Two-Degree-of-Freedom Controller Design for ... - Semantic Scholar

Proceedings of the 6th WSEAS International Conference on Robotics, Control and Manufacturing Technology, Hangzhou, China, April 16-18, 2006 (pp87-92)

Two-Degree-of-Freedom Controller Design for Takagi-Sugeno Fuzzy Systems WEI XIE and TOSHIO EISAKA Computer Sciences Kitami Institute of Technology 165 Koencho, Kitami JAPAN Abstract: - Controller design strategy for Takagi-Sugeno (T-S) fuzzy systems is considered in the two-degree-of-freedom (TDOF) framework. Firstly, coprime factorization described in the state space formulas for T-S fuzzy systems is introduced based on a common Lyapunov function. Secondly, based on this coprime factorization, TDOF framework for LTI systems is extended to be applied to T-S fuzzy systems. Consequently, good tracking performance and good disturbance rejection (and robustness) are compatibly achieved by a feedforward controller and a feedback controller, respectively. Furthermore, each controller design problem can be formulated with dynamic parallel distributed compensation in terms of linear matrix inequality related to L2 gain performance. Key-Words: T-S fuzzy system, L2 gain performance, Two-Degree-of-Freedom, Dynamic parallel distributed compensation (DPDC), Nonlinear dynamic control, Linear matrix inequality, Controller design.

1 Introduction Takagi-Sugeno (T-S) fuzzy systems can be formalized from a large class of nonlinear systems [1,2]. Despite the fact that the global T-S model is nonlinear due to the dependence of the membership functions on the fuzzy variables, it has a special formulation, known as Polytopic Linear Differential Inclusions (PLDI) [3], in which the coefficients are normalized membership functions. That is, local dynamics in different state-space regions are represented by linear models; and the nonlinear system is approximated by the overall fuzzy linear models. Most of the existing control techniques for T-S model utilized the parallel distributed compensation (PDC) law [4]. With quadratic Lyapunov functions and PDC law, a great deal of attention has been focused on analysis and synthesis of these systems [4-9]. In particular, in [9] sufficient linear matrix inequality (LMI) conditions are provided for the existence of a quadratically stabilizing dynamic compensator or the performance-oriented controller based on the notion of dynamic parallel distributed compensator (DPDC). On the other hand, two-degree-of-freedom (TDOF) control scheme is fundamental strategy for design of linear time invariant (LTI) systems to deal with both command tracking and disturbance rejection

independently [10-13]. It is noted that not only control performance but also controller structures can be independently treated in TDOF framework. To the authors’ knowledge, however, general TDOF control scheme including command tracking issue of T-S fuzzy systems has not been discussed explicitly. As for T-S fuzzy systems, it is no use treating transfer functions or eigenvalues of state matrices, thus, the TDOF methodology for LTI systems can not be applied to T-S fuzzy systems straightforwardly. In the present paper, we extend TDOF control framework for LTI systems to be applied to T-S fuzzy systems. Firstly, coprime factorization for T-S fuzzy systems is described in state space formulas based on a common Lyapunov function. Secondly, TDOF control framework for T-S fuzzy systems is presented based on this coprime factorization. Then two-step controller design approach is proposed. First, a feedforward T-S fuzzy controller that achieves good tracking performance is designed by model matching strategy with L2 gain performance. Second, a feedback controller that rejects disturbances and/or model uncertainties and does not affect tracking performance is designed with another L2 gain performance. Each controller design problem can be formulated with dynamic parallel distributed compensation in terms of linear matrix inequality expressions [14].

Proceedings of the 6th WSEAS International Conference on Robotics, Control and Manufacturing Technology, Hangzhou, China, April 16-18, 2006 (pp87-92)

We will assume that at least one wi ( z ) is always r

nonzero so that

2 Preliminaries In this section, firstly notations regarding T-S fuzzy systems are introduced. And then, definition of coprime factorization for T-S fuzzy systems is newly introduced . L2 gain performance with L2-norm is also recapped. Definition 1 [9]. The T-S fuzzy model G consists of a finite set of fuzzy IF-THEN rules. Each rule has the following form: Dynamic part: Rule i = 1, 2,L, r : IF z1 (t ) is M i1,L and z p (t ) is M ip ,

Each variable z i (t ) is a known parameter that is a function of the state variables x(t ) , external disturbances, and/or time. The symbols M ij represent membership functions for fuzzy sets. Using the center of gravity method for defuzzification, we can express the aggregated fuzzy model G ( z ) as:

B( z ) = 0  C ( z )

A

hi ( z )  i ∑ Ci i=1

Bi  0 

(1)

Here, z denotes the vector containing all the individual parameters z i (t ) . The hi ( z ) is the normalized possibility for the ith rule to fire given by

hi ( z ) =

wi ( z )

(2)

r

∑

wi ( z )

i =1

Where, possibility for the ith rule to fire: wi ( z ) is given by the product of all the membership functions associated with the ith rule as:

wi ( z ) =

p

∏M j =1

ij

(z j )

It is noted that the

normalized possibility hi ( z ) satisfies

hi ( z ) > 0 and

conditions

r

∑ h ( z) = 1 . i =1

i

Definition 2. The T-S fuzzy plant G ( z ) is said to have doubly coprime factorization if there exit right coprime factorization G ( z ) = N ( z ) D ( z ) −1 and left coprime factorization G ( z ) = D% ( z ) −1 N% ( z ) , where a set of

(4)

A particular set of realizations can be chosen such that

THEN y (t ) = C i x(t )

 A( z )

j

 V% ( z ) U% ( z )   D( z ) −U ( z )   I 0  % =  %   − N ( z ) D( z )   N ( z ) V ( z )   0 I 

Output part: Rule i = 1, 2,L, r : IF z1 (t ) is M i1,L and z p (t ) is M ip ,

G ( z )  

i

i =1

realizations for stable T-S fuzzy systems (z), U (z), V (z), U(z) and V(z) N (z), D(z), N (z), D can be chosen such that

THEN x& (t ) = Ai x(t ) + Bi u (t )

r

∑w (z ) ≠ 0 .

(3)

 A( z ) − B ( z ) F  D ( z ) −U ( z )    −F  N ( z) V ( z)      C ( z)   Ai − Bi F Bi L  r   = hi ( z )  −F I 0  i =1  C 0 I  i 

B( z ) L  I 0  0 I 

(5)

∑

 A( z ) − LC ( z ) B( z ) L   V( z ) U ( z )    F I 0  −N ( z ) D ( z )     −C ( z )  0 I    Ai − LCi Bi L r   = hi ( z )  F I 0   i =1  0 I   −Ci

(6)

∑

in which F and L are chosen such that both Ai − Bi F and Ai − LCi are quadratically stable. That is, F = −VP1−1 and L = − P2−1W should satisfy the following LMIs, respectively: P1 > 0 ; P1 AiT + Ai P1 + BiV + V T BiT < 0, i = 1, 2,L r and P2 > 0 ;

(7)

AiT P2 + P2 Ai + WCi + CiT W T < 0 , i = 1, 2,L r

(8)

Proceedings of the 6th WSEAS International Conference on Robotics, Control and Manufacturing Technology, Hangzhou, China, April 16-18, 2006 (pp87-92)

Definition 3. The control system satisfying the property b2 sup < γ is said to have L2 gain performance a ≠ 0, a 2 0 .   11

4 Conclusions We have developed two-step nonlinear dynamic controller design strategy for T-S fuzzy plant in TDOF framework. The first step is to design a feedforward controller that achieves good tracking performance by model matching strategy with L2 gain performance. The second step is to design a feedback controller that rejects disturbances while does not affect tracking performance with another L2 gain performance. Each controller design problem is formulated in terms of linear matrix inequality and both problems can be solved by DPDC methodology. Concerning feedback controllers, it is well known that all proper stabilizing controllers are parameterized in terms of arbitrary Q ∈ RH ∞ for LTI control system. The configuration of TDOF control system is deeply considered with the Q-parameter and its consequences － sophisticated design framework has been developed. In the present paper, we have newly developed general TDOF control scheme for T-S fuzzy system, however, we just focused on L2 gain performance to design controllers. The Q-parameter approach will give us more practical validity to deduce the solution and covers more general control system designs including multi-objective and/or switching system for T-S fuzzy plants. Based on our results, Q-parameter approach can be applicable to T-S fuzzy control system and it will be treated in another paper. References:

Proceedings of the 6th WSEAS International Conference on Robotics, Control and Manufacturing Technology, Hangzhou, China, April 16-18, 2006 (pp87-92)

[1] J. Castro, “Fuzzy Logic controllers are Universal Approximators,” IEEE Trans. Syst. Man and Cybern., vol. 25, pp. 629-635, 1995. [2] K. Tanaka and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, 15(1), 116-132, 1985. [3] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory. SIAM Studies in Applied Mathematics, Vol.15, SIAM, Philadelphia, 1994. [4] H. O. Wang, K. Tanaka, and M. F. Griffin, “An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues,” IEEE Trans. Fuzzy Sys, Vol.4, No.1, pp.14-23, 1996. [5] M. Johansson, A. Rantzer, and K. E. Arzen, “Piecewise Quadratic Stability for Affine Sugeno Systems,” Proc. 7th IEEE Int. Conf. Fuzzy Syst. Anchorage, AK, vol. 1, pp. 55-60, 1998. [6] S. Petterson and B. Lennartson, “An LMI Approach for Stability Analysis of Nonlinear Systems,” Proc. Of European Control Conference, ECC’98, Brussels, Belgium, 1997. [7] K. Tanaka and M. Sugeno, “Stability Analysis and Design of Fuzzy Control systems,” Fuzzy Sets and systems, Vol.45, No.2, pp.135-156, 1992. [8] K. Tanaka, T. Ikeda, and H. O. Wang, “Design of Fuzzy Control Systems Based on Relaxed LMI Stability Conditions,” Proc.35th CDC, pp.598-603, 1996. [9] J. Li, H. O. Wang, D. Niemann and K. Tanaka, “Dynamic parallel distributed compensation for Takagi-Sugeno fuzzy systems: An LMI approach,” Inf. Sci. 123(3-4): 201-221, 2000. [10] I. M Horowitz, Synthesis of Feedback Systems, Academic Press, 1963. [11] W. A. Wolovich, Linear Multivariable Systems, Springer-Verlag, 1974. [12] K. Zhou and Z. Ren, “A New Controller Architecture for High Performance, Robust, and Fault-tolerant Control,” IEEE Trans. on Automatic Control, , Vol. 46, No. 10, pp.1613-1618, 2001 [13]M. Vidyasagar, Control system synthesis, The MIT Press, 1985. [14] P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, LMI control toolbox for use with matlab, User’s guide, The Math Works Inc Natick, MA, USA, 1995.