improvements on stabilization and robust tracking control for t-s fuzzy control system

(1)

Improvements on Stabilization and Robust Tracking

Control for T-S Fuzzy Control Systems

Shih-Wei Kau, Xin-Yuan Huang, Sheng-Yu Shiu, and Chun-Hsiung Fang*

Department of Electrical Engineering, National Kaohsiung University of Applied Sciences

Kaohsiung, Taiwan e-mail: [email protected]

* Corresponding author

Abstract—By using the non-PDC control and the multiple

Lyapunov function, a set of LMI-based stabilization conditions for continuous-time T-S fuzzy systems is developed in the paper. In comparison with existing results, not only the derived condition is more relaxed but the approach used is simpler and more realizable in dealing with the time derivatives of membership functions. Some drawback existing in current literature is removed in the paper. The proposed idea is also extended to solve the robust tracking control problem and obtain a more satisfactory tracking performance. At the end, practical examples are given to illustrate the effectiveness of the proposed approach.

Keywords—T-S fuzzy systems, multiple Lyapunov function,

non-PDC controller, stabilization, robust fuzzy tracking control.

I. INTRODUCTION

Takagi-Sugeno (T-S) fuzzy models provide a powerful basis for development of systematic approaches to stability analysis and controller design of fuzzy control systems [4,6,12,13,14]. In the development of fuzzy control, the various forms of Lyapunov function play a key role [1,2,15] for stability analysis and controller design. Recently, the multiple Lyapunov functions have been frequently used to relax the conservativeness of stabilization conditions [3,5,15]. However, by using this kind of function, it usually requires to calculate the derivatives of membership functions, which is quit difficult in practice. In [15], the authors assumed there is a bound on the derivative. However, the bound may not be easily evaluated and it always leads to a more conservative result. In order to relax the conservativeness, the reference [7] further subdivided the lower and upper bounds for the derivatives. For both papers, evaluating a suitable bound is difficult. A more serious drawback of these two papers is that after solving the controllers, it has to verify if the assumption is satisfied for the closed-loop system. This is quite unrealistic in the design of controllers. For avoiding computing the derivative, a line-integral fuzzy Lyapunov function approach is presented in [10]. Although the assumption on the bound of derivatives is no more, the

proposed condition is represented in the form of bilinear matrix inequalities and needs a two-step procedure to find the controller.

There are four advantages in our approach. (a) The present result does not need the assumptions on the bounds of derivatives. Thus it does not have to verify if the assumption is satisfied for the closed-loop systems. (b) The result is expressed in the form of linear matrix inequality. All conditions and controllers can be solved easily in one step. (c) The proposed stabilization conditions are more relaxed than related results. (d) The proposed idea can be extended to solve the robust tracking problem. A more satisfactory tracking performance in comparison with existing results is obtained.

This paper is organized as follows. Section Ⅱ presents a more relaxed stabilization conditions for T-S fuzzy systems. Section Ⅲ gives the extension result on the robust tracking problem. An inverted pendulum example is given to demonstrate the improved performance on tracking. Finally, some conclusions are summarized in Section Ⅳ.

Ⅱ. THE RELAXED STABILIZATION CONDITIONS

Consider a T-S fuzzy system described by :

1 1

Rule : IF ( ) isi _θ t M_i and…and_θ_s( ) ist M_is

( )

Thenx t =A x ti ( )+B w twi +B u tui ( ) (1)

( )

zi ( ) zwi

( )

zui ( )

z t =C x t +D w t +D u t (2) where _θ_i( )t , i=1, 2, ,… s , are measurable premise

variables. r is number of IF-THEN

rules. M iij( =1, 2, , ,… r j=1, 2, , )… s are fuzzy sets.

( )

n

x t ∈ is the state vector. u t

( )

∈ mu is the input

control vector. n n i A_∈ × _, n mw wi B _∈ × _, n mu ui B _∈ × _, p n zi C _∈ × _, p mw zwi D _∈ × _, p mu zui

D _∈ × _{are known constant}

(2)

matrices that describe the nominal system. Given a pair of (x t u t( ), ( )), the final outputs of the fuzzy system are

inferred as follows: ( )

( )

(

( ) ( ) ( )

)

1 r i i wi ui i t h t A x t B w t B u t x θ = =

∑

+ + (3) ( )

( )

(

( ) ( ) ( )

)

1 r i zi zwi zui i z t h θ t C x t D w t D u t = =

∑

+ + (4) where Note that

( )

( ) 1 0 , r 0 , 0 , i i i i t t h t ψ θ ψ θ θ = >

∑

> ≥ and ( )

( )

1 1 r i i h θ t = =

∑

In this paper, we employ the following

candidate Lyapunov function: ( )

( )

( )T 1

( )

( ) ( ) V x t ₌x t P− x t x t where ( )( ) ( ( )) 1 r i i i P x t h θ t P = =∑ is a positive-definite matrix. For the convenience of notation, hi

( )

θ( )t is denoted by

i

h. In this paper, consider a non-PDC controller [7,11].

Control Rule j: 1 1 IF_θ( ) ist M_j and… and_θ_s( ) ist M_js

( )

Then F x t =F P x tj, =P jj, =1, 2, ,…r

where F P_j , _j are feedback gains. The non-PDC fuzzy controller can be described as follows

( ) 1 1 1 ( ) r j j r i i j i u t h F h P x t − = = ⎛ ⎞ =

∑

⎜_⎝

∑

⎟_⎠ (5) From (3) (whenw t

_{( )}

=0), the closed-loop system can be

expressed as ( ) 1

( )

( ) ( ) 1 1 r r i i ui j j i j t h A B h F P x t x t x − = = ⎧ ⎫ = ⎨ + ⎬ ⎩ ⎭

∑

₍₆₎

The following Theorem provides with a sufficient condition for the stability of (6).

Theorem 1: If there exist Pi>0, i=1, 2, , ,… r Q> 0,

0, S> F_j, j=1, 2, , ,… r Mii,i=1, 2, , ,… r , T ij ji M =M i> j,i j, =1, 2, , ,… r _{( )} T_{( )}, r i j j r i M ₊ =M ₊ , 1, 2, , , i j= … r M_{( )( )}_{r i r i}₊ ₊ , i=1, 2, , ,… r ( )(r i r j) M ₊ ₊ ₍T _{)( )}, r j r i M ₊ ₊ = i> j, i j, =1, 2, , ,… r such that , 1, 2, , ; ii ii R <M i= … r (7) 2 T, , 1, 2, , ij ij ij R <M +M i j= … r i> j (8) ( ) ( )T , , 1, 2, , i r i j r i j P M ₊ M ₊ i j r − < + = … (9) ( )( )r i r i, 1, 2, , ; S M ₊ ₊ i r − < = … (10) ( )( ) ( )( ) 2 T , , 1, 2, , ; r i r j r i r j S M ₊ ₊ M ₊ ₊ i j r i j − < + = … > (11) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) 11 12 1 21 22 2 1 2 1 1 1 2 1 1 1 1 2 1 2 1 2 2 2 2 1 2 2 2 1 2 1 2 * * * * * * * * * * * * * * * * 0 r r r r rr r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r M M M M M M M M M M M M M M M M M M M M M M M M M M M + + + + + + + + + + + + + + + + + + + + + + + + + + + ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ≤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , ₍₁₂₎ where , 1, 2, , T T T ii i i i i i i i i R =A P P A+ +B F F B+ +S i= … r

(

)

1 2 T T T ij i j ui j j i j ui T T T j i uj i i j i uj R A P B F P A F B A P B F P A F B S = + + + + + + + + , 1, 2, , , i j= … r i> j then the closed-loop system (6) is asymptotically stable.

Example 1 [7]:This example illustrates the effectiveness on the realization of the proposed conditions in comparison with [7,10]. Consider the following T-S fuzzy system

( )

1

( )

1 1 1 1 Rule 1: IF x t is M Then x t =A x t +B u t_u

( )

2

( )

1 1 2 2 Rule 2 : IF x t is M Then x t =A x t +B u t_u

( )

3

( )

1 1 3 3 Rule 3: IF x t is M Then x t = A x t +B u t_u where 1 1 2 2 3 3 1.59 7.29 1 ; ; 0.01 0 0 0.02 4.64 2 ; ; 0.35 0.21 0 4.33 2.5 ; . 0.1 0.05 0 A B A B a b A B − ⎡ ⎤ ⎡ ⎤ =_⎢ _⎥ =_{⎢ ⎥} ⎣ ⎦ ⎣ ⎦ − ⎡ ⎤ ⎡ ⎤ =⎢ ⎥ =⎢ ⎥ ⎣ ⎦ ⎣ ⎦ − − − + ⎡ ⎤ ⎡ ⎤ =_⎢ _⎥ =_⎢ _⎥ ⎣ ⎦ ⎣ ⎦

The parameters

a

and b are adjusted to test the feasibility of the conditions. In Fig. 1, the symbol “o” denotes the feasible area.

( )

( ) ( )

( )

( ) 1

(

( )

)

1 , s i i r i j ij j i i t h t t M t t ψ θ θ ψ θ θ ψ θ = = =

∏

∑

3182

(3)

and robust tracking control for T-S fuzzy control systems is proposed. Four improvements are achieved in this paper. (a) The present result does not need the assumptions on the bounds of derivatives. Thus it does not have to verify if the assumption is satisfied for the closed-loop systems. (b) The result is derived in the form of linear matrix inequality. All conditions and controllers can be solved easily in one step. (c) Comparing with related result, the proposed stabilization conditions are more relaxed. (d) The proposed idea can be extended to solve the robust tracking problem. A more satisfactory tracking performance is obtained.

Acknowledgements

This work was supported in part by the National Science Council, Taiwan, under Grants NSC-96-2221-E-151-056 and NSC-95- 2622- E-151-021-CC3

References

[1] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design of a class of continuous time fuzzy control systems,” Int. J. Control, vol. 64, pp. 1069–1087, 1996.

[2] S. G. Cao, N. W. Rees, and G. Feng, “Analysis and design for a class of complex control systems, Part II: Fuzzy controller design,” Automatica, vol. 33, no. 6, pp. 1029–1039, 1997.

[3] B. C. Ding, H. X. Sun, and P. Yang, “Further studies on LMI-based stabilization conditions for nonlinear systems in Takagi-Sugeno’s form,”

Automatica, vol. 42, no. 4, pp. 503-508, 2006.

[4] C. H. Fang, Y. S. Liu, S. W. Kau, L. Hong, and C. H. Lee, “A New LMI-Based Approach to Relaxed Quadratic Stabilization of T–S Fuzzy Control Systems,” IEEE Trans. Fuzzy Systems, vol. 14, no. 3, pp. 386-397, Jun. 2006.

[5] T. M. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,”

Automatica, vol. 40, pp. 823–829, 2003.

[6] X. Liu and Q. Zhang, “New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI, ”Automatica, vol. 39, no. 9, pp. 1571-1582, Sep. 2003.

[7] H. K. Lam and L. D. Seneviratne, “Stability analysis of fuzzy-model- based control systems using fuzzy Lyapunov function,” in review 2008. [8] H. K. Lam and L. D. Seneviratne, “LMI-based

stability design of fuzzy controller for nonlinear systems,”IET Control Theory Appl., vol. 1, no. 1, pp. 393-401, Jan. 2007.

[9] X. J. Ma, Z. Q. Sun, and Y. Y. He, “Analysis and Design of Fuzzy Controller and Fuzzy Observer,”

IEEE Trans. Fuzzy Systems, vol. 6 no. 1, pp. 41-51,

Feb. 1998.

[10] B. J. Rhee and S. Won, “A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design,” Fuzzy Sets and Systems, vol. 157, pp.1211-1228, Jan. 2006.

[11] K. Tanaka, H. Ohtake and H. O. Wang, “A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions,”

IEEE Trans. Fuzzy Systems, vol. 15, no. 3,

pp.333-341, Feb. 2007.

[12] K. Tanaka and H. O. Wang, “Fuzzy Control Systems

Design and Analysis: A Linear Matrix Inequality Approach,” John Wiley & Sons Inc. New York,

2001.

[13] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1,pp. 116-132, Jan. 1985.

[14] K. Tanaka, T. Ikeda and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic Stabilizability, H∞ control theory, and linear matrix inequalities,”

IEEE Trans. Fuzzy Systems, vol. 4, no. 1, pp. 1-13,

Feb. 1996.

[15] K. Tanaka, T. Hori and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Trans. Fuzzy

Systems, vol. 11, no. 4, pp. 582-589, 2003.