# Control Law Design for Twin Rotor MIMO System with Nonlinear Control Strategy

Authors: M. Ilyas; N. Abbas; M. UbaidUllah; Waqas A. Imtiaz; M. A. Q. Shah; K. Mahmood
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

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 a helicopter 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 of main and tail rotor of the TRMS under parametric uncertainty. The proposed control strategy with SMC and backstepping achieves all mentioned limitations of TRMS. Result analysis of SMC and backstepping control schemes elucidates that backstepping provides efficient behavior with the parametric uncertainty for twin rotor system. Chattering and oscillating behaviors of SMC are removed with the backstepping control scheme considering the pitch and yaw angle for TRMS.


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## Body

## 1. Introduction

Recent times have witnessed the evolution of various approaches for proper flight of air vehicles such as helicopter. Modeling of air vehicles dynamics is difficult owing to the significant coupling effect among rotors and the unavailability of some system states. The laboratory setup, twin rotor MIMO system (TRMS), is readily utilized, which resembles the flight of a helicopter [1]. It has gained much importance among the control community by serving as a tool for different experiments and providing real time environment of an air vehicle.It is difficult to design a controller for TRMS due to its nonlinear behavior between two axes [2, 3]. TRMS consists of a beam with two rotors connected at its ends which are driven by separate DC motors and the beam is counterbalanced by an arm having weight at its end [4]. It has two degrees of freedom, which facilitate movements in both horizontal and vertical direction. TRMS is basically a prototype model of a helicopter; however, there is significant difference in control of helicopter and its prototype. In order to control TRMS in a desired way, the speed of rotors is altered, while in helicopter it is done by changing the angles of rotors. There is no cylindrical control in TRMS while in helicopter it is used in directional control [5].The control problem of TRMS has gained much attention, owing to the high coupling effect between two propellers, unstable and nonlinear dynamics. Several techniques like observer based and hybrid adaptive fuzzy output feedback control approaches are developed to solve the nonlinear MIMO system with unknown control direction and dead zones [6, 7]. Genetic algorithms to control the unstable and nonlinear dynamics in TRMS are designed using PID control [8]. Adaptive fuzzy sliding mode control is developed for a class of MIMO nonlinear system which estimates the states from a semi high gain observer to construct the output feedback fuzzy controller by incorporating the dynamic sliding mode [9]. References [10, 11] developed the observer based adaptive fuzzy backstepping dynamic surface control (DSC) for nonlinear MIMO with immeasurable states. [12] performs a comparative analysis between intelligent control and classical control for TRMS. Adaptive fuzzy, neural network, and feedback linearization based controllers are also designed for the tracking of yaw and pitch angles in TRMS [4, 13–22]. However, the proposed state variables are assumed measurable, which is practically not feasible to control the pitch and yaw angle of TRMS.This paper proposes the first-order sliding mode control and backstepping control scheme for the nonlinear TRMS. In our proposed methodology, the mathematical model of TRMS is linearized and the cross coupling effect between the main rotors is considered as disturbance. The main advantage of SMC is that it mitigates the parametric uncertainty present in TRMS while backstepping control algorithm performs better in case of external disturbance, which in this case is the coupling effect of the rotors, because of its recursive structure. The proposed approaches are investigated for TRMS keeping in view the need for cancelling the strong coupling between rotors and finally providing the desired tracking response of both the controllers. Simulation results show the effectiveness of control algorithms but comparatively the backstepping scheme gives the best performance in terms of stability and reference tracking.The remainder of the paper is arranged as follows. In Section2, the mathematical model of TRMS system is introduced and the parameters of the system are specified. The proposed SMC and backstepping controller along with their simulation results are given in Sections 3 and 4, respectively. Comparison of proposed controllers is introduced in Section 5 followed by the concluding remarks.


## 2. Mathematical Modeling

Figure1 shows the TRMS laboratory setup, which is used to develop the mathematical model to compare the operation of SMC and backstepping controllers.Figure 1
TRMS laboratory setup.TRMS system is designed with two rotors (main rotor and tail rotor) as shown in Figure1 encompassing the effect of forces like gravitational, propulsion, centrifugal, frictional, and disturbance torque on movement of the propellers. To overcome the effects of these forces we provide control input through motors. In the given case, only the pitch and azimuth angles are the measureable outputs and its stability is the main objective of designing the controller. As far as the mechanical unit is concerned the following nonlinear momentum equations can be derived for the pitch movement of TRMS [1]. Consider(1)
I
1
Ψ
¨
=
M
1
-
M
F
G
-
M
B
Ψ
-
M
G
,where(2)
M
1
=
a
1
τ
1
2
+
b
1
τ
1
,
(3)
M
F
G
=
M
g
sin
⁡
ψ
,
(4)
M
B
ψ
=
B
1
Ψ
Ψ
˙
+
B
2
Ψ
sin
⁡
2
Ψ
φ
˙
2
,
(5)
M
G
=
K
g
y
M
1
φ
˙
cos
⁡
Ψ
,where a
1 and b
1 are constants. Equation (5) is derived based on law of conservation of angular momentum of main rotor:(6)
I
2
φ
¨
=
M
2
-
M
B
φ
-
M
R
.The momentum equations in the vertical plane of motion are written as(7)
M
2
=
a
2
τ
2
2
+
b
2
τ
2
,
(8)
M
B
φ
=
B
1
φ
φ
˙
,
(9)
M
R
=
K
c
T
o
s
+
1
T
p
s
+
1
M
1
.Similar momentum equation can be used for the horizontal plane motion as well.Equation (9) is derived based on law of angular conservation of momentum of main rotor. The state space equations are as follows:(i)
For main motor,(10)
τ
1
˙
=
T
10
T
11
τ
1
+
k
1
T
11
u
1
.
(ii)
For tail motor,(11)
τ
2
˙
=
T
20
T
21
τ
2
+
k
2
T
21
u
2
,where k
1 and k
2 are the motor gain and T
20, T
21, T
10, and T
11 are the motor parameter. τ
1 and τ
2 are momentum of main motor and tail motor. The linearization of the nonlinear model of TRMS is given in the following section.(A) Linearization of TRMS. The state space equation of nonlinear system along with the parameters is given by the following equations: (12)
d
Ψ
˙
d
t
=
a
1
I
1
τ
1
2
+
b
1
I
1
τ
1
-
M
g
I
1
sin
⁡
Ψ
+
0.0326
2
I
1
sin
⁡
2
Ψ
φ
˙
2
-
B
1
Ψ
I
1
Ψ
˙
-
k
g
y
I
1
cos
⁡
Ψ
φ
a
1
τ
1
2
+
b
1
τ
1
,
d
φ
˙
d
t
=
a
2
I
2
τ
2
2
+
b
2
I
2
τ
2
-
B
1
φ
I
2
φ
˙
-
k
c
I
2
1.75
a
1
˙
τ
1
2
+
b
1
τ
1
,
d
τ
1
d
t
=
-
T
10
T
11
τ
1
+
k
1
T
11
u
1
,
d
τ
2
d
t
=
-
T
20
T
21
τ
2
+
k
2
T
21
u
2
.The state and output vectors are given by(13)
x
=
Ψ
Ψ
˙
φ
φ
˙
τ
1
τ
2
T
,
y
=
Ψ
φ
T
,where variables are as follows (Table 1):ψ: pitch (elevation) angle.
ϕ: yaw (azimuth) angle.
τ
1: momentum of main rotor.
τ
2: momentum of tail rotor.Table 1
Values of tuning parameters for SMC.
Pitch angle (rad)
Yaw angle (rad)
Case 1
C
1 = 2.102
C
2 = 3.5
K
1 = 1.5
C
3 = 6.305
C
4 = 0.0875
K
2 = 4.9
Case 2
C
1 = 1.1
C
2 = 3.5
K
1 = 1.95
C
3 = 1.3
C
4 = 0.088
K
2 = 2.897
Case 3
C
1 = 2.5
C
2 = 3.5
K
1 = 1.915
C
3 = 3.3
C
4 = 0.088
K
2 = 2.89Here all the variables of system are expressed in term of “x.” So (14)
x
1
=
Ψ
,
x
2
=
Ψ
˙
,
x
3
=
φ
,
x
4
=
φ
˙
,
x
5
=
τ
1
,
x
6
=
τ
2
.Now the state space of the system in term of variable “x” will become (15)
x
˙
1
=
x
2
,
x
˙
2
=
a
1
I
1
x
5
2
+
b
1
I
1
x
5
-
M
g
I
1
sin
⁡
x
1
+
0.0326
2
I
1
sin
⁡
2
x
1
x
4
2
-
B
1
Ψ
I
1
x
˙
1
-
k
g
y
I
1
cos
⁡
x
1
x
3
a
1
˙
x
5
2
+
b
1
x
5
,
x
˙
3
=
x
4
,
x
˙
4
=
a
2
I
2
x
6
2
+
b
2
I
2
x
6
-
B
1
φ
I
2
x
5
-
k
c
I
2
1.75
a
1
˙
x
5
2
+
b
1
x
5
,
x
˙
5
=
-
T
10
T
11
x
5
+
k
1
T
11
u
1
,
x
˙
6
=
-
T
20
T
21
x
6
+
k
2
T
21
u
2
.To linearize the system, let the system be represented as(16)
x
˙
=
A
x
+
B
u
,
y
=
C
x
,where x
∈
R as states, u
∈
R as the control input, and y
∈
R as the measured output.Consider(17)
f
1
=
x
˙
1
,
f
2
=
x
˙
2
,
f
3
=
x
˙
3
,
f
4
=
x
˙
4
,
f
5
=
x
˙
5
,
f
6
=
x
˙
6
.After taking Jacobean and putting point (
0,0
), then resulting system matrices are given below. Consider(18)
A
=
0
1
0
0
0
0
-
M
g
I
1
-
B
1
Ψ
I
1
0
0
b
1
I
1
0
0
0
0
1
0
0
0
0
0
-
B
1
φ
I
2
-
k
c
I
2
1.75
b
2
I
2
0
0
0
0
-
T
10
T
11
0
0
0
0
0
0
-
T
20
T
21
,
B
=
0
0
0
0
0
0
0
0
k
1
T
11
0
0
k
2
T
21
,
C
=
1
0
0
0
0
0
0
0
1
0
0
0
.(B) State Space Equations of Linearized Model. Values of constants are given in the Abbreviation section [1]. By putting values of all these constants, the state space equations can be given as(19)
x
˙
1
=
x
2
,
x
˙
2
=
-
4.7059
x
1
-
0.0882
x
2
+
1.3588
x
5
,
x
˙
3
=
x
4
,
x
˙
4
=
-
5
x
4
+
1.617
x
5
+
4.5
x
6
,
x
˙
5
=
-
0.9091
x
5
+
u
1
,
x
˙
6
=
-
x
6
+
0.8
u
2
.


## 3. Proposed SMC Controller

(A) Choosing Sliding Surface. SMC is a nonlinear control technique, which deals with the capability of controlling the uncertainties of nonlinear systems [23, 24]. The primary advantage of the SMC technique is the low sensitivity to system disturbances. Moreover, it accredits the decoupling of the lower dimensions, and consequently, it scales down the complication of feedback design [25]. SMC generally consist of two phases: reaching phase and the sliding phase. The reaching phase converges the system states to desired surface and sliding phase handles the oscillations. Sliding surface can be designed as(20)
S
=
c
T
x
=
c
1
x
1
+
c
2
x
2
+
⋯
+
c
n
-
1
x
n
-
1
+
x
n
=
0
.Now the control input consist of two parts:(i)
Equivalent controller,U
e
q.
(ii)
Discontinuous controller,U
^.Consequently, the required controller can be determines as(21)
U
=
U
e
q
+
U
^
.(B) SMC Design for Linearized Model. This section outlines the SMC design for linearized model. Sliding surface of the system is designed at first to facilitate the process. TRMS is a MIMO system so we will design two sliding surfaces.Sliding surface for the vertical plane is as follows:(22)
s
1
=
x
5
+
c
2
x
2
+
c
1
e
1
,
e
1
=
x
1
-
x
1
d
.Lyapunov condition is satisfied as(23)
V
˙
1
=
-
s
1
2
-
k
1
sign
⁡
s
1
.Sliding surface for horizontal plane is as follows:(24)
s
2
=
x
6
+
c
4
x
4
+
c
3
e
2
,
e
2
=
x
3
-
x
2
d
,where we have the following.Lyapunov condition is satisfied as(25)
V
˙
2
=
-
s
2
2
-
k
2
sign
⁡
s
2
.


### 3.1. Simulation Results of SMC

Figures2 and 3 show the pitch and yaw position of TRMS obtained after implementation of SMC in Simulink MATLAB on linearized model. It is clear that the desired objective of regulating the system for two degrees of freedom has been achieved under the robust control action of SMC. It is shown that settling time for pitch and yaw 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 position with respect to pitch and yaw movement under applied control action.Figure 2
Pitch angle for linearized system using SMC.Figure 3
Yaw angle for linearized system using SMC.The control inputsu
a and u
b for pitch and yaw movements of TRMS, respectively, are in volts and provided by two independent 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 from Figures 2 and 3 that the corresponding equivalent control efforts successfully drive the system dynamics to corresponding sliding surfaces in a short period of time.Moreover, the discontinuous control parts efficiently maintain the system states on sliding manifolds for all subsequent times and are responsible for system robustness against uncertainties. However, chattering in control inputs “u
a” and “u
b” can be clearly seen from Figures 4 and 5, which arises due to fast switching of discontinuous control action around the sliding manifolds. Since the amplitude of chattering is small, both yaw and pitch movement of TRMS are not affected by this undesired phenomenon.Figure 4
Control input for pitch angle.Figure 5
Control input for yaw angle.The sliding manifoldss
1 and s
2 have been designed by linearly combining system states for regulation purpose of TRMS under system uncertainties and significant coupling but in the absence of external perturbation. The tuning parameters have been suitably adjusted for sliding surface. Figures 6 and 7 show chattering phenomenon of sliding surface for pitch and yaw angle of TRMS. It is observed that the chattering phenomenon in the sliding surface is miniscule. Moreover, the settling time of sliding surface for both vertical and horizontal planes is under 1 second, which is desirable for the system under consideration. Now another discussion is provided about the implementation of SMC with tracking.Figure 6
Chattering in sliding surface for pitch angle.Figure 7
Chattering in sliding surface for yaw angle.


### 3.2. Simulation Results of TRMS with Tracking

The pitch and yaw angles are obtained after implementation of SMC on a TRMS in Simulink MATLAB. Figure8 shows the response of pitch angle at different values of tuning parameters and Figure 9 shows the response of yaw angle at different values of tuning parameters. Figure 8 shows that the tuning parameter for case 1 has approximately 20% overshoot from the desired position. Case  3 is showing undershoot from the desired position due the difference tuning parameters. It is observed that case 2 is the most suitable set of tuning parameters for achieving the desired results without showing over- and undershoot. Thus, the desired objective of regulating the system for two degrees of freedom has been achieved under the robust control action of SMC. Same phenomenon is obtained for yaw angle in Figure 9, where case 2 is most preferable with desired tuning parameters.Figure 8
Response of pitch angle for TRMS.Figure 9
Response of yaw angle for TRMS.Different values of tuning parameters show how we can obtain different responses of pitch and yaw angles according to requirements. Overshoot problem is faced in case of sharp response, and if we need a slower response then settling time is increased.


## 3.1. Simulation Results of SMC

Figures2 and 3 show the pitch and yaw position of TRMS obtained after implementation of SMC in Simulink MATLAB on linearized model. It is clear that the desired objective of regulating the system for two degrees of freedom has been achieved under the robust control action of SMC. It is shown that settling time for pitch and yaw 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 position with respect to pitch and yaw movement under applied control action.Figure 2
Pitch angle for linearized system using SMC.Figure 3
Yaw angle for linearized system using SMC.The control inputsu
a and u
b for pitch and yaw movements of TRMS, respectively, are in volts and provided by two independent 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 from Figures 2 and 3 that the corresponding equivalent control efforts successfully drive the system dynamics to corresponding sliding surfaces in a short period of time.Moreover, the discontinuous control parts efficiently maintain the system states on sliding manifolds for all subsequent times and are responsible for system robustness against uncertainties. However, chattering in control inputs “u
a” and “u
b” can be clearly seen from Figures 4 and 5, which arises due to fast switching of discontinuous control action around the sliding manifolds. Since the amplitude of chattering is small, both yaw and pitch movement of TRMS are not affected by this undesired phenomenon.Figure 4
Control input for pitch angle.Figure 5
Control input for yaw angle.The sliding manifoldss
1 and s
2 have been designed by linearly combining system states for regulation purpose of TRMS under system uncertainties and significant coupling but in the absence of external perturbation. The tuning parameters have been suitably adjusted for sliding surface. Figures 6 and 7 show chattering phenomenon of sliding surface for pitch and yaw angle of TRMS. It is observed that the chattering phenomenon in the sliding surface is miniscule. Moreover, the settling time of sliding surface for both vertical and horizontal planes is under 1 second, which is desirable for the system under consideration. Now another discussion is provided about the implementation of SMC with tracking.Figure 6
Chattering in sliding surface for pitch angle.Figure 7
Chattering in sliding surface for yaw angle.


## 3.2. Simulation Results of TRMS with Tracking

The pitch and yaw angles are obtained after implementation of SMC on a TRMS in Simulink MATLAB. Figure8 shows the response of pitch angle at different values of tuning parameters and Figure 9 shows the response of yaw angle at different values of tuning parameters. Figure 8 shows that the tuning parameter for case 1 has approximately 20% overshoot from the desired position. Case  3 is showing undershoot from the desired position due the difference tuning parameters. It is observed that case 2 is the most suitable set of tuning parameters for achieving the desired results without showing over- and undershoot. Thus, the desired objective of regulating the system for two degrees of freedom has been achieved under the robust control action of SMC. Same phenomenon is obtained for yaw angle in Figure 9, where case 2 is most preferable with desired tuning parameters.Figure 8
Response of pitch angle for TRMS.Figure 9
Response of yaw angle for TRMS.Different values of tuning parameters show how we can obtain different responses of pitch and yaw angles according to requirements. Overshoot problem is faced in case of sharp response, and if we need a slower response then settling time is increased.


## 4. Backstepping Controller

In control system theory, backstepping controller scheme is introduced by Krstic in 1995 and his companions for designing stability control system for a special class of linear and nonlinear dynamical system [26]. Backstepping is a systematic, Lyapunov-based method for nonlinear control which refers to the recursive nature of the design procedure which starts at the scalar equation separated by the largest number of integrations from the control input and steps back towards the control input [27, 28].In the theory of Ordinary Differential Equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of equilibrium of an ODE. The basic idea behind the Lyapunov function method consists of (I) choosing a radially unbounded positive definite Lyapunov function candidateV
(
x
) and (II) evaluating its derivative V
(
x
) along system dynamics and checking its negativeness for stability analysis [25, 26].The recursitivity terminates when the final control phase is reached. The process which receives its stability through recursitivity is called backstepping [27, 28]. Backstepping can be used for tracking and regulation problem. With the aid of Lyapunov stability, this control approach for asymptotic tracking can be achieved.


### 4.1. Design Steps

The controller is designed using backstepping control technique on the proposed control problem. The standard backstepping control is based on step-by-step construction of Lyapunov function. Here we design controller, based on Lyapunov function.(A) For Vertical Plane. First of all in 1st step we introduce a new state (26)
z
1
=
x
1
-
x
1
d
,where “z
1” is the new state and “x
1” is state variable.Lyapunov candidate function (LCF) for new state is(27)
V
1
=
1
2
z
1
2
,where V
>
0, ∀
x
≠
0, and V
(
0
)
=
0.By taking time derivative of Lyapunov function we get(28)
V
˙
1
=
z
1
z
˙
1
,
(29)
z
˙
1
=
z
2
+
a
1
-
x
˙
1
d
,where “a
1” is virtual control input to control the system and z
2 is the new state (30)
a
1
=
-
c
1
z
1
+
x
1
d
.Now another new state is introduced that is given by “z
2”:(31)
z
2
=
x
2
-
a
1
,where “x
2” is the second state variable of the system. By putting values in (28) we get(32)
V
˙
1
=
-
z
1
2
+
z
1
z
2
.Now again we repeat the previous step to calculate the virtual control input.So,(33)
z
2
=
x
2
-
a
1
.We know(34)
a
˙
1
=
-
c
1
z
˙
1
+
x
¨
1
d
.New Lyapunov candidate function (LCF) is as follows:(35)
V
2
=
V
1
+
1
2
z
2
2
.Taking derivative of (35),(36)
V
˙
2
=
V
˙
1
+
z
2
z
˙
2
.By putting the value of V
˙
1 and z
˙
2 we get(37)
V
˙
2
=
-
z
1
2
-
z
2
2
+
z
2
z
3
,where (38)
z
˙
2
=
x
˙
2
-
a
˙
1
,
V
˙
1
=
-
z
1
2
+
z
1
z
2
.Now again we introduce new state(39)
z
3
=
x
5
-
a
2
.By taking derivative we get(40)
z
˙
3
=
x
˙
5
-
a
˙
2
.Now LCF will be as(41)
V
3
=
V
2
+
1
2
z
3
2
.Taking derivative of above equation, we get(42)
V
˙
3
=
V
˙
2
+
z
˙
3
z
3
.By putting the values in above equation we obtain another virtual control input for the second state of the system as given below:(43)
a
2
=
1
1.3588
-
z
1
+
4.7051
x
1
+
0.0882
x
2
-
c
1
z
˙
1
+
x
¨
1
d
.After differentiation,(44)
a
˙
2
=
1
1.3588
-
z
˙
1
+
4.7051
x
˙
1
+
0.0882
x
˙
2
-
c
1
z
¨
1
+
x
⃛
1
d
.Now the control input is given by(45)
u
1
=
-
c
3
z
3
-
z
2
+
0.909
x
5
+
a
˙
2
.We get the control input u
1 for vertical plane of TRMS. Now putting the value of control input we get(46)
V
˙
3
=
-
z
1
2
-
z
2
2
-
z
3
2
.Hence condition is satisfied and system is asymptotically stable.(B) For horizontal Plane. First of all in 1st step we introduce a new state (47)
z
4
=
x
3
-
x
2
d
,where z
4 is the new state and x
3 is the state variable.LCF for new state is(48)
V
4
=
1
2
z
4
2
.By taking derivative and putting the value of variables we get(49)
V
˙
4
=
-
z
4
2
+
z
4
z
5
.Now, the abovementioned steps are repeated to calculate the other virtual control input.Consequently,(50)
z
˙
4
=
z
5
+
a
3
-
x
˙
2
d
,
a
3
=
-
c
4
z
4
+
x
˙
2
d
.Here a
3 is arbitrary control input to converge the state z
4 towards stability.Introduce another new state to calculate another arbitrary control input.So,(51)
z
5
=
x
4
-
a
3
,where z
5 is new state for state variable x
4. By taking time derivative, (52)
z
˙
5
=
x
˙
4
-
a
˙
3
.Also we take time derivative of previous arbitrary control input as(53)
a
˙
3
=
-
c
4
z
˙
4
+
x
¨
2
d
.Now LCF will be as(54)
V
5
=
V
4
+
1
2
z
5
2
.By taking derivative and putting the values of z
˙
5 and V
˙
3 we get(55)
V
˙
5
=
-
z
4
2
+
z
4
z
5
+
z
5
x
˙
4
-
-
c
4
z
˙
4
+
x
¨
2
d
.After some algebraic calculations we get(56)
V
˙
5
=
-
z
4
2
-
z
5
2
+
z
6
z
5
,where z
6 is the new state:(57)
z
6
=
x
6
-
a
4
.Now LCF will be as (58)
V
6
=
V
5
+
1
2
z
6
2
.After taking derivative,(59)
V
˙
6
=
V
˙
5
+
z
˙
6
z
6
,where(60)
a
4
=
1
4.5
-
z
4
-
c
5
z
5
+
5
x
4
-
1.617
x
5
-
a
˙
3
+
x
¨
2
d
.By taking derivative of (60),(61)
a
˙
4
=
1
4.5
-
z
˙
4
-
c
5
z
˙
5
+
5
x
˙
4
-
1.617
x
˙
5
-
a
¨
3
+
x
⃛
2
d
.By putting the values of z
˙
6, a
˙
4, and V
˙
5 we get(62)
V
˙
6
=
-
z
4
2
-
z
5
2
-
z
6
2
,where V
˙
6 is negative definite and system will be asymptotically stable. Now we get control input for the horizontal plane. Finally we get the control law, which will regulate all the states of the system to the origin. The system is asymptotically stable by using Backstepping design method:(63)
u
2
=
1
0.8
-
c
6
z
6
-
z
5
+
x
6
+
1
4.5
-
z
˙
4
-
c
5
z
˙
5
+
5
x
˙
4
-
1.617
x
˙
5
-
a
¨
3
+
x
⃛
2
d
.After mathematical calculations of backstepping controller, we use user defined block from MATLAB Simulink library. Required equations are used in this function. On the basis of simulation results we will elaborate the performance of controller which is given below


### 4.2. Simulation Results of Backstepping Controller

On the basis of backstepping control design, simulation results in Figures10 and 11 show the stability response of pitch angle and the control input for pitch angle, respectively. Similarly Figures 12 and 13 show the stability of yaw angle and control input for the pitch angle, respectively. It is observed that settling time for pitch and yaw angle in case of backstepping technique is less as compared to sliding mode control. The controller shows very promising results and it is found that backstepping controller is capable of tracking and little variation in control inputs of both pitch and yaw angle.Figure 10
Pitch angle (rad) for backstepping.Figure 11
Control input for pitch angle.Figure 12
Yaw angle (rad) for backstepping.Figure 13
Control input for yaw angle.The performance of SMC is limited due to chattering in control inputs but this issue is resolved through backstepping, which is shown in Figures11 and 13. An extensive overview is given below for comparison of backstepping and SMC on the basis of simulation results.


## 4.1. Design Steps

The controller is designed using backstepping control technique on the proposed control problem. The standard backstepping control is based on step-by-step construction of Lyapunov function. Here we design controller, based on Lyapunov function.(A) For Vertical Plane. First of all in 1st step we introduce a new state (26)
z
1
=
x
1
-
x
1
d
,where “z
1” is the new state and “x
1” is state variable.Lyapunov candidate function (LCF) for new state is(27)
V
1
=
1
2
z
1
2
,where V
>
0, ∀
x
≠
0, and V
(
0
)
=
0.By taking time derivative of Lyapunov function we get(28)
V
˙
1
=
z
1
z
˙
1
,
(29)
z
˙
1
=
z
2
+
a
1
-
x
˙
1
d
,where “a
1” is virtual control input to control the system and z
2 is the new state (30)
a
1
=
-
c
1
z
1
+
x
1
d
.Now another new state is introduced that is given by “z
2”:(31)
z
2
=
x
2
-
a
1
,where “x
2” is the second state variable of the system. By putting values in (28) we get(32)
V
˙
1
=
-
z
1
2
+
z
1
z
2
.Now again we repeat the previous step to calculate the virtual control input.So,(33)
z
2
=
x
2
-
a
1
.We know(34)
a
˙
1
=
-
c
1
z
˙
1
+
x
¨
1
d
.New Lyapunov candidate function (LCF) is as follows:(35)
V
2
=
V
1
+
1
2
z
2
2
.Taking derivative of (35),(36)
V
˙
2
=
V
˙
1
+
z
2
z
˙
2
.By putting the value of V
˙
1 and z
˙
2 we get(37)
V
˙
2
=
-
z
1
2
-
z
2
2
+
z
2
z
3
,where (38)
z
˙
2
=
x
˙
2
-
a
˙
1
,
V
˙
1
=
-
z
1
2
+
z
1
z
2
.Now again we introduce new state(39)
z
3
=
x
5
-
a
2
.By taking derivative we get(40)
z
˙
3
=
x
˙
5
-
a
˙
2
.Now LCF will be as(41)
V
3
=
V
2
+
1
2
z
3
2
.Taking derivative of above equation, we get(42)
V
˙
3
=
V
˙
2
+
z
˙
3
z
3
.By putting the values in above equation we obtain another virtual control input for the second state of the system as given below:(43)
a
2
=
1
1.3588
-
z
1
+
4.7051
x
1
+
0.0882
x
2
-
c
1
z
˙
1
+
x
¨
1
d
.After differentiation,(44)
a
˙
2
=
1
1.3588
-
z
˙
1
+
4.7051
x
˙
1
+
0.0882
x
˙
2
-
c
1
z
¨
1
+
x
⃛
1
d
.Now the control input is given by(45)
u
1
=
-
c
3
z
3
-
z
2
+
0.909
x
5
+
a
˙
2
.We get the control input u
1 for vertical plane of TRMS. Now putting the value of control input we get(46)
V
˙
3
=
-
z
1
2
-
z
2
2
-
z
3
2
.Hence condition is satisfied and system is asymptotically stable.(B) For horizontal Plane. First of all in 1st step we introduce a new state (47)
z
4
=
x
3
-
x
2
d
,where z
4 is the new state and x
3 is the state variable.LCF for new state is(48)
V
4
=
1
2
z
4
2
.By taking derivative and putting the value of variables we get(49)
V
˙
4
=
-
z
4
2
+
z
4
z
5
.Now, the abovementioned steps are repeated to calculate the other virtual control input.Consequently,(50)
z
˙
4
=
z
5
+
a
3
-
x
˙
2
d
,
a
3
=
-
c
4
z
4
+
x
˙
2
d
.Here a
3 is arbitrary control input to converge the state z
4 towards stability.Introduce another new state to calculate another arbitrary control input.So,(51)
z
5
=
x
4
-
a
3
,where z
5 is new state for state variable x
4. By taking time derivative, (52)
z
˙
5
=
x
˙
4
-
a
˙
3
.Also we take time derivative of previous arbitrary control input as(53)
a
˙
3
=
-
c
4
z
˙
4
+
x
¨
2
d
.Now LCF will be as(54)
V
5
=
V
4
+
1
2
z
5
2
.By taking derivative and putting the values of z
˙
5 and V
˙
3 we get(55)
V
˙
5
=
-
z
4
2
+
z
4
z
5
+
z
5
x
˙
4
-
-
c
4
z
˙
4
+
x
¨
2
d
.After some algebraic calculations we get(56)
V
˙
5
=
-
z
4
2
-
z
5
2
+
z
6
z
5
,where z
6 is the new state:(57)
z
6
=
x
6
-
a
4
.Now LCF will be as (58)
V
6
=
V
5
+
1
2
z
6
2
.After taking derivative,(59)
V
˙
6
=
V
˙
5
+
z
˙
6
z
6
,where(60)
a
4
=
1
4.5
-
z
4
-
c
5
z
5
+
5
x
4
-
1.617
x
5
-
a
˙
3
+
x
¨
2
d
.By taking derivative of (60),(61)
a
˙
4
=
1
4.5
-
z
˙
4
-
c
5
z
˙
5
+
5
x
˙
4
-
1.617
x
˙
5
-
a
¨
3
+
x
⃛
2
d
.By putting the values of z
˙
6, a
˙
4, and V
˙
5 we get(62)
V
˙
6
=
-
z
4
2
-
z
5
2
-
z
6
2
,where V
˙
6 is negative definite and system will be asymptotically stable. Now we get control input for the horizontal plane. Finally we get the control law, which will regulate all the states of the system to the origin. The system is asymptotically stable by using Backstepping design method:(63)
u
2
=
1
0.8
-
c
6
z
6
-
z
5
+
x
6
+
1
4.5
-
z
˙
4
-
c
5
z
˙
5
+
5
x
˙
4
-
1.617
x
˙
5
-
a
¨
3
+
x
⃛
2
d
.After mathematical calculations of backstepping controller, we use user defined block from MATLAB Simulink library. Required equations are used in this function. On the basis of simulation results we will elaborate the performance of controller which is given below


## 4.2. Simulation Results of Backstepping Controller

On the basis of backstepping control design, simulation results in Figures10 and 11 show the stability response of pitch angle and the control input for pitch angle, respectively. Similarly Figures 12 and 13 show the stability of yaw angle and control input for the pitch angle, respectively. It is observed that settling time for pitch and yaw angle in case of backstepping technique is less as compared to sliding mode control. The controller shows very promising results and it is found that backstepping controller is capable of tracking and little variation in control inputs of both pitch and yaw angle.Figure 10
Pitch angle (rad) for backstepping.Figure 11
Control input for pitch angle.Figure 12
Yaw angle (rad) for backstepping.Figure 13
Control input for yaw angle.The performance of SMC is limited due to chattering in control inputs but this issue is resolved through backstepping, which is shown in Figures11 and 13. An extensive overview is given below for comparison of backstepping and SMC on the basis of simulation results.


## 5. Comparison of Backstepping and SMC Controller

Figures14 and 15 show the comparison between SMC and backstepping controller for pitch and yaw angle of TRMS. It is observed that backstepping controller shows good results compared to SMC in terms of handling oscillation and chattering. By using backstepping controller, the settling time for the pitch and yaw angle is less as compared to SMC as shown in Figures 14 and 15.Figure 14
Pitch angle (rad) for SMC and backstepping.Figure 15
Yaw angle (rad) for SMC and backstepping.


## 6. Conclusion

This work outlines the design analysis of robust controller techniques by implementing them in TRMS, which is a nonlinear system. SMC and backstepping are implemented and analyzed for handling the oscillation and chattering in pitch and yaw angles of TRMS. It is observed that backstepping shows better performance in terms of less settling time and handling perturbation as compared to SMC, owing to the recursive structure for controller design. This philosophy is the core idea that has been followed for developing robust controller. The controller was implemented in the Simulink environment where the state space model of the controller was engaged with system to achieve the desired result. Implementing the sliding mode control via backstepping control can be considered as a recommended future work, since parametric uncertainty and external disturbance can be mitigated within a single model, which can stabilize the TRMS in a more robust way.



---

*Source: 2952738-2016-07-20.xml*