robust adaptive control for a class of nonlinear systems using

International Journal of Advanced Robotic Systems Robust Adaptive Control for a Class of Nonlinear Systems Using the Backstepping Method Regular Paper

Farouk Zouari1,*, Kamel Ben Saad1 and Mohamed Benrejeb1

1 Unité de Recherche LARA-Automatique, Ecole Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, Tunis, Tunisia * Corresponding author E-mail: [email protected] Received 17 Oct 2012; Accepted 7 Nov 2012 DOI: 10.5772/54932 © 2013 Zouari et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract This paper develops a robust adaptive control for a class of nonlinear systems using the backstepping method. The proposed robust adaptive control is a recursive method based on the Lyapunov synthesis approach. It ensures that, for any initial conditions, all the signals of the closed‐loop system are regularly bounded and the tracking errors converge to zero. The results are illustrated with simulation examples. Keywords Robust Adaptive Control System, Nonlinear Systems, Backstepping Method, Closed‐Loop States

1. Introduction

In recent decades, a large number of papers have studied the problem of robust adaptive control of nonlinear systems (see, e.g., [1‐15] and references therein). In [1], a new adaptive law based on an optimal control formulation for the minimization of the 2L norm of the tracking bounded error is considered. A method for designing a global adaptive neural network controller for a class of uncertain non‐linear systems is proposed in [2]. In [3‐9], adaptive control of uncertain nonlinear systems

using backstepping is developed. In these papers, the backstepping method guarantees global stabilities, tracking and transient performance for a broad class of strict‐feedback system. New adaptive feedforward cancellations (AFC) control providing periodic tracking and/or periodic disturbance rejection is proposed in [10]. In [11], a robust control system combining backstepping and sliding mode control techniques is used to realize the synchronization of two gap junction coupled chaotic FitzHugh–Nagumo (FHN) neurons in the external electrical stimulation. The paper [12] introduces an optimal H adaptive PID (OHAPID) control scheme for a class of nonlinear chaotic system with uncertainties and external disturbances. In [13], an adaptive backstepping control scheme is proposed for task‐space trajectory tracking of robot manipulators. In [14], an adaptive integral backstepping algorithm is proposed as a means to effectuate the attitude control of a 3‐DOF helicopter. In [15], a backstepping approach is used for the design of a discontinuous state feedback controller for the controller. The main contribution of this paper is the design of an adaptive backstepping control method for a class of

1Farouk Zouari, Kamel Ben Saad and Mohamed Benrejeb: Robust Adaptive Control for a Class of Nonlinear Systems Using the Backstepping Method

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ARTICLE

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uncertain single input single output nonlinear systems which can be transformed into a triangular form. Many systems, such as AC motors, spacecraft, magnetic suspension and robot manipulators, possess this structure [14]. These systems can be built from subsystems that can be stabilized [14, 15]. In this paper, we assume that the bounds of the uncertainty system parameters are available. The adaptive calculating procedure of the control is a recursive procedure based on the Lyapunov approach. It is composed of several steps. It can start at the known‐stable system and back out new controllers that gradually stabilize each outer subsystem. The procedure stops when the final external control is reached. Compared with the adaptive control scheme, the proposed control approach has the advantages of adaptive technique and robust control, which makes this approach attractive for a wide class of nonlinear systems with both uncertain nonlinearities and disturbances. This paper is organized as follows: the formulation of the problem is introduced in Section 2; the controller design and stability analysis are presented in Section 3; the results of the simulation, illustrating the efficiency of the proposed controller, are presented in Section 4.

2. Problem statement and preliminaries

Let us consider the following nonlinear system

i i ii i i i 1 i

n n nn n n n

1

x h x x x x ,i 1, ,n 1

x h x x u x

y x

(1)

where: T i

i 1 ix x , ,x is the state of the i‐th subsystem;

T nn 1 nx x , ,x , u and y are

the state, the input and the output of the overall system, respectively; iih x , i 1, ,n are the known functions; ii x are unknown Lipschitz continuous functions such that:

ii i i

ii

x l , l 0

x ,i 1, ,n

(2)

where: il are known constants. The known functions ii x are defined as

ii

ii

x 0

x ,i 1, ,n

(3)

The purpose of this paper is the design of a control u that ensures the tracking output y sticks to a specified trajectory dy so that all system variables are bounded.

The desired reference signal dy t is of class C , t 0well as n 1 n

d d d dy ,y , ,y ,y are known and bounded. The tracking error is defined as

de y y (4)

The determination procedure of the control u is presented in the next section.

3. Robust adaptive controller design and stability analysis

The control u is calculated using the backstepping method. The calculation procedure involves n steps. From step 1 to step n 1 , the virtual control inputs

i ,i 1, ,n 1 are designed, respectively. The practice control input u is built at step n . The detailed design is described in the following steps [3‐9, 11, 13‐15]. Step 1: In this step, the design objective is to choose the virtual controller 1 so that the tracking error

1 1 de x y is as small as possible. The time derivative of 1 is

1 1 d 1 1 1 1 2 d 1 1x y h x x x y x (5)

Introducing the variable:

2 2 1 1 1 dx x ,c y (6)

We then choose:

1 1 1 1 1 1 1 1 1 d d 1 11 1

1x ,c c h x x y y l sgnx

(7)

where:

1

1 1

1

1 if 0sgn 0 if 0

1 if 0

(8)

The parameter 1c is selected such that 1c 0 . For 1 , a smooth approximation of the sgn function is

1 1sgn tanh (9)

The relationship (9) is usually used to reduce the chattering which is caused by the sgn function [16]. Therefore, the equation (7) can be written in the following form

1 1 1 1 1 1 1 1 1 d d 1 11 1

1x ,c c h x x y y l tanhx

(10)

Consider the following Lyapunov function candidate

21 1

1V2 (11)

2 Int J Adv Robotic Sy, 2013, Vol. 10, 166:2013 www.intechopen.com

By using (1), (5), (6), (9) and (10), the time derivative of

1V is

21 1 1 1 1 1 1 1 2V c x (12)

where the coupling term 1 1 1 2x will be cancelled in the next step. Step i 2 i n 1 : Similar to step 1, the virtual controller i will be chosen to make the error variable

i 1i 1i i i 1 1 i 1 dx x ,c , ,c y as small as possible.

The time derivative of i is expressed as:

ii 1i i i 1 1 i 1 d

ii i ii i i 1 d i

i 1 i 1i 1 1 i 1

j 1 j

j j jj j j 1 j

x x ,c , ,c y

h x x x y x

x ,c , ,c

x

h x x x x

(13)

The variable i 1 is as follows

iii 1 i 1 i 1 i dx x ,c , ,c y (14)

ii 1 ix ,c , ,c and ic are chosen such that

ii i ii 1 i i i i i d

ii

i 1 i 1i 1 1 i 1

j 1 j

j jj j j 1

1x ,c , ,c c h x x yx

x ,c , ,c

x

h x x x

i 1i 1 1 i 1j i

j

ii 1d i 1 i 1 i i

i

x ,c , ,cl tanh

x

y x l tanh

c 0

(15)

The Lyapunov function candidate is defined as:

2i i 1 i

1V V2 (16)

Its derivative is:

i i 1 i i

i2

ii i i i i 1j 1

V V

c x

(17)

When i n 1 , (17) can be rewritten as:

n 1

2n 1n 1 i i n 1 n 1 n

j 1V c x

(18)

Step n : In this step the practical control input u will be constructed to make

n 1n 1n n n 1 1 n 1 dx x ,c , ,c y as small as

possible. The time derivative of n is given by:

nn 1n n n 1 1 n 1 d

n n nn n n

n 1 n 1n 1 1 n 1

j 1 j

j j jj j j 1 j

nd

x x ,c , ,c y

h x x u x

x ,c , ,c

x

h x x x x

y

(19)

We choose nc such that

nc 0 (20)

We consider the Lyapunov function candidate:

2n n 1 n

1V V2 (21)

The time derivative of nV is

n n 1 n nn 1

2ni i n n

j 1

n n nn

nn n d n

n 1n 1 n 1 n

n 1 n 1n 1 1 n 1

j 1 j

V V

c h x

l tanh

x u y

x

x ,c , ,c

x

n

j jj j j 1

n 1n 1 1 n 1j n

j

h x x x

x ,c , ,cl tanh

x

(22)

At the end of the backstepping procedure, we take:

n 1 n 1n 1 1 n 1

j 1n jn

j jj j j 1

n 1n 1 1 n 1j n

j

x ,c , ,c1uxx

h x x x

x ,c , ,cl tanh

x

nn n n

nn 1n n d n 1 n 1

c h x

l tanh y x

(23)

Then, the time‐derivative of nV satisfies the following condition:

n

2n i i

j 1V c

(24)


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Let us consider:

ic 02

i 1, ,n

(25)

Therefore, we obtain:

n nV V 0 (26)

Theorem 1. Suppose that the proposed control method in this section is applied to the system (1). Then, for any initial conditions, the closed‐loop system is globally stable for t 0, . Moreover, the tracking error converges to zero, i.e., e t 0 as t . Proof. By means of the Barbalat lemma [17, 18], we have

ntlimV t 0

(27)

This implies that:

1tlim t 0

(28)

From (28), it is easy to see that

dt t

1t

lime t lim y t y t

lim t

0

(29)

This completes the proof.

4. Simulation examples and discussion

In this section, the feasibility of the proposed method and the control performances are illustrated with three examples. The simulation results are carried out using the software MATLAB. Example 1 For the simulation example 1, we consider the following nonlinear system

21 1 2 1 1

2 22 1 1 2 1 2

1

x 1 4x 4x x

x 2 x 1 x u x ,x

y x

(30)

where: u and y are the control signal and measured output, respectively; 1 1x 0.1 and 2 1 2x ,x 0.2 . The aim of the control is to force the output 1y x to asymptotically track a reference signal dy . Using the controller design procedure described in section 3, we can write:

n 2 , 21 1 1h x 1 4x , 2

2 1 2 1h x ,x 2 x , 1 1x 4 ,

22 1 2 1x ,x 1 x , 1 1 dx y , 1l 0.1 , 2l 0.2 ,

21 1 1 1 d 1 1 1 d 1 d

1x ,c c y c x 1 4x 3y 0.1tanh x y4

,

22 2 1 d 1 1 1 d 1 d

1x c y c x 1 4x y 0.1tanh x y4

.

We choose 1c 1 , 2c 2 and 510 . Then, the control input becomes:

21 1 11 22

11

5 1 1 12

1

22 1

52

x ,c1u 1 4x 4x

x1 x

x ,c0.1tanh 10

x

2 2 x

0.2 tanh 10 y

d 1 d4x 4y

(31)

Such as

1 1 1 2 5 51 1 d

1

x ,c0.25 2x 2500 2500 tanh 10 x 10 y

x

(32)

The simulation results for the initial states 1 2x 0 20,x 0 25 are shown in figures 1‐2. It can

be seen that the output of the closed‐loop system tracks the reference signals.

(a) (b)

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(c) (d)

Figure 1. Simulation results if dy 40 : (a) system output 1y x and the reference signal dy ; (b) State variable 2x ; (c) control input u; (d) tracking error dy y

(a) (b)

(c) (d)

Figure 2. Simulation results if d2y t 40sin t10

: (a) system output 1y x and the reference signal dy ; (b) State variable 2x ; (c) control input u ; (d) tracking error dy y

Example 2 For the simulation example 2, we consider the nonlinear system, which is described as

21 1 2 1 1

2 2 2 22 1 2 1 2 2 1 2

1

x 1 6x 5x x

x 2 x 3x 1 x 3x u x ,x

y x

(33)

where: u and y are the control signal and measured output, respectively; 1 1x 10 and 2 1 2x ,x 20 . The objective of the control is to force the output 1y x to asymptotically track a reference signal dy . According to the described controller design procedure in section 3, we have:

n 2 , 21 1 1h x 1 6x , 2 2

2 1 2 1 2h x ,x 2 x 3x ,

1 1x 5 , 2 22 1 2 1 2x ,x 1 x 3x , 1l 10 , 2l 20 ,

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1400

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1 1 dx y ,

21 1 1 1 d 1 1 1 d 1 d

1x ,c c y c x 1 6x 4y 10 tanh x y5

,

22 2 1 d 1 1 1 d 1 d

1x c y c x 1 6x y 10 tanh x y5

.

We choose 1c 1 and 2c 2 . Then, we can obtain the control input

1 1 1 21 22 2

11 2

1 1 12

1

2 22 1 2

x ,c1u 1 6x 5xx1 x 3x

x ,c10tanh

x

2 2 x 3x

2 d 1 d20tanh y 5x 5y

(34)

Such that

1 1 1 211 d21 1

x ,c 12x1 1 10 10 tanh x yx 5 1 6x

(35)

For the initial states 1 2x 0 50,x 0 50 , simulation results are presented in figures 3‐6. From the results, we find that the output of the closed‐loop system tracks the reference signals. It can be seen that the chattering amplitudes have been effectively reduced where 100 compared to where 510 .

(a) (b)

(c) (d)

Figure 3. Simulation results if d2

y t 10sin t20

and 510 : (a) system output 1y x and the reference signal dy ; (b) State variable 2x ; (c) control input u ; (d) tracking error dy y

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-20

-10

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30

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track

ing

erro

r


(a) (b)

(c) (d)

Figure 4. Simulation results if d2

y t 10sin t20

and 100 : (a) system output 1y x and the reference signal dy ; (b) State variable 2x ; (c) control input u ; (d) tracking error dy y

(a) (b)

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(c) (d)

Figure 5. Simulation results if dy t 20 and 510 : (a) system output 1y x and the reference signal dy ; (b) State variable 2x ; (c) control input u ; (d) tracking error dy y

(a) (b)

(c) (d)


Example 3 For simulation example 3, we consider the following nonlinear system, which is defined as

1 1 1 2 1 1

2 22 1 2 1 2 1 2

1

1x 2 cos x 10 5cos x x x5

x 1 cos x x 2 sin x u x ,x

y x

(36)

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0

50

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nput

u

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-20

-10

0

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track

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0 5 10 15 20 25 30 35 40 45 50

-50

-40

-30

-20

-10

0

10

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x2

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-1

-0.5

0

0.5

1

Time (sec)

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trol i

nput

u

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-20

-10

0

10

20

30

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track

ing

erro

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where: u and y are the control signal and measured output, respectively; 1 1x 20 and 2 1 2x ,x 30 . The objective of the control is to force the output 1y x to asymptotically track a reference signal dy . According to the described controller design procedure in section 3, we have:

n 2 , 1 1 11h x 2 cos x5

,

22 1 2 1 2h x ,x 1 cos x x , 1 1 1x 10 5cos x ,

22 1 2 1x ,x 2 sin x , 1l 20 , 2l 30 , 1 1 dx y ,

1 1 1 1 1 11

1 d

d 1

1 1x ,c c 2 cos x510 5cos x

10 5cos x y

y 20 tanh

,

2 2 1 1 1 d 1

1

1 1x c 2 cos x y 20tanh510 5cos x

.

We choose 1c 1 and 2c 2 . Then, the control input is designed as

1 1 11 1 22

11

1 1 12

1

22 1 2

x ,c1 1u 2 cos x 10 5cos x xx 52 sin x

x ,c20 tanh

x

2 1 cos x x

2 d 1 130 tanh y 10 5cos x

(37)

Such that

1 1 1 11 1 1 d d 12

1 1

21 1 d 1 1

1

x ,c 5sin x 12 cos x 10 5cos x y y 20 tanhx 510 5cos x

1 1x sin x 5y sin x 20 1 tanh510 5cos x

(38)

(a) (b)

(c) (d)


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For the initial states 1 2x 0 10,x 0 10 , the results of the simulation are shown in figures 7‐10. It can be concluded that the output of the closed‐loop system tracks the reference signals very well. The value of the parameter has an influence on the chattering

amplitudes. Furthermore in the three examples, we notice that all the signals of the resulting closed‐loop systems are regularly bounded and the tracking error converges to zero.

(a) (b)

(c) (d)


(a) (b)

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(c) (d)

Figure 9. Simulation results if dy t sin t and 100 : (a) system output 1y x and the reference signal dy ; (b) State variable2x ; (c) control input u ; (d) tracking error dy y

(a) (b)

(c) (d)

Figure 10. Simulation results if dy t sin t and 510 : (a) system output 1y x and the reference signal dy ; (b) State variable 2x ; (c) control input u ; (d) tracking error dy y

5. Conclusion

In this paper, we propose a method of designing an adaptive controller for a class of nonlinear systems using the backstepping technique. The on‐line calculation of the control input is obtained using the Lyapunov synthesis

approach. The proposed approach guarantees that all the signals of the resulting closed‐loop systems are regularly bounded and the tracking error converges to zero. It has advantages, such as a simple structure, easy realization, a good control effect and strong robustness. The efficiency of the proposed control has been demonstrated by

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simulation studies. Future works could expand the method to be used for a more general class of uncertain nonlinear systems.

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[5] H.‐Yan Li, Y.‐An Hu, J.‐Cun Ren and M. Zhu, “Dynamic Feedback Backstepping Control for a Class of MIMO Nonaffine Block Nonlinear Systems,” Mathematical Problems in Engineering, vol. 2012(2012), Article ID 493715, 18 pages..

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