Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P10 ppt

Wang Copyright 䊚 2001 John Wiley & Sons, Inc.A motivating example for using the fuzzy descriptor system instead of theoriginal Takagi-Sugeno fuzzy model is presented.. The descriptor sys

Kazuo Tanaka, Hua O Wang Copyright 䊚 2001 John Wiley & Sons, Inc.

A motivating example for using the fuzzy descriptor system instead of theoriginal Takagi-Sugeno fuzzy model is presented An LMI-based designapproach is employed to find stabilizing feedback gains and a commonLyapunov function

The descriptor system, which differs from a state-space representation, hasgenerated a great deal of interest in control systems design The descriptorsystem describes a wider class of systems including physical models and

w xnondynamic constraints 1 It is well known that the descriptor system ismuch tighter than the state-space model for representing real independentparametric perturbations There exist a large number of papers on thestability analysis of the T-S fuzzy systems based on the state-space represen-tation In contrast, the definition of a fuzzy descriptor system and its stability

w x w xanalysis have not been discussed until recently 2 In 2 we introduced thefuzzy descriptor systems and analyzed the stability of such systems This

w xchapter presents both the basic framework of 2, 3 as well as some newdevelopments on this topic

10.1 FUZZY DESCRIPTOR SYSTEM

fuzzy descriptor system, where the E matrix in the fuzzy descriptor system

is assumed to be not always nonsingular The fuzzy descriptor system isdefined as

Here x g R n is the descriptor vector, u g R m is the input vector, y g R q is

the output vector, E g R k n =n , A g R i n =n , B g R i n =m , and C g R i q =n The

Ž Ž known premise variables z t1 ; z t may be functions of the states, external p

disturbances, andror time

THEOREM 33 The fuzzy descriptor system 10.3 is quadratically stable if

there exists a common matrix X such that

condition 10.5 for the pairs i, k such that h z t ® z t s 0 for all z t i k

Remark 34 In Theorem 33, X is not required to be positive definite.

Corollary 5 is needed to discuss the stability of closed-loop systems

where S is a positi®e definite matrix:1

Remark 35 It is stated in Remark 34 that X is not required to be tive definite However, in Corollary 5, X is assumed to be invertible since

where F* s i k i k The fuzzy controller design problem is to determine

the local feedback gains F i k

Proof. Consider a candidate of a quadratic function

Ž

Equation 10.13 can be rewritten as

XyT E*TsE *Xy 1G0.The above inequality is

descriptor system 10.16 can be stabilized ®ia the PDC fuzzy controller 10.17 if

there exist Z , Z , and M such that1 3 i

Now consider the common B matrix case, that is, B s B s1 2 ⭈⭈⭈ s B in r

Ž10.2 The stability analysis for the common B matrix case is simpler and.easier in comparison with that of the general case Keep this in mind because

we will refer to this when discussing the motivation behind the introduction

of the fuzzy descriptor system

In the common B matrix case, the stability conditions of Theorems 34 and

35 can be simplified as Theorems 36 and 37, respectively Theorem 37 gives

The feedback gains F i k are obtained as F s M Z i k i k y11

Proof. Consider a candidate of quadratic function

0

B

Therefore, the fuzzy control system is stable if

10.3 RELAXED STABILITY CONDITIONS

This section derives relaxed stability conditions by utilizing properties ofmembership functions Theorem 38 is a relaxed stability condition forTheorem 34

THEOREM 38 Assume that the number of rules that fire for all t is less than

or equal to s, where 1 - s F r The fuzzy descriptor system 10.2 can be

stabilized ®ia the PDC fuzzy controller 10.8 if there exist a common matrix Z ,1

Z3, Y , Y ,1 2 and Y such that3

The feedback gains are obtained as F s M Z i k i k y11

Proof. Consider a candidate of quadratic function

From the above assumption, we have

Ž given below Equation 10.27 can be rewritten as well:

Y s Z Q Z y Z1 1 1 1 T3Q Z y Z Q3 1 1 T3Z q Z3 T3Q Z2 3,

Y s Z Q Z2 1 2 1,

Theorem 38 is reduced to Theorem 34 when Y s Y s Y s 0 This1 2 3

means that Theorem 38 gives more relaxed conditions

Ž Ž Next, we derive stability conditions for Theorem 38 in the case of h z t i

The feedback gains are obtained as F s M Z i i y11.

Proof. Consider a candidate of a quadratic function

T

q2Ý Ýh z t iŽ Ž h z t jŽ Ž x* Ž t Ux* tŽ

is1 i -j r

F Ýh iŽzŽ t .x* Ž t ŽG X q X G q i i i i Žs y 1 U x* t Ž

Theorem 39 is reduced to Theorem 35 when Y s Y s Y s 0 This1 2 3

means that Theorem 39 gives more relaxed conditions

Consider the common B matrix case, that is, B s B s1 2 ⭈⭈⭈ s B Itr

should be emphasized that stability conditions for the common B matrix case

become very easy

The feedback gains F are obtained as F s M Z i i i y11.

Proof. Theorem 41 is derived in the same way as in the proof of Theorem 37.Note that Theorems 40 and 41 are the same as Theorems 36 and 37,respectively

We present a motivating example of the need of the fuzzy descriptor systeminstead of the ordinary fuzzy model Consider a simple nonlinear system,

Note that the fuzzy descriptor system has the common B matrix The simpler

stability condition, Theorem 36 or 41, is applicable for designing a stable

Fig 10.1 Control result 1.

Fig 10.2 Control result 2.

Note that five LMI conditions are required to find feedback gains F i k

Therefore the fuzzy descriptor system is suitable for modeling and analysis

of complex systems represented in the form 10.31 The form is often

w xobserved in nonlinear mechanical systems 6, 7

Figures 10.1 and 10.2 show the control results for the fuzzy descriptorsystem The fuzzy controller is designed using Theorem 41 The designed

matic Control, Vol AC-22, No 3, pp 312᎐321 1977

2 T Taniguchi, K Tanaka, K Yamafuji, and H O Wang, ‘‘Fuzzy Descriptor Systems: Stability Analysis and Design via LMIs,’’ 1999 American Control Confer- ence, San Diego, June 1999, pp 1827 ᎐1831.

3 T Taniguchi, K Tanaka, and H O Wang, ‘‘Fuzzy Descriptor Systems and Fuzzy Controller Designs,’’ Eighth International Fuzzy Systems Association World Congress, Taipei, Vol 2, Aug 1999, pp 655 ᎐659.

4 T Taniguchi, K Tanaka, and H O Wang, ‘‘Universal Trajectory Tracking Control Using Fuzzy Descriptor Systems,’’ 38th IEEE Conference on Decision and Con- trol, Phoenix, Dec 1999, pp 4852 ᎐4857.

5 T Taniguchi, K Tanaka, and H O Wang, ‘‘Fuzzy Descriptor Systems and Nonlinear Model Following Control,’’IEEE Trans on Fuzzy Syst., Vol 8, No 4,

pp 442 ᎐452 2000

6 A Bedford and W Fowler, Statics ᎏEngineering Mechanics, Addison-Wesley

Pub-lishing Company, Reading, MA, 1995.

7 A Bedford and W Fowler, Dynamics ᎏEngineering Mechanics, Addison-Wesley

Publishing Company, Reading, MA, 1995.