Fuzzy Adaptive Control for a Class of Nonlinear Systems with ...

Fuzzy Adaptive Control for a Class of Nonlinear Systems with Unknown Control Gain Salim Labiod

Thierry Marie Guerra

Faculty of Science and Engineering, University of Jijel, 18000, Jijel, Algeria. e-mail: [email protected]

LAMIH, UMR CNRS 8530, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313, Valenciennes Cedex 9, France. e-mail: [email protected]

Abstract—This paper deals with direct adaptive control using fuzzy systems of a class of uncertain SISO nonlinear systems with unknown control gain sign. Within this scheme, a fuzzy system is used to generate directly the control input signal without dynamic system estimation, and the Nussbaum-type function is used to deal with the unknown control gain sign. The stability of the closed-loop system is performed using a Lyapunov approach. Simulation results are provided to verify the effectiveness of the proposed design. Keywords-fuzzy control, adaptive control, Nonlinear systems, Nussbaum function.

I.

INTRODUCTION

In recent years, applications of fuzzy systems to adaptive control have been intensively conducted and some developments have been achieved in the literature [1, 2]. This is due to the fact that fuzzy systems have the ability to approximate continuous nonlinear functions. The stability study in such adaptive methods is performed by using the Lyapunov design approach. Most of the proposed adaptive control methods using fuzzy systems can be roughly classified into two types: the direct fuzzy adaptive control and the indirect fuzzy adaptive control. In the direct approach, the fuzzy system is used to approximate an unknown ideal controller [1-5]. On the other hand, the indirect approach uses fuzzy systems to estimate the plant dynamics and then synthesizes a control law based on these estimates [1-4, 6]. However, in the aforementioned papers, the control direction is assumed known a priori, i.e., the sign of the control gain is assumed known for the designer. Without this assumption, adaptive controllers design becomes much more difficult, because in this case, one cannot decide the direction along which the control operates and/or the direction of the search of controller parameters. In the adaptive control literature, the unknown control direction problem is mainly solved by using the Nussbaum-type function [7, 8], and it has been successfully applied for robust control design [9, 10] and adaptive control design for nonlinear systems [11-16]. In the literature, other approaches solved the unknown control gain sign problem by using: the correction vector method for first order nonlinear systems [17]; a switching scheme based on a monitoring function [18]; a hysteresis-type function for indirect

adaptive fuzzy control [19, 20]; or by estimating unknown parameters involved in the control gain [21]. In this work we present a direct adaptive fuzzy control scheme for a class of uncertain nonlinear systems. In the design; instead of using fuzzy systems to approximate the unknown nonlinearities in the system, a fuzzy system is used to approximate an unknown desired controller that can achieve control objectives, and a Nussbaum gain function is introduced to solve the problem of unknown control gain sign. Using the Lyapunov approach, it is proved that the proposed adaptive fuzzy controller can guarantee the boundedness of all the signals of the closed-loop system and the asymptotic convergence of the tracking error. Compared with the existing results in the adaptive controller design for systems with unknown control high frequency gain sign, a new adaptive parameter update law is proposed within this scheme to meet control objectives. This paper is organized as follows. The problem formulation and preliminaries are given in section II. The proposed direct adaptive control scheme is presented in section III with its adaptive law and stability analysis. In section IV, the proposed adaptive control algorithm is used to control a simple nonlinear system. II.

PROBLEM FORMULATION AND PRELIMINARIES

Consider the class of single-input single-output (SISO) nonlinear systems modeled by



 xi  xi 1 , i  1, , n  1   xn  f  x   g  x  u  y  x 1 



y  n  f  x   g  x  u 



or, equivalently 

where x   x1 , , xn    n , is the state vector of the system which is assumed available for measurement, u   is the T

scalar control input, y   is the scalar system output, f  x  and g  x  are unknown smooth nonlinear functions. In respect of the dynamic system (1), the following assumption will be made:

positive constants. The objective is to design an adaptive fuzzy controller for system (1) such that the system output y  t  follows a desired trajectory yd  t  while all signals in the closed-loop system remain bounded. Regarding the development of the control law, the following assumption should also be made: Assumption 2:

The desired trajectory yd  t  and its time

derivatives yd i   t  , i  1, , n , are smooth and bounded. Now, let us define the tracking error as 

e  yd  y 



and let s be a sliding surface defined as d  s    dt 





s  k s 



From which one can conclude that s  t   0 as t   and, therefore, e  t  and all its time derivatives up to n  1 converge to zero [22]. However, in this paper, the nonlinear functions f  x  and g  x  are unknown, so control law (7) is not implementable. In this case, the purpose is to use a fuzzy system to identity the entire unknown control function (7).

In this paper, we use the zero-order Takagi-Sugeno fuzzy system that performs a mapping from an input vector T z   z1 , , zm    z   m to a scalar output variable y f   , where  z   z1     zm and  zi   . If we define M i fuzzy sets Fi j , j  1, , M i , for each input zi , then the fuzzy system will be characterized by a set of if-then rules of the form [1, 2]

n 1

e,   0 



R k : If z1 is G1k and  and zm is Gmk



whose solution implies that the tracking error e  t  converges to zero with a time constant

 n  1 

. In addition, the

derivatives of e  t  up to n  1 also converge to zero [22]. Thus, the control objective becomes the design of a controller to keep s  t  at zero. The time derivative of s is given by s    f  x   g  x  u 

 k  1, , N 

Then y f is y kf



of the k -th rule, and N is the total number of rules. By using the singleton fuzzifier and the product inference engine, the final output of the fuzzy system is given by [1, 2]

 z   N

yf



k 1 N

where m

k i

k  z  y kf

k 1

k  z 





(10)

Now, if the nonlinear functions f  x  and g  x  are known, the following ideal controller can be derived to achieve the tracking control objective

k i

1 i

i

Mi

,

with  F j  xi  is the membership function of the fuzzy set



with  j  (n  1)!  (n  j )!( j  1)!  n  j , j  1, , n  1 .





k  z    i 1 G  zi  , with G   F , ,  F

  yd( n )   n 1e( n 1)    1e 



where Gik  Fi1 , , Fi M i , i  1, , n , y kf is the crisp output



where 



As a result, the closed-loop error dynamic becomes

From (5), s  t   0 represents a linear differential equation



1   f  x   k s   g  x

where k is a positive design parameter.

Assumption 1: The control gain g  x  and its sign are

unknown with 0  g  g  x   g , where g and g are

u  u* 



i

Fi j . By introducing the concept of fuzzy basis functions [1], the output given by (10) can be rewritten in the following compact form 

y f  z   wT  z 

(11)

T

where    y1f , , y Nf  is a vector grouping all consequent

Proof: For the proof, please refer to [10].

parameters, and w  z    w1  z  , , wN  z   is a set of fuzzy T

basis functions defined as wk  z  



k  z 



N j 1

 j z

, k  1,  , N 

(12)

The fuzzy system (11) is assumed to be well-defined so that



N j 1

 j  z   0 for all z   z .

It has been proved in [1] that fuzzy systems in the form of (12) with Gaussian membership functions can approximate continuous functions over a compact set to an arbitrary degree of accuracy provided that enough number of rules are considered. In order to deal with the unknown control direction (sign of g  x  ), the Nussbaum gain technique is used in this paper. The Nussbaum function technique was originally proposed in [7] and has been effectively used in controller design in solving the difficulty of unknown control directions [8-16]. A function N   is called a Nussbaum-type function if it has the following properties [7]

III.

FUZZU ADAPTIVE CONTROL DESIGN

In the precedent section we have established that there exists an ideal control law u * given by (7) that can achieve the control objective. However, this nonlinear controller cannot be used since it depends on unknown functions. In this section, to circumvent this problem, we propose to use adaptive fuzzy systems for approximating this ideal controller. To develop the control law, we assume that the ideal controller u * can be approximated by a fuzzy system in the form of (11) as follows



u*  z   wT  z  * 

(15)

where z is the input vector of the fuzzy system, w  z  is a fuzzy basis function vector fixed by the designer, and  * is an bounded unknown time-varying parameter vector with bounded time derivative, i.e.  * and * are assumed upper bounded. Since the ideal parameter vector  * is unknown, let us use its estimate  instead to form the adaptive control



u  z   wT  z  

(16)





1  lim sup   N   d      0 



1  lim inf   N   d        0 



 



For example, the continuous functions e cos   and 2

 2 cos   are Nussbaum-type functions. In this paper, the even Nussbaum function  cos   is used.

After the specification of the controller, the next step should be the design of an adaptive law for the free parameters  to achieve control objectives. However, since the control direction (sign of g  x  ) is unknown, we have to use a Nussbaum-type function in designing the parameter adaptation law as follows

   N   w T  z  s   

(17)

     s  k s 

(18)

    s  k s  s   

(19)

N     2 cos  

(20)

2

The following Lemma regarding to the property of Nussbaum functions is used in the controller design and theorem proof of next section.

Lemma 1: Le V   and    be smooth functions defined on

0, t f  with V  t   0 ,  t  0, t f  , N   be an even smooth Nussbaum-type function. If the following inequality holds:





V  t   c0   g   N      c1    d  for  t   0, t f t

0

,

where g   is a piecewise continuous time function which takes values in the unknown closed interval I   g , g  with 0  I , c1 any positive number and c0 represents some suitable constant,

V t 

then

  g   N      c    d t

0

1

,

 t 

are bounded on 0, t f  .

and

where  ,  and  are strictly positive constants.

Theorem 1: For nonlinear system (1), under Assumptions 1 and 2 and Lemma 1, the control law (16) with adaptation laws (17)–(20) guarantee the boundedness of all the signals in the closed-loop system and the asymptotic convergence of the tracking error. Proof: By adding and subtracting g  x  u * to the right-hand side of (5), one obtains



s    f  x   g  x  u *  g  x   u *  u  



Using (7), (21) can be expressed as



s   k s  g  x   u *  u  



With (15) and (16), one can rewrite (22) as



s  k s  g  x  w T  z   



1 2 1 T  1 2 s      2 2 2



The time derivative of V is



1 1 V  ss   T       T * 







1 V  k s 2  s g w T    s    N   w T      T * 



s  k s  g  x  w T  z   



Thus, (26) can be rewritten as N   1 V  k s 2  s  s  k s    s    s  k s      T *   g  x

Using (19), (28) can be further rewritten as follows

 1 V  k s 2   h  x  N     

 1  T *     s  k s          

with h  x   1  g  x   .

 1 V  k s 2   h  x  N     

2  2 1 2                

t 1 V  t   c0    h   N      0  



 1  T * 2         

Now, we assume that the following assumption holds

1





0

    d   



 2  t  dt .

According to Assumption 1, we can consider h   as a time-varying parameter which takes values in the interval 1 1 I h     g ,   g   with 0  I h . Thus, Using Lemma 1,   we can conclude from (33) the boundedness of the signals

 

V t 

N t 

,

t  0, t f



  h   N    1     d t

and

0

for

. According to [9], since no finite-time escape

s  t  is square integrable and s  t  is bounded. Moreover, by invoking Barbalat’s lemma, we can conclude the asymptotic convergence of s  t  , which further implies the asymptotic

convergence of the tracking error e  t  and its first  n  1 derivatives. Remark 1: It is worth noticing that because of the integral structure of adaptive laws (18) and (19), these parameter updating laws are implementable despite the presence of the time derivative s  t  . Actually,   t  and   t  can be expressed as  

Using (18), (29) can be simplified to



    1 V    k  1  s 2   h  x  N        

phenomenon may happen, then t f   . Therefore, s  t  ,   t  , x  t  , and u  t  are bounded. As an intermediate result,

From (23) one can obtain



With (31), (30) can be upper bounded by

where c0  V  0  

Substituting (17) and (23) into (25), and using the equality w T  z    T w  z  , it follows that



where 1 and  2 are positive constants, and   t  is a bounded square integrable signal.

(32) over  0,t  , one can obtain the following inequality

Now, consider the following positive definite function V



Assuming that k  1  and    2  , then by integrating

where    *   is the parameter error vector.



T *  1 s 2   2 2   2  t  





t

  t     0   s 0      k s  d  t

0



  t     0     s  0    s  1  k  s   k      dv  t

t

0

IV.

SIMULATION RESULTS

To demonstrate the effectiveness of the above proposed adaptive controller, the following second-order nonlinear system is used for simulation

 x1  x2    x2  f  x   g  x  u





10

where

5

f  x   1.5 1  x12  x2  x1 , g  x   1  0.5cos  x1 x2  .

0

The control objective is to force the system output y  t   x1  t  to track the desired trajectory yd  t   sin  t  . Within this simulation, the unknown ideal controller is approximated by a fuzzy system in the form of (15). The input T variables of the fuzzy system are chosen as z   x1 , x2  . For

-5

-10

0

each variable zi , i  1, 2 , five Gaussian membership functions are used with centers 1 , 0.5 , 0, 0.5 and 1, and a variance equal to 4.

In this simulation, the system initial conditions are T x  0    0.5, 0 , the adjustable parameters  are initialized with zero values, and the design parameters are   1 , k  10 ,   30 ,   30 and   50 . Simulation results for the case of an unknown positive control direction ( sgn  g  x    1 ) are shown isFigures 1 and 2. To show the controller behavior in the case of a negative control direction, we consider the case g  x    1  0.5cos  x1 x2   . Using the same simulation design parameters, simulation results are shown in Figures 3 and 4. We can see that actual trajectories converge to the desired ones and that the control input signal is bounded.

5

10 Time (sec)

15

20

Figure 2. Control input signal ( g  x   0 ).

1

x1

x2

0.5

0

-0.5

-1 0

5

10 Time (sec)

15

20

Figure 3. System’s response ( g  x   0 ): actual (solid lines); desired (dotted lines). 1

x1

x2

10

0.5

5 0

0 -0.5

-5

-1 0

5

10 Time (sec)

15

20

Figure 1. System’s response ( g  x   0 ): actual (solid lines); desired (dotted lines).

-10

0

5

10 Time (sec)

15

Figure 4. Control input signal ( g  x   0 ).

20

V. CONCLUSION This paper proposed direct adaptive fuzzy control for a class of uncertain SISO nonlinear systems with unknown control gain sign. The scheme consists of an adaptive fuzzy controller with a new adaptive parameter update law. The fuzzy system is used to construct adaptively an unknown ideal controller that can achieve control objectives, and the Nussbaum gain technique is employed for dealing with the unknown control direction. Using the Lyapunov synthesis method, all signals in the closed-loop control system have been theoretically shown to be bounded and the tracking error has been shown to be asymptotically stable. Simulation results have verified the effectiveness of the proposed adaptive control approach. REFERENCES [1] [2]

[3]

[4]

[5]

[6]

[7] [8] [9]

L.X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall, Englewood Clifs, NJ, 1994. J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable adaptive control and estimation for nonlinear systems. John Wiley & Sons, 2002. J.T. Spooner and K.M. Passino, “Stable adaptive control using fuzzy systems and neural networks,” IEEE Trans. Fuzzy Systems, vol. 4, pp. 339-359, 1996. N. Essounbouli and A. Hamzaoui, “Direct and indirect robust adaptive fuzzy controllers for a class of nonlinear systems,” International Journal of Control, Automation, and Systems, vol. 4, no. 2, pp. 146-154, 2006. S. Labiod and T.M. Guerra, “Direct adaptive fuzzy control for a class of MIMO nonlinear systems,” International Journal. of Systems Science, vol. 38, no. 8, pp. 665-675, 2007. S. Labiod, M.S. Boucherit, and T.M. Guerra, “Adaptive fuzzy control of a class of MIMO nonlinear systems,” Fuzzy Sets and Systems, vol. 151, no. 1, pp. 59-77, 2005. R. D. Nussbaum, “Some Remarks on the Conjecture in parameter Adaptive Control,” Systems Control Letter, vol. 3, pp. 243-246, 1983. Ioannou and J. Sun, Robust adaptive control. Prentice Hall, Englewood Cliffs, NJ, 1996. L. Liu and J. Huang, “Global robust stabilization of cascade-connected systems with dynamic uncertainties without knowing the control direction,” IEEE Trans. Automat. Contr., vol. 51, pp. 1693-1699, 2006.

[10] L. Liu and J. Huang, “Global robust output regulation of lower triangular systems with unknown control direction,” Automatica, vol. 44, pp. 1278–1284, 2008. [11] X. Ye and J. Jiang, “Adaptive nonlinear design without a priori knowledge of control directions,” IEEE Trans. Automatic Control, vol. 43, pp. 1617–1621, 1998. [12] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients,” IEEE Trans. Syst., Man, Cybern. B, vol. 34, no. 1, pp. 499-516, 2004. [13] T.P. Zhang, and S.S. Ge, “Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, pp. 1021-1033, 2007. [14] Y. J. Liu and Z. F. Wang, “Adaptive fuzzy controller design of nonlinear systems with unknown gain sign,” Nonlinear Dynamics, vol. 58, pp. 687–695, 2009. [15] A. Boulkroune, M. Tadjine, M. M’Saad, and M. Farza, “Fuzzy adaptive controller for MIMO nonlinear systems with known and unknown control direction,” Fuzzy Sets and Systems, vol. 161, pp. 797–820, 2010. [16] X. Li, D. Wang, Y. Yao, W. Lan, Z. Peng, G. Sun, Adaptive Fuzzy Control for a Class of Nonlinear Time-delay Systems with Unknown Control Direction, In Proc. 29th Chinese Control Conference, 2010, pp. 2593-2597. [17] B. Brogliato and R. Lozano, “Adaptive control of first-order nonlinear systems with reduced knowledge of the plant parameters,” IEEE Trans. on Automatic Control, vol. 39, pp. 1764-1768, 1994. [18] W. H. Dong, X. X. Sun and Y. Lin, “Variable Structure Model Reference Adaptive Control with Unknown High Frequency Gain Sign,” Acta Automatica Sinica, vol. 33, no. 4, pp. 404-408, 2007. [19] S. Labiod and T. M. Guerra, “Indirect adaptive fuzzy control for a class of nonaffine nonlinear systems with unknown control directions,” International Journal of Control, Automation, and Systems, vol. 8, no. 4, 903-907, 2010. [20] J. H. Park, S. H. Kim, and C. J. Moon, “Adaptive fuzzy controller for the nonlinear system with unknown sign of the input gain,” International Journal of Control, Automation, and Systems, vol. 4, no.2, 178-186, 2006. [21] J. Kaloust and Z. Qu, “Robust Control Design for Nonlinear Uncertain Systems with an Unknown Time-Varying Control Direction,” IEEE Trans. Automat. Contr., vol. 42, no. 3, 393-399, 1997. [22] J.E. Slotine and W. Li, Applied nonlinear control. Prentice Hall, Englewood Cliffs, NJ, 1991.