Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P14 pdf

Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O. Wang Copyright ᮊ 2001 John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic CHAPTER 14 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR In this chapter, we present two results concerning the fuzzy modeling and wx control of nonlinear systems 1 . First, we prove that any smooth nonlinear control systems can be approximated by Takagi-Sugeno fuzzy models with linear rule consequence. Then, we prove that any smooth nonlinear state feedback controller can be approximated by the parallel distributed compen- Ž. sation PDC controller. Ž. wx Among various fuzzy modeling themes, the Takagi-Sugeno T-S model 2 has been one of the most popular modeling frameworks. A general T-S model employs an affine model with a constant term in the consequent part for each rule. This is often referred as an affine T-S model. In this book, we focus on the special type of T-S fuzzy model in which the consequent part for Ž. each rule is represented by a linear model without a constant term . We refer to this type of T-S fuzzy model as a T-S model with linear rule consequence, or simply a linear T-S model. As evident throughout this book, the appeal of a T-S model with linear rule consequence is that it renders itself naturally to Lyapunov based system analysis and design techniques wx 12, 15 . A commonly held view is that a T-S model with linear rule conse- quence has limited capability in representing a nonlinear system in compari- wx son with an affine T-S model 9 . wx In Chapter 2, the PDC controller structure was introduced 11, 12 . This structure utilizes a fuzzy state feedback controller which mirrors the struc- ture of the associated T-S model with linear rule consequence. As shown throughout this book, T-S models together with PDC controllers form a powerful framework for fuzzy control systems resulting in many successful wx applications 10, 13, 14 . 277 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR 278 In this chapter, we attempt to address the fundamental capabilities of T-S models with linear rule consequence and PDC controllers. To this end, two results are presented. The first result is that a linear Takagi-Sugeno fuzzy model can be a universal approximator of any smooth nonlinear control system. It has been known that smooth nonlinear dynamic systems can be approximated by T-S models with affine models as fuzzy rule consequences wx 4, 7 . However, most results on stability analysis and controller design of T-S models are based on T-S models with linear rule consequence. The question needed to be addressed is: ‘‘Is it possible to approximate any smooth nonlinear systems with Takagi-Sugeno models having linear models as rule wx consequences?’’ Reference 6 gave an answer to this question for the simple one-dimensional case. This chapter tries to answer this question for the n-dimensional nonlinear dynamic system by constructing T-S model to ap- proximate the original nonlinear system. The answer is yes. That is, the original vector field plus its velocity can be accurately approximated if enough fuzzy rules are used. The second result is that the PDC controller can be a universal approxi- mator of any nonlinear state feedback controller. Therefore linear T-S models and PDC controllers together provide a universal framework for the modeling and control of nonlinear control systems. In this chapter, ޒ n is used to denote the n-dimensional vector spaces of real vectors; C m is used to represent the set of n-dimension functions whose n mth derivative is continuous on the defined region; x stands for the ith i 55 component of vector x and stands for the standard vector norm or matrix Ž. << norm; Ox is the set of numbers y such that yrx - M, where M is a constant.; and Ý is used to represent the summation with all jj . j 12 n the possible combinations of j , j , ., j . We will often drop the x and 12 n just write h , but it should be kept in mind that h ’s are functions of the ii variable x. 14.1 APPROXIMATION OF NONLINEAR FUNCTIONS USING LINEAR T-S SYSTEMS 14.1.1 Linear T-S Fuzzy Systems The main feature of linear Takagi-Sugeno fuzzy systems is to express the Ž. local properties of each fuzzy implication rule by a linear function. The overall fuzzy system is achieved by fuzzy ‘‘blending’’ of these linear functions. Specifically, the linear Takagi-Sugeno fuzzy system is of the following form: Rule i IF x is M иии and x is M , 1 i1 nin THEN y s ax, i APPROXIMATION OF NONLINEAR FUNCTIONS USING LINEAR T-S SYSTEMS 279 T wx where x s x , x , ., x are the function variables; i s 1, 2, . . . , r and r is 12 n the number of IF-THEN rules; and M are fuzzy sets. The linear function ij y s axis the consequence of the ith IF-THEN rule, where a g ޒ 1=n . i i The possibility that the ith rule will fire is given by the product of all the membership functions associated with the ith rule: n hxs ⌸ Mx. Ž. Ž . iijj j s1 Ž. We will assume that h ’s have already been normalized, that is, hxG 0 and ii r Ž. Ý hxs 1. Then by using the center-of-gravity method for defuzzifica- is1 i tion, we can represent the T-S system as r ˆ y s fxs hxax.14.1 Ž. Ž. Ž . Ý ii i s1 The summation process associated with the center of gravity defuzzifica- Ž. tion in system 14.1 can also be viewed as an interpolation between the functions axbased on the value of the parameter x. i 14.1.2 Construction Procedure of T-S Fuzzy Systems Ž. n Suppose that the nonlinear function fx: ޒ ™ ޒ is defined over the compact region D ; ޒ n with the following assumptions: Ž. 1. f 0 s 0. 2. f g C 2 . Therefore, f, Ѩ fr Ѩ x, and Ѩ 2 fr Ѩ x 2 are continuous and there- 1 fore bounded over D. ˆ r Ž. Ž. Next, we will construct the T-S system fxs Ý hxaxto approxi- is1 ii Ž. Ž. Ž. mate fx. The objective is to make the approximation error exs fxy ˆ Ž. fx and its derivative Ѩ er Ѩ x small for all x g D. Construction Procedures: Ä4 1. In region D s xx- ⑀ where ⑀ is a chosen positive number, 0 i 00 choose a s Ѩ fr Ѩ x xs0 . 0 n 2. Define the projection operator P mapping ޒ to n y 1 dimensional x subspace ޒ n rx as ²: y, x Pys y y x. x 2 x wx T In region D _ D , choose x as j ⑀ j ⑀ . j ⑀ , where ⑀ is a 0 jj . j 12 n 12 n positive number and j are integers. Build the linear model a as ijj . j 12 n T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR 280 the solution of the following linear equations: ax s fx ,14.2 Ž. Ž. jj . jjj . jjj . j 12 n 12 n 12 n Ѩ f aP s P .14.3 Ž. xx jj . jjj . j jj . j 12 n 12 n 12 n x Ѩ x jj . j 12 n Ž.Ž. For fixed x , 14.2 ᎐ 14.3 are n linear equations with the compo- jj . j 12 n ˆ Ž. nent of a as the variables. Equation 14.2 implies that f and f jj . j 12 n Ž. have the same value at point x . Equation 14.3 implies that jj . j 12 n a agree with Ѩ fr Ѩ x in the n y 1 dimensional space ޒ n rx . jj . j jj . j 12 n 12 n They are always solvable since x and P are independent of each other, wx that is, the matrices xPx are always invertible. jj . jjj . j 12 n 12 n 3. Choose the fuzzy rules as following: Rule 0 IF x is about 0 иии and x is about 0, 1 n ˆ Ž. THEN fxs ax. 0 Rule j j . . . j 12 n IF x is about j ⑀ иии and x is about j ⑀ , 11nn ˆ Ž. THEN fxs ax. jj . j 12 n Ž. For Rule 0, choose the possibility of firing hxas 1 inside D and 0 00 Ž. outside. The possibility of firing for the jj . j th rule is given by the 12 n Ž. product of all the membership functions associated with the jj . j th 12 n rule: n hxs ⌸ Mx,14.4 Ž. Ž . Ž . jj . jji 12 ni i s1 where the membership function for x is given as i x y j ⑀ ° ii 1 y , x y j ⑀ - ⑀ , ~ ii Mxs 14.5 Ž. Ž . ⑀ ji i ¢ 0, elsewhere. Ž. Ž. It is noted that hxhave already been normalized, that is, hx jj . jjj . j 12 n 12 n Ž. G 0 and Ý hxs 1. jj . jjj . j 12 n 12 n ˆ Ž. Therefore, we can write fx as ˆ fxs haxq hax.14.6 Ž. Ž . Ý 00 jj . jjj . j 12 n 12 n jj . j 12 n APPROXIMATION OF NONLINEAR FUNCTIONS USING LINEAR T-S SYSTEMS 281 Remark 42 It should be pointed out that the specific membership function constructed above is only needed when we want to approximate both the nonlinear function and its derivative. There will be much more freedom if we only want to approximate the function itself. 14.1.3 Analysis of Approximation In this subsection, we will prove the fact that any smooth nonlinear function satisfying the assumptions outlined in the previous subsection can be approxi- mated, to any degree of accuracy, using the linear T-S fuzzy systems con- structed above. This fact forms the foundation of the two statements in this chapter. First, we divide region D _ D into many small regions: 0 D s xxg D, j ⑀ F x F j q 1 ⑀ ᭙i . Ž. Ä4 jj . jiii 12 n Ž. In the following discussions, we concentrate on one such region D , jj . j 12 n which is shown in Figure 14.1, by assuming that x g D . From the jj, ., j 12 n construction procedure above, we know that only the fuzzy rules centered at Ž. the vertices of D can be activated at x. That is, hx/ 0 only if jj . jll .l 12 n 12 n x is one of the vertex points of D . ll .ljj . j 12 n 12 n ˆ Ž. Ž. Ž. Consider ex, the approximation error between fx and fx: ex s fxy hxax Ž. Ž. Ž. Ý jj . jjj . j 12 n 12 n jj . j 12 n s fxy hxax Ž. Ž. Ý jj . jjj . jjj . j 12 n 12 n 12 n jj . j 12 n y hxaxy x Ž. Ž . Ý jj . jjj . jjj . j 12 n 12 n 12 n jj . j 12 n Fig. 14.1 Projection of D on xx plane. jj . jii 12 n 12 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR 282 s fxy hxfx Ž. Ž.Ž . Ý jj . jjj . j 12 n 12 n jj . j 12 n y hxaxy x Ž. Ž . Ý jj . jjj . jjj . j 12 n 12 n 12 n jj . j 12 n F hxfxy fx Ž. Ž. Ž . Ý jj . jjj . j 12 n 12 n jj . j 12 n q hxaxy x Ž. Ž . Ý jj . jjj . jjj . j 12 n 12 n 12 n jj . j 12 n F max fxy fx q max axy x . Ž. Ž . Ž . ll .lll .lll .l 12 n 12 n 12 n ll .lll .l 11 n 12 n Note that axy x Ž. ll .lll .l 12 n 12 n ²: x y x , x Ѩ f Ž. ll .lll .l 12 n 12 n s x y x y x Ž. ll .lll .l 2 12 n 12 n x ž/ Ѩ x ll .l x 12 n ll .l 12 n ²: x y x , x Ž. ll .lll .l 12 n 12 n q fx . Ž. ll .l 2 12 n x ll .l 12 n Since x g D , the distance between x and any vertex point of jj . j 12 n '' << Ž. D is less than n ⑀ , that is, x y x F n ⑀ , we can make ex jj . jll .l 12 n 12 n arbitrarily small by just reducing ⑀ . Now consider the approximation of Ѩ fr Ѩ x. Before doing that, three facts for the membership functions are presented. LEMMA 6 Define Ѩ h Ѩ h Ѩ h Ѩ h jj . jjj . jjj . jjj, ., j 12 n 12 n 12 n 12 n s иии Ѩ x Ѩ x Ѩ x Ѩ x x 12 n xx x where it exists; then Ѩ h jj . j 12 n s 0. 14.7 Ž. Ý Ѩ x x jj . j 12 n Proof. Take the derivatives of Ý h . Since Ý h s 1, its jj . jjj . jjj . jjj . j 12 n 12 n 12 n 12 n Ž. derivatives with respect to x will be 0. Q.E.D. i APPROXIMATION OF NONLINEAR FUNCTIONS USING LINEAR T-S SYSTEMS 283 LEMMA 7 Ѩ h jj . j 12 n x y x syI. Ž. Ý jj . j 12 n Ѩ x x jj . j 12 n Proof. For vertex point x g D , define l s 2 j q 1 y l ; then it ll .ljj . jiii 12 n 12 n can be proven that Ѩ h Ѩ h ll .i .lll .l 12 in 12 n x y x q x y x Ž. Ž. ll .i .lll .l 12 in 12 n ii Ѩ x Ѩ x ii x x sy h q h , Ž. ll .lll .i .l 12 n 12 in Ѩ h Ѩ h ll .i .lll .l 12 in 12 n x y x q x y x s 0, Ž. Ž. ll .i .lll .l 12 in 12 n ii Ѩ xx jj xx i / j. Summing up these equations for all the rules ll .l that are effective in 12 n Ž. region D , the fact is proved. Q.E.D. jj . j 12 n LEMMA 8 Define a as the solution of the following linear equations: x axs fx,14.8 Ž. Ž . x Ѩ f aP s P.14.9 Ž. x x Ѩ x x 5555 Then ᭙ ␦ , ᭚ ⑀ such that a y a F ␦ if x y x F ⑀ < 1. xjj . jjj . j 12 n 12 n Ž. Ž. Proof. Since a is the solution of the linear equations 14.8 and 14.9 and x ŽŽ. < . all the parameters of the equations fx, Ѩ fr Ѩ x, and P are continuous x 55 functions of x, a will depend continuously on x. Consequently, a y a x jj . j 12 n can be made arbitrarily small by choosing a small enough value for ⑀ . Ž. Q.E.D. Now consider Ѩ er Ѩ x, the difference between Ѩ fr Ѩ x and Ѩ fr Ѩ x. Ѩ Ý hax Ѩ e Ѩ f Ž. jj . jjj . jjj . j 12 n 12 n 12 n sy Ѩ x Ѩ x Ѩ x x Ѩ h Ѩ f jj . j 12 n sy ax Ý jj . j 12 n Ѩ x Ѩ x x jj . j 12 n x y hxa Ž. Ý jj . jjj . j 12 n 12 n jj . j 12 n T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR 284 Ѩ h Ѩ f jj . j 12 n sy axy x Ž. Ý jj . jjj . j 12 n 12 n Ѩ x Ѩ x x jj . j 12 n x Ѩ h jj . j 12 n y ax Ý jj . jjj . j 12 n 12 n Ѩ x jj . j 12 n x y hxa Ž. Ý jj . jjj . j 12 n 12 n jj . j 12 n Ѩ h Ѩ f jj . j 12 n sy axy x Ž. Ý jj . jjj . j 12 n 12 n Ѩ x Ѩ x x jj . j 12 n x Ѩ h jj . j 12 n y fx y hxa Ž. Ž. ÝÝ jj . jjj . jjj . j 12 n 12 n 12 n Ѩ x jj . jjj . j 12 n 12 n x Ѩ h Ѩ f jj . j 12 n sy axy x Ž. Ý jj . jjj . j 12 n 12 n Ѩ x Ѩ x x jj . j 12 n x Ѩ h Ѩ f jj . j 12 n 2 y fxq x y x q O ⑀ Ž. Ž . Ž . Ý jj . j 12 n ž/ Ѩ x Ѩ x x x jj . j 12 n y hxa Ž. Ý jj . jjj . j 12 n 12 n jj . j 12 n Ѩ h Ѩ f jj . j 12 n sy axy x Ž. Ý jj . jjj . j 12 n 12 n Ѩ x Ѩ x x jj . j 12 n x Ѩ h Ѩ f jj . j 12 n y x y x Ž. Ý jj . j 12 n Ѩ x Ѩ x x x jj . j 12 n y hxa q O ⑀ from Fact 6 Ž. Ž. Ž . Ý jj . jjj . j 12 n 12 n jj . j 12 n Ѩ h jj . j 12 n sy axy x Ž. Ý jj . jjj . j 12 n 12 n Ѩ x x jj . j 12 n y hxa q O ⑀ from Fact 7 Ž. Ž. Ž . Ý jj . jjj . j 12 n 12 n jj . j 12 n Ѩ h jj . j 12 n s axy x q a Ž. Ý jj . jjj . jx 12 n 12 n Ѩ x x jj . j 12 n q ha y hxaq O ⑀ Ž. Ž. ÝÝ jj . jjj . jjj . jx 12 n 12 n 12 n jj . jjj . j 12 n 12 n APPROXIMATION OF NONLINEAR FUNCTIONS USING LINEAR T-S SYSTEMS 285 Ѩ h jj . j 12 n F a y axy x Ž.Ž. Ý jj . jx jj . j 12 n 12 n Ѩ x x jj . j 12 n q hay a q O ⑀ from Fact 7 . Ž.Ž.Ž. Ý jj . jjj . jx 12 n 12 n jj . j 12 n From Fact 8, it is known that Ѩ er Ѩ x can be made arbitrarily small by reducing ⑀ . Next consider region D . In region D , it is known from Taylor series that 00 Ž. ex and Ѩ er Ѩ x can also be made arbitrarily small by reducing ⑀ . There- 0 fore, we have the following theorem by summarizing the results above: Ž. n 1 THEOREM 56 For any smooth nonlinear function f x : ޒ ™ ޒ defined on Ž. 2 a compact region, satisfying f 0 s 0 and f g C , both the function and its n deri®ati®es can be approximated, to any degree of accuracy, by linear T-S fuzzy systems. Ž. Remark 43 It may be argued that the condition f 0 s 0 is too restrictive. Ž. However, in the case of f 0 / 0, we argue that f can still be approximated by a linear T-S model through a simple coordination transformation, that is, the function f is now represented by a linear T-S model in the new coordinate system. A coordination transformation might puzzle the mind of a purist of function approximation. However, for control system analysis and design, which is the sole focus of this book, this is not a problem at all. It is well known that for the stability analysis and design of nonlinear control systems, it can be assumed without loss of generality that the origin is an equilibrium point of the system. Ž. Ž. 3 Fig. 14.2 Nonlinear function fx, x s 8 x q 10 x sin 4 x q x y 4 xx. 12 1 2 1 1 12 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR 286 Remark 44 It may be argued that the membership function is not continu- ous on the boundary between D and D . To overcome the discontinu- 0 jj . j 12 n ity, some bumper functions can be included to smooth the membership wx function without affecting the approximation accuracy 16 . 14.1.4 Example An example is given in this subsection for illustration. Consider the approxi- Ž. mation of a two-dimensional nonlinear function fx, x s 8 x q 12 1 Ž. 3 10 x sin 4 x q x y 4 xx as shown in Figure 14.2. The constructed T-S 21112 fuzzy model is shown in Figure 14.3. A 25 = 40 grid is used. The maximum approximation error is 1.38. We also plot the approximation error in Figure 14.4. It should be pointed out that the approximation error could be further reduced by using more fuzzy rules. Fig. 14.3 Constructed T-S fuzzy model. Fig. 14.4 Approximation error of nonlinear function. [...]... the system states; i s 1, 2, , r and r is the number of IF-THEN rules; Mi j are fuzzy sets; and ˙Ž t s A i x Ž t x are the consequences of the ith IF-THEN rule By using the center-of-gravity method for defuzzification, we can represent the T-S model as x ˙ s fˆŽ x s r Ý h i Ž x A i x, Ž 14.10 is1 where h i Ž x is the possibility for the ith rule to fire Consider the nonlinear system x ˙s f... any degree of accuracy, by a T-S model Ž14.10 288 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR Similarly, a smooth nonlinear control system ˙ s f Ž x q g Ž x u can also x be approximated using a T-S fuzzy model ˙ s Ý r h i Ž x Ž A i x q Bi u By x is1 treating u as an extraneous system state, we can also approximate the smooth nonlinear control system ˙ s f Ž x, u by a T-S fuzzy model ˙ s x x Ý r ˆ i... Based on TSK Fuzzy Model Using Pole Placement,’’ Proc FUZZ-IEEE’98, 1998, pp 246᎐251 10 S K Hong and R Langari, ‘‘Synthesis of an LMI-Based Fuzzy Control System with Guaranteed Optimal H ϱ performance,’’ Proc FUZZ-IEEE’98, 1998, pp 422᎐427 11 H O Wang, K Tanaka, and M F Griffin, ‘‘Parallel Distributed Compensation of Nonlinear Systems by Takagi-Sugeno Fuzzy Model,’’ in Proc FUZZIEEErIFES’95, 1995, pp... High-RiserHigh-Speed Elevators,’’ Proc 1998 American Control Conference, 1998, pp 3445᎐3449 14 T Tanaka and M Sano, ‘‘A Robust Stabilization Problem of Fuzzy Control Systems and Its Applications to Backing Up Control of a Truck-Trailer,’’ IEEE Trans Fuzzy Syst., Vol 2, No 3, pp 119᎐134 Ž1994 15 J Zhao, V Wertz, and R Gorez, ‘‘ Fuzzy Gain Scheduling Controllers Based on Fuzzy Models,’’ Proc Fuzzy-IEEE’96,... Nonlinear Dynamic Systems Using Linear Takagi-Sugeno Fuzzy Models The following dynamic linear Takagi-Sugeno fuzzy model is used to describe dynamic systems: Rule i IF x 1Ž t is Mi1 , иии and x nŽ t is Mi n , THEN ˙Ž t s A i x Ž t , x where x T Ž t s w x 1Ž t , x 2 Ž t , , x nŽ t x are the system states; i s 1, 2, , r and r is the number of IF-THEN rules; Mi j are fuzzy sets; and ˙Ž t s... Cybernet., Vol 25, No 4, pp 629᎐635 Ž1998 6 C Fantuzzi and R Rovatti, ‘‘On the Approximation Capabilities of the Homogeneous Takagi-Sugeno Model,’’ Proc FUZZ-IEEE’96, 1996, pp 1067᎐1072 7 H Ying, ‘‘Sufficient Conditions on Uniform Approximation of Multivariate Functions by General Takagi-Sugeno Fuzzy Systems with Linear Rule Consequence,’’ IEEE Trans Syst Man Cybern Vol 28, No 4, pp 515᎐521 Ž1998 8 X J et and... approximated, to any degree of accuracy, by a PDC controller Ž14.12 BIBLIOGRAPHY 289 BIBLIOGRAPHY 1 H O Wang, J Li, D Niemann, and K Tanaka,‘‘T-S Fuzzy Model with Linear Rule Consequence and PDC Controller: A Universal Framework for Nonlinear Control Systems,’’ Proc FUZZ-IEEE’2000, San Antonio, TX, 2000, pp 549᎐554 2 T Takagi and M Sugeno, ‘‘Fuzzy Identification of Systems and Its Applications to Modeling... , x nŽ t x are the system states and uT Ž t s w u1Ž t , u 2 Ž t , , u mŽ t x are the system inputs; i s 1, 2, , r and r is the number of IF-THEN rules; Mi j , Ni j are fuzzy sets and ˙Ž t s A i x Ž t q Bi uŽ t x is the consequence of the ith IF-THEN rule; and n m js1 ks1 ˆi Ž x, u s ⌸ Mi j Ž x i Ž t ⌸ Ni k Ž u k Ž t h is the possibility for the ith rule to fire 14.2.2 Approximation of... equilibrium point 2 2 f g Cn Therefore, f, Ѩ frѨ x, and Ѩ 2 frѨ x 2 are continuous and bounded over D Suppose f Ž x can be written as w f 1Ž x f nŽ x xT What we mean by approxiˆ ˆ ˆ mation is finding a T-S fuzzy model f Ž x s w f 1Ž x f nŽ x xT such that ˆ ˆ 5 f Ž x y f Ž x 5 is small Since 5 f Ž x y f Ž x 5 is small if and only if each of its components Žwhich are nonlinear functions are small, . 2001 John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-4 7 1-3 232 4-1 Hardback ; 0-4 7 1-2 245 9-6 Electronic CHAPTER 14 T-S FUZZY MODEL AS UNIVERSAL APPROXIMATOR In this. is that a T-S model with linear rule conse- quence has limited capability in representing a nonlinear system in compari- wx son with an affine T-S model