship used in the simulation are given as[119906 V 119903]119879 = [01 0 0]

119879 [119909 119910 120595]119879 = [10 30 minus1205874]

119879 is theinitial position and orientation vector of ship and the desiredvelocity of ship is given as 119906

119889= 1 (ms) We choose the

reference trajectory as 10 cos120596119905In order to further verify the validity of the proposed

control method the algorithm of this paper is compared withthe simulation results in [12] So the robustness of trajec-tory tracking controller against the disturbance and modeluncertainties can be evaluated All the simulation resultsare depicted in Figures 1ndash4 Figure 1 shows the trajectorytracking of ship with the given path and the ship can trackand converge to the reference path with more accuracy in[12] Figure 2 plots the position tracking errors the along-track and cross-track errors asymptotically converge to zerofaster Figure 3 gives the control inputs response Surge swayyaw velocities and orientation of ship during the trajectorytracking control process are plotted in Figure 4 which givesa clear insight into the model response involved in nonlineardynamics

5 Conclusions

In this paper we proposed a solution to the course controlof underactuated surface vessel Firstly the direct adaptiveneural network control and its application are introducedThen the backstepping controller with robust neural networkis designed to deal with the uncertain and underactuatedcharacteristics for the ship Neural networks are adopted todetermine the parameters of the unknown part of the idealvirtual control and the ideal control even the weights ofneural network are updated by using adaptive techniqueFinally uniform stability for the convergence of trackingerrors has been proven through Lyapunov stability theory

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Computational Intelligence and Neuroscience

0 20 40 60 80 100 120 140 160 180 200minus5

0

5

10

0 20 40 60 80 100 120 140 160 180 200minus10

0

10

20

30xe

(m)

ye

(m)

t (s)t (s)

Figure 2 Tracking errors of surge and sway

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

0 20 40 60 80 100 120 140 160 180 200minus500

0

500

t (s)t (s)

F(N

)

T(N

lowastm

)Figure 3 Control force and torque of ship

0 20 40 60 80 100 120 140 160 180 200012

0 20 40 60 80 100 120 140 160 180 200minus05

005

0 20 40 60 80 100 120 140 160 180 200minus20

020

0 20 40 60 80 100 120 140 160 180 2000

200400

t (s)

t (s)

t (s)

t (s)

u(m

s)

(m

s)

r(∘

s)

120595(∘

)

Figure 4 State changing curves of ship

The simulation results illustrate the performance of theproposed course tracking controller with good precision

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China under Grant 51309067E091002

References

[1] T I Fossen ldquoA survey on nonlinear ship control from theoryto practicerdquo in Proceedings of the 5th IFAC Conference onManoeuvring and Control of Marine Craft pp 1ndash16 AalborgDenmark 2000

[2] L Lapierre and D Soetanto ldquoNonlinear path-following controlof an AUVrdquoOcean Engineering vol 34 no 11-12 pp 1734ndash17442007

[3] J He-ming SWen-long andC Zi-yin ldquoBottom following con-trol of underactuated AUV based on nonlinear backsteppingmethodrdquoAdvances in Information Sciences and Service Sciencesvol 4 no 12 pp 362ndash369 2012

[4] B Sun D Zhu and S X Yang ldquoA bio-inspired cascadedapproach for three-dimensional tracking control of unmannedunderwater vehiclesrdquo International Journal of Robotics andAutomation vol 29 no 4 2014

[5] T I Fossen ldquoHigh performance ship autopilot with wave filterrdquoin Proceedings of the 10th International Ship Control SystemsSymposium (SCSS rsquo93) pp 2271ndash2285 Ottawa Canada 1993

[6] C Y Tzeng G C Goodwin and S Crisafulli ldquoFeedback lin-earization design of a ship steering autopilot with saturating andslew rate limiting actuatorrdquo International Journal of AdaptiveControl and Signal Processing vol 13 no 1 pp 23ndash30 1999

[7] A Witkowska and R Smierzchalski ldquoNonlinear backsteppingship course controllerrdquo International Journal of Automation andComputing vol 6 no 3 pp 277ndash284 2009

[8] Y S Yang ldquoRobust adaptive control algorithm applied to shipsteering autopilot with uncertain nonlinear systemrdquo Shipbuild-ing of China vol 41 no 1 pp 21ndash25 2000 (Chinese)

[9] J He-Ming S Wen-Long and C Zi-Yin ldquoNonlinear backstep-ping control of underactuated AUV in diving planerdquo Advancesin Information Sciences and Service Sciences vol 4 no 9 pp214ndash221 2012

[10] J-H Li P-M Lee B-H Jun and Y-K Lim ldquoPoint-to-pointnavigation of underactuated shipsrdquo Automatica vol 44 no 12pp 3201ndash3205 2008

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience 11

[11] M Bao-li ldquoGlobal K-exponential asymptotic stabilization ofunderactuated surface vesselsrdquo Systems amp Control Letters vol58 no 3 pp 194ndash201 2009

[12] L-J Zhang H-M Jia and X Qi ldquoNNFFC-adaptive outputfeedback trajectory tracking control for a surface ship at highspeedrdquo Ocean Engineering vol 38 no 13 pp 1430ndash1438 2011

[13] K D Do Z P Jiang and J Pan ldquoRobust adaptive path followingof underactuated shipsrdquoAutomatica vol 40 no 6 pp 929ndash9442004

[14] K D Do and J Pan ldquoState- and output-feedback robust path-following controllers for underactuated ships using Serret-Frenet framerdquo Ocean Engineering vol 31 no 5-6 pp 587ndash6132004

[15] K D Do ldquoPractical control of underactuated shipsrdquo OceanEngineering vol 37 no 13 pp 1111ndash1119 2010

[16] Y-L Liao L Wan and J-Y Zhuang ldquoBackstepping dynamicalsliding mode control method for the path following of theunderactuated surface vesselrdquo Procedia Engineering vol 15 pp256ndash263 2011

[17] K D Do and J Pan ldquoGlobal robust adaptive path following ofunderactuated shipsrdquo Automatica vol 42 no 10 pp 1713ndash17222006

[18] V Sakhre S Jain V S Sapkal and D P Agarwal ldquoFuzzycounter propagation neural network control for a class ofnonlinear dynamical systemsrdquo Computational Intelligence andNeuroscience vol 2015 Article ID 719620 12 pages 2015

[19] C-Z Pan S X Yang X-Z Lai and L Zhou ldquoAn efficient neuralnetwork based tracking controller for autonomous underwatervehicles subject to unknown dynamicsrdquo in Proceedings of the26th Chinese Control and Decision Conference (CCDC rsquo14) pp3300ndash3305 IEEE Changsha China June 2014

[20] L A Wulandhari A Wibowo and M I Desa ldquoImprovementof adaptive GAs and back propagation ANNs performance incondition diagnosis of multiple bearing system using grey rela-tional analysisrdquo Computational Intelligence and Neurosciencevol 2014 Article ID 419743 11 pages 2014

[21] M M Polycarpou ldquoStable adaptive neural control scheme fornonlinear systemsrdquo IEEE Transactions on Automatic Controlvol 41 no 3 pp 447ndash451 1996

[22] L Moreira T I Fossen and C Guedes Soares ldquoPath followingcontrol system for a tanker ship modelrdquoOcean Engineering vol34 no 14-15 pp 2074ndash2085 2007

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014