Stability Analysis of the Simplest Takagi-Sugeno Fuzzy Control - PowerPoint PPT Presentation

Stability Analysis of the Simplest Takagi-Sugeno Fuzzy Control System Using Popov Criterion Xiaojun Ban 1 , X. Z. Gao 2 , Xianlin Huang 3 , and Hang Yin 4 1 Department of Control Theory and Engineering, Harbin Institute of Technology, Harbin, China, westzebra@hit.edu.cn 2 Institute of Intelligent Power Electronics, Helsinki University of Technology, Espoo, Finland, gao@cc.hut.fi. 3 Department of Control Theory and Engineering, Harbin Institute of Technology, Harbin, China, xlinhuang@hit.edu.cn 4 Department of Control Theory and Engineering, Harbin Institute of Technology, Harbin, China, yinhang@hit.edu.cn

• Abstract — In our paper, the properties of the simplest Takagi-Sugeno (T-S) fuzzy controller are first investigated. Next, based on the well-known Popov criterion with graphical interpretation, a sufficient condition in the frequency domain is proposed to guarantee the globally asymptotical stability of the simplest T-S fuzzy control system. Since this sufficient condition is presented in the frequency domain, it is of great significance in designing the simplest T-S fuzzy controller.

1. Introduction 1. The T-S fuzzy controller is a nonlinear controller. Hence, it is inherently suitable for nonlinear objects. 2. On the other hand, the frequency-response method has been well developed and widely used in industrial applications, which is straightforward and easy to follow by practicing engineers.

3. Therefore, fusion of the T-S fuzzy model and the frequency-response method is of great significance in the perspective of control engineering. 4. It is apparently necessary to analyze the stability of T-S fuzzy control systems in the frequency domain, when the frequency response methods are utilized in designing T-S fuzzy controllers.

2. Configuration of the simplest T-S fuzzy control system Fig. 1. Structure of the simplest T-S fuzzy control system.

This simplest T-S fuzzy controller can be described by the following two rules: u If e is A , then e k = 1 1 u k If e is B , then e = 2 2 u where e is the input of this T-S fuzzy controller, , i i , =1,2 are the outputs of the local consequent controllers, which are both proportional controllers here. It should = be pointed out , k i , i 1 , 2

gains of these local controllers, are assumed to be positive in this paper. Both A and B are fuzzy sets, and we use the triangular membership functions to quantify them, as shown in Fig. 2. μ B A B 1 a e -a 0 Fig. 2. Membership functions of A and B .

3 . Popov criterion Fig. 3. Structure of nonlinear system for Popov criterion.

Popov Criterion: Consider the above system, and suppose (i) matrix A is Hurwitz, (ii) pair ( A , b ) is controllable, (iii) pair ( c , A ) is observable, (iv) d >0, and ( ) the nonlinear element belongs to sector , where k >0 0 , k is a finite number. Under these conditions, this system is globally asymptotically stable, if there exists a number r >0, such that 1 + ω ω + > inf Re[( 1 j r ) h ( j )] 0 ω ∈ k R

The graphical interpretation of the Popov criterion can be ω ω given as follows: suppose we plot Im h ( j ) ω ∞ ω vs. , when varies from 0 to , which is Re h ( j ) known as the Popov plot of h ( s ), the nonlinear system is globally asymptotically stable, if there exists a nonnegative number r , such that the Popov plot of h ( s ) lies to the right of a straight line passing through point (-1/ k ,0) with a slope of 1/ r .

4. Analysis of the Simplest Takagi-Sugeno Fuzzy Control System Several theorems and lemmas are proven to demonstrate if certain hypotheses are satisfied, the stability of the nonlinear system illustrated in Fig. 1 can be analyzed by using the Popov criterion.

Φ Theorem 1: Let represent the functional mapping ( e ) achieved by the simplest T-S fuzzy control system, the following equation holds: Φ − = − Φ ( e ) ( e )

Lemma 1: The following two statements are equivalent: Φ − Φ > ∀ ≠ Φ = (i). , ( y )( ky ( y )) 0 , y 0 , and , ( 0 ) 0 < Φ < ∀ ≠ Φ = 2 (ii). , 0 y ( y ) ky , y 0 , and , ( 0 ) 0 where k is a positive number.

Φ Theorem 2: Let denote the functional mapping of the ( e ) Φ T-S fuzzy controller in Figs. 4 or 1, belongs to sector ( e ) ε + ε ( 0 , k ) , where is a sufficiently small positive 2 number, i.e., the following inequality holds: Φ = (i). , ( 0 ) 0 Φ + ε − Φ > ∀ ≠ (ii). . ( e )[( k ) e ( e )] 0 , e 0 2

Theorem 3: The fuzzy control system shown in Fig. 1 is globally asymptotically stable, if the following set of conditions hold: (i). G ( s ) can be represented by the state equations from (3) to (6), (ii). matrix A is Hurwitz, (iii). pair ( A, b ) is controllable, and pair ( c, A ) is observable, (iv). d > 0, (v). there exists a number r > 0, such that, 1 + ω ω + > inf Re[( 1 j r ) h ( j )] 0 + ε ω ∈ k R 2 ε where is a sufficiently small positive number.

Corollary 1: The fuzzy control system shown in Fig. 1 is globally asymptotically stable, if the following set of conditions hold: (i) - (iv) are the same with those in Theorem 3. (v). there exists a number r>0, such that 1 + ω ω + > inf Re[( 1 j r ) h ( j )] 0 ω ∈ k R 2 In the next section, a numerical example is presented to demonstrate how to employ Theorem 3 or Corollary 1 in analyzing the stability of our T-S fuzzy control system.

5. Simulations Example. In this example, a stable plant to be controlled is: 1 = G ( s ) + 2 s ( s 1 ) = = k 0 . 2 , k 0 . 5 Two suitable proportional gains, , 1 2 are obtained based on the Bode plot of G ( s ) . A simplest T-S fuzzy controller with the following two rules is constructed: u If e is A , then = 0.2 e , 1 If e is B , then = 0.5 e . u 2

Both the Popov plot of G ( s ) and a straight line passing 1 through point ( ) with the slope of 0.5 are shown in Fig. , 0 k 2 6. It is argued in Theorem 3 that the nonlinear system is globally asymptotically stable, if there exists a nonnegative number r , such that the Popov plot of h (s) lies to the right ⎛ − ⎞ 1 ⎜ ⎟ of a straight line passing through point with a slope of , 0 ⎜ ⎟ ⎝ ⎠ k 2 1/ r . Hence, the T-S fuzzy control system in this example is globally asymptotically stable, since we can easily find such ⎛ − ⎞ 1 ⎜ ⎟ a straight line passing through point , provided that k , 0 ⎜ ⎟ ⎝ ⎠ k 2 2 is less than 2.

0.2 0 -0.2 ω Im G(j ω ) -0.4 -0.6 -0.8 -1 -2 -1.5 -1 -0.5 0 Re G(j ω ) Fig. 5. Popov plot of G ( s ) and a straight line (solid line represents Popov plot of G ( s ), and thin line represents straight line passing 1 through point ( ) with a slope of 0.5). , 0 k 2

6. Conclusions • 1. A sufficient condition is derived to guarantee the globally asymptotical stability of the equilibrium point of the simplest T-S fuzzy control system by using the well-known Popov criterion. • 2. The theorem derived based on the Popov theorem has a good graphical interpretation. Thus, it can be employed in designing a T-S fuzzy controller in the frequency domain.

• 3. Additionally, it can be observed that a , which is the characteristics parameter of the input membership functions, has no effect on the stability of the fuzzy control system. However, a does affect its dynamical control performance. • 4. We also emphasize although only two fuzzy rules are examined here, the proposed stability analysis method is still applicable to the single- input T-S fuzzy controllers with multiple rules.

Future research • In the future research, we are going to explore new stability theorems with graphical interpretation in the frequency domain for a wider class of plants as well as general T-S fuzzy controllers.

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