Research ArticleControl Law Design for Twin Rotor MIMO System withNonlinear Control Strategy
M. Ilyas,1 N. Abbas,2 M. UbaidUllah,2 Waqas A. Imtiaz,1 M. A. Q. Shah,2 and K. Mahmood1
1Department of Electrical Engineering, Iqra National University, Peshawar 25000, Pakistan2Department of Electrical Engineering, CIIT, Islamabad 45550, Pakistan
Correspondence should be addressed to Waqas A. Imtiaz; [email protected]
Received 13 May 2016; Revised 10 June 2016; Accepted 15 June 2016
Academic Editor: Jean J. Loiseau
Copyright Β© 2016 M. Ilyas et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Modeling of complex air vehicles is a challenging task due to high nonlinear behavior and significant coupling effect between rotors.Twin rotor multi-input multioutput system (TRMS) is a laboratory setup designed for control experiments, which resembles ahelicopter with unstable, nonlinear, and coupled dynamics. This paper focuses on the design and analysis of sliding mode control(SMC) and backstepping controller for pitch and yaw angle control ofmain and tail rotor of theTRMSunder parametric uncertainty.The proposed control strategy with SMC and backstepping achieves all mentioned limitations of TRMS. Result analysis of SMCand backstepping control schemes elucidates that backstepping provides efficient behavior with the parametric uncertainty for twinrotor system. Chattering and oscillating behaviors of SMC are removed with the backstepping control scheme considering the pitchand yaw angle for TRMS.
1. Introduction
Recent times have witnessed the evolution of variousapproaches for proper flight of air vehicles such as helicopter.Modeling of air vehicles dynamics is difficult owing to the sig-nificant coupling effect among rotors and the unavailability ofsome system states. The laboratory setup, twin rotor MIMOsystem (TRMS), is readily utilized, which resembles the flightof a helicopter [1]. It has gained much importance among thecontrol community by serving as a tool for different experi-ments and providing real time environment of an air vehicle.
It is difficult to design a controller for TRMS due to itsnonlinear behavior between two axes [2, 3]. TRMS consists ofa beamwith two rotors connected at its ends which are drivenby separate DC motors and the beam is counterbalanced byan arm having weight at its end [4]. It has two degrees offreedom, which facilitate movements in both horizontal andvertical direction. TRMS is basically a prototype model of ahelicopter; however, there is significant difference in controlof helicopter and its prototype. In order to control TRMS in adesired way, the speed of rotors is altered, while in helicopterit is done by changing the angles of rotors. There is no
cylindrical control in TRMS while in helicopter it is used indirectional control [5].
The control problem of TRMS has gainedmuch attention,owing to the high coupling effect between two propellers,unstable and nonlinear dynamics. Several techniques likeobserver based and hybrid adaptive fuzzy output feedbackcontrol approaches are developed to solve the nonlinearMIMO system with unknown control direction and deadzones [6, 7]. Genetic algorithms to control the unstable andnonlinear dynamics in TRMS are designed using PID control[8]. Adaptive fuzzy sliding mode control is developed for aclass of MIMO nonlinear system which estimates the statesfrom a semi high gain observer to construct the output feed-back fuzzy controller by incorporating the dynamic slidingmode [9]. References [10, 11] developed the observer basedadaptive fuzzy backstepping dynamic surface control (DSC)for nonlinearMIMOwith immeasurable states. [12] performsa comparative analysis between intelligent control and classi-cal control for TRMS. Adaptive fuzzy, neural network, andfeedback linearization based controllers are also designedfor the tracking of yaw and pitch angles in TRMS [4, 13β22]. However, the proposed state variables are assumed
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016, Article ID 2952738, 10 pageshttp://dx.doi.org/10.1155/2016/2952738
2 Discrete Dynamics in Nature and Society
π
I2 MBπ + MR
I1
MFG + MΨ + MG
Ξ¨
Figure 1: TRMS laboratory setup.
measurable, which is practically not feasible to control thepitch and yaw angle of TRMS.
This paper proposes the first-order sliding mode controland backstepping control scheme for the nonlinear TRMS.In our proposed methodology, the mathematical model ofTRMS is linearized and the cross coupling effect between themain rotors is considered as disturbance.Themain advantageof SMC is that it mitigates the parametric uncertainty presentin TRMS while backstepping control algorithm performsbetter in case of external disturbance, which in this case is thecoupling effect of the rotors, because of its recursive structure.The proposed approaches are investigated for TRMS keepingin view the need for cancelling the strong coupling betweenrotors and finally providing the desired tracking response ofboth the controllers. Simulation results show the effectivenessof control algorithms but comparatively the backsteppingscheme gives the best performance in terms of stability andreference tracking.
The remainder of the paper is arranged as follows. In Sec-tion 2, themathematicalmodel of TRMS system is introducedand the parameters of the system are specified.The proposedSMC and backstepping controller along with their simulationresults are given in Sections 3 and 4, respectively. Comparisonof proposed controllers is introduced in Section 5 followed bythe concluding remarks.
2. Mathematical Modeling
Figure 1 shows the TRMS laboratory setup, which is used todevelop the mathematical model to compare the operation ofSMC and backstepping controllers.
TRMS system is designed with two rotors (main rotorand tail rotor) as shown in Figure 1 encompassing the effectof forces like gravitational, propulsion, centrifugal, frictional,and disturbance torque on movement of the propellers. Toovercome the effects of these forces we provide controlinput through motors. In the given case, only the pitch andazimuth angles are the measureable outputs and its stabilityis the main objective of designing the controller. As far asthe mechanical unit is concerned the following nonlinear
momentum equations can be derived for the pitchmovementof TRMS [1]. Consider
πΌ1Ξ¨ = π
1βππΉπΊβππ΅Ξ¨βππΊ, (1)
where
π1= π1π2
1+ π1π1, (2)
ππΉπΊ= ππsinπ, (3)
ππ΅π= π΅1ΨΨ + π΅
2Ξ¨sin (2Ξ¨) οΏ½οΏ½2, (4)
ππΊ= πΎππ¦π1οΏ½οΏ½ cos (Ξ¨) , (5)
where π1and π1are constants. Equation (5) is derived based
on law of conservation of angular momentum of main rotor:πΌ2οΏ½οΏ½ = π
2βππ΅πβππ . (6)
Themomentum equations in the vertical plane of motion arewritten as
π2= π2π2
2+ π2π2, (7)
ππ΅π= π΅1ποΏ½οΏ½, (8)
ππ = πΎπ
πππ + 1
πππ + 1
π1. (9)
Similar momentum equation can be used for the horizontalplane motion as well.
Equation (9) is derived based on law of angular conserva-tion of momentum of main rotor. The state space equationsare as follows:
(i) For main motor,
π1=π10
π11
π1+π1
π11
π’1. (10)
(ii) For tail motor,
π2=π20
π21
π2+π2
π21
π’2, (11)
where π1and π
2are the motor gain and π
20, π21, π10, and
π11
are the motor parameter. π1and π
2are momentum of
mainmotor and tail motor.The linearization of the nonlinearmodel of TRMS is given in the following section.
(A) Linearization of TRMS. The state space equation ofnonlinear system along with the parameters is given by thefollowing equations:
πΞ¨
ππ‘=π1
πΌ1
π2
1+π1
πΌ1
π1β
ππ
πΌ1
sin (Ξ¨)
+0.0326
2πΌ1
sin (2Ξ¨) οΏ½οΏ½2 βπ΅1Ξ¨I1Ξ¨
β
πππ¦
πΌ1
cos (Ξ¨) π (π1π2
1+ π1π1) ,
ποΏ½οΏ½
ππ‘=π2
πΌ2
π2
2+π2
πΌ2
π2β
π΅1π
πΌ2
οΏ½οΏ½ βππ
πΌ2
1.75 ( π1π2
1+ π1π1) ,
Discrete Dynamics in Nature and Society 3
Table 1: Values of tuning parameters for SMC.
Pitch angle(rad)
Yaw angle(rad)
Case 1πΆ1 = 2.102πΆ2 = 3.5πΎ1 = 1.5
πΆ3 = 6.305πΆ4 = 0.0875πΎ2 = 4.9
Case 2πΆ1 = 1.1πΆ2 = 3.5πΎ1 = 1.95
πΆ3 = 1.3πΆ4 = 0.088πΎ2 = 2.897
Case 3πΆ1 = 2.5πΆ2 = 3.5πΎ1 = 1.915
πΆ3 = 3.3πΆ4 = 0.088πΎ2 = 2.89
ππ1
ππ‘= β
π10
π11
π1+π1
π11
π’1,
ππ2
ππ‘= β
π20
π21
π2+π2
π21
π’2.
(12)
The state and output vectors are given by
x = [Ξ¨ Ξ¨ π οΏ½οΏ½π1π2]π
,
y = [Ξ¨ π]π
,
(13)
where variables are as follows (Table 1):π: pitch (elevation) angle.π: yaw (azimuth) angle.π1: momentum of main rotor.
π2: momentum of tail rotor.
Here all the variables of system are expressed in term ofβπ₯.β So
π₯1= Ξ¨,
π₯2= Ξ¨,
π₯3= π,
π₯4= οΏ½οΏ½,
π₯5= π1,
π₯6= π2.
(14)
Now the state space of the system in term of variable βπ₯β willbecome
οΏ½οΏ½1= π₯2,
οΏ½οΏ½2=π1
πΌ1
π₯2
5+π1
πΌ1
π₯5β
ππ
πΌ1
sin (π₯1)
+0.0326
2πΌ1
sin (2π₯1) π₯2
4βπ΅1Ξ¨
πΌ1
οΏ½οΏ½1
β
πππ¦
πΌ1
cos (π₯1) π₯3( π1π₯2
5+ π1π₯5) ,
οΏ½οΏ½3= π₯4,
οΏ½οΏ½4=π2
πΌ2
π₯2
6+π2
πΌ2
π₯6β
π΅1π
πΌ2
π₯5βππ
πΌ2
1.75 ( π1π₯2
5+ π1π₯5) ,
οΏ½οΏ½5= β
π10
π11
π₯5+π1
π11
π’1,
οΏ½οΏ½6= β
π20
π21
π₯6+π2
π21
π’2.
(15)
To linearize the system, let the system be represented as
οΏ½οΏ½ = Aπ₯ + Bπ’,
π¦ = Cπ₯,(16)
where π₯ β π as states, π’ β π as the control input, and π¦ β π as the measured output.
Consider
π1= οΏ½οΏ½1,
π2= οΏ½οΏ½2,
π3= οΏ½οΏ½3,
π4= οΏ½οΏ½4,
π5= οΏ½οΏ½5,
π6= οΏ½οΏ½6.
(17)
After taking Jacobean and putting point (0, 0), then resultingsystem matrices are given below. Consider
A=
[[[[[[[[[[[[[[[[[[[[[
[
0 1 0 0 0 0
β
ππ
πΌ1
βπ΅1Ξ¨
πΌ1
0 0π1
πΌ1
0
0 0 0 1 0 0
0 0 0 β
π΅1π
πΌ2
βππ
πΌ2
1.75π2
πΌ2
0 0 0 0 βπ10
π11
0
0 0 0 0 0 βπ20
π21
]]]]]]]]]]]]]]]]]]]]]
]
,
B =
[[[[[[[[[[[[[[[
[
0 0
0 0
0 0
0 0
π1
π11
0
0π2
π21
]]]]]]]]]]]]]]]
]
,
C = [
1 0 0 0 0 0
0 0 1 0 0 0] .
(18)
4 Discrete Dynamics in Nature and Society
(B) State Space Equations of Linearized Model. Values ofconstants are given in the Abbreviation section [1]. By puttingvalues of all these constants, the state space equations can begiven as
οΏ½οΏ½1= π₯2,
οΏ½οΏ½2= β4.7059π₯
1β 0.0882π₯
2+ 1.3588π₯
5,
οΏ½οΏ½3= π₯4,
οΏ½οΏ½4= β5π₯
4+ 1.617π₯
5+ 4.5π₯
6,
οΏ½οΏ½5= β0.9091π₯
5+ π’1,
οΏ½οΏ½6= βπ₯6+ 0.8π’
2.
(19)
3. Proposed SMC Controller
(A) Choosing Sliding Surface. SMC is a nonlinear controltechnique, which deals with the capability of controlling theuncertainties of nonlinear systems [23, 24]. The primaryadvantage of the SMC technique is the low sensitivity tosystem disturbances. Moreover, it accredits the decoupling ofthe lower dimensions, and consequently, it scales down thecomplication of feedback design [25]. SMC generally consistof two phases: reaching phase and the sliding phase. Thereaching phase converges the system states to desired surfaceand sliding phase handles the oscillations. Sliding surface canbe designed as
π = πππ₯ = π1π₯1+ π2π₯2+ β β β + π
πβ1π₯πβ1
+ π₯π= 0. (20)
Now the control input consist of two parts:(i) Equivalent controller, πeq.
(ii) Discontinuous controller, οΏ½οΏ½.Consequently, the required controller can be determines as
π = πeq + οΏ½οΏ½. (21)
(B) SMC Design for Linearized Model. This section outlinesthe SMC design for linearized model. Sliding surface of thesystem is designed at first to facilitate the process. TRMS is aMIMO system so we will design two sliding surfaces.
Sliding surface for the vertical plane is as follows:
π 1= π₯5+ π2π₯2+ π1π1,
π1= π₯1β π₯1π.
(22)
Lyapunov condition is satisfied as
οΏ½οΏ½1= βπ 2
1β π1sign (π
1) . (23)
Sliding surface for horizontal plane is as follows:
π 2= π₯6+ π4π₯4+ π3π2,
π2= π₯3β π₯2π,
(24)
where we have the following.Lyapunov condition is satisfied as
οΏ½οΏ½2= βπ 2
2β π2sign (π
2) . (25)
2 4 6 8 10 12 14 16 18 200Time (s)
β0.2
0
0.2
0.4
0.6
0.8
1
Pitc
h an
gle (
rad)
SMC controller
Pitch angle (rad)
Figure 2: Pitch angle for linearized system using SMC.
SMC controller
Yaw angle (rad)
2 4 6 8 10 12 14 16 18 200Time (s)
β0.2
0
0.2
0.4
0.6
0.8
1
Yaw
angl
e (ra
d)
Figure 3: Yaw angle for linearized system using SMC.
3.1. Simulation Results of SMC. Figures 2 and 3 show the pitchand yaw position of TRMS obtained after implementation ofSMC in Simulink MATLAB on linearized model. It is clearthat the desired objective of regulating the system for twodegrees of freedom has been achieved under the robust con-trol action of SMC. It is shown that settling time for pitch andyaw angles is under 3 and 5 seconds, respectively. Moreover,it is observed that steady state error is approximately zero.Therefore, the proposed TRMS attains the equilibrium posi-tion with respect to pitch and yaw movement under appliedcontrol action.
The control inputs π’πand π’
πfor pitch and yaw move-
ments of TRMS, respectively, are in volts and provided by twoindependent DC motors connected to corresponding rotors.The control inputs contain two types of control action, that is,the equivalent and discontinuous control. It is evident fromFigures 2 and 3 that the corresponding equivalent controlefforts successfully drive the system dynamics to correspond-ing sliding surfaces in a short period of time.
Moreover, the discontinuous control parts efficientlymaintain the system states on sliding manifolds for all
Discrete Dynamics in Nature and Society 5
SMC controller
2 4 6 8 10 12 14 16 18 200Time (s)
β4
β2
02468
10121416
Con
trol i
nput
(V) f
or p
itch
angl
e
u1
Figure 4: Control input for pitch angle.
SMC controller
2 4 6 8 10 12 14 16 18 200Time (s)
β10
β5
0
5
10
15
Con
trol i
nput
(V) f
or y
aw an
gle
u2
Figure 5: Control input for yaw angle.
subsequent times and are responsible for system robustnessagainst uncertainties. However, chattering in control inputsβπ’πβ and βπ’
πβ can be clearly seen from Figures 4 and 5,
which arises due to fast switching of discontinuous controlaction around the sliding manifolds. Since the amplitude ofchattering is small, both yaw and pitch movement of TRMSare not affected by this undesired phenomenon.
The sliding manifolds π 1and π 2have been designed by
linearly combining system states for regulation purpose ofTRMS under system uncertainties and significant couplingbut in the absence of external perturbation. The tuningparameters have been suitably adjusted for sliding surface.Figures 6 and 7 show chattering phenomenon of sliding sur-face for pitch and yaw angle of TRMS. It is observed that thechattering phenomenon in the sliding surface is miniscule.Moreover, the settling time of sliding surface for both verticaland horizontal planes is under 1 second, which is desirablefor the system under consideration. Now another discussionis provided about the implementation of SMC with tracking.
8 9 10
2 4 6 8 10 12 14 16 18 200Time (s)
β0.1
0
0.1
0.2
0.3
0.4
0.5
Slid
ing
surfa
ce
SMC controller
β10
β5
05Γ10β3
Sliding surface for vertical plane
Figure 6: Chattering in sliding surface for pitch angle.
8 9 10
2 4 6 8 10 12 14 16 18 200Time (s)
β0.5
0
0.5
1
1.5
2
2.5
Slid
ing
surfa
ce fo
r yaw
angl
e
SMC controller
0.01
0
β0.01
Sliding surface for horizontal plane
Figure 7: Chattering in sliding surface for yaw angle.
3.2. Simulation Results of TRMS with Tracking. The pitch andyaw angles are obtained after implementation of SMC on aTRMS in Simulink MATLAB. Figure 8 shows the responseof pitch angle at different values of tuning parameters andFigure 9 shows the response of yaw angle at different values oftuning parameters. Figure 8 shows that the tuning parameterfor case 1 has approximately 20% overshoot from the desiredposition. Case 3 is showing undershoot from the desiredposition due the difference tuning parameters. It is observedthat case 2 is the most suitable set of tuning parametersfor achieving the desired results without showing over- andundershoot. Thus, the desired objective of regulating thesystem for two degrees of freedom has been achieved underthe robust control action of SMC. Same phenomenon isobtained for yaw angle in Figure 9, where case 2 is mostpreferable with desired tuning parameters.
Different values of tuning parameters show how we canobtain different responses of pitch and yaw angles accordingto requirements. Overshoot problem is faced in case of sharpresponse, and if we need a slower response then settling timeis increased.
6 Discrete Dynamics in Nature and Society
SMC controller
Pitch angle (rad) for case 1Pitch angle (rad) for case 2Pitch angle (rad) for case 3
3010 15 20 2550Time (s)
β0.2
0
0.2
0.4
0.6
0.8
1
1.2
Pitc
h an
gle (
rad)
Figure 8: Response of pitch angle for TRMS.
SMC controller
Yaw angle (rad) for case 3Yaw angle (rad) for case 2Yaw angle (rad) for case 1
5 10 15 20 25 300 Time (s)
0
0.2
0.4
0.6
0.8
1
1.2
Yaw
angl
e (ra
d)
Figure 9: Response of yaw angle for TRMS.
4. Backstepping Controller
In control system theory, backstepping controller schemeis introduced by Krstic in 1995 and his companions fordesigning stability control system for a special class of linearand nonlinear dynamical system [26]. Backstepping is asystematic, Lyapunov-based method for nonlinear controlwhich refers to the recursive nature of the design procedurewhich starts at the scalar equation separated by the largestnumber of integrations from the control input and steps backtowards the control input [27, 28].
In the theory of Ordinary Differential Equations (ODEs),Lyapunov functions are scalar functions that may be usedto prove the stability of equilibrium of an ODE. The basicidea behind the Lyapunov function method consists of (I)choosing a radially unbounded positive definite Lyapunovfunction candidate π(π₯) and (II) evaluating its derivative
π(π₯) along system dynamics and checking its negativenessfor stability analysis [25, 26].
The recursitivity terminates when the final control phaseis reached. The process which receives its stability throughrecursitivity is called backstepping [27, 28]. Backstepping canbe used for tracking and regulation problem. With the aidof Lyapunov stability, this control approach for asymptotictracking can be achieved.
4.1. Design Steps. The controller is designed using backstep-ping control technique on the proposed control problem.The standard backstepping control is based on step-by-step construction of Lyapunov function. Here we designcontroller, based on Lyapunov function.
(A) For Vertical Plane. First of all in 1st step we introduce anew state
π§1= π₯1β π₯1π, (26)
where βπ§1β is the new state and βπ₯
1β is state variable.
Lyapunov candidate function (LCF) for new state is
π1= (
1
2) π§2
1, (27)
where π > 0, βπ₯ = 0, and π(0) = 0.By taking time derivative of Lyapunov function we get
οΏ½οΏ½1= π§1οΏ½οΏ½1, (28)
οΏ½οΏ½1= π§2+ π1β οΏ½οΏ½1π, (29)
where βπ1β is virtual control input to control the system and
π§2is the new state
π1= βπ1π§1+ π₯1π. (30)
Now another new state is introduced that is given by βπ§2β:
π§2= π₯2β π1, (31)
where βπ₯2β is the second state variable of the system. By
putting values in (28) we get
οΏ½οΏ½1= βπ§2
1+ π§1π§2. (32)
Now again we repeat the previous step to calculate the virtualcontrol input.
So,
π§2= π₯2β π1. (33)
We know
οΏ½οΏ½1= βπ1οΏ½οΏ½1+ οΏ½οΏ½1π. (34)
New Lyapunov candidate function (LCF) is as follows:
π2= π1+ (
1
2) π§2
2. (35)
Discrete Dynamics in Nature and Society 7
Taking derivative of (35),
οΏ½οΏ½2= οΏ½οΏ½1+ π§2οΏ½οΏ½2. (36)
By putting the value of οΏ½οΏ½1and οΏ½οΏ½2we get
οΏ½οΏ½2= βπ§2
1β π§2
2+ π§2π§3, (37)
where
οΏ½οΏ½2= οΏ½οΏ½2β οΏ½οΏ½1,
οΏ½οΏ½1= βπ§2
1+ π§1π§2.
(38)
Now again we introduce new state
π§3= π₯5β π2. (39)
By taking derivative we get
οΏ½οΏ½3= οΏ½οΏ½5β οΏ½οΏ½2. (40)
Now LCF will be as
π3= π2+ (
1
2) π§2
3. (41)
Taking derivative of above equation, we get
οΏ½οΏ½3= οΏ½οΏ½2+ οΏ½οΏ½3π§3. (42)
By putting the values in above equation we obtain anothervirtual control input for the second state of the system asgiven below:
π2= (
1
1.3588)
β [βπ§1+ 4.7051π₯
1+ 0.0882π₯
2β π1οΏ½οΏ½1+ οΏ½οΏ½1π] .
(43)
After differentiation,
οΏ½οΏ½2= (
1
1.3588)
β [βοΏ½οΏ½1+ 4.7051οΏ½οΏ½
1+ 0.0882οΏ½οΏ½
2β π1οΏ½οΏ½1+...π₯1π] .
(44)
Now the control input is given by
π’1= βπ3π§3β π§2+ 0.909π₯
5+ οΏ½οΏ½2. (45)
We get the control input π’1for vertical plane of TRMS. Now
putting the value of control input we get
οΏ½οΏ½3= βπ§2
1β π§2
2β π§2
3. (46)
Hence condition is satisfied and system is asymptoticallystable.
(B) For horizontal Plane. First of all in 1st step we introduce anew state
π§4= π₯3β π₯2π, (47)
where π§4is the new state and π₯
3is the state variable.
LCF for new state is
π4= (
1
2) π§2
4. (48)
By taking derivative and putting the value of variables we get
οΏ½οΏ½4= βπ§2
4+ π§4π§5. (49)
Now, the abovementioned steps are repeated to calculate theother virtual control input.
Consequently,
οΏ½οΏ½4= π§5+ π3β οΏ½οΏ½2π,
π3= βπ4π§4+ οΏ½οΏ½2π.
(50)
Here π3is arbitrary control input to converge the state π§
4
towards stability.Introduce another new state to calculate another arbitrary
control input.So,
π§5= π₯4β π3, (51)
where π§5is new state for state variable π₯
4. By taking time
derivative,
οΏ½οΏ½5= οΏ½οΏ½4β οΏ½οΏ½3. (52)
Also we take time derivative of previous arbitrary controlinput as
οΏ½οΏ½3= βπ4οΏ½οΏ½4+ οΏ½οΏ½2π. (53)
Now LCF will be as
π5= π4+ (
1
2) π§2
5. (54)
By taking derivative and putting the values of οΏ½οΏ½5and οΏ½οΏ½
3we
get
οΏ½οΏ½5= βπ§2
4+ π§4π§5+ π§5[οΏ½οΏ½4β (βπ4οΏ½οΏ½4+ οΏ½οΏ½2π)] . (55)
After some algebraic calculations we get
οΏ½οΏ½5= βπ§2
4β π§2
5+ π§6π§5, (56)
where π§6is the new state:
π§6= π₯6β π4. (57)
Now LCF will be as
π6= π5+ (
1
2) π§2
6. (58)
After taking derivative,
οΏ½οΏ½6= οΏ½οΏ½5+ οΏ½οΏ½6π§6, (59)
8 Discrete Dynamics in Nature and Society
Backstepping controller
Pitch angle (rad)
5 10 150Time (s)
β0.2
0
0.2
0.4
0.6
0.8
1
Pitc
h an
gle (
rad)
Figure 10: Pitch angle (rad) for backstepping.
where
π4= (
1
4.5) [βπ§4β π5π§5+ 5π₯4β 1.617π₯
5β οΏ½οΏ½3+ οΏ½οΏ½2π] . (60)
By taking derivative of (60),
οΏ½οΏ½4= (
1
4.5) [βοΏ½οΏ½4β π5οΏ½οΏ½5+ 5οΏ½οΏ½4β 1.617οΏ½οΏ½
5β οΏ½οΏ½3+...π₯2π] . (61)
By putting the values of οΏ½οΏ½6, οΏ½οΏ½4, and οΏ½οΏ½
5we get
οΏ½οΏ½6= βπ§2
4β π§2
5β π§2
6, (62)
where οΏ½οΏ½6is negative definite and system will be asymptoti-
cally stable. Nowwe get control input for the horizontal plane.Finally we get the control law, whichwill regulate all the statesof the system to the origin.The system is asymptotically stableby using Backstepping design method:
π’2= (
1
0.8) [βπ6π§6β π§5+ π₯6] + (
1
4.5)
β [βοΏ½οΏ½4β π5οΏ½οΏ½5+ 5οΏ½οΏ½4β 1.617οΏ½οΏ½
5β οΏ½οΏ½3+...π₯2π] .
(63)
After mathematical calculations of backstepping controller,we use user defined block from MATLAB Simulink library.Required equations are used in this function. On the basisof simulation results we will elaborate the performance ofcontroller which is given below
4.2. Simulation Results of Backstepping Controller. On thebasis of backstepping control design, simulation results inFigures 10 and 11 show the stability response of pitch angleand the control input for pitch angle, respectively. SimilarlyFigures 12 and 13 show the stability of yaw angle and controlinput for the pitch angle, respectively. It is observed thatsettling time for pitch and yaw angle in case of backsteppingtechnique is less as compared to sliding mode control. Thecontroller shows very promising results and it is foundthat backstepping controller is capable of tracking and littlevariation in control inputs of both pitch and yaw angle.
Control input (V)
Backstepping controller
5 10 150Time (s)
β2
0
2
4
6
8
10
Con
trol i
nput
(V) f
or p
itch
angl
e
Figure 11: Control input for pitch angle.
Backstepping controller
Yaw angle (rad)
5 10 150Time (s)
β0.2
0
0.2
0.4
0.6
0.8
1
Yaw
angl
e (ra
d)
Figure 12: Yaw angle (rad) for backstepping.
Control input (V)
Backstepping controller
5 10 150Time (s)
β20
β15
β10
β5
0
5
Con
trol i
nput
(V) f
or y
aw an
gle
Figure 13: Control input for yaw angle.
The performance of SMC is limited due to chattering incontrol inputs but this issue is resolved through backstepping,which is shown in Figures 11 and 13. An extensive overview is
Discrete Dynamics in Nature and Society 9
SMC and backstepping
RefSMCBackstepping
5 10 150Time (s)
β0.2
0
0.2
0.4
0.6
0.8
1
1.2
Pitc
h an
gle (
rad)
Figure 14: Pitch angle (rad) for SMC and backstepping.
SMC and backstepping
SMCRefBackstepping
5 10 150Time (s)
0
0.2
0.4
0.6
0.8
1
1.2
Yaw
angl
e (ra
d)
Figure 15: Yaw angle (rad) for SMC and backstepping.
given below for comparison of backstepping and SMC on thebasis of simulation results.
5. Comparison of Backstepping andSMC Controller
Figures 14 and 15 show the comparison between SMC andbackstepping controller for pitch and yaw angle of TRMS. Itis observed that backstepping controller shows good resultscompared to SMC in terms of handling oscillation andchattering. By using backstepping controller, the settling timefor the pitch and yaw angle is less as compared to SMC asshown in Figures 14 and 15.
6. Conclusion
This work outlines the design analysis of robust controllertechniques by implementing them in TRMS, which is a non-linear system. SMC and backstepping are implemented andanalyzed for handling the oscillation and chattering in pitchand yaw angles of TRMS. It is observed that backsteppingshows better performance in terms of less settling time andhandling perturbation as compared to SMC, owing to therecursive structure for controller design. This philosophy isthe core idea that has been followed for developing robustcontroller. The controller was implemented in the Simulinkenvironment where the state space model of the controllerwas engaged with system to achieve the desired result.Implementing the sliding mode control via backsteppingcontrol can be considered as a recommended future work,since parametric uncertainty and external disturbance canbe mitigated within a single model, which can stabilize theTRMS in a more robust way.
System Parameters
πΌ1: Moment of inertia of vertical rotor (6.8 Γ 10β2 kgm2)
πΌ2: Moment of inertia of horizontal rotor (2 Γ 10β2 kgm2)
π΅2π: Friction momentum function parameter(1 Γ 10β2Nms2/rad)
πππ¦: Gyroscopic momentum parameter (0.05 s/rad)
π20: Motor denominator parameter (1)
ππ: Cross reaction momentum gain (2)
π΅2Ξ¨: Friction momentum function parameter(1 Γ 10β3Nms2/rad)
π΅1Ξ¨: Friction momentum function parameter(6 Γ 10β3Nmβ s/rad)
π1: Static characteristic parameter (0.0135)
π1: Static characteristic parameter (0.0924)
π2: Static characteristic parameter (0.02)
π10: Motor 1 denominator parameter (1)
ππ: Gravity momentum (0.32Nm)
π1: Motor 1 gain 1 2 3.5 0.2 (1.1)
π2: Motor 2 gain (0.8)
π11: Motor 1 denominator parameter (1.1).
Competing Interests
The authors declare that they have no competing interests.
References
[1] M. Twin Rotor, System Manual, Feedback Instruments, Crow-borough, UK, 2002.
[2] M. Chen, S. S. Ge, and B. V. E. How, βRobust adaptive neuralnetwork control for a class of uncertain MIMO nonlinearsystems with input nonlinearities,β IEEE Transactions on NeuralNetworks, vol. 21, no. 5, pp. 796β812, 2010.
[3] A. Boulkroune,M.MβSaad, andH. Chekireb, βDesign of a fuzzyadaptive controller for MIMO nonlinear time-delay systemswith unknown actuator nonlinearities and unknown controldirection,β Information Sciences, vol. 180, no. 24, pp. 5041β5059,2010.
10 Discrete Dynamics in Nature and Society
[4] A. Rahideh, A. H. Bajodah, and M. H. Shaheed, βReal timeadaptive nonlinear model inversion control of a twin rotorMIMO systemusing neural networks,β Engineering Applicationsof Artificial Intelligence, vol. 25, no. 6, pp. 1289β1297, 2012.
[5] A. Rahideh,M.H. Saheed, A. Safavi, and J. C. Huijberts, βModelpredictive control of a twin rotorMIMO system,β in Proceedingsof the 21st International Conference on Methods and Models inAutomation and Robotics, p. 2831, MiΔdzyzdroje, Poland, 2006.
[6] Y. Li, S. Tong, and T. Li, βObserver-based adaptive fuzzytracking control of MIMO stochastic nonlinear systems withunknown control directions and unknown dead zones,β IEEETransactions on Fuzzy Systems, vol. 23, no. 4, pp. 1228β1241, 2015.
[7] Y. M. Li, S. C. Tong, and T. S. Li, βHybrid fuzzy adaptiveoutput feedback control design for uncertain MIMO nonlinearsystems with time-varying delays and input saturation,β IEEETransactions on Fuzzy Systems, 2015.
[8] J.-G. Juang, M.-T. Huang, and W.-K. Liu, βPID control usingpresearched genetic algorithms for a MIMO system,β IEEETransactions on Systems, Man and Cybernetics Part C: Applica-tions and Reviews, vol. 38, no. 5, pp. 716β727, 2008.
[9] S. Tong and H.-X. Li, βFuzzy adaptive sliding-mode control forMIMOnonlinear systems,β IEEETransactions on Fuzzy Systems,vol. 11, no. 3, pp. 354β360, 2003.
[10] S. C. Tong, Y. M. Li, and P. Shi, βObserver-based adaptive fuzzybackstepping output feedback control of uncertain MIMOpure-feedback nonlinear systems,β IEEE Transactions on FuzzySystems, vol. 20, no. 4, pp. 771β785, 2012.
[11] S.-C. Tong, Y.-M. Li, G. Feng, and T.-S. Li, βObserver-basedadaptive fuzzy backstepping dynamic surface control for a classof MIMO nonlinear systems,β IEEE Transactions on Systems,Man, and Cybernetics, Part B: Cybernetics, vol. 41, no. 4, pp.1124β1135, 2011.
[12] J.-G. Juang, R.-W. Lin, and W.-K. Liu, βComparison of classicalcontrol and intelligent control for a MIMO system,β AppliedMathematics and Computation, vol. 205, no. 2, pp. 778β791,2008.
[13] J.-G. Juang, W.-K. Liu, and R.-W. Lin, βA hybrid intelligentcontroller for a twin rotor MIMO system and its hardwareimplementation,β ISA Transactions, vol. 50, no. 4, pp. 609β619,2011.
[14] C.-W. Tao, J.-S. Taur, Y.-H. Chang, and C.-W. Chang, βA novelfuzzy-sliding and fuzzy-integral-sliding controller for the twin-rotor multi-inputβmulti-output system,β IEEE Transactions onFuzzy Systems, vol. 18, no. 5, pp. 893β905, 2010.
[15] C. W. Tao, J. S. Taur, and Y. C. Chen, βDesign of a paralleldistributed fuzzy LQR controller for the twin rotor multi-inputmulti-output system,β Fuzzy Sets and Systems, vol. 161, no. 15, pp.2081β2103, 2010.
[16] M. Sacki, J. Imura, and Y. Wada, βFlight control design oftwin rotorhelicopter model by 2 step exact linearization,β inProceedings of the IEEE International Conference on ControlApplications, August 1999.
[17] G. Mustafa and N. Iqbal, βController design for a twin rotorhelicopter model via exact state feedback linearization,β in Pro-ceedings of the 8th International Multitopic Conference (INMICβ04), pp. 706β711, Lahore, Pakistan, December 2004.
[18] C. Yang, S. S. Ge, C. Xiang, T. Chai, and T. H. Lee, βOutputfeedback NN control for two classes of discrete-time systemswith unknown control directions in a unified approach,β IEEETransactions on Neural Networks, vol. 19, no. 11, pp. 1873β1886,2008.
[19] M. Chen, C.-S. Jiang, and Q.-X. Wu, βRobust adaptive controlof time delay uncertain systems with FLS,β International Journalof Innovative Computing, Information and Control, vol. 4, no. 8,pp. 1995β2004, 2008.
[20] M. Chen, S. S. Ge, and B. Ren, βRobust attitude control ofhelicopters with actuator dynamics using neural networks,β IETControl Theory & Applications, vol. 4, no. 12, pp. 2837β2854,2010.
[21] Y. Li, C. Yang, S. S. Ge, and T. H. Lee, βAdaptive output feed-back NN control of a class of discrete-time MIMO nonlinearsystems with unknown control directions,β IEEE Transactionson Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 41,no. 2, pp. 507β517, 2011.
[22] Z. J. Li and C. Yang, βNeural-adaptive output feedback controlof a class of transportation vehicles based on wheeled invertedpendulum models,β IEEE Transactions on Control SystemsTechnology, vol. 20, no. 6, pp. 1583β1591, 2012.
[23] H. A. Hashim and M. A. Abido, βFuzzy controller design usingevolutionary techniques for twin rotor MIMO system: a com-parative study,β Computational Intelligence and Neuroscience,vol. 2015, Article ID 704301, 11 pages, 2015.
[24] P.Wen andT.-W. Lu, βDecoupling control of a twin rotorMIMOsystem using robust deadbeat control technique,β IET ControlTheory & Applications, vol. 2, no. 11, pp. 999β1007, 2008.
[25] S. Mondal and C. Mahanta, βSecond order sliding modecontroller for twin rotor MIMO system,β in Proceedings ofthe Annual IEEE India Conference: Engineering SustainableSolutions (INDICON β11), pp. 1β5, December 2011.
[26] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, NonlinearandAdaptive Control Design, JohnWiley& Sons, NewYork, NY,USA, 1995.
[27] A. A. Ezzabi, K. C. Cheok, and F. A. Alazabi, βA nonlinearbackstepping control design for ball and beam system,β inProceedings of the IEEE 56th International Midwest Symposiumon Circuits and Systems (MWSCAS β13), pp. 1318β1321, IEEE,Columbus, Ohio, USA, August 2013.
[28] Y. Yang, G. Feng, and J. Ren, βA combined backstepping andsmall-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems,β IEEE Transactions on Systems,Man, and Cybernetics, Part A: Systems and Humans, vol. 34, no.3, pp. 406β420, 2004.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of
Top Related