x i2 j i2 ε + ε i2 ε ji1 ε j i1

Transcription

1 T-S Fuzzy Model with Linear Rule Consequence and PDC Controller: A Universal Framewor for Nonlinear Control Systems Hua O. Wang Jing Li David Niemann Laboratory for Intelligent and Nonlinear Control (LINC) Department of Electrical and Computer Engineering Due University, Durham, NC 2778, U.S.A Kazuo Tanaa Department of Mechanical and Control Engineering University of Electro-Communications Chofugaoa, Chofu, Toyo 182, Japan Abstract In this paper, we present two results concerning the fuzzy modeling and control of nonlinear control systems. First, we prove that any smooth nonlinear control systems can be approximated by Taagi-Sugeno fuzzy models with linear rule consequence. Then, we prove that any smooth nonlinear state-feedbac controller can be approximated by the parallel distributed compensation (PDC) controller. 1 Introduction Among various fuzzy modeling themes, the so-called Taagi-Sugeno (T-S) model [1] has been one of the most popular modeling framewor. A general T-S model employs an ane model with a constant term in the consequent part for each rule. This is often referred as an ane T-S model. In this paper, we focus on a special type of T-S fuzzy model in which the consequent part for each rule is represented by a linear model (without a constant term). We refer this type of T-S fuzzy model as a T-S model with linear rule consequence, or simply a linear T-S model. It is worthy to point out that the overall linear T-S model is actually a nonlinear model. The appeal of a linear T-S model is that it renders itself naturally to Lyapunov based system analysis and design techniques [11] [14]. A commonly held view is that a T-S model with linear rule consequence has limited capability in representing a nonlinear control system [8] in comparison with an ane T-S model. In our earlier wor[1, 11], a controller structure called parallel distributed compensation (PDC) is introduced. This structure utilizes a fuzzy state feedbac controller which mirrors the structure of the associated linear T-S model. The idea is that for each local linear model, a linear feedbac control is designed. The resulting overall controller, which is nonlinear in general, is a fuzzy blending of each individual linear controller. Applications of T-S model together with PDC controller have been successful in many practical applications [9], [12], [13].

2 In this paper, we attempt to address the fundamental capabilities of linear T-S models and PDC controllers. Two results are given in this paper. The rst result is that a linear Taagi-Sugeno fuzzy model can be an universal approximator of any smooth nonlinear control system. The second result is that PDC controller can be an universal approximator of any nonlinear state-feedbac controller. Linear T-S models and PDC controllers together provide a universal framewor for the modeling and control of nonlinear control systems. In this paper, R n is used to denote the n-dimensional vector spaces of real vectors. Cn m is used to represent the set of n-dimension functions whose m-th derivative is continuous on the dened region. x i stands for the i-th component of vector x and stands for the standard vector norm or P matrix norm. O(x) is the set of numbers y such that j y x j< M, where M is a constant. is used to represent the summation with all the possible combinations of j 1, j 2 : : : and j n. The paper is organized as follows: Section 2 gives a construction procedure of T-S system and the proof of the fact that the constructed T-S system can approximate any smooth nonlinear function. Section 3 presents the two statements fore-mentioned for T-S model and PDC controller. Concluding remars are collected in Section 4. 2 Approximation of Nonlinear Functions Using Linear T-S systems 2.1 Linear T-S Fuzzy Systems The main feature of linear Taagi-Sugeno fuzzy systems is to express the local properties of each fuzzy implication (rule) by a linear function. The overall fuzzy system is achieved by fuzzy `blending' of these linear functions. Specically, the linear Taagi-Sugeno fuzzy system is of the following form: Rule i: IF x 1 (t) is M i1 and x n (t) is M in THEN y = a i x(t), where x T (t) = [x 1 (t); x 2 (t); ; x n (t)] are the function variables. i = 1; 2; ; r and r is the number of IF-THEN rules. M ij are fuzzy sets. The linear function y = a i x(t) is the consequence of the i-th IF-THEN rule, where a i 2 R 1n. The possibility that the ith rule will re is given by the product of all the membership functions associated with the ith rule. h i (x(t)) = n j=1m ij (x j (t)): We will assume that h i 's have already been normalized, i.e. h i (x) > and r h i (x) = 1. Then by using center of gravity method for defuzzication, we can represent the T-S system as: y = ^f(x) = r h i (x)a i x (1) The summation process associated with the center of gravity defuzzication in system (1) can also be viewed as an interpolation between the functions a i x based on the value of the parameter x. 2

3 2.2 Construction Procedure of T-S Fuzzy Systems Suppose that the nonlinear function f(x) : R n! R is dened over compact region D R n with the following assumptions: 1. f() =. 2. f 2 C 1. Therefore, f, f are continuous and therefore bounded over D. 2 Next, we will construct T-S system ^f(x) = r h i (x)a i x to approximate f(x). The objective is to mae the approximation error e(x) = f(x)? ^f(x) and its small for all x 2 D. Construction Procedures: 1. In region D = fx j jx i j < g where is a chosen positive number, choose a j x=. 2. Dene the projection operator P j x mapping R n to n? 1 dimensional subspace R n =x as P j x y = y? < y; x > x x 2 In region DnD, choose x j1 j 2 ::::j n as [ j 1 j 2 : : : j n ] T, where is a positive number and j i are integers. Build the linear model as the solution of the following linear equations: x j1 j 2 :::j n = f(x j1 j 2 ::::j n ) (2) P j xj 1 j 2 :::jn j x j 1 j 2 :::jn P j x j 1 j 2 :::jn (3) For xed x j1 j 2 :::j n, (2)-(3) are n linear equations with the component of as the variables. (2) implies that f and ^f have the same value at point x j1 j 2 :::j n. (3) implies that agree in the n? 1 dimensional space Rn =x j1 j 2 :::j n. They are always solvable since x and P are independent of each other, i.e., the matrices xj1 j 2 :::j n P jx j1 j 2 :::j n are always invertible. 3. Choose the fuzzy rules as following: Rule : IF x 1 (t) is about and x n (t) is about THEN ^f(x) = a x, Rule : If x 1 (t) is about j 1 and x n (t) is about j n THEN ^f(x) = x For Rule, choose the possibility of ring h (x) as 1 inside D and outside. The possibility of ring for the th rule is given by the product of all the membership functions associated with the -th rule. (x(t)) = n M j i (x i (t)) (4) 3

4 x i2 j i2 ε + ε j i2 ε ji1 ε j i1 ε + ε x i1 Figure 1: Projection of D j1 j 2 :::j n on x i1 x i2 plane where the membership function for x i is given as M ji (x i ) = f 1? jx i?j i j jx i? j i j < else where (5) PIt is noted that (x) have already been normalized, i.e. (x) and (x) = 1. Therefore, we can write ^f(x) as: ^f(x) = h a x + x (6) Remar: It should be pointed out that the specic 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 need to approximate the function itself. 2.3 Analysis of Approximation In this subsection, we will prove the fact that any smooth nonlinear function can be approximated, to any degree of accuracy, using the linear T-S fuzzy systems constructed above. This fact will form the foundation of the two statements in this paper. First, we will divide region DnD into many small regions D j1 j 2 :::j n = fxj x 2 D; j i x i (j i + 1) 8ig In the following discussions, we will only concentrate on one of such regions (D j1 j 2 :::j n ), which is shown in Fig. 1, by assuming that x 2 D j1 j 2 :::j n. From the construction procedure above, we now that only the fuzzy rules centered at the vertices of D j1 j 2 :::j n can be activated at x. That is h l1 l 2 :::l n (x) 6= only if x l1 l 2 :::l n is one of the vertex points of D j1 j 2 :::j n. Consider e(x), the approximation error between f(x) and ^f(x) e(x) = f(x)? x = f(x)? x j1 j 2 :::j n 4

5 Note that? ) = f(x)? (x)f(x j1 j 2 :::j n )? ) (x)f(x)? f(x j1 j 2 :::j n ) j 1 j 2 :::j n + (x) ) max f(x)? f(x l1 l 2 :::l n ) + max a l1 l 2 :::l n (x? x l1 l 2 :::l n ) l 1 l 2 :::l n l 1 l 2 :::l n a l1 l 2 :::l n (x? x l1 l 2 :::l n ) j x l 1 l 2 :::ln (x? x l1 l 2 :::l n )? h(x? x l 1 l 2 :::l n ); x l1 l 2 :::l n i x l1 l 2 :::l n 2 + h(x? x l 1 l 2 :::l n ); x l1 l 2 :::l n i x l1 l 2 :::l n 2 f(x l1 l 2 :::l n ) x l1 l 2 :::l n Since x 2 D j1 j 2 :::j n, the distance between x and any vertex point of D j1 j 2 :::j n is less than p n, i.e. j x? xl1 l 2 :::l n j p n, we can mae e(x) arbitrarily small by just reducing. Now consider the approximation Before doing that, three facts for the membership functions are presented. h Fact 1 1 j 2 j x = 1 j 2 1 j j i 1 j 2 :::jn x 2 j x : : 1 j 2 :::jn n j x where it exists, j1j2:::jn j x = (7) Proof: Tae the derivatives of P. Since P = 1, its derivatives with respect to x i will be. Fact 2 j x =?I (8) Proof: For vertex point x l1 l 2 :::l n 2 D j1 j 2 :::j n, dene l i = 2j i + 1? l i, then it can be proven that (x? x l1 l 2 ::: l i :::l n ) l1 l 2 ::: l i :::l n i j x +(x? x l1 l 2 :::l n ) l1 l 2 :::l n i j x =?(h l1 l 2 :::l n + h l1 l 2 ::: l i :::l n ) (x? x l1 l 2 ::: l i :::l n ) l1 l 2 ::: l i :::l n j j x +(x? x l1 l 2 :::l n ) l1 l 2 :::l n j j x = ; i 6= j Summing up these equations for all the rules l 1 l 2 :::l n that are eective in region D j1 j 2 :::j n, the fact is proved. 5

6 Fact 3 Dene a x as the solution of the following linear equations a x x = f(x) (9) a x P j x j x P (1) Then 8, 9 such that a x? if x? x j1 j 2 :::j n 1, Proof: Since a x is the solution of the linear equations (9) (1) and all the parameters of the equations and P j x) are continuous functions of x, a x will depend continuously on x. Consequently, a? can be made arbitrarily small by choosing a small enough value for. the dierence j j x? 1 j 2 :::j n x) j x? j1j2:::jn j x? j x? j j1j2:::jn? x j1 j 2 :::j n j x? h j1 j 2 :::j n j x? j x? f(x j1 j 2 :::j n j1j2:::jn j x? h j1 j 2 :::j n j j 2 :::j n? f(x) j x (x j1 j 2 :::j n? x) + O( 2 )? j j x (x j1 j 2 :::j n? j x? h j1 j 2 :::j n +O() (From Fact 1) =? j1j2:::jn j x? h j1 j 2 :::j n +O() (From Fact 2) j x j x j x 6

7 f(x 1,x 2 )=8x 1 +1x 2 sin(4x 1 )+x 1 3 4x1 x Figure 2: Nonlinear function f(x 1 ; x 2 ) = 8x 1 + 1x 2 sin(4x 1 ) + x 3 1? 4x 1x 2 = j x +a x j 1 j 2 :::j n +? (x)a x + O() j 1 j 2 :::j n (? a x ) j x j 1 j 2 :::j n + (? a x ) + O() (From Fact 2) From Fact (3), it is can be made arbitrarily small by reducing. Next consider region D. In region D, it is nown from Taylor series that e(x) can also be made arbitrarily small by reducing. Therefore, we have the following theorem by summarizing the results above: Theorem 4 For any smooth nonlinear function f(x) : R n! R 1 dened on a compact region, both the function and its derivatives can be approximated, to any degree of accuracy, by linear T-S fuzzy systems. Remar: It may be argued that the membership function is not continuous on the boundary between D and D j1 j 2 :::j n. To overcome the discontinuity, some bumper functions can be included to smooth the membership function without aecting the approximation accuracy [15]. 2.4 Example An example is given in this subsection for illustration. Consider the approximation of two dimensional nonlinear function f(x 1 ; x 2 ) = 8x 1 + 1x 2 sin(4x 1 ) + x 3? 4x 1 1x 2 shown in Figure 2. A 25 4 grid is used. The maximum approximation error is We also plot the approximation error in Figure 4. It should be pointed out the approximation error could be further reduced by using more fuzzy rule. 7

8 Constructed T S fuzzy model Figure 3: Constructed T-S fuzzy model Approximation error Figure 4: Approximation error of nonlinear function 8

9 3 Applications to Modeling and Control of Nonlinear Control Systems 3.1 Approximation of Nonlinear Control Systems using Linear Taagi- Sugeno Fuzzy Models The linear Taagi-Sugeno fuzzy model is used to describe dynamic systems. It is of the following form: Rule i: IF x 1 (t) is M i1, and x n (t) is M in THEN _x(t) = A i x(t), where x T (t) = [x 1 (t); x 2 (t); ; x n (t)] are the system states. i = 1; 2; ; r and r is the number of IF-THEN rules. M ij are fuzzy sets and _x(t) = A i x(t) are the consequences of the i-th IF-THEN rule. By using center of gravity method for defuzzication, we can represent the T-S model as: _x = ^f(x) = r where h i (x) is the possibility for the i-th rule to re. Consider the nonlinear control system: h i (x)a i x (11) _x = f(x) (12) where f(x) is a vector eld dened over compact region D R n with the following assumptions: 1. f() =, i.e. the origin is an equilibrium point. 2. f 2 C 2 n. Therefore, f 2 are continuous and bounded over D. Suppose f(x) can be written as f 1 (x) : : : f n (x) T. What we mean by approximation is nding a T-S fuzzy model ^f(x) = ^f1 (x) : : : ^fn (x) T such that f(x)? ^f(x) is small. Since f(x)? ^f(x) is small if and only if each of its components (which are nonlinear functions) is small, then by applying Theorem 4 proven in the previous section, we obtain the following corollary: Corollary 5 For any smooth nonlinear control system (12) satisfying the above assumptions, it can be approximated, to any degree of accuracy, by a T-S model (11). Similarly, smooth nonlinear control system _x = f(x) + g(x)u can also be approximated using a T-S fuzzy model _x = r h i (x)(a i x + B i u). By treating u as extraneous system state, we can also approximate the smooth nonlinear control system _x = f(x; u) by T-S fuzzy model _x = r ^h i (x; u)(a i x+b i u). In this case, the fuzzy rule is of the following form: Rule i: IF x 1 (t) is M i1,, x n (t) is M in, u 1 (t) is N i1, and u m (t) is N im THEN _x(t) = A i x(t) + B i u(t), 9

10 where x T (t) = [x 1 (t); x 2 (t); ; x n (t)] are the system states and u T (t) = [u 1 (t); u 2 (t); ; u m (t)] are the system inputs. i = 1; 2; ; r and r is the number of IF-THEN rules. M ij, N ij are fuzzy sets and _x(t) = A i x(t) + B i u(t) is the consequence of the i-th IF-THEN rule. ^h i (x; u) = n j=1 M ij(x i (t)) m =1 N i(u (t)) is the possibility for the i-th rule to re. Remar: There are many results on the approximation of a nonlinear control system using a T-S model with ane models as rule consequences. Instead of using linear models _x = A i x(t) + B i u(t) as rule consequences, ane models _x = A i x(t) + B i u(t) + C i are used. To do that, the state space is rst divided into many small regions and rst-order Taylor expansion around a point in that region is adopted as the rule consequence. By including the constant term in the output of the fuzzy rules, more exibility can be obtained in the choice of regions and membership functions, but the stability analysis and synthesis become more involved. 3.2 Approximation of Nonlinear State-Feedbac Controller using PDC Controller In this paper, we will consider a special form of fuzzy controller introduced in [11] where it was termed parallel distributed compensation (PDC). The PDC controller structure consists of the fuzzy rules: Rule j: IF x 1 (t) is M j1 and x n (t) is M jn THEN u(t) = K j x(t) where j = 1; 2; ; s. The output of the PDC controller is u = s j=1 h j (x)k j x (13) Following similar argument as in the above subsection, we obtain the following theorem: Theorem 6 Any smooth nonlinear state feedbac controller u = K(x) where x is dened over a compact region can be approximated, to any degree of accuracy, by PDC controller (13). 4 Conclusions In this paper, we discussed the approximation of nonlinear control system using T-S models with linear models as rule consequences. We presented a construction procedures of T-S models and proved that any smooth nonlinear system plus its velocity can be approximated using this constructed T-S model with any desired accuracy. We also showed that PDC controller can be an universal approximator of any smooth nonlinear state-feedbac controller. References [1] T. Taagi and M. Sugeno, \Fuzzy identication of systems and its applications to modeling and control," IEEE Trans. Syst., Man, and Cybern., vol. 15, pp ,

11 [2] J. J. Bucley, \Universal fuzzy controllers," Automatica, vol. 28, pp , [3] S. G. Cao, N. W. Rees and G. Feng, \Fuzzy control of nonlinear continuous-time systems," in Proc. 35th IEEE Conf. Decision and Control, Kobe, Japan, 1996, pp [4] J. L. Castro, \Fuzzy logic controllers are universal approximators," IEEE Trans. Syst., Man, Cybern., vol. 25, no. 4, pp , [5] C. Fantuzzi and R. Rovatti, \On the approximation capabilities of the homogeneous Taagi-Sugeno model," Proc. FUZZ-IEEE'96, 1996, pp [6] Hao Ying, \Sucient conditions on uniform approximation of multivariate functions by general Taagi-Sugeno fuzzy systems with linear rule consequence," IEEE Trans. Syst., Man, Cybern., vol. 28, no. 4, pp , [7]. J. Zeng and M. G. Singh, \Approximation theory of fuzzy systems - SISO case," IEEE Trans. Fuzzy Systems, vol. 2, pp , [8] G. Kang, W. Lee and M. Sugeno, \Design of TSK fuzzy controller based on TSK fuzzy model using pole placement," Proc. FUZZ-IEEE'98, 1998, pp [9] S. K. Hong and R. Langari, \Synthesis of an LMI-based fuzzy control system with guaranteed optimal H 1 performance," Proc. FUZZ-IEEE'98, 1998, pp [1] H. O. Wang, K. Tanaa, and M.F. Grin, \Parallel distributed compensation of nonlinear systems by Taagi-Sugeno fuzzy model," in Proc. FUZZ-IEEE/IFES'95, 1995, pp [11] H.O. Wang, K. Tanaa and M. Grin, \An approach to fuzzy control of nonlinear systems: stability and design issues," IEEE Trans. on Fuzzy Systems, vol. 4, no. 1, pp , [12] J. Li, D. Niemann and H. O. Wang, \Robust tracing for high-rise/high-speed elevators," Proc American Control Conference, 1998, pp [13] T. Tanaa and M. Sano, \A robust stabilization problem of fuzzy control systems and its applications to bacing up control of a truc-trailer", IEEE Trans. on Fuzzy Systems,, vol. 2, no. 3, pp , [14] J. Zhao, V. Wertz and R. Gorez, \ Fuzzy gain scheduling controllers based on fuzzy models," Proc. Fuzzy-IEEE'96, [15] Michael Spiva, Comprehensive Introduction to Dierential Geometry (Volume 1), Addison Wesley, [16] R. R. Yager and P. F. Dimitar, Essential on Fuzzy Modeling and Control, John Wiley & Son,