Research Article Combination of Two Nonlinear Techniques...
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Research Article Combination of Two Nonlinear Techniques Applied to a 3-DOF Helicopter P. Ahmadi, 1 M. Golestani, 1 S. Nasrollahi, 2 and A. R. Vali 1 1 Department of Electrical and Electronic Engineering, Malek Ashtar University of Technology, Tehran, Iran 2 Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran Correspondence should be addressed to P. Ahmadi; [email protected] Received 23 December 2013; Accepted 6 March 2014; Published 17 April 2014 Academic Editors: S. Aus Der Wiesche, P.-C. Chen, and R. K. Sharma Copyright © 2014 P. Ahmadi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A combination of two nonlinear control techniques, fractional order sliding mode and feedback linearization control methods, is applied to 3-DOF helicopter model. Increasing of the convergence rate is obtained by using proposed controller without increasing control effort. Because the proposed control law is robust against disturbance, so we only use the upper bound information of disturbance and estimation or measurement of the disturbance is not required. e performance of the proposed control scheme is compared with integer order sliding mode controller and results are justified by the simulation. 1. Introduction Helicopters are versatile flight vehicles that can perform aggressive maneuvers because of their unique thrust gen- eration and operation principle. ey can perform many missions that are dangerous or impossible for human to per- form them. Helicopter is a multi-input multioutput (MIMO) highly nonlinear dynamical system, so most of the existing results to date have been based on the linearization model or through several linearization techniques [1, 2]. e lin- earization method provides local stability. In presence of disturbance or uncertainties the linearization can lead state variables of system to instability. In recent years, many papers have been published about control design of helicopter. e sliding mode approach has been employed for helicopter’s altitude regulation at hovering [3, 4]. H-infinity approach has been also used to design a robust control scheme for helicopter [5, 6]. In [5] a robust H-infinity controller has been presented using augmented plant and the performance and robustness of the proposed controller have been investigated in both time and frequency domain. e proposed controller in [6] is based on H- infinity loop shaping approach and it has been shown that the proposed controller is more efficient than classical controller such as PI and PID controller. A robust linear time-invariant controller based on signal compensation has been presented in [7]. By suitably combining feedforward control actions and high-gain and nested saturation feedback laws, a new control scheme has been presented in [8]. Intelligent methods such as fuzzy [9] and neural network theory [10] have been used to design controller. Furthermore, in [11] a new intelligent control approach based on emotional model of human brain has been presented. e history of fractional calculus goes back 300 years ago. For many years, it has remained with no applications. Recently, this branch of science has become an attractive discussion among control scholars [12]. Good references on fractional calculus have been presented in [13, 14]. e sliding mode control (SMC) has been also extended in [12, 15– 19]. In [15] a PID controller based on sliding mode strategy is designed for linear fractional order systems. In [17]a single input fractional order model, described by a chain of integrators, is considered for nonlinear systems. In [16– 18] sliding mode method has been applied to synchronize fractional order nonlinear chaotic systems. In [12, 19] sliding surface has been defined as an expressed manifold with fractional order integral. In this paper, we propose a combination of two nonlinear control techniques, fractional order (FO) sliding mode and Hindawi Publishing Corporation ISRN Aerospace Engineering Volume 2014, Article ID 436072, 8 pages http://dx.doi.org/10.1155/2014/436072
Transcript of Research Article Combination of Two Nonlinear Techniques...
Research ArticleCombination of Two Nonlinear Techniques Applied toa 3-DOF Helicopter
P Ahmadi1 M Golestani1 S Nasrollahi2 and A R Vali1
1 Department of Electrical and Electronic Engineering Malek Ashtar University of Technology Tehran Iran2Department of Aerospace Engineering Sharif University of Technology Tehran Iran
Correspondence should be addressed to P Ahmadi peymanahmadi1366gmailcom
Received 23 December 2013 Accepted 6 March 2014 Published 17 April 2014
Academic Editors S Aus Der Wiesche P-C Chen and R K Sharma
Copyright copy 2014 P Ahmadi 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
A combination of two nonlinear control techniques fractional order sliding mode and feedback linearization control methods isapplied to 3-DOF helicopter model Increasing of the convergence rate is obtained by using proposed controller without increasingcontrol effort Because the proposed control law is robust against disturbance so we only use the upper bound information ofdisturbance and estimation or measurement of the disturbance is not required The performance of the proposed control schemeis compared with integer order sliding mode controller and results are justified by the simulation
1 Introduction
Helicopters are versatile flight vehicles that can performaggressive maneuvers because of their unique thrust gen-eration and operation principle They can perform manymissions that are dangerous or impossible for human to per-form them Helicopter is a multi-input multioutput (MIMO)highly nonlinear dynamical system so most of the existingresults to date have been based on the linearization modelor through several linearization techniques [1 2] The lin-earization method provides local stability In presence ofdisturbance or uncertainties the linearization can lead statevariables of system to instability
In recent years many papers have been published aboutcontrol design of helicopter The sliding mode approach hasbeen employed for helicopterrsquos altitude regulation at hovering[3 4] H-infinity approach has been also used to design arobust control scheme for helicopter [5 6] In [5] a robustH-infinity controller has been presented using augmentedplant and the performance and robustness of the proposedcontroller have been investigated in both time and frequencydomain The proposed controller in [6] is based on H-infinity loop shaping approach and it has been shown that theproposed controller is more efficient than classical controllersuch as PI and PID controller A robust linear time-invariant
controller based on signal compensation has been presentedin [7] By suitably combining feedforward control actions andhigh-gain and nested saturation feedback laws a new controlscheme has been presented in [8] Intelligent methods suchas fuzzy [9] and neural network theory [10] have been usedto design controller Furthermore in [11] a new intelligentcontrol approach based on emotional model of human brainhas been presented
The history of fractional calculus goes back 300 yearsago For many years it has remained with no applicationsRecently this branch of science has become an attractivediscussion among control scholars [12] Good references onfractional calculus have been presented in [13 14]The slidingmode control (SMC) has been also extended in [12 15ndash19] In [15] a PID controller based on sliding mode strategyis designed for linear fractional order systems In [17] asingle input fractional order model described by a chainof integrators is considered for nonlinear systems In [16ndash18] sliding mode method has been applied to synchronizefractional order nonlinear chaotic systems In [12 19] slidingsurface has been defined as an expressed manifold withfractional order integral
In this paper we propose a combination of two nonlinearcontrol techniques fractional order (FO) sliding mode and
Hindawi Publishing CorporationISRN Aerospace EngineeringVolume 2014 Article ID 436072 8 pageshttpdxdoiorg1011552014436072
2 ISRN Aerospace Engineering
feedback linearization control methods The performance ofthe proposed control scheme is compared with integer order(IO) sliding mode controller and results are justified by thesimulation
This paper is organized as follows in Section 2 theinteger order SMC and feedback linearization techniquesare briefly reviewed and in Section 3 some preliminariesand definitions of fractional order calculus are introducedIn Section 4 the 3-DOF helicopter model description ispresented In Section 5 the proposed controller scheme isgiven and in Section 6 the proposed approach is appliedto 3-DOF helicopter model and experimental results areprovided Finally conclusion is addressed in Section 7
2 Review of IO SMC andFeedback Linearization
In this section definitions of sliding mode and feedbacklinearization control methods are briefly reviewed for essen-tial preparation of the combination of two nonlinear controltechniques FO SMC and feedback linearization
21 Integer Order Sliding Mode Control Consider a second-order nonlinear dynamical system as
1= 119909
2
2= 119891 (119909) + 119892 (119909) 119906 (1)
where 119909 = [119909
1119909
2]
119879 is the system state vector 119891(119909) and119892(119909) are nonlinear functions of 119909 and 119906 = 0 is the scalarinput Sliding mode control approach consists of two partscorrectivecontrol law (119906
119888) that compensates the deviations
from the sliding surface to reach the sliding surfaceandequivalent control law (119906eq) that makes the derivative of thesliding surface equal zero to stay on the sliding surface Thiscontrol law is represented as
119906 = 119906eq + 119906119888 (2)
In general the sliding surface is 119878 = 1199092+ 120573119909
1 where 120573 gt 0
To guarantee the existence of sliding mode the controllaw must satisfy the condition
1
2
119889
119889119905
119904
2lt 0
(3)
To have a fast convergence it is sufficient tomodify the slidingsurface
22 Input-Output Linearization Control Consider the fol-lowing nonlinear dynamical system
= 119891 (119909) + 119892 (119909) 119906 119910 = ℎ (119909) (4)
The control method consists of the following steps
(a) Differentiate 119910 until 119906 appears in one of the equationsfor the derivatives of 119910 Consider
119910
(119903)= 120572 (119909) + 120573 (119909) 119906
(5)
(b) Choose 119906 to give 119910(119903) = V where V is the syntheticinput Consider
119906 =
1
120573 (119909)
[minus120572 (119909) + V] (6)
(c) Then the system has the following form
119884 (119904)
119881 (119904)
=
1
119904
119903 (7)
Design a linear control law for this 119903-integrator linearsystem
(d) Check internal dynamics
3 Basic Description of Fractional Calculus
There exist many definitions of fractional derivative Twoof the most commonly used definitions are the Riemann-Liouville (RL) and Caputo (C) definitions [14]
Definition 1 (see [20]) The left RL fractional derivative isdescribed by
RL119886119863
120572
119905119891 (119905) =
1
Γ (119899 minus 120572)
119889
119899
119889119905
119899int
119905
119886
119891 (120591)
(119905 minus 120591)
120572minus119899+1119889120591 (8)
where 119899 is the first integer which is not less than 120572 that is119899 minus 1 le 120572 lt 119899 and 120572 gt 0 and Γ(sdot) is the gamma function
Definition 2 (see [21]) The Laplace transform of the RLdefinition is described as
int
infin
0
119890
minusst119886119863
120572
119905119891 (119905) 119889119905 = 119904
120572119865 (119904) minus
119899minus1
sum
119896=0
119904
119896
119886119863
120572minus119896minus1
119905119891 (119905)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816119905=0
(9)
Lemma 3 (see [20]) The following equality is satisfied for all120572 gt 0 and 119899 is a natural number Consider
119889
119899
119889119905
119899
RL119863
120572119891 (119905) =
RL119863
119899+120572119891 (119905)
C119863
120572
119889
119899
119889119905
119899119891 (119905) =
C119863
119899+120572119891 (119905)
(10)
Lemma 4 (see [21]) Fractional differentiation and fractionalintegration are linear operations Consider
0119863
120572
119905(119886119891 (119905) + 119887119892 (119905)) = 119886
0119863
120572
119905119891 (119905) + 119887
0119863
120572
119905119892 (119905) (11)
Lemma 5 (see [20]) The following equality for left RL defini-tion is satisfied for all 120572 isin (0 1)
RL119863
120572119868
120572119891 (119905) = 119891 (119905) 120572 isin (0 1)
(12)
However the opposite is not true since
119868
120572
RL119863
120572119891 (119905) = 119891 (119905) minus
119891
1minus120572(0)
Γ (120572)
119905
120572minus1 (13)
where 1198911minus120572
(0) = lim119905rarr0
119868
1minus120572119891(119905)
ISRN Aerospace Engineering 3
Figure 1 The 3-DOF helicopter system
4 System Description
In this work a 3-DOF helicopter model is considered (seeFigure 1)
The following notations are used to describe the 3-DOFhelicopter system dynamics
120579 is angular position of the pitch axis rad120593 is angular position of the roll axis rad120595 is angular position of the yaw axis rad119869
120579 is moment of inertia of the system around the pitchaxis kgm2
119869
120593 is moment of inertia of the helicopter body about theroll axis kgm2
119869
120595 is moment of inertia of the helicopter body about theyaw axis kgm2
119881
119897 is voltage applied to the left motor V
119881
119903 is voltage applied to the right motor V
119870
119891 is force constant of the motor combination N
119897
119886 is distance between the base and the helicopter bodym
119897
ℎ is distance from the pitch axis to either motor m
119879
119892 is effective gravitational torque Nm
119870
119901 is constant of proportionality of the gravitationalforce N
The 3-DOF helicopter control system consists of twoDC motors at the end of the arm as shown in Figure 1Figure 2 shows a physical model of the 3-DOF helicopterThe following equations (see Figure 2) describe the 3-DOFhelicopter dynamics
120595 = minus
119870
119901119897
119886
119869
120595
sin (120593) (14)
120579 = minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119881
119897+ 119881
119903+ 120585
120593) (15)
=
119870
119891119897
ℎ
119869
120593
(119881
119897minus 119881
119903) (16)
119910 = [120595 120579 120593]
119879
(17)
Fl = KfVl
120601 lh
lh
Fg
la
la
Fr = KfVr
120579
Left motor
Right motor120595120595
Figure 2 Schematic diagram of the 3-DOF helicopter
0 5 10 15 20 25 30 35 40
005
115
2
Time (s)El
evat
ion
(rad
)IO SMCREFFL FO SMC
minus05
minus1
minus15
Figure 3 Stability curve of elevation
0 5 10 15 20 25 30 35 40Time (s)
IO SMCREFFL FO SMC
005
115
2
Rota
tion
(rad
)
minus05
minus1
minus15
Figure 4 Stability curve of rotation
Let define 1199061= 119881
119897+ 119881
119903and 119906
2= 119881
119897minus 119881
119903 for the sake of
simplicity we define state vector as 119909 = [120595
120595 120579
120579 120593 ]
119879 andthe model (14)ndash(16) can be rewritten as
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(18)
4 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
FL FO SMC
012
Pitc
h (r
ad)
minus1
(a)
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
012
Pitc
h (r
ad)
minus1
(b)
Figure 5 Stability curve of pitch
Furthermore let vectors 1198831and 119883
2represent the partitions
of the state vector respectivelyThus 119909 = [1198831
119879 119883
2
119879]
119879 where119883
1isin 119877
2 and 119883
2isin 119877
4 Hence
119883
1-subsystem and
119883
2-
subsystem are presented as follows
119883
1-subsystem
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
119883
2-subsystem
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(19)
5 Design of Controller
It will be seen that
119883
1-subsystem and
119883
2-subsystem have
no coupling We know feedback linearization scheme issuitable approach for multivariable systems but is not robustinherently Sliding mode control is well known as a robustnonlinear control scheme by disturbance rejection capabilityBecause feedback linearization is a simple method andthe FO SMC method is robust against disturbance so wecombine these two approaches Firstly feedback linearizationmethod is applied secondly FO SMC method is used Ourcontrol approach is formed in two steps In the first step afractional order sliding mode controller for
119883
1-subsystem
is designed then since the
119883
2-subsystem is multivariable
and nonlinear feedback linearization control is applied For
119883
2-subsystem controller must be robust against disturbance
After linearization a fractional order sliding mode controlleris proposed for linearized subsystem This helicopter modelis an underactuated system by three outputs and two inputsso we can control two outputs and third output only is keptlimited
Consider
119883
1-subsystem as
1= 119891 + 119892119906
1 (20)
Let define sliding surface as 1198781=
RL119863
120572119890
1+ 120582
1119890
1and the error
signal is defined as 1198901= 119909
1119889minus119909
1 Also disturbance is imposed
to roll and yaw channelThe fractional order slidingmode control input signal can
be defined as follows for subsystem (20) [12]
119906
1(119905) = 119892
minus1[
119889(119905) minus 119891 +
RL119863
minus1minus120572(120582
1119890
1)
+
RL119863
minus1minus120572(119896
1119878
1+ 119896
1sign (119878
1))]
(21)
where 1198961and 120582
1are positive constants
In [12] it has been shown that this control input signalguarantees the stability of the closed-loop
119883
1-subsystem and
the tracking error converges to zero in finite time Then inorder to control
119883
2-subsystem at first we use input-output
feedback linearization method Feedback linearization con-trol law is defined as
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
[119881 minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(22)
119881 is new control law that will be designed for linearizedsystem So we want to obtain overall control law that ithas robustness behavior and good performance We usefractional order control instead of integer order to designcontrol input 119881
The sliding mode control law 119881 is designed as follows
119881 = 119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus3120582
2
2
RL119863
120573119890
2
minus3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
(23)
Let the sliding surface equation be proposed as follows
119878
2= 119890
(3)
2+ 120582
3
2
RL119863
120573119890
2+3120582
2
2
RL119863
120573119890
2+3120582
2
RL119863
120573119890
2
(24)
The error signal is defined as 1198902= 119909
3minus 119909
3119889and the Lyapunov
function is to be defined as
119881 =
1
2
119878
2
2 (25)
ISRN Aerospace Engineering 5
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 6 Force control inputs of proposed controller (stability)
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 7 Force control inputs of IO SMC (stability)
0 5 10 15 20 25 30 35 40
1
0
152
25
353
Time (s)
Elev
atio
n (r
ad)
IO SMCREFFL FO SMC
05
minus05
Figure 8 Tracking curve of elevation
Consider the sufficient condition for the existence finite timeconvergence is
1
2
119889
119889119905
119878
2
2(119905) le minus119896
2
1003816
1003816
1003816
1003816
119878
2(119905)
1003816
1003816
1003816
1003816
(26)
where 1198962is a positive constant Taking time derivative of both
sides of (25) according to Lemmas 3 and 5 then we have
119881 = 119878
2
119878
2= 119878
2(119890
(4)
2+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
0 5 10 15 20 25 30 35 40
1
0
15
2
25
3
Time (s)
Rota
tion
(rad
)
IO SMCREFFL FO SMC
05
minus05
Figure 9 Tracking curve of rotation
= 119878
2(119881 minus 119909
(4)
3119889+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
(27)
Replacing (23) into (27) gives
119881 = 119878
2
119878
2= 119878
2(minus119896
2sgn (119878
2)) le minus (119896
2+ 119865)
1003816
1003816
1003816
1003816
119878
2
1003816
1003816
1003816
1003816
(28)
where 119865 is maximum of disturbance So total feedbacklinearization fractional order sliding mode control (FL FO
6 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
0
1Pi
tch
(rad
)
minus1
(a)
012
Pitc
h (r
ad)
minus10 5 10 15 20 25 30 35 40
Time (s)
FL FO SMC
(b)
Figure 10 Tracking curve of pitch
SMC) law is achieved by combination of (22) and (23) thatis represented as follows
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
times [119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus 3120582
2
2
RL119863
120573119890
2
minus 3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(29)
These two control laws 1199061and 119906
2 are applied to 3-DOF
helicoptermodel and simulation results are shown in the nextsection
6 Simulation Results
To see the performance of the proposed controller wehave simulated the controlled system The results have beencompared with integer order sliding mode controller Thehelicopter nominal parameters are shown in Table 1 Wechoose 120572 = 01 120573 = 015 in this simulation Furthermorea step disturbance is applied in 119905 = 12 sec
To investigate the stability of the closed-loop systemthe initial conditions are applied and the reference input isconsidered to be zero The time response of the states of thesystem (18) in the presence of the control laws (21) and (29) isillustrated in Figures 3 4 and 5 It can be seen that under theproposed controller the output converges to zero and when adisturbance applies the proposed controller is able to nullifythe output again It is clear that the convergence rate in theproposed controller is more than the integer order controller
The force control inputs are given in Figures 6 and7 Although the proposed controller is faster than integerorder controller in states convergence rate it requires lesscontrol signals energy compared with the integer order SMCcontroller
The simulation results of the reference tracking underthe disturbance are illustrated in Figures 8 9 and 10 Thesefigures show that the helicopter can follow the referencetrajectory under the proposed control method and theoutput response under the proposed control scheme is faster
Table 1 Parametersrsquo values
Parameter Value Unit119869
120579091 (kgm2
)
119869
12059300364 (kgm2
)
119869
120595091 (kgm2
)
119870
11989105 (N)
119870
11990105 (N)
119897
119886066 (m)
119897
ℎ0177 (m)
119879
119892033 (Nm)
than another one in spite of disturbance Also the pitch angleremained limited
The control efforts have been shown in Figures 11 and 12According to these figures it can be seen that although theproposed controller is faster than integer order controller itneeds less control effort To implement the controller it isessential that the control effort has an acceptable value
According to these simulations it can be seen that thetracking performance of proposed controller ismore efficientwith less control effort compared with the integer ordersliding mode controller in spite of disturbances
7 Conclusion
In this paper we presented a combination of two nonlinearcontrol techniques applied to 3-DOF helicopter Also we con-sidered disturbance in roll and yaw channel In fact we haveused feedback linearization fractional order sliding modetheory to control roll and yaw channel It is desirable thatthe outputs track the reference input in less time In practicalapplication a high-gain controller may be undesirable Soin this paper a fractional order controller was developed toincrease the convergence rate in less control signal energyTo verify the performance of the proposed controller itwas compared with integer order sliding mode theory Theresults show that the proposed method also simplifies thedesign process due to the use of feedback linearizationcontrol mechanism for multivariable systems the slidingmodemethod advantages such as robustness is retainedThe
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
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2 ISRN Aerospace Engineering
feedback linearization control methods The performance ofthe proposed control scheme is compared with integer order(IO) sliding mode controller and results are justified by thesimulation
This paper is organized as follows in Section 2 theinteger order SMC and feedback linearization techniquesare briefly reviewed and in Section 3 some preliminariesand definitions of fractional order calculus are introducedIn Section 4 the 3-DOF helicopter model description ispresented In Section 5 the proposed controller scheme isgiven and in Section 6 the proposed approach is appliedto 3-DOF helicopter model and experimental results areprovided Finally conclusion is addressed in Section 7
2 Review of IO SMC andFeedback Linearization
In this section definitions of sliding mode and feedbacklinearization control methods are briefly reviewed for essen-tial preparation of the combination of two nonlinear controltechniques FO SMC and feedback linearization
21 Integer Order Sliding Mode Control Consider a second-order nonlinear dynamical system as
1= 119909
2
2= 119891 (119909) + 119892 (119909) 119906 (1)
where 119909 = [119909
1119909
2]
119879 is the system state vector 119891(119909) and119892(119909) are nonlinear functions of 119909 and 119906 = 0 is the scalarinput Sliding mode control approach consists of two partscorrectivecontrol law (119906
119888) that compensates the deviations
from the sliding surface to reach the sliding surfaceandequivalent control law (119906eq) that makes the derivative of thesliding surface equal zero to stay on the sliding surface Thiscontrol law is represented as
119906 = 119906eq + 119906119888 (2)
In general the sliding surface is 119878 = 1199092+ 120573119909
1 where 120573 gt 0
To guarantee the existence of sliding mode the controllaw must satisfy the condition
1
2
119889
119889119905
119904
2lt 0
(3)
To have a fast convergence it is sufficient tomodify the slidingsurface
22 Input-Output Linearization Control Consider the fol-lowing nonlinear dynamical system
= 119891 (119909) + 119892 (119909) 119906 119910 = ℎ (119909) (4)
The control method consists of the following steps
(a) Differentiate 119910 until 119906 appears in one of the equationsfor the derivatives of 119910 Consider
119910
(119903)= 120572 (119909) + 120573 (119909) 119906
(5)
(b) Choose 119906 to give 119910(119903) = V where V is the syntheticinput Consider
119906 =
1
120573 (119909)
[minus120572 (119909) + V] (6)
(c) Then the system has the following form
119884 (119904)
119881 (119904)
=
1
119904
119903 (7)
Design a linear control law for this 119903-integrator linearsystem
(d) Check internal dynamics
3 Basic Description of Fractional Calculus
There exist many definitions of fractional derivative Twoof the most commonly used definitions are the Riemann-Liouville (RL) and Caputo (C) definitions [14]
Definition 1 (see [20]) The left RL fractional derivative isdescribed by
RL119886119863
120572
119905119891 (119905) =
1
Γ (119899 minus 120572)
119889
119899
119889119905
119899int
119905
119886
119891 (120591)
(119905 minus 120591)
120572minus119899+1119889120591 (8)
where 119899 is the first integer which is not less than 120572 that is119899 minus 1 le 120572 lt 119899 and 120572 gt 0 and Γ(sdot) is the gamma function
Definition 2 (see [21]) The Laplace transform of the RLdefinition is described as
int
infin
0
119890
minusst119886119863
120572
119905119891 (119905) 119889119905 = 119904
120572119865 (119904) minus
119899minus1
sum
119896=0
119904
119896
119886119863
120572minus119896minus1
119905119891 (119905)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816119905=0
(9)
Lemma 3 (see [20]) The following equality is satisfied for all120572 gt 0 and 119899 is a natural number Consider
119889
119899
119889119905
119899
RL119863
120572119891 (119905) =
RL119863
119899+120572119891 (119905)
C119863
120572
119889
119899
119889119905
119899119891 (119905) =
C119863
119899+120572119891 (119905)
(10)
Lemma 4 (see [21]) Fractional differentiation and fractionalintegration are linear operations Consider
0119863
120572
119905(119886119891 (119905) + 119887119892 (119905)) = 119886
0119863
120572
119905119891 (119905) + 119887
0119863
120572
119905119892 (119905) (11)
Lemma 5 (see [20]) The following equality for left RL defini-tion is satisfied for all 120572 isin (0 1)
RL119863
120572119868
120572119891 (119905) = 119891 (119905) 120572 isin (0 1)
(12)
However the opposite is not true since
119868
120572
RL119863
120572119891 (119905) = 119891 (119905) minus
119891
1minus120572(0)
Γ (120572)
119905
120572minus1 (13)
where 1198911minus120572
(0) = lim119905rarr0
119868
1minus120572119891(119905)
ISRN Aerospace Engineering 3
Figure 1 The 3-DOF helicopter system
4 System Description
In this work a 3-DOF helicopter model is considered (seeFigure 1)
The following notations are used to describe the 3-DOFhelicopter system dynamics
120579 is angular position of the pitch axis rad120593 is angular position of the roll axis rad120595 is angular position of the yaw axis rad119869
120579 is moment of inertia of the system around the pitchaxis kgm2
119869
120593 is moment of inertia of the helicopter body about theroll axis kgm2
119869
120595 is moment of inertia of the helicopter body about theyaw axis kgm2
119881
119897 is voltage applied to the left motor V
119881
119903 is voltage applied to the right motor V
119870
119891 is force constant of the motor combination N
119897
119886 is distance between the base and the helicopter bodym
119897
ℎ is distance from the pitch axis to either motor m
119879
119892 is effective gravitational torque Nm
119870
119901 is constant of proportionality of the gravitationalforce N
The 3-DOF helicopter control system consists of twoDC motors at the end of the arm as shown in Figure 1Figure 2 shows a physical model of the 3-DOF helicopterThe following equations (see Figure 2) describe the 3-DOFhelicopter dynamics
120595 = minus
119870
119901119897
119886
119869
120595
sin (120593) (14)
120579 = minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119881
119897+ 119881
119903+ 120585
120593) (15)
=
119870
119891119897
ℎ
119869
120593
(119881
119897minus 119881
119903) (16)
119910 = [120595 120579 120593]
119879
(17)
Fl = KfVl
120601 lh
lh
Fg
la
la
Fr = KfVr
120579
Left motor
Right motor120595120595
Figure 2 Schematic diagram of the 3-DOF helicopter
0 5 10 15 20 25 30 35 40
005
115
2
Time (s)El
evat
ion
(rad
)IO SMCREFFL FO SMC
minus05
minus1
minus15
Figure 3 Stability curve of elevation
0 5 10 15 20 25 30 35 40Time (s)
IO SMCREFFL FO SMC
005
115
2
Rota
tion
(rad
)
minus05
minus1
minus15
Figure 4 Stability curve of rotation
Let define 1199061= 119881
119897+ 119881
119903and 119906
2= 119881
119897minus 119881
119903 for the sake of
simplicity we define state vector as 119909 = [120595
120595 120579
120579 120593 ]
119879 andthe model (14)ndash(16) can be rewritten as
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(18)
4 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
FL FO SMC
012
Pitc
h (r
ad)
minus1
(a)
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
012
Pitc
h (r
ad)
minus1
(b)
Figure 5 Stability curve of pitch
Furthermore let vectors 1198831and 119883
2represent the partitions
of the state vector respectivelyThus 119909 = [1198831
119879 119883
2
119879]
119879 where119883
1isin 119877
2 and 119883
2isin 119877
4 Hence
119883
1-subsystem and
119883
2-
subsystem are presented as follows
119883
1-subsystem
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
119883
2-subsystem
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(19)
5 Design of Controller
It will be seen that
119883
1-subsystem and
119883
2-subsystem have
no coupling We know feedback linearization scheme issuitable approach for multivariable systems but is not robustinherently Sliding mode control is well known as a robustnonlinear control scheme by disturbance rejection capabilityBecause feedback linearization is a simple method andthe FO SMC method is robust against disturbance so wecombine these two approaches Firstly feedback linearizationmethod is applied secondly FO SMC method is used Ourcontrol approach is formed in two steps In the first step afractional order sliding mode controller for
119883
1-subsystem
is designed then since the
119883
2-subsystem is multivariable
and nonlinear feedback linearization control is applied For
119883
2-subsystem controller must be robust against disturbance
After linearization a fractional order sliding mode controlleris proposed for linearized subsystem This helicopter modelis an underactuated system by three outputs and two inputsso we can control two outputs and third output only is keptlimited
Consider
119883
1-subsystem as
1= 119891 + 119892119906
1 (20)
Let define sliding surface as 1198781=
RL119863
120572119890
1+ 120582
1119890
1and the error
signal is defined as 1198901= 119909
1119889minus119909
1 Also disturbance is imposed
to roll and yaw channelThe fractional order slidingmode control input signal can
be defined as follows for subsystem (20) [12]
119906
1(119905) = 119892
minus1[
119889(119905) minus 119891 +
RL119863
minus1minus120572(120582
1119890
1)
+
RL119863
minus1minus120572(119896
1119878
1+ 119896
1sign (119878
1))]
(21)
where 1198961and 120582
1are positive constants
In [12] it has been shown that this control input signalguarantees the stability of the closed-loop
119883
1-subsystem and
the tracking error converges to zero in finite time Then inorder to control
119883
2-subsystem at first we use input-output
feedback linearization method Feedback linearization con-trol law is defined as
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
[119881 minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(22)
119881 is new control law that will be designed for linearizedsystem So we want to obtain overall control law that ithas robustness behavior and good performance We usefractional order control instead of integer order to designcontrol input 119881
The sliding mode control law 119881 is designed as follows
119881 = 119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus3120582
2
2
RL119863
120573119890
2
minus3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
(23)
Let the sliding surface equation be proposed as follows
119878
2= 119890
(3)
2+ 120582
3
2
RL119863
120573119890
2+3120582
2
2
RL119863
120573119890
2+3120582
2
RL119863
120573119890
2
(24)
The error signal is defined as 1198902= 119909
3minus 119909
3119889and the Lyapunov
function is to be defined as
119881 =
1
2
119878
2
2 (25)
ISRN Aerospace Engineering 5
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 6 Force control inputs of proposed controller (stability)
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 7 Force control inputs of IO SMC (stability)
0 5 10 15 20 25 30 35 40
1
0
152
25
353
Time (s)
Elev
atio
n (r
ad)
IO SMCREFFL FO SMC
05
minus05
Figure 8 Tracking curve of elevation
Consider the sufficient condition for the existence finite timeconvergence is
1
2
119889
119889119905
119878
2
2(119905) le minus119896
2
1003816
1003816
1003816
1003816
119878
2(119905)
1003816
1003816
1003816
1003816
(26)
where 1198962is a positive constant Taking time derivative of both
sides of (25) according to Lemmas 3 and 5 then we have
119881 = 119878
2
119878
2= 119878
2(119890
(4)
2+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
0 5 10 15 20 25 30 35 40
1
0
15
2
25
3
Time (s)
Rota
tion
(rad
)
IO SMCREFFL FO SMC
05
minus05
Figure 9 Tracking curve of rotation
= 119878
2(119881 minus 119909
(4)
3119889+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
(27)
Replacing (23) into (27) gives
119881 = 119878
2
119878
2= 119878
2(minus119896
2sgn (119878
2)) le minus (119896
2+ 119865)
1003816
1003816
1003816
1003816
119878
2
1003816
1003816
1003816
1003816
(28)
where 119865 is maximum of disturbance So total feedbacklinearization fractional order sliding mode control (FL FO
6 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
0
1Pi
tch
(rad
)
minus1
(a)
012
Pitc
h (r
ad)
minus10 5 10 15 20 25 30 35 40
Time (s)
FL FO SMC
(b)
Figure 10 Tracking curve of pitch
SMC) law is achieved by combination of (22) and (23) thatis represented as follows
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
times [119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus 3120582
2
2
RL119863
120573119890
2
minus 3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(29)
These two control laws 1199061and 119906
2 are applied to 3-DOF
helicoptermodel and simulation results are shown in the nextsection
6 Simulation Results
To see the performance of the proposed controller wehave simulated the controlled system The results have beencompared with integer order sliding mode controller Thehelicopter nominal parameters are shown in Table 1 Wechoose 120572 = 01 120573 = 015 in this simulation Furthermorea step disturbance is applied in 119905 = 12 sec
To investigate the stability of the closed-loop systemthe initial conditions are applied and the reference input isconsidered to be zero The time response of the states of thesystem (18) in the presence of the control laws (21) and (29) isillustrated in Figures 3 4 and 5 It can be seen that under theproposed controller the output converges to zero and when adisturbance applies the proposed controller is able to nullifythe output again It is clear that the convergence rate in theproposed controller is more than the integer order controller
The force control inputs are given in Figures 6 and7 Although the proposed controller is faster than integerorder controller in states convergence rate it requires lesscontrol signals energy compared with the integer order SMCcontroller
The simulation results of the reference tracking underthe disturbance are illustrated in Figures 8 9 and 10 Thesefigures show that the helicopter can follow the referencetrajectory under the proposed control method and theoutput response under the proposed control scheme is faster
Table 1 Parametersrsquo values
Parameter Value Unit119869
120579091 (kgm2
)
119869
12059300364 (kgm2
)
119869
120595091 (kgm2
)
119870
11989105 (N)
119870
11990105 (N)
119897
119886066 (m)
119897
ℎ0177 (m)
119879
119892033 (Nm)
than another one in spite of disturbance Also the pitch angleremained limited
The control efforts have been shown in Figures 11 and 12According to these figures it can be seen that although theproposed controller is faster than integer order controller itneeds less control effort To implement the controller it isessential that the control effort has an acceptable value
According to these simulations it can be seen that thetracking performance of proposed controller ismore efficientwith less control effort compared with the integer ordersliding mode controller in spite of disturbances
7 Conclusion
In this paper we presented a combination of two nonlinearcontrol techniques applied to 3-DOF helicopter Also we con-sidered disturbance in roll and yaw channel In fact we haveused feedback linearization fractional order sliding modetheory to control roll and yaw channel It is desirable thatthe outputs track the reference input in less time In practicalapplication a high-gain controller may be undesirable Soin this paper a fractional order controller was developed toincrease the convergence rate in less control signal energyTo verify the performance of the proposed controller itwas compared with integer order sliding mode theory Theresults show that the proposed method also simplifies thedesign process due to the use of feedback linearizationcontrol mechanism for multivariable systems the slidingmodemethod advantages such as robustness is retainedThe
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Aerospace Engineering 3
Figure 1 The 3-DOF helicopter system
4 System Description
In this work a 3-DOF helicopter model is considered (seeFigure 1)
The following notations are used to describe the 3-DOFhelicopter system dynamics
120579 is angular position of the pitch axis rad120593 is angular position of the roll axis rad120595 is angular position of the yaw axis rad119869
120579 is moment of inertia of the system around the pitchaxis kgm2
119869
120593 is moment of inertia of the helicopter body about theroll axis kgm2
119869
120595 is moment of inertia of the helicopter body about theyaw axis kgm2
119881
119897 is voltage applied to the left motor V
119881
119903 is voltage applied to the right motor V
119870
119891 is force constant of the motor combination N
119897
119886 is distance between the base and the helicopter bodym
119897
ℎ is distance from the pitch axis to either motor m
119879
119892 is effective gravitational torque Nm
119870
119901 is constant of proportionality of the gravitationalforce N
The 3-DOF helicopter control system consists of twoDC motors at the end of the arm as shown in Figure 1Figure 2 shows a physical model of the 3-DOF helicopterThe following equations (see Figure 2) describe the 3-DOFhelicopter dynamics
120595 = minus
119870
119901119897
119886
119869
120595
sin (120593) (14)
120579 = minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119881
119897+ 119881
119903+ 120585
120593) (15)
=
119870
119891119897
ℎ
119869
120593
(119881
119897minus 119881
119903) (16)
119910 = [120595 120579 120593]
119879
(17)
Fl = KfVl
120601 lh
lh
Fg
la
la
Fr = KfVr
120579
Left motor
Right motor120595120595
Figure 2 Schematic diagram of the 3-DOF helicopter
0 5 10 15 20 25 30 35 40
005
115
2
Time (s)El
evat
ion
(rad
)IO SMCREFFL FO SMC
minus05
minus1
minus15
Figure 3 Stability curve of elevation
0 5 10 15 20 25 30 35 40Time (s)
IO SMCREFFL FO SMC
005
115
2
Rota
tion
(rad
)
minus05
minus1
minus15
Figure 4 Stability curve of rotation
Let define 1199061= 119881
119897+ 119881
119903and 119906
2= 119881
119897minus 119881
119903 for the sake of
simplicity we define state vector as 119909 = [120595
120595 120579
120579 120593 ]
119879 andthe model (14)ndash(16) can be rewritten as
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(18)
4 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
FL FO SMC
012
Pitc
h (r
ad)
minus1
(a)
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
012
Pitc
h (r
ad)
minus1
(b)
Figure 5 Stability curve of pitch
Furthermore let vectors 1198831and 119883
2represent the partitions
of the state vector respectivelyThus 119909 = [1198831
119879 119883
2
119879]
119879 where119883
1isin 119877
2 and 119883
2isin 119877
4 Hence
119883
1-subsystem and
119883
2-
subsystem are presented as follows
119883
1-subsystem
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
119883
2-subsystem
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(19)
5 Design of Controller
It will be seen that
119883
1-subsystem and
119883
2-subsystem have
no coupling We know feedback linearization scheme issuitable approach for multivariable systems but is not robustinherently Sliding mode control is well known as a robustnonlinear control scheme by disturbance rejection capabilityBecause feedback linearization is a simple method andthe FO SMC method is robust against disturbance so wecombine these two approaches Firstly feedback linearizationmethod is applied secondly FO SMC method is used Ourcontrol approach is formed in two steps In the first step afractional order sliding mode controller for
119883
1-subsystem
is designed then since the
119883
2-subsystem is multivariable
and nonlinear feedback linearization control is applied For
119883
2-subsystem controller must be robust against disturbance
After linearization a fractional order sliding mode controlleris proposed for linearized subsystem This helicopter modelis an underactuated system by three outputs and two inputsso we can control two outputs and third output only is keptlimited
Consider
119883
1-subsystem as
1= 119891 + 119892119906
1 (20)
Let define sliding surface as 1198781=
RL119863
120572119890
1+ 120582
1119890
1and the error
signal is defined as 1198901= 119909
1119889minus119909
1 Also disturbance is imposed
to roll and yaw channelThe fractional order slidingmode control input signal can
be defined as follows for subsystem (20) [12]
119906
1(119905) = 119892
minus1[
119889(119905) minus 119891 +
RL119863
minus1minus120572(120582
1119890
1)
+
RL119863
minus1minus120572(119896
1119878
1+ 119896
1sign (119878
1))]
(21)
where 1198961and 120582
1are positive constants
In [12] it has been shown that this control input signalguarantees the stability of the closed-loop
119883
1-subsystem and
the tracking error converges to zero in finite time Then inorder to control
119883
2-subsystem at first we use input-output
feedback linearization method Feedback linearization con-trol law is defined as
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
[119881 minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(22)
119881 is new control law that will be designed for linearizedsystem So we want to obtain overall control law that ithas robustness behavior and good performance We usefractional order control instead of integer order to designcontrol input 119881
The sliding mode control law 119881 is designed as follows
119881 = 119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus3120582
2
2
RL119863
120573119890
2
minus3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
(23)
Let the sliding surface equation be proposed as follows
119878
2= 119890
(3)
2+ 120582
3
2
RL119863
120573119890
2+3120582
2
2
RL119863
120573119890
2+3120582
2
RL119863
120573119890
2
(24)
The error signal is defined as 1198902= 119909
3minus 119909
3119889and the Lyapunov
function is to be defined as
119881 =
1
2
119878
2
2 (25)
ISRN Aerospace Engineering 5
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 6 Force control inputs of proposed controller (stability)
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 7 Force control inputs of IO SMC (stability)
0 5 10 15 20 25 30 35 40
1
0
152
25
353
Time (s)
Elev
atio
n (r
ad)
IO SMCREFFL FO SMC
05
minus05
Figure 8 Tracking curve of elevation
Consider the sufficient condition for the existence finite timeconvergence is
1
2
119889
119889119905
119878
2
2(119905) le minus119896
2
1003816
1003816
1003816
1003816
119878
2(119905)
1003816
1003816
1003816
1003816
(26)
where 1198962is a positive constant Taking time derivative of both
sides of (25) according to Lemmas 3 and 5 then we have
119881 = 119878
2
119878
2= 119878
2(119890
(4)
2+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
0 5 10 15 20 25 30 35 40
1
0
15
2
25
3
Time (s)
Rota
tion
(rad
)
IO SMCREFFL FO SMC
05
minus05
Figure 9 Tracking curve of rotation
= 119878
2(119881 minus 119909
(4)
3119889+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
(27)
Replacing (23) into (27) gives
119881 = 119878
2
119878
2= 119878
2(minus119896
2sgn (119878
2)) le minus (119896
2+ 119865)
1003816
1003816
1003816
1003816
119878
2
1003816
1003816
1003816
1003816
(28)
where 119865 is maximum of disturbance So total feedbacklinearization fractional order sliding mode control (FL FO
6 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
0
1Pi
tch
(rad
)
minus1
(a)
012
Pitc
h (r
ad)
minus10 5 10 15 20 25 30 35 40
Time (s)
FL FO SMC
(b)
Figure 10 Tracking curve of pitch
SMC) law is achieved by combination of (22) and (23) thatis represented as follows
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
times [119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus 3120582
2
2
RL119863
120573119890
2
minus 3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(29)
These two control laws 1199061and 119906
2 are applied to 3-DOF
helicoptermodel and simulation results are shown in the nextsection
6 Simulation Results
To see the performance of the proposed controller wehave simulated the controlled system The results have beencompared with integer order sliding mode controller Thehelicopter nominal parameters are shown in Table 1 Wechoose 120572 = 01 120573 = 015 in this simulation Furthermorea step disturbance is applied in 119905 = 12 sec
To investigate the stability of the closed-loop systemthe initial conditions are applied and the reference input isconsidered to be zero The time response of the states of thesystem (18) in the presence of the control laws (21) and (29) isillustrated in Figures 3 4 and 5 It can be seen that under theproposed controller the output converges to zero and when adisturbance applies the proposed controller is able to nullifythe output again It is clear that the convergence rate in theproposed controller is more than the integer order controller
The force control inputs are given in Figures 6 and7 Although the proposed controller is faster than integerorder controller in states convergence rate it requires lesscontrol signals energy compared with the integer order SMCcontroller
The simulation results of the reference tracking underthe disturbance are illustrated in Figures 8 9 and 10 Thesefigures show that the helicopter can follow the referencetrajectory under the proposed control method and theoutput response under the proposed control scheme is faster
Table 1 Parametersrsquo values
Parameter Value Unit119869
120579091 (kgm2
)
119869
12059300364 (kgm2
)
119869
120595091 (kgm2
)
119870
11989105 (N)
119870
11990105 (N)
119897
119886066 (m)
119897
ℎ0177 (m)
119879
119892033 (Nm)
than another one in spite of disturbance Also the pitch angleremained limited
The control efforts have been shown in Figures 11 and 12According to these figures it can be seen that although theproposed controller is faster than integer order controller itneeds less control effort To implement the controller it isessential that the control effort has an acceptable value
According to these simulations it can be seen that thetracking performance of proposed controller ismore efficientwith less control effort compared with the integer ordersliding mode controller in spite of disturbances
7 Conclusion
In this paper we presented a combination of two nonlinearcontrol techniques applied to 3-DOF helicopter Also we con-sidered disturbance in roll and yaw channel In fact we haveused feedback linearization fractional order sliding modetheory to control roll and yaw channel It is desirable thatthe outputs track the reference input in less time In practicalapplication a high-gain controller may be undesirable Soin this paper a fractional order controller was developed toincrease the convergence rate in less control signal energyTo verify the performance of the proposed controller itwas compared with integer order sliding mode theory Theresults show that the proposed method also simplifies thedesign process due to the use of feedback linearizationcontrol mechanism for multivariable systems the slidingmodemethod advantages such as robustness is retainedThe
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
FL FO SMC
012
Pitc
h (r
ad)
minus1
(a)
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
012
Pitc
h (r
ad)
minus1
(b)
Figure 5 Stability curve of pitch
Furthermore let vectors 1198831and 119883
2represent the partitions
of the state vector respectivelyThus 119909 = [1198831
119879 119883
2
119879]
119879 where119883
1isin 119877
2 and 119883
2isin 119877
4 Hence
119883
1-subsystem and
119883
2-
subsystem are presented as follows
119883
1-subsystem
1= 119909
2
2= minus
119879
119892
119869
120579
+
119870
119891119897
119886
119869
120579
(119906
1+ 120585
120593)
119883
2-subsystem
3= 119909
4
4= minus
119870
119901119897
119886
119869
120595
sin (1199095)
5= 119909
6
6=
119870
119891119897
ℎ
119869
120593
119906
2
(19)
5 Design of Controller
It will be seen that
119883
1-subsystem and
119883
2-subsystem have
no coupling We know feedback linearization scheme issuitable approach for multivariable systems but is not robustinherently Sliding mode control is well known as a robustnonlinear control scheme by disturbance rejection capabilityBecause feedback linearization is a simple method andthe FO SMC method is robust against disturbance so wecombine these two approaches Firstly feedback linearizationmethod is applied secondly FO SMC method is used Ourcontrol approach is formed in two steps In the first step afractional order sliding mode controller for
119883
1-subsystem
is designed then since the
119883
2-subsystem is multivariable
and nonlinear feedback linearization control is applied For
119883
2-subsystem controller must be robust against disturbance
After linearization a fractional order sliding mode controlleris proposed for linearized subsystem This helicopter modelis an underactuated system by three outputs and two inputsso we can control two outputs and third output only is keptlimited
Consider
119883
1-subsystem as
1= 119891 + 119892119906
1 (20)
Let define sliding surface as 1198781=
RL119863
120572119890
1+ 120582
1119890
1and the error
signal is defined as 1198901= 119909
1119889minus119909
1 Also disturbance is imposed
to roll and yaw channelThe fractional order slidingmode control input signal can
be defined as follows for subsystem (20) [12]
119906
1(119905) = 119892
minus1[
119889(119905) minus 119891 +
RL119863
minus1minus120572(120582
1119890
1)
+
RL119863
minus1minus120572(119896
1119878
1+ 119896
1sign (119878
1))]
(21)
where 1198961and 120582
1are positive constants
In [12] it has been shown that this control input signalguarantees the stability of the closed-loop
119883
1-subsystem and
the tracking error converges to zero in finite time Then inorder to control
119883
2-subsystem at first we use input-output
feedback linearization method Feedback linearization con-trol law is defined as
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
[119881 minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(22)
119881 is new control law that will be designed for linearizedsystem So we want to obtain overall control law that ithas robustness behavior and good performance We usefractional order control instead of integer order to designcontrol input 119881
The sliding mode control law 119881 is designed as follows
119881 = 119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus3120582
2
2
RL119863
120573119890
2
minus3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
(23)
Let the sliding surface equation be proposed as follows
119878
2= 119890
(3)
2+ 120582
3
2
RL119863
120573119890
2+3120582
2
2
RL119863
120573119890
2+3120582
2
RL119863
120573119890
2
(24)
The error signal is defined as 1198902= 119909
3minus 119909
3119889and the Lyapunov
function is to be defined as
119881 =
1
2
119878
2
2 (25)
ISRN Aerospace Engineering 5
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 6 Force control inputs of proposed controller (stability)
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 7 Force control inputs of IO SMC (stability)
0 5 10 15 20 25 30 35 40
1
0
152
25
353
Time (s)
Elev
atio
n (r
ad)
IO SMCREFFL FO SMC
05
minus05
Figure 8 Tracking curve of elevation
Consider the sufficient condition for the existence finite timeconvergence is
1
2
119889
119889119905
119878
2
2(119905) le minus119896
2
1003816
1003816
1003816
1003816
119878
2(119905)
1003816
1003816
1003816
1003816
(26)
where 1198962is a positive constant Taking time derivative of both
sides of (25) according to Lemmas 3 and 5 then we have
119881 = 119878
2
119878
2= 119878
2(119890
(4)
2+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
0 5 10 15 20 25 30 35 40
1
0
15
2
25
3
Time (s)
Rota
tion
(rad
)
IO SMCREFFL FO SMC
05
minus05
Figure 9 Tracking curve of rotation
= 119878
2(119881 minus 119909
(4)
3119889+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
(27)
Replacing (23) into (27) gives
119881 = 119878
2
119878
2= 119878
2(minus119896
2sgn (119878
2)) le minus (119896
2+ 119865)
1003816
1003816
1003816
1003816
119878
2
1003816
1003816
1003816
1003816
(28)
where 119865 is maximum of disturbance So total feedbacklinearization fractional order sliding mode control (FL FO
6 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
0
1Pi
tch
(rad
)
minus1
(a)
012
Pitc
h (r
ad)
minus10 5 10 15 20 25 30 35 40
Time (s)
FL FO SMC
(b)
Figure 10 Tracking curve of pitch
SMC) law is achieved by combination of (22) and (23) thatis represented as follows
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
times [119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus 3120582
2
2
RL119863
120573119890
2
minus 3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(29)
These two control laws 1199061and 119906
2 are applied to 3-DOF
helicoptermodel and simulation results are shown in the nextsection
6 Simulation Results
To see the performance of the proposed controller wehave simulated the controlled system The results have beencompared with integer order sliding mode controller Thehelicopter nominal parameters are shown in Table 1 Wechoose 120572 = 01 120573 = 015 in this simulation Furthermorea step disturbance is applied in 119905 = 12 sec
To investigate the stability of the closed-loop systemthe initial conditions are applied and the reference input isconsidered to be zero The time response of the states of thesystem (18) in the presence of the control laws (21) and (29) isillustrated in Figures 3 4 and 5 It can be seen that under theproposed controller the output converges to zero and when adisturbance applies the proposed controller is able to nullifythe output again It is clear that the convergence rate in theproposed controller is more than the integer order controller
The force control inputs are given in Figures 6 and7 Although the proposed controller is faster than integerorder controller in states convergence rate it requires lesscontrol signals energy compared with the integer order SMCcontroller
The simulation results of the reference tracking underthe disturbance are illustrated in Figures 8 9 and 10 Thesefigures show that the helicopter can follow the referencetrajectory under the proposed control method and theoutput response under the proposed control scheme is faster
Table 1 Parametersrsquo values
Parameter Value Unit119869
120579091 (kgm2
)
119869
12059300364 (kgm2
)
119869
120595091 (kgm2
)
119870
11989105 (N)
119870
11990105 (N)
119897
119886066 (m)
119897
ℎ0177 (m)
119879
119892033 (Nm)
than another one in spite of disturbance Also the pitch angleremained limited
The control efforts have been shown in Figures 11 and 12According to these figures it can be seen that although theproposed controller is faster than integer order controller itneeds less control effort To implement the controller it isessential that the control effort has an acceptable value
According to these simulations it can be seen that thetracking performance of proposed controller ismore efficientwith less control effort compared with the integer ordersliding mode controller in spite of disturbances
7 Conclusion
In this paper we presented a combination of two nonlinearcontrol techniques applied to 3-DOF helicopter Also we con-sidered disturbance in roll and yaw channel In fact we haveused feedback linearization fractional order sliding modetheory to control roll and yaw channel It is desirable thatthe outputs track the reference input in less time In practicalapplication a high-gain controller may be undesirable Soin this paper a fractional order controller was developed toincrease the convergence rate in less control signal energyTo verify the performance of the proposed controller itwas compared with integer order sliding mode theory Theresults show that the proposed method also simplifies thedesign process due to the use of feedback linearizationcontrol mechanism for multivariable systems the slidingmodemethod advantages such as robustness is retainedThe
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Aerospace Engineering 5
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 6 Force control inputs of proposed controller (stability)
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 7 Force control inputs of IO SMC (stability)
0 5 10 15 20 25 30 35 40
1
0
152
25
353
Time (s)
Elev
atio
n (r
ad)
IO SMCREFFL FO SMC
05
minus05
Figure 8 Tracking curve of elevation
Consider the sufficient condition for the existence finite timeconvergence is
1
2
119889
119889119905
119878
2
2(119905) le minus119896
2
1003816
1003816
1003816
1003816
119878
2(119905)
1003816
1003816
1003816
1003816
(26)
where 1198962is a positive constant Taking time derivative of both
sides of (25) according to Lemmas 3 and 5 then we have
119881 = 119878
2
119878
2= 119878
2(119890
(4)
2+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
0 5 10 15 20 25 30 35 40
1
0
15
2
25
3
Time (s)
Rota
tion
(rad
)
IO SMCREFFL FO SMC
05
minus05
Figure 9 Tracking curve of rotation
= 119878
2(119881 minus 119909
(4)
3119889+ 120582
3
2
RL119863
120573+1119890
2
+3120582
2
2
RL119863
120573+1119890
2+ 3120582
2
RL119863
120573+1119890
2)
(27)
Replacing (23) into (27) gives
119881 = 119878
2
119878
2= 119878
2(minus119896
2sgn (119878
2)) le minus (119896
2+ 119865)
1003816
1003816
1003816
1003816
119878
2
1003816
1003816
1003816
1003816
(28)
where 119865 is maximum of disturbance So total feedbacklinearization fractional order sliding mode control (FL FO
6 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
0
1Pi
tch
(rad
)
minus1
(a)
012
Pitc
h (r
ad)
minus10 5 10 15 20 25 30 35 40
Time (s)
FL FO SMC
(b)
Figure 10 Tracking curve of pitch
SMC) law is achieved by combination of (22) and (23) thatis represented as follows
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
times [119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus 3120582
2
2
RL119863
120573119890
2
minus 3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(29)
These two control laws 1199061and 119906
2 are applied to 3-DOF
helicoptermodel and simulation results are shown in the nextsection
6 Simulation Results
To see the performance of the proposed controller wehave simulated the controlled system The results have beencompared with integer order sliding mode controller Thehelicopter nominal parameters are shown in Table 1 Wechoose 120572 = 01 120573 = 015 in this simulation Furthermorea step disturbance is applied in 119905 = 12 sec
To investigate the stability of the closed-loop systemthe initial conditions are applied and the reference input isconsidered to be zero The time response of the states of thesystem (18) in the presence of the control laws (21) and (29) isillustrated in Figures 3 4 and 5 It can be seen that under theproposed controller the output converges to zero and when adisturbance applies the proposed controller is able to nullifythe output again It is clear that the convergence rate in theproposed controller is more than the integer order controller
The force control inputs are given in Figures 6 and7 Although the proposed controller is faster than integerorder controller in states convergence rate it requires lesscontrol signals energy compared with the integer order SMCcontroller
The simulation results of the reference tracking underthe disturbance are illustrated in Figures 8 9 and 10 Thesefigures show that the helicopter can follow the referencetrajectory under the proposed control method and theoutput response under the proposed control scheme is faster
Table 1 Parametersrsquo values
Parameter Value Unit119869
120579091 (kgm2
)
119869
12059300364 (kgm2
)
119869
120595091 (kgm2
)
119870
11989105 (N)
119870
11990105 (N)
119897
119886066 (m)
119897
ℎ0177 (m)
119879
119892033 (Nm)
than another one in spite of disturbance Also the pitch angleremained limited
The control efforts have been shown in Figures 11 and 12According to these figures it can be seen that although theproposed controller is faster than integer order controller itneeds less control effort To implement the controller it isessential that the control effort has an acceptable value
According to these simulations it can be seen that thetracking performance of proposed controller ismore efficientwith less control effort compared with the integer ordersliding mode controller in spite of disturbances
7 Conclusion
In this paper we presented a combination of two nonlinearcontrol techniques applied to 3-DOF helicopter Also we con-sidered disturbance in roll and yaw channel In fact we haveused feedback linearization fractional order sliding modetheory to control roll and yaw channel It is desirable thatthe outputs track the reference input in less time In practicalapplication a high-gain controller may be undesirable Soin this paper a fractional order controller was developed toincrease the convergence rate in less control signal energyTo verify the performance of the proposed controller itwas compared with integer order sliding mode theory Theresults show that the proposed method also simplifies thedesign process due to the use of feedback linearizationcontrol mechanism for multivariable systems the slidingmodemethod advantages such as robustness is retainedThe
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 ISRN Aerospace Engineering
0 5 10 15 20 25 30 35 40Time (s)
IO SMC
0
1Pi
tch
(rad
)
minus1
(a)
012
Pitc
h (r
ad)
minus10 5 10 15 20 25 30 35 40
Time (s)
FL FO SMC
(b)
Figure 10 Tracking curve of pitch
SMC) law is achieved by combination of (22) and (23) thatis represented as follows
119906
2(119905) = (minus
119870
119901119897
119886
119869
120595
cos (1199095))
minus1
times [119909
(4)
3119889minus 120582
3
2
RL119863
120573119890
2minus 3120582
2
2
RL119863
120573119890
2
minus 3120582
2
RL119863
120573119890
(3)
2minus119896
2Sgn (119878
2)
minus
119870
119901119897
119886
119869
120595
119909
6sin (119909
5)]
(29)
These two control laws 1199061and 119906
2 are applied to 3-DOF
helicoptermodel and simulation results are shown in the nextsection
6 Simulation Results
To see the performance of the proposed controller wehave simulated the controlled system The results have beencompared with integer order sliding mode controller Thehelicopter nominal parameters are shown in Table 1 Wechoose 120572 = 01 120573 = 015 in this simulation Furthermorea step disturbance is applied in 119905 = 12 sec
To investigate the stability of the closed-loop systemthe initial conditions are applied and the reference input isconsidered to be zero The time response of the states of thesystem (18) in the presence of the control laws (21) and (29) isillustrated in Figures 3 4 and 5 It can be seen that under theproposed controller the output converges to zero and when adisturbance applies the proposed controller is able to nullifythe output again It is clear that the convergence rate in theproposed controller is more than the integer order controller
The force control inputs are given in Figures 6 and7 Although the proposed controller is faster than integerorder controller in states convergence rate it requires lesscontrol signals energy compared with the integer order SMCcontroller
The simulation results of the reference tracking underthe disturbance are illustrated in Figures 8 9 and 10 Thesefigures show that the helicopter can follow the referencetrajectory under the proposed control method and theoutput response under the proposed control scheme is faster
Table 1 Parametersrsquo values
Parameter Value Unit119869
120579091 (kgm2
)
119869
12059300364 (kgm2
)
119869
120595091 (kgm2
)
119870
11989105 (N)
119870
11990105 (N)
119897
119886066 (m)
119897
ℎ0177 (m)
119879
119892033 (Nm)
than another one in spite of disturbance Also the pitch angleremained limited
The control efforts have been shown in Figures 11 and 12According to these figures it can be seen that although theproposed controller is faster than integer order controller itneeds less control effort To implement the controller it isessential that the control effort has an acceptable value
According to these simulations it can be seen that thetracking performance of proposed controller ismore efficientwith less control effort compared with the integer ordersliding mode controller in spite of disturbances
7 Conclusion
In this paper we presented a combination of two nonlinearcontrol techniques applied to 3-DOF helicopter Also we con-sidered disturbance in roll and yaw channel In fact we haveused feedback linearization fractional order sliding modetheory to control roll and yaw channel It is desirable thatthe outputs track the reference input in less time In practicalapplication a high-gain controller may be undesirable Soin this paper a fractional order controller was developed toincrease the convergence rate in less control signal energyTo verify the performance of the proposed controller itwas compared with integer order sliding mode theory Theresults show that the proposed method also simplifies thedesign process due to the use of feedback linearizationcontrol mechanism for multivariable systems the slidingmodemethod advantages such as robustness is retainedThe
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
ISRN Aerospace Engineering 7
0 5 10 15 20 25 30 35 40Time (s)
024
minus2
Vf
(V)
Vf control effortmdashIO SMC
(a)
0 5 10 15 20 25 30 35 40Time (s)
0
5
minus5
Vb
(V)
Vb control effortmdashIO SMC
(b)
Figure 11 Force control inputs of proposed controller (tracking)
0 5 10 15 20 25 30 35 40Time (s)
0
1
2
minus1
Vf
(V)
Vf control effortmdashFL FO SMC
(a)
0
2
4
minus2
0 5 10 15 20 25 30 35 40Time (s)
Vb
(V)
Vb control effortmdashFL FO SMC
(b)
Figure 12 Force control inputs of pitch (tracking)
reason is that after linearization a fractional order slidingmode controller is designed for linearized subsystem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K H Kienitz Q-HWu andMMansour ldquoRobust stabilizationof a helicopter modelrdquo in Proceedings of the 29th IEEE Confer-ence on Decision and Control pp 2607ndash2612 December 1990
[2] W Garrad and E Low ldquoEigenspace design of helicopter flightcontrol systemsrdquo Tech Rep Department of Aerospace Engi-neering and Mechanics University of Minnesota 1990
[3] K M Im and W Ham ldquoSliding mode control for helicopterattitude regulation at hoveringrdquo Journal of Control Automationand Systems Engineering vol 3 no 6 pp 563ndash568 1997
[4] A Ferreira de Loza H Rıos and A Rosales ldquoRobust regulationfor a 3-DOF helicopter via sliding-mode observation andidentificationrdquo Journal of the Franklin Institute vol 349 no 2pp 700ndash718 2012
[5] D Jeong T Kang H Dharmayanda and A BudiyonoldquoH-infinity attitude control system design for a small-scaleautonomous helicopter with nonlinear dynamics and uncer-taintiesrdquo Journal of Aerospace Engineering vol 25 no 4 pp 501ndash518 2011
[6] M Boukhnifer A Chaibet and C Larouci ldquoH-infinity robustcontrol of 3-DOF helicopterrdquo in Proceedings of the 9th Interna-tional Multi-Conference on Systems Signals and Devices (SSDrsquo12) 2012
[7] Y Yu and Y Zhong ldquoRobust attitude control of a 3DOFhelicopter with multi-operation pointsrdquo Journal of SystemsScience and Complexity vol 22 no 2 pp 207ndash219 2009
[8] L Marconi and R Naldi ldquoRobust full degree-of-freedomtracking control of a helicopterrdquo Automatica vol 43 no 11 pp1909ndash1920 2007
[9] H T Nguyen and N R Prasad ldquoDevelopment of an intelligentunmanned helicopterrdquo in Fuzzy Modeling and Control SelectedWorks of M Sugeno pp 13ndash43 CRC Press NewYork NY USA1999
[10] A J Calise and R T Rysdyk ldquoNonlinear adaptive flight controlusing neural networksrdquo IEEE Control SystemsMagazine vol 18no 6 pp 14ndash24 1998
[11] S Jafarzadeh R Mirheidari andM JahedMotlagh ldquoIntelligentautopilot control design for a 2-DOF helicopter modelrdquo Inter-national Journal of Computers Communications amp Control vol3 pp 337ndash342 2008
[12] S Dadras andH RMomeni ldquoFractional terminal slidingmodecontrol design for a class of dynamical systems with uncer-taintyrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 1 pp 367ndash377 2012
[13] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999
[14] A Kilbas H Srivastava and J TrujilloTheory and Applicationof Fractional Differential Equations North HollandMathematicsStudies Edited by J van Mill Elsevier 2006
[15] A J Calderon B M Vinagre and V Feliu ldquoFractional ordercontrol strategies for power electronic buck convertersrdquo SignalProcessing vol 86 no 10 pp 2803ndash2819 2006
[16] R El-Khazali W Ahmad and Y Al-Assaf ldquoSliding modecontrol of generalized fractional chaotic systemsrdquo InternationalJournal of Bifurcation and Chaos vol 16 no 10 pp 3113ndash31252006
[17] MO Efe andC A Kasnakoglu ldquoA fractional adaptation law forsliding mode controlrdquo International Journal of Adaptive Controland Signal Processing vol 22 no 10 pp 968ndash986 2008
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 ISRN Aerospace Engineering
[18] M S Tavazoei and M Haeri ldquoSynchronization of chaoticfractional-order systems via active sliding mode controllerrdquoPhysica A vol 387 no 1 pp 57ndash70 2008
[19] A Si-Ammour S Djennoune and M Bettayeb ldquoA slidingmode control for linear fractional systems with input and statedelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2310ndash2318 2009
[20] A Pisano M Rapaic Z Jelicic and E Usai ldquoNonlinearfractional PI control of a class of fractional order systemsrdquo inIFACConference on Advances in PIDControl pp 637ndash642 2012
[21] C A Monje Y Q Chen B M Vinagre D Xue and VFeliu Fractional Order Systems and Control Fundamentals andApplications Springer 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
