Research Article Adaptive Fuzzy Tracking Control for a...

Research ArticleAdaptive Fuzzy Tracking Control for a Class of UncertainNonlinear Time-Delayed Systems with Saturation Constrains

Yu-Jun Zhang1 Li-Bing Wu2 Hong-Yang Zhao3 Xiao-Dong Hu3

Wen-Yu Zhang4 and Dong-Ying Ju3

1Software School University of Science and Technology Liaoning Anshan Liaoning 114051 China2School of Science University of Science and Technology Liaoning Anshan Liaoning 114051 China3School of Material and Metallurgy University of Science and Technology Liaoning Anshan Liaoning 114051 China4College of International of Finance and Banking University of Science and Technology Liaoning Anshan Liaoning 114051 China

Correspondence should be addressed to Dong-Ying Ju dyjusitacjp

Received 28 September 2016 Revised 15 November 2016 Accepted 22 November 2016

Academic Editor Xian Zhang

Copyright copy 2016 Yu-Jun Zhang 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

In this paper the problem of adaptive fuzzy tracking control is considered for a class of uncertain nonaffine nonlinear systems withexternal disturbances multiple time delays and nonsymmetric saturation constrains First the mean value theorem is employedto deal with the nonaffine term with input nonlinearity Then a new adaptive fuzzy tracking controller with parameter updatinglaws is designed by using fuzzy approximation technique Moreover it is shown that all the closed-loop signals are bounded andthe tracking errors can asymptotically converge to zero via the Lyapunov stability analysis Finally the simulation example for vander Pol oscillator system is worked out to verify the effectiveness of the proposed adaptive fuzzy design approach

1 Introduction

Over the past decades the modeling and control designproblem of nonlinear systems have attracted considerableattentions because of the extensively practical applicationsConsequently a large number of successful control schemesfor uncertain nonlinear systems with dead-zone time delaysand actuator failures have been developed in this area see[1ndash13] and the references therein More specifically Tong andLi [1] studied the adaptive fuzzy tracking control problem fora class of nonlinear systems with dead-zone nonlinearitiesIn [9] by employing a nonlinear fault estimator the outputfault-tolerant tracking control was developed using theadaptive backstepping technique Zhou et al [13] consideredthe adaptive output tracking control problem for a class ofnonlinear systems with stochastic disturbances and timedelays Recently the authors in [14] proposed the adaptivetracking control approaches for a class of nonlinear time-delayed systems with dead-zone nonlinearities The globalstabilization problem for a class of nonlinear time-delayed

systems [15] was considered by using multiswitching-basedadaptive neural network control method Besides by com-bining fuzzy approximation and adaptive backsteppingtechnique a novel robust fault-tolerant control scheme [16]was developed for a class of non-lower-triangular nonlinearsystems with actuator failures

It should be pointed out that all the abovementionedcontrol strategies are merely suitable for the considerednonlinear systems in affine form rather than nonaffine formIt is well known that the nonaffine nonlinear systems repre-sent more general cases which can describe many practicalprocesses Li andTong [17] proposed an adaptive fuzzy outputcontrol approach for a class of pure-feedback nonlinearsystems with dead-zone constrains In [18] an adaptivefuzzy asymptotic tracking controller was designed for a classof uncertain nonaffine nonlinear systems with dead-zoneinputs Meanwhile the actuator saturation constrain controlimplies that the input signals are always bounded for most ofpractical systems The saturation problem is very importantfor the actuator control design and the performance and

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 2931524 8 pageshttpdxdoiorg10115520162931524

2 Journal of Control Science and Engineering

stability of the closed-loop systems will be severely effectiveif the input constrain in the design process is ignoredThus the adaptive control design for nonlinear systems withsaturation constrains is a challenging topic Wen et al [19]studied the adaptive control problem for a class of uncertainnonlinear systems with saturation constrains Based on thedisturbance observer direct adaptive NNs control strategiesin [20] were developed for a class of uncertain nonaffinenonlinear systems with saturation inputs Additionally theadaptive fuzzy tracking control scheme for a class of nonaffinenonlinear systems with saturation constrains and stochasticdisturbances in [21] was proposed

Motivated by the above considerations this paper studiesthe problem of adaptive fuzzy tracking control for a class ofuncertain nonaffinenonlinear systemswith input saturationsCompared with the existing results the main contributionsof this paper are as follows (1) The approximation-basedadaptive tracking control scheme is extended to nonaffinenonlinear systems with multiple time delays and saturationconstrains (2) Different from the design methods proposedin [20 21] it is obtained that the tracking errors canasymptotically converge to zero rather than being uniformlyultimately bounded (3) The mean value theorem is used todeal with the nonaffine term with input saturation and thusthe desired asymptotic tracking performance of the closed-loop systems can be achieved by using Lyapunov stabilityanalysis

The rest of the paper is organized as follows Section 2gives the problem statement and preliminaries A noveladaptive fuzzy asymptotic tracking controller is designed inSection 3 Simulation studies are then provided in Section 4to verify the effectiveness of the proposed control methodand Section 5 draws the conclusions

2 Problem Statement and Preliminaries

Consider the nonaffine nonlinear systems described as fol-lows

1 = 11990922 = 1199093

119899 = 119891 (119909 sat (119906)) + 119903sum

119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)

(1)

where 119909 = [1199091 1199092 119909119899]119879 isin R119899 is the system state119906 isin R is the actual control input and 119889(119905) isin R is theunknown bounded external disturbanceThe unknown non-linear function 119891(sdot sdot) R119899 times R rarr R is sufficiently smoothWithout loss of generalityΔ119864119894(sdot) 119894 = 1 2 119903 represent thesystem uncertainties and are assumed to be continuous withthe approximate dimensions 120591119894(119905) is the time-varying timedelay satisfying 0 le 120591119894(119905) le 120591lowast119894 119894(119905) le 120591lowast119894 lt 1 119894 = 1 2 119903119909(119905) = 120601(119905) 119905 isin [0 120591lowast] where 120601(119905) is the initial condition

and 120591lowast = max1le119894le119903120591lowast119894 In addition sat(119906) denotes the inputsaturation which is defined by

sat (119906 (119905)) =

119906119872 119906 (119905) ge 119906119872119906 (119905) 119906119898 le 119906 (119905) le 119906119872119906119898 119906 (119905) le 119906119898

(2)

where 119906119872 and 119906119898 are known upper and lower bounds of thecontrol output 119906(119905) respectively

Moreover as in [21] (2) can be written as

sat (119906 (119905)) = 119878 (119906 (119905)) + 119863 (119906 (119905)) (3)

where the piecewise smooth function 119878(119906) is defined as

119878 (119906) =

119906119872 lowast tanh( 119906119906119872) 119906 ge 0119906119898 lowast tanh( 119906119906119898) 119906 lt 0

=

119906119872 lowast 119890119906119906119872 minus 119890minus119906119906119872119890119906119906119872 + 119890minus119906119906119872 119906 ge 0119906119898 lowast 119890119906119906119898 minus 119890minus119906119906119898119890119906119906119898 + 119890minus119906119906119898 119906 lt 0

(4)

and 119863(119906) = sat(119906) minus 119878(119906) Furthermore the bound of 119863(119906)can be obtained as

|119863 (119906)| le |sat (119906) minus 119878 (119906)|le max 119906119872 (1 minus tanh (1)) 119906119898 (tanh (1) minus 1)= 119863

(5)

with 119863 being the upper bound between the continuousfunction sat(119906) and the smooth function 119878(119906) Consequentlyit follows from the mean value theorem that 119878(119906) = (120597119878(119906120585)120597119906)119906 + 119878(0) with 119906120585 = 120582119906 and 0 lt 120582 lt 1 Noting (3) and (4)we can obtain that

sat (119906) = 120597119878 (119906120585)120597119906 119906 + 119863 (119906) (6)

The control objective is to design an adaptive fuzzyasymptotic tracking controller 119906(119905) to ensure that all theclosed-loop signals are bounded and the system states asymp-totically stable in the presence of external disturbancesmultiple time delays and saturation constrains Besides inorder to design adaptive state feedback controller119909 = [1199091 1199092 119909119899]119879 is assumed to be measurable and the state trackingerror is defined as 119890 = 119909 minus 119909119889 with the reference signal119909119889 being 119909119889 = [119910119889 119889 119910(119899minus1)119889 ]119879 Moreover the followingassumptions are given for system (1)

Assumption 1 There exist unknown positive constants 119889 and119889119896 such that |119889(119905)| le 1198890 and |119910(119896)119889 | le 119889119896 for 119896 = 0 1 2 119899respectively

Journal of Control Science and Engineering 3

Assumption 2 The uncertain function Δ119864119894(119909 119905) is assumedto satisfy Δ119864119894(119909 119905) le 120598lowast119896 with 120598lowast119894 119894 = 1 2 119903 beingunknown positive constants

Assumption 3 (see [18 20]) For all 119909 isin R119899 and 119906 isin R insystem (5) there always exist positive constants 1198721 and 1198722such that the following inequality holds

0 lt 1198721 le 120597119891 (119909 119906)120597119906 le 1198722 (7)

Assumption 4 Given the practical system described by (1)satisfying the input saturation (2) there exists feasible actualcontrol input 119906 which can achieve the desired controlobjective

Remark 5 Clearly Assumption 1 is quite standard andmeansthat the external disturbance the reference output signaland its time derivatives are bounded respectively It followsfrom Assumption 2 that the change rate of the input gainis bounded Particularly different from [20 21] the trackingerror of this paper can asymptotically converge to zero ratherthan to a desired compact set

Similar to [10 11 16ndash18] the following fuzzy approxima-tion lemma is given by the following lemma

Lemma 6 Let 119865(119883) be a continuous function that is definedon a compact set Ω119883 For any given positive constant 120598 therealways exists a fuzzy logic system 119910(119883) in the form of (7) suchthat

sup119883isinΩ119883

1003816100381610038161003816119865 (119883) minus 119910 (119883)1003816100381610038161003816 = 10038161003816100381610038161003816119865 (119883) minus 120579119879120585 (119883)10038161003816100381610038161003816 lt 120598 (8)

Consequently the optimal parameter vectors 120579lowast of fuzzy logicsystem (FLS) is defined as

120579lowast = argmin120579isinΩ120579

[ sup119883isinΩ119883

10038161003816100381610038161003816119865 (119883) minus 120579119879120585 (119883)10038161003816100381610038161003816] (9)

whereΩ120579 andΩ119883 are compact regions for 120579 and119883 respectivelyIn addition the fuzzy approximation error 120575lowast(119883) is defined as

119865 (119883) = 120579lowast119879120585 (119883) + 120575lowast (119883) forall119883 isin Ω119883 120579lowast isin Ω120579 (10)

3 Adaptive Fuzzy Tracking Controller Design

In this section the adaptive fuzzy asymptotic tracking controlscheme will be developed for the nonlinear system (1) withexternal disturbance multiple time delays and saturationconstrain For this purpose taking the time derivative of thetracking error 119890 = 119909 minus 119909119889 with respect to 119905 yields

119890 = 119860119890 + 119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 ) (11)

where

119860 =[[[[[[[[[

0 1 0 sdot sdot sdot 00 0 1 sdot sdot sdot 0

d d


]]]]]]]]]

119861 =[[[[[[[[[[

0001

]]]]]]]]]]

(12)

Then from (12)119860+119861119870 is a stablematrix by properly choosinga gain vector 119870 Moreover for any given 119876 = 119876119879 gt 0 thereexists 119875 = 119875119879 gt 0 such that the Lyapunov equation (119860 +119861119870)119879119875+119875(119860+119861119870)+(119903120578)119868 = minus119876 holds where 120578 is a positivedesign parameter

For the nonaffine term 119891(119909 sat(119906)) using the meanvalue theorem again gives 119891(119909 sat(119906)) = 119891(119909 0) + (120597119891(119909119906120577)120597119906)sat(119906) with 119906120577 = 120583119906 and 0 lt 120583 lt 1 From (6) it iseasy to see that

119891 (119909 sat (119906)) = 119891 (119909 0)+ 120597119891 (119909 119906120577)120597119906 (120597119878 (119906120585)120597119906 119906 + 119863 (119906))

= 120591119904119906 + Ψ (119909 119906) (13)

whereΨ(119909 119906) = ((120597119891(119909 119906120577)120597119906)(120597119878(119906120585)120597119906)minus120591119904)119906+119891(119909 0)+119863(119906) with 120591119904 being a positive design parameter In additionby using the expression in (10) (13) can be expressed as

119891 (119909 sat (119906)) = 120591119904119906 + 120579lowast119879120585 (119909 119890) + 120575lowast (119909 119890) forall (119909 119890) isin Ω sub R

2119899 (14)

where the approximation error 120575lowast(119909 119890) satisfies |120575lowast(119909 119890)| le120575lowast with 120575lowast being any small positive constant and Ω is anappropriate compact set Without loss of generality basedon Assumptions 1 and 2 we introduce the notions 119870lowast1 =sup119905ge0(|119909(119899)119889 (119905)| + |119889(119905)| + sum119896119894=1 119909119889(119905 minus 120591119894(119905)) + 120575lowast) and 119870lowast2 =sum119896119894=1(120598lowast2119894 (1minus120591lowast119894 )) where119870lowast1 and119870lowast2 are unknown constantsTherefore the adaptive fuzzy controller is designed as

119906 = 120591minus1119904 (119870119890 minus 119879120585 (119909 119890) minus 2111986111987911987511989010038161003816100381610038161198901198791198751198611003816100381610038161003816 1 + 120590 (119905)minus 121205781198611198791198751198902)

(15)


with the corresponding adaptive control laws

120579 = minusΓ120590 minus Γ119890119879119875119861120585 (119909 119890) 1198701 = minus12057411205901 + 21205741 1003817100381710038171003817100381711986111987911987511989010038171003817100381710038171003817 1198702 = minus12057421205902 + 1205742120578 10038171003817100381710038171003817119861119879119875119890100381710038171003817100381710038172

(16)

where 119894 and are the estimates of 119870lowast119894 and 120579lowast respectivelyΓ = Γ119879 gt 0 and 120574119894 are positive design parameters Besidesthe continuous function 120590(119905) is subject to 120590(119905) gt 0 andint1199050120590(120591)119889120591 le 120590 lt infin forall119905 ge 0 with any constant 120590 gt 0 119894 = 1 2

Remark 7 The adaptive fuzzy controller (15) mainly consistsof four terms Concretely 120591minus1119904 is the positive design parameterof the adaptive control gain and the first term 119870119909 of theright hand side plays a key role for stabilizing system Thesecond term is used to decouple the nonaffine term withsaturation nonlinearity The third term and the fourth termwith adaptation laws (16) are used to deal with the effects ofmultiple time delays and external disturbance respectively

Now the stability of the resulting closed-loop system isgiven in the following theorem

Theorem 8 Consider the uncertain nonaffine nonlinear sys-tem (1) satisfying Assumptions 1ndash4 With the application ofadaptive fuzzy controller (15) and parameter updated laws (16)the tracking error of the closed-loop system can asymptoticallyconverge to zero that is lim119905rarrinfin119890(119905) = 0 for any (119909 119890) isin Ωwhich is a proper compact set

Proof For the closed-loop error system (11) choose a Lya-punov function candidate as follows

119881(119890 1 2) = 119890119879119875119890 + 122sum119894=1

120574minus1119894 2119894 + 12 119879Γminus1

+ 120578minus1 119903sum119894=1

int119905119905minus120591119894(119905)

119890119879 (119904) 119890 (119904) 119889119904(17)

where = minus120579lowast and 119894 = 119894minus119870lowast119894 119894 = 1 2 are the parameterestimation errors Then taking the time derivative of 119881 withrespect to 119905 yields

= 119890119879 (119875119860 + 119860119879119875) 119890 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

+ 2119890119879119875119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 )

+ 120578minus1 119903sum119894=1

(119890 (119905)2 minus (1 minus 119894 (119905)) 1003817100381710038171003817119890 (119905 minus 120591119894 (119905))10038171003817100381710038172)

(18)

By invoking (14) we obtain that

= 119890119879 (119875119860 + 119860119879119875) 119890+ 2119890119879119875119861 119903sum

119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 2119890119879119875119861 (120591119904119906+ 120579lowast119879120585 (119909 119890) + 120575lowast (119909 119890) + 119889 (119905) minus 119909(119899)119889 )+ 120578minus1 119903sum119894=1


+ 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579 le 119890119879 (119875119860 + 119860119879119875) 119890+ 2120591119904119890119879119875119861119906 + 2119890119879119875119861120579lowast119879120585 (119909 119890) + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816sdot 1003816100381610038161003816100381610038161003816100381610038161003816120575lowast (119909 119890) + 119889 (119905) minus 119909(119899)119889

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909119889 (119905 minus 120591119894 (119905))1003816100381610038161003816100381610038161003816100381610038161003816 +

119903120578 119890 (119905)2

minus 119903sum119894=1

(1 minus 120591lowast119894 ) 1003817100381710038171003817119890 (119905 minus 120591119894 (119905))10038171003817100381710038172 + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816sdot 119903sum119894=1

120598lowast119894 1003817100381710038171003817119890 (119905 minus 120591119894 (119905))1003817100381710038171003817 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

(19)

Using triangle inequality and according to the definitions of119870lowast119894 119894 = 1 2 le 119890119879 (119875119860 + 119860119879119875) 119890 + 2120591119904119890119879119875119861119906

+ 2119890119879119875119861120579lowast119879120585 (119909 119890) + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast1 + 119903120578 119890 (119905)2

+ 120578 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast2 +2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579(20)

Then from the adaptive controller (15) and the parameterupdated laws (16) we can obtain that

le minus119890119879119876119890 + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 112059010038161003816100381610038161198901198791198751198611003816100381610038161003816 1 + 120590 minus 120590119879 minus 120590 2sum119894=1

119894119894 (21)

Using the inequality 0 le 119909(119909 + 119910) lt 1 forall119909 ge 0 119910 gt 0 and119886119879119887 le 119886119887 forall119886 119887 isin R119899 (21) becomes

le minus119890119879119876119890 + 120590(120579lowast24 + 119870lowast214 + 119870lowast224 + 2)= minus119890119879119876119890 + 120590120587

(22)

where 120587 = 120579lowast24 + 119870lowast21 4 + 119870lowast22 4 + 2Integrating (22) from 0 to 119905 yields

119881 (119905) + int1199050119890119879 (120591) 119876119890 (120591) 119889120591 le 119881|119905=0 + 120587120590 (23)


Thus it further implies that int1199050119890119879(120591)119890(120591)119889120591 le (1120582min(119876))(119881|119905=0 + 120587120590) forall119905 gt 0 where 120582min(sdot) denotes the

minimum eigenvalue of amatrix that is 119890 isin 1198712 According toBarbalatrsquos lemma [22] it can be concluded that lim119905rarrinfin119890(119905) =0 The proof is completed

Remark 9 It should be pointed out that the control methodsproposed in [20 21] can guarantee that the tracking errorsconverge to the desired compact sets The tracking error ofthe closed-loop system can asymptotically converge to zeroby employing the adaptive control scheme in [18] howeverthis control scheme cannot deal with nonaffine nonlinearsystems with multiple time delays and saturation constrainsIn this paper based on fuzzy approximation technique andthe mean value theorem the proper nonlinear functions itis proved that the desired asymptotic tracking performanceof the closed-loop systems can be achieved via Lyapunovstability analysis

4 Simulation Studies

In this section a third-order van der Pol oscillator systemfrom [23] is used for simulation study of this paper Besidesby adding the external disturbances 119889(119905) and the multipletime-delayed perturbation sum119903119894=1 Δ119864119894(119909 119905)119909(119905 minus 120591119894(119905)) the cor-responding nonaffine nonlinear systemmodel is described asfollows

1 = 11990922 = 11990933 = minus1199091 minus 051199092 + 120573 (1 minus (1199091 + 051199092)2) 1199093

+ (2 + sin (119909111990921199093)) (119906 + 131199063 + sin (119906))+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)119910 = 1199091

(24)

where 120573 = 07 the time delays and parameter uncertainfunctions are 1205911(119905) = 08 cos(119905) 1205912(119905) = 05 sin(119905) 1205913(119905) =05 sin(2119905) and 1198641(119909 119905) = [1 0 0] 1198642(119909 119905) = [0 2 0]1198643(119909 119905) = [0 0 3] and the external disturbance119889(119905) is chosenas 119889(119905) = 05 sin(119905) respectively and the reference signal isgiven as 119910119889 = 01 + sin(05119905)

Take the membership functions of fuzzy logic system asfollows

1205831198651119895 = exp[[minus(119909119895 + 15)2

2 ]]

1205831198652119895 = exp[[minus(119909119895 + 125)2

2 ]]

1205831198653119895 = exp[[minus(119909119895 + 1)2

2 ]]

1205831198654119895 = exp[[minus(119909119895 + 075)2

2 ]]

1205831198655119895 = exp[[minus(119909119895 + 05)2

2 ]]

1205831198656119895 = exp[[minus(119909119895 + 025)2

2 ]]

1205831198657119895 = exp(minus11990921198952 )

1205831198658119895 = exp[[minus(119909119895 minus 025)2

2 ]]

1205831198659119895 = exp[[minus(119909119895 minus 05)2

2 ]]

12058311986510119895 = exp[[minus(119909119895 minus 075)2

2 ]]

12058311986511119895 = exp[[minus(119909119895 minus 1)2

2 ]]

12058311986512119895 = exp[[minus(119909119895 minus 125)2

2 ]]

12058311986513119895 = exp[[minus(119909119895 minus 15)2

2 ]]

12058311986514119895 = exp[[minus(119909119895 minus 175)2

2 ]]

119895 = 1 2 3

(25)

Define fuzzy basis functions as

120585119894 (119909) = prod3119895=1120583119865119894119895 (119909119895)sum14119894=1 [prod3119895=1120583119865119894119895 (119909119895)]

119894 = 1 2 14 (26)

where 119909 = [1199091 1199092 1199093]119879By choosing gain 119870 = [minus6 minus11 minus6] and 119876 = 119868 gt 0 it

is easy to obtain that 119875 = [ 23617 14950 0108314950 26108 0195001083 01950 01408

] Based onTheo-rem8 the adaptive controller and the parameter updated lawsare designed as (15) and (16) respectivelyThe correspondingsimulation parameters are selected as 119903 = 3 120578 = 10 120598 = 02120583119894 = 1 Γ0 = 5119868 1205900(119905) = 05119890minus01119905 1205741 = 1205742 = 1 and the initial


0 10 20 30 40 50 60Time (sec)

Trac

king

erro

r e(t)

minus15

minus1

minus05

0

05

1

15

2

Tracking error e1(t)



Figure 1 Trajectory of the tracking errors 1198901 = 119910minus119909119889 and 1198902 = minus119889

minus1

minus05

0

05

1

15

2

System output yReference signal yd

10 20 30 40 50 600Time (sec)

Figure 2 System output 119910 and desired output 119909119889

values are chosen as 119909(0) = [2 minus05 08]119879 (0) = [0 1 0 10 1 0 1 0 1 0 1 0 1]119879 1(0) = 2 2(0) = 6 The simu-lation results are shown in Figures 1ndash7 From Figures 1 and2 it can be seen that the state tracking errors converge to zerofor the uncertain nonaffinenonlinear system (1)withmultipletime delays saturation constrain and external disturbancesimultaneously Moreover the boundedness of parameterestimations and 119895 119895 = 1 2 is shown in Figures 3ndash5 InFigures 6 and 7 we can see that the actual control input andthe saturation output signal are also bounded respectively

10 20 30 40 50 600Time (sec)

minus02

0

02

04

06

08

1

12

Estim

ates

of t

he p

aram

eter

vec

tor120579

lowast

Figure 3 Response curves of

10 20 30 40 50 600Time (sec)

0

02

04

06

08

1

12

14

16

18

2Es

timat

e of t

he p

aram

eter

klowast 1

Figure 4 Response curve of 1

0

1

2

3

4

5

6

Estim

ate o

f the

par

amet

erklowast 2

10 20 30 40 50 600Time (sec)



minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)

Figure 6 Output respond curve of the saturation function sat(119906)

10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)

Figure 7 Respond curve of the control input 119906(119905)

5 Conclusion

This paper studies a novel adaptive fuzzy asymptotic trackingcontrol scheme for a class of uncertain nonaffine nonlinearsystems with multiple time delays saturation constrains andexternal disturbances By using the mean value theorem andfuzzy logic system (FLS) the parameter updated laws areconstructed to estimate the unknown adaptive controllerparameters online It is also shown that the proposed controlmethod guarantees all the closed-loop system signals to beuniformly bounded and the tracking error can asymptoticallyconverge to zero based on Lyapunov-based analysis Numer-ical simulation results are provided to show the effectivenessof the proposed adaptive fuzzy tracking control designapproach

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported in part by the FundamentalResearch Funds of Anshan Municipal Government

References

[1] S Tong and Y Li ldquoAdaptive fuzzy output feedback trackingbackstepping control of strict-feedback nonlinear systems withunknown dead zonesrdquo IEEE Transactions on Fuzzy Systems vol20 no 1 pp 168ndash180 2012

[2] X Zhang L G Wu and S C Cui ldquoAn improved integralinequality to stability analysis of genetic regulatory networkswith interval time-varying delaysrdquo IEEEACM Transactions onComputational Biology and Bioinformatics vol 12 no 2 pp398ndash409 2015

[3] X Lin X Zhang and Y Wang ldquoRobust passive filtering forneutral-type neural networks with time-varying discrete andunbounded distributed delaysrdquo Journal of the Franklin InstituteEngineering and Applied Mathematics vol 350 no 5 pp 966ndash989 2013

[4] XWang and G-H Yang ldquoDistributed fault-tolerant control fora class of cooperative uncertain systems with actuator failuresand switching topologiesrdquo Information Sciences vol 370-371pp 650ndash666 2016

[5] X Wang and G-H Yang ldquoCooperative adaptive fault-toleranttracking control for a class of multi-agent systems with actuatorfailures and mismatched parameter uncertaintiesrdquo IET ControlTheory amp Applications vol 9 no 8 pp 1274ndash1284 2015

[6] J-W Zhu andG-H Yang ldquoFault accommodation for linear sys-tems with time-varying delayrdquo International Journal of SystemsScience vol 48 no 2 pp 316ndash323 2017

[7] S Tong and Y Li ldquoAdaptive fuzzy output feedback control ofMIMO nonlinear systems with unknown dead-zone inputsrdquoIEEE Transactions on Fuzzy Systems vol 21 no 1 pp 134ndash1462013

[8] Z Zhang S Xu and B Zhang ldquoAsymptotic tracking control ofuncertain nonlinear systemswith unknown actuator nonlinear-ityrdquo IEEE Transactions on Automatic Control vol 59 no 5 pp1336ndash1341 2014

[9] M Chen B Jiang and W W Guo ldquoFault-tolerant control fora class of non-linear systems with dead-zonerdquo InternationalJournal of Systems Science vol 47 no 7 pp 1689ndash1699 2016

[10] S Tong B Huo and Y Li ldquoObserver-based adaptive decentral-ized fuzzy fault-tolerant control of nonlinear large-scale systemswith actuator failuresrdquo IEEE Transactions on Fuzzy Systems vol22 no 1 pp 1ndash15 2014

[11] S C Tong T Wang and Y M Li ldquoFuzzy adaptive actuator fail-ure compensation control of uncertain stochastic nonlinear sys-tems with unmodeled dynamicsrdquo IEEE Transactions on FuzzySystems vol 22 no 3 pp 563ndash574 2014

[12] Z Zhang S Xu and B Zhang ldquoAsymptotic tracking control ofuncertain nonlinear systemswith unknown actuator nonlinear-ityrdquo Institute of Electrical and Electronics Engineers Transactionson Automatic Control vol 59 no 5 pp 1336ndash1341 2014

[13] Q Zhou P Shi S Xu and H Li ldquoObserver-based adaptiveneural network control for nonlinear stochastic systems withtime delayrdquo IEEETransactions onNeural Networks and LearningSystems vol 24 no 1 pp 71ndash80 2013

[14] Z Q Zhang S Y Xu and B Y Zhang ldquoExact tracking controlof nonlinear systems with time delays and dead-zone inputrdquoAutomatica vol 52 pp 272ndash276 2015


[15] J Wu W Chen F Yang J Li and Q Zhu ldquoGlobal adaptiveneural control for strict-feedback time-delay systems with pre-defined output accuracyrdquo Information Sciences vol 301 pp 27ndash43 2015

[16] H Wang X Liu P X Liu and S Li ldquoRobust adaptivefuzzy fault-tolerant control for a class of non-lower-triangularnonlinear systems with actuator failuresrdquo Information Sciencesvol 336 pp 60ndash74 2016

[17] Y Li and S Tong ldquoAdaptive fuzzy output-feedback controlof pure-feedback uncertain nonlinear systems with unknowndead zonerdquo IEEE Transactions on Fuzzy Systems vol 22 no 5pp 1341ndash1347 2014

[18] L-B Wu G-H Yang H Wang and F Wang ldquoAdaptive fuzzyasymptotic tracking control of uncertain nonaffine nonlinearsystems with non-symmetric dead-zone nonlinearitiesrdquo Infor-mation Sciences vol 348 pp 1ndash14 2016

[19] C Wen J Zhou Z Liu and H Su ldquoRobust adaptive control ofuncertain nonlinear systems in the presence of input saturationand external disturbancerdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1672ndash1678 2011

[20] M Chen and S S Ge ldquoDirect adaptive neural control for a classof uncertain nonaffine nonlinear systems based on disturbanceobserverrdquo IEEE Transactions on Cybernetics vol 43 no 4 pp1213ndash1225 2013

[21] H Wang B Chen X Liu K Liu and C Lin ldquoRobust adap-tive fuzzy tracking control for pure-feedback stochastic nonlin-ear systemswith input constraintsrdquo IEEETransactions onCyber-netics vol 43 no 6 pp 2093ndash2104 2013

[22] M Krstic I Kanellakopoulos and P V Kokotovic Nonlinearand Adaptive Control Design Wiley New York NY USA 1995

[23] S-L Dai C Wang and M Wang ldquoDynamic learning fromadaptive neural network control of a class of nonaffine nonlin-ear systemsrdquo IEEE Transactions on Neural Networks and Learn-ing Systems vol 25 no 1 pp 111ndash123 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



stability of the closed-loop systems will be severely effectiveif the input constrain in the design process is ignoredThus the adaptive control design for nonlinear systems withsaturation constrains is a challenging topic Wen et al [19]studied the adaptive control problem for a class of uncertainnonlinear systems with saturation constrains Based on thedisturbance observer direct adaptive NNs control strategiesin [20] were developed for a class of uncertain nonaffinenonlinear systems with saturation inputs Additionally theadaptive fuzzy tracking control scheme for a class of nonaffinenonlinear systems with saturation constrains and stochasticdisturbances in [21] was proposed

Motivated by the above considerations this paper studiesthe problem of adaptive fuzzy tracking control for a class ofuncertain nonaffinenonlinear systemswith input saturationsCompared with the existing results the main contributionsof this paper are as follows (1) The approximation-basedadaptive tracking control scheme is extended to nonaffinenonlinear systems with multiple time delays and saturationconstrains (2) Different from the design methods proposedin [20 21] it is obtained that the tracking errors canasymptotically converge to zero rather than being uniformlyultimately bounded (3) The mean value theorem is used todeal with the nonaffine term with input saturation and thusthe desired asymptotic tracking performance of the closed-loop systems can be achieved by using Lyapunov stabilityanalysis

The rest of the paper is organized as follows Section 2gives the problem statement and preliminaries A noveladaptive fuzzy asymptotic tracking controller is designed inSection 3 Simulation studies are then provided in Section 4to verify the effectiveness of the proposed control methodand Section 5 draws the conclusions

2 Problem Statement and Preliminaries

Consider the nonaffine nonlinear systems described as fol-lows

1 = 11990922 = 1199093

119899 = 119891 (119909 sat (119906)) + 119903sum

119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)

(1)

where 119909 = [1199091 1199092 119909119899]119879 isin R119899 is the system state119906 isin R is the actual control input and 119889(119905) isin R is theunknown bounded external disturbanceThe unknown non-linear function 119891(sdot sdot) R119899 times R rarr R is sufficiently smoothWithout loss of generalityΔ119864119894(sdot) 119894 = 1 2 119903 represent thesystem uncertainties and are assumed to be continuous withthe approximate dimensions 120591119894(119905) is the time-varying timedelay satisfying 0 le 120591119894(119905) le 120591lowast119894 119894(119905) le 120591lowast119894 lt 1 119894 = 1 2 119903119909(119905) = 120601(119905) 119905 isin [0 120591lowast] where 120601(119905) is the initial condition

and 120591lowast = max1le119894le119903120591lowast119894 In addition sat(119906) denotes the inputsaturation which is defined by

sat (119906 (119905)) =

119906119872 119906 (119905) ge 119906119872119906 (119905) 119906119898 le 119906 (119905) le 119906119872119906119898 119906 (119905) le 119906119898

(2)

where 119906119872 and 119906119898 are known upper and lower bounds of thecontrol output 119906(119905) respectively

Moreover as in [21] (2) can be written as

sat (119906 (119905)) = 119878 (119906 (119905)) + 119863 (119906 (119905)) (3)

where the piecewise smooth function 119878(119906) is defined as

119878 (119906) =

119906119872 lowast tanh( 119906119906119872) 119906 ge 0119906119898 lowast tanh( 119906119906119898) 119906 lt 0

=

119906119872 lowast 119890119906119906119872 minus 119890minus119906119906119872119890119906119906119872 + 119890minus119906119906119872 119906 ge 0119906119898 lowast 119890119906119906119898 minus 119890minus119906119906119898119890119906119906119898 + 119890minus119906119906119898 119906 lt 0

(4)

and 119863(119906) = sat(119906) minus 119878(119906) Furthermore the bound of 119863(119906)can be obtained as

|119863 (119906)| le |sat (119906) minus 119878 (119906)|le max 119906119872 (1 minus tanh (1)) 119906119898 (tanh (1) minus 1)= 119863

(5)

with 119863 being the upper bound between the continuousfunction sat(119906) and the smooth function 119878(119906) Consequentlyit follows from the mean value theorem that 119878(119906) = (120597119878(119906120585)120597119906)119906 + 119878(0) with 119906120585 = 120582119906 and 0 lt 120582 lt 1 Noting (3) and (4)we can obtain that

sat (119906) = 120597119878 (119906120585)120597119906 119906 + 119863 (119906) (6)

The control objective is to design an adaptive fuzzyasymptotic tracking controller 119906(119905) to ensure that all theclosed-loop signals are bounded and the system states asymp-totically stable in the presence of external disturbancesmultiple time delays and saturation constrains Besides inorder to design adaptive state feedback controller119909 = [1199091 1199092 119909119899]119879 is assumed to be measurable and the state trackingerror is defined as 119890 = 119909 minus 119909119889 with the reference signal119909119889 being 119909119889 = [119910119889 119889 119910(119899minus1)119889 ]119879 Moreover the followingassumptions are given for system (1)

Assumption 1 There exist unknown positive constants 119889 and119889119896 such that |119889(119905)| le 1198890 and |119910(119896)119889 | le 119889119896 for 119896 = 0 1 2 119899respectively




0 lt 1198721 le 120597119891 (119909 119906)120597119906 le 1198722 (7)






1003816100381610038161003816119865 (119883) minus 119910 (119883)1003816100381610038161003816 = 10038161003816100381610038161003816119865 (119883) minus 120579119879120585 (119883)10038161003816100381610038161003816 lt 120598 (8)



[ sup119883isinΩ119883

10038161003816100381610038161003816119865 (119883) minus 120579119879120585 (119883)10038161003816100381610038161003816] (9)





119890 = 119860119890 + 119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 ) (11)

where

119860 =[[[[[[[[[


d d


]]]]]]]]]

119861 =[[[[[[[[[[

0001

]]]]]]]]]]

(12)



119891 (119909 sat (119906)) = 119891 (119909 0)+ 120597119891 (119909 119906120577)120597119906 (120597119878 (119906120585)120597119906 119906 + 119863 (119906))

= 120591119904119906 + Ψ (119909 119906) (13)



2119899 (14)



(15)




(16)






119881(119890 1 2) = 119890119879119875119890 + 122sum119894=1

120574minus1119894 2119894 + 12 119879Γminus1

+ 120578minus1 119903sum119894=1

int119905119905minus120591119894(119905)

119890119879 (119904) 119890 (119904) 119889119904(17)


= 119890119879 (119875119860 + 119860119879119875) 119890 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

+ 2119890119879119875119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 )

+ 120578minus1 119903sum119894=1


(18)


= 119890119879 (119875119860 + 119860119879119875) 119890+ 2119890119879119875119861 119903sum

119894=1



+ 2sum119894=1


+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909119889 (119905 minus 120591119894 (119905))1003816100381610038161003816100381610038161003816100381610038161003816 +

119903120578 119890 (119905)2

minus 119903sum119894=1


120598lowast119894 1003817100381710038171003817119890 (119905 minus 120591119894 (119905))1003817100381710038171003817 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

(19)


+ 2119890119879119875119861120579lowast119879120585 (119909 119890) + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast1 + 119903120578 119890 (119905)2

+ 120578 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast2 +2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579(20)



119894119894 (21)



(22)


119881 (119905) + int1199050119890119879 (120591) 119876119890 (120591) 119889120591 le 119881|119905=0 + 120587120590 (23)








+ (2 + sin (119909111990921199093)) (119906 + 131199063 + sin (119906))+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)119910 = 1199091

(24)



1205831198651119895 = exp[[minus(119909119895 + 15)2

2 ]]

1205831198652119895 = exp[[minus(119909119895 + 125)2

2 ]]

1205831198653119895 = exp[[minus(119909119895 + 1)2

2 ]]

1205831198654119895 = exp[[minus(119909119895 + 075)2

2 ]]

1205831198655119895 = exp[[minus(119909119895 + 05)2

2 ]]

1205831198656119895 = exp[[minus(119909119895 + 025)2

2 ]]

1205831198657119895 = exp(minus11990921198952 )

1205831198658119895 = exp[[minus(119909119895 minus 025)2

2 ]]

1205831198659119895 = exp[[minus(119909119895 minus 05)2

2 ]]

12058311986510119895 = exp[[minus(119909119895 minus 075)2

2 ]]

12058311986511119895 = exp[[minus(119909119895 minus 1)2

2 ]]

12058311986512119895 = exp[[minus(119909119895 minus 125)2

2 ]]

12058311986513119895 = exp[[minus(119909119895 minus 15)2

2 ]]

12058311986514119895 = exp[[minus(119909119895 minus 175)2

2 ]]

119895 = 1 2 3

(25)


120585119894 (119909) = prod3119895=1120583119865119894119895 (119909119895)sum14119894=1 [prod3119895=1120583119865119894119895 (119909119895)]

119894 = 1 2 14 (26)





0 10 20 30 40 50 60Time (sec)

Trac

king

erro

r e(t)

minus15

minus1

minus05

0

05

1

15

2





minus1

minus05

0

05

1

15

2


10 20 30 40 50 600Time (sec)



10 20 30 40 50 600Time (sec)

minus02

0

02

04

06

08

1

12

Estim

ates

of t

he p

aram

eter

vec

tor120579

lowast


10 20 30 40 50 600Time (sec)

0

02

04

06

08

1

12

14

16

18

2Es

timat

e of t

he p

aram

eter

klowast 1


0

1

2

3

4

5

6

Estim

ate o

f the

par

amet

erklowast 2

10 20 30 40 50 600Time (sec)



minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)


10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)


5 Conclusion


Competing Interests


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0 lt 1198721 le 120597119891 (119909 119906)120597119906 le 1198722 (7)






1003816100381610038161003816119865 (119883) minus 119910 (119883)1003816100381610038161003816 = 10038161003816100381610038161003816119865 (119883) minus 120579119879120585 (119883)10038161003816100381610038161003816 lt 120598 (8)



[ sup119883isinΩ119883

10038161003816100381610038161003816119865 (119883) minus 120579119879120585 (119883)10038161003816100381610038161003816] (9)





119890 = 119860119890 + 119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 ) (11)

where

119860 =[[[[[[[[[


d d


]]]]]]]]]

119861 =[[[[[[[[[[

0001

]]]]]]]]]]

(12)



119891 (119909 sat (119906)) = 119891 (119909 0)+ 120597119891 (119909 119906120577)120597119906 (120597119878 (119906120585)120597119906 119906 + 119863 (119906))

= 120591119904119906 + Ψ (119909 119906) (13)



2119899 (14)



(15)




(16)






119881(119890 1 2) = 119890119879119875119890 + 122sum119894=1

120574minus1119894 2119894 + 12 119879Γminus1

+ 120578minus1 119903sum119894=1

int119905119905minus120591119894(119905)

119890119879 (119904) 119890 (119904) 119889119904(17)


= 119890119879 (119875119860 + 119860119879119875) 119890 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

+ 2119890119879119875119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 )

+ 120578minus1 119903sum119894=1


(18)


= 119890119879 (119875119860 + 119860119879119875) 119890+ 2119890119879119875119861 119903sum

119894=1



+ 2sum119894=1


+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909119889 (119905 minus 120591119894 (119905))1003816100381610038161003816100381610038161003816100381610038161003816 +

119903120578 119890 (119905)2

minus 119903sum119894=1


120598lowast119894 1003817100381710038171003817119890 (119905 minus 120591119894 (119905))1003817100381710038171003817 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

(19)


+ 2119890119879119875119861120579lowast119879120585 (119909 119890) + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast1 + 119903120578 119890 (119905)2

+ 120578 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast2 +2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579(20)



119894119894 (21)



(22)


119881 (119905) + int1199050119890119879 (120591) 119876119890 (120591) 119889120591 le 119881|119905=0 + 120587120590 (23)








+ (2 + sin (119909111990921199093)) (119906 + 131199063 + sin (119906))+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)119910 = 1199091

(24)



1205831198651119895 = exp[[minus(119909119895 + 15)2

2 ]]

1205831198652119895 = exp[[minus(119909119895 + 125)2

2 ]]

1205831198653119895 = exp[[minus(119909119895 + 1)2

2 ]]

1205831198654119895 = exp[[minus(119909119895 + 075)2

2 ]]

1205831198655119895 = exp[[minus(119909119895 + 05)2

2 ]]

1205831198656119895 = exp[[minus(119909119895 + 025)2

2 ]]

1205831198657119895 = exp(minus11990921198952 )

1205831198658119895 = exp[[minus(119909119895 minus 025)2

2 ]]

1205831198659119895 = exp[[minus(119909119895 minus 05)2

2 ]]

12058311986510119895 = exp[[minus(119909119895 minus 075)2

2 ]]

12058311986511119895 = exp[[minus(119909119895 minus 1)2

2 ]]

12058311986512119895 = exp[[minus(119909119895 minus 125)2

2 ]]

12058311986513119895 = exp[[minus(119909119895 minus 15)2

2 ]]

12058311986514119895 = exp[[minus(119909119895 minus 175)2

2 ]]

119895 = 1 2 3

(25)


120585119894 (119909) = prod3119895=1120583119865119894119895 (119909119895)sum14119894=1 [prod3119895=1120583119865119894119895 (119909119895)]

119894 = 1 2 14 (26)





0 10 20 30 40 50 60Time (sec)

Trac

king

erro

r e(t)

minus15

minus1

minus05

0

05

1

15

2





minus1

minus05

0

05

1

15

2


10 20 30 40 50 600Time (sec)



10 20 30 40 50 600Time (sec)

minus02

0

02

04

06

08

1

12

Estim

ates

of t

he p

aram

eter

vec

tor120579

lowast


10 20 30 40 50 600Time (sec)

0

02

04

06

08

1

12

14

16

18

2Es

timat

e of t

he p

aram

eter

klowast 1


0

1

2

3

4

5

6

Estim

ate o

f the

par

amet

erklowast 2

10 20 30 40 50 600Time (sec)



minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)


10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)


5 Conclusion


Competing Interests


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(16)






119881(119890 1 2) = 119890119879119875119890 + 122sum119894=1

120574minus1119894 2119894 + 12 119879Γminus1

+ 120578minus1 119903sum119894=1

int119905119905minus120591119894(119905)

119890119879 (119904) 119890 (119904) 119889119904(17)


= 119890119879 (119875119860 + 119860119879119875) 119890 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

+ 2119890119879119875119861(119891 (119909 sat (119906))

+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905) minus 119909(119899)119889 )

+ 120578minus1 119903sum119894=1


(18)


= 119890119879 (119875119860 + 119860119879119875) 119890+ 2119890119879119875119861 119903sum

119894=1



+ 2sum119894=1


+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909119889 (119905 minus 120591119894 (119905))1003816100381610038161003816100381610038161003816100381610038161003816 +

119903120578 119890 (119905)2

minus 119903sum119894=1


120598lowast119894 1003817100381710038171003817119890 (119905 minus 120591119894 (119905))1003817100381710038171003817 + 2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579

(19)


+ 2119890119879119875119861120579lowast119879120585 (119909 119890) + 2 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast1 + 119903120578 119890 (119905)2

+ 120578 1003816100381610038161003816100381611989011987911987511986110038161003816100381610038161003816 119870lowast2 +2sum119894=1

120574minus1119894 119894 119870119894 + 119879Γminus1 120579(20)



119894119894 (21)



(22)


119881 (119905) + int1199050119890119879 (120591) 119876119890 (120591) 119889120591 le 119881|119905=0 + 120587120590 (23)








+ (2 + sin (119909111990921199093)) (119906 + 131199063 + sin (119906))+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)119910 = 1199091

(24)



1205831198651119895 = exp[[minus(119909119895 + 15)2

2 ]]

1205831198652119895 = exp[[minus(119909119895 + 125)2

2 ]]

1205831198653119895 = exp[[minus(119909119895 + 1)2

2 ]]

1205831198654119895 = exp[[minus(119909119895 + 075)2

2 ]]

1205831198655119895 = exp[[minus(119909119895 + 05)2

2 ]]

1205831198656119895 = exp[[minus(119909119895 + 025)2

2 ]]

1205831198657119895 = exp(minus11990921198952 )

1205831198658119895 = exp[[minus(119909119895 minus 025)2

2 ]]

1205831198659119895 = exp[[minus(119909119895 minus 05)2

2 ]]

12058311986510119895 = exp[[minus(119909119895 minus 075)2

2 ]]

12058311986511119895 = exp[[minus(119909119895 minus 1)2

2 ]]

12058311986512119895 = exp[[minus(119909119895 minus 125)2

2 ]]

12058311986513119895 = exp[[minus(119909119895 minus 15)2

2 ]]

12058311986514119895 = exp[[minus(119909119895 minus 175)2

2 ]]

119895 = 1 2 3

(25)


120585119894 (119909) = prod3119895=1120583119865119894119895 (119909119895)sum14119894=1 [prod3119895=1120583119865119894119895 (119909119895)]

119894 = 1 2 14 (26)





0 10 20 30 40 50 60Time (sec)

Trac

king

erro

r e(t)

minus15

minus1

minus05

0

05

1

15

2





minus1

minus05

0

05

1

15

2


10 20 30 40 50 600Time (sec)



10 20 30 40 50 600Time (sec)

minus02

0

02

04

06

08

1

12

Estim

ates

of t

he p

aram

eter

vec

tor120579

lowast


10 20 30 40 50 600Time (sec)

0

02

04

06

08

1

12

14

16

18

2Es

timat

e of t

he p

aram

eter

klowast 1


0

1

2

3

4

5

6

Estim

ate o

f the

par

amet

erklowast 2

10 20 30 40 50 600Time (sec)



minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)


10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)


5 Conclusion


Competing Interests


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















+ (2 + sin (119909111990921199093)) (119906 + 131199063 + sin (119906))+ 119903sum119894=1

Δ119864119894 (119909 119905) 119909 (119905 minus 120591119894 (119905)) + 119889 (119905)119910 = 1199091

(24)



1205831198651119895 = exp[[minus(119909119895 + 15)2

2 ]]

1205831198652119895 = exp[[minus(119909119895 + 125)2

2 ]]

1205831198653119895 = exp[[minus(119909119895 + 1)2

2 ]]

1205831198654119895 = exp[[minus(119909119895 + 075)2

2 ]]

1205831198655119895 = exp[[minus(119909119895 + 05)2

2 ]]

1205831198656119895 = exp[[minus(119909119895 + 025)2

2 ]]

1205831198657119895 = exp(minus11990921198952 )

1205831198658119895 = exp[[minus(119909119895 minus 025)2

2 ]]

1205831198659119895 = exp[[minus(119909119895 minus 05)2

2 ]]

12058311986510119895 = exp[[minus(119909119895 minus 075)2

2 ]]

12058311986511119895 = exp[[minus(119909119895 minus 1)2

2 ]]

12058311986512119895 = exp[[minus(119909119895 minus 125)2

2 ]]

12058311986513119895 = exp[[minus(119909119895 minus 15)2

2 ]]

12058311986514119895 = exp[[minus(119909119895 minus 175)2

2 ]]

119895 = 1 2 3

(25)


120585119894 (119909) = prod3119895=1120583119865119894119895 (119909119895)sum14119894=1 [prod3119895=1120583119865119894119895 (119909119895)]

119894 = 1 2 14 (26)





0 10 20 30 40 50 60Time (sec)

Trac

king

erro

r e(t)

minus15

minus1

minus05

0

05

1

15

2





minus1

minus05

0

05

1

15

2


10 20 30 40 50 600Time (sec)



10 20 30 40 50 600Time (sec)

minus02

0

02

04

06

08

1

12

Estim

ates

of t

he p

aram

eter

vec

tor120579

lowast


10 20 30 40 50 600Time (sec)

0

02

04

06

08

1

12

14

16

18

2Es

timat

e of t

he p

aram

eter

klowast 1


0

1

2

3

4

5

6

Estim

ate o

f the

par

amet

erklowast 2

10 20 30 40 50 600Time (sec)



minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)


10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)


5 Conclusion


Competing Interests


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20 30 40 50 60Time (sec)

Trac

king

erro

r e(t)

minus15

minus1

minus05

0

05

1

15

2





minus1

minus05

0

05

1

15

2


10 20 30 40 50 600Time (sec)



10 20 30 40 50 600Time (sec)

minus02

0

02

04

06

08

1

12

Estim

ates

of t

he p

aram

eter

vec

tor120579

lowast


10 20 30 40 50 600Time (sec)

0

02

04

06

08

1

12

14

16

18

2Es

timat

e of t

he p

aram

eter

klowast 1


0

1

2

3

4

5

6

Estim

ate o

f the

par

amet

erklowast 2

10 20 30 40 50 600Time (sec)



minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)


10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)


5 Conclusion


Competing Interests


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus6

minus5

minus4

minus3

minus2

minus1

0

1

Out

put s

igna

l of t

he sa

tura

tion

func

tion

sat(u

)

10 20 30 40 50 600Time (sec)


10 20 30 40 50 600Time (sec)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

Con

trol i

nput

sign

al u(t)


5 Conclusion


Competing Interests


Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Adaptive Fuzzy Tracking Control for a...

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