1、 中文 2094 字 外文资料翻译 Static Output Feedback Control for Discrete-time Fuzzy Bilinear System Abstract The paper addressed the problem of designing fuzzy static output feedback controller for T-S discrete-time fuzzy bilinear system (DFBS). Based on parallel distribute compensation method, some sufficient
2、 conditions are derived to guarantee the stability of the overall fuzzy system. The stabilization conditions are further formulated into linear matrix inequality (LMI) so that the desired controller can be easily obtained by using the Matlab LMI toolbox. In comparison with the existing results, the
3、drawbacks such as coordinate transformation, same output matrices have been eliminated. Finally, a simulation example shows that the approach is effective. Keywords discrete-time fuzzy bilinear system (DFBS)； static output feedback control； fuzzy control； linear matrix inequality (LMI) 1 Introductio
4、n It is well known that T-S fuzzy model is an effective tool for control of nonlinear systems where the nonlinear model is approximated by a set of linear local models connected by IF-THEN rules. Based on T-S model, a great number of results have been obtained on concerning analysis and controller d
5、esign1-11. Most of the above results are designed based on either state feedback control or observer-based control1-7.Very few results deal with fuzzy output feedback8-11. The scheme of static output feedback control is very important and must be used when the system states are not completely availa
6、ble for feedback. The static output feedback control for fuzzy systems with time-delay was addressed 910 and a robust H controller via static output feedback was designed11. But the derived conditions are not solvable by the convex programming technique since they are bilinear matrix inequality prob
7、lems. Moreover, it is noted that all of the aforementioned fuzzy systems were based on the T-S fuzzy model with linear rule consequence. Bilinear systems exist between nonlinear and linear systems, which provide much better approximation of the original nonlinear systems than the linear systems 12.T
8、he research of bilinear systems has been paid a lot of attention and a series of results have been obtained1213.Considering the advantages of bilinear systems and fuzzy control, the fuzzy bilinear system (FBS) based on the T-S fuzzy model with bilinear rule consequence was attracted the interest of
9、researchers14-16. The paper 14 studied the robust stabilization for the FBS, then the result was extended to the FBS with time-delay15. The problem of robust stabilization for discrete-time FBS (DFBS) was considered16. But all the above results are obtained via state feedback controller. In this pap
10、er, a new approach for designing a fuzzy static output feedback controller for the DFBS is proposed. Some sufficient conditions for synthesis of fuzzy static output feedback controller are derived in terms of linear matrix inequality (LMI) and the controller can be obtained by solving a set of LMIs.
11、 In comparison with the existing literatures, the drawbacks such as coordinate transformation and same output matrices have been eliminated. Notation: In this paper, a real symmetric matrix 0P denotes P being a positive definite matrix. In symmetric block matrices, an asterisk (*) is used to represe
12、nt a symmetric term and .diag stands for a block-diagonal matrix. The notion , , 1si j l means 1 1 1s s si j l . 2 Problem formulations Consider a DFBS that is represented by T-S fuzzy bilinear model. The i th rule of the DFBS is represented by the following form 11 ( ) . . . ( ) ( 1) ( ) ( ) ( ) (
13、)( ) ( ) 1 , 2 , . . . ,ii v v ii i iiR i f t i s M a n d a n d t i s Mt h e n x t A x t B u t N x t u ty t C x t i s (1) Where iR denotes the fuzzy inference rule, s is the number of fuzzy rules. , 1, 2.jiM j v is fuzzy set and ()j t is premise variable. () nx t R Is the state vector, ()ut R is the
14、 control input and T12( ) ( ) , ( ) , . . , ( ) qqy t y t y t y t R is the system output. The matrices , , ,i i i iA B N C are known matrices with appropriate dimensions. Since the static output feedback control is considered in this paper, we simply setvq and 11( ) ( ) , . . . , ( ) ( )vqt y t t y
15、t. By using singleton fuzzifier, product inference and center-average defuzzifier, the fuzzy model (1) Can be expressed by the following global model 11( 1) ( ( ) ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( )si i i iisiiix t h y t A x t B u t N x t u ty t h y t C x t (2)Where 1 1( ( ) ) ( ( ) ) / ( ( ) ) , ( (
16、) ) ( ( ) )qsi i i i i ji jh y t y t y t y t y t . ( ( )ij yt is the grade of Membership of ()jyt in jiM . We assume that ( ( ) 0i yt and 1 ( ( ) 0s ii yt . Then we have the following conditions: 1( ( ) ) 0 , ( ( ) ) 1siiih y t h y t .Based on parallel distribute compensation, the fuzzy controller s
17、hares the same premise parts with (1)； that is, the static output controller for fuzzy rule i is written as 11TT ( ) . . . ( ) ( ) ( ) / 1i i v v ii i iR i f y t i s M a n d a n d y t i s Mt h e n u t F y t y F F y (3)Hence, the overall fuzzy control law can be represented as 1 1 1()( ) s i n c o s
18、( )1s s sii i i i i ii i iTTiiF y tu t h h h F y ty F F y (4) Where T T T T() 1s i n , c o s , , , 1 , 2 , . . . ,2211ii i ii i i iF y t isy F F y y F F y .1 qiFR is a vector to be determined and 0 is a scalar to be assigned. By substituting (4) into (2), the closed-loop fuzzy systems can be represe
19、nted as , , 11( 1) ( )( ) ( )si j l ijli j lsiiix t h h h x ty t h C x t (5) where c o s s i ni j l i i j l j i jA B F C N . The objective of this paper is to design fuzzy controller (4) such that the DFBS (5) is asymptotically stable. 3 Main results Now we introduce the following Lemma which will b
20、e used in our main results. Lemma 1 Given any matrices ,MNand 0P with appropriate dimensions such that 0 , the inequality T T T 1 TM P N N P M M P M N P N holds. Proof: Note that 1 1 1 12 2 2 2( ) ( ) ( ) ( )T T T TM P N N P M P M P N P N P M Applying Lemma 1 in 1: 1T T T TM N N M M M N N , the inequality T T T 1 TM P N N P M M P M N P N can be obtained. Thus the proof is completed. Theorem 1 For given scalar 0 and 0 , , 1, 2 ,.,ij i j s , the DFBS (5) is asymptotically stable in the large, if there exist matrices 0Q and , 1, 2,.,iF i s satisfying the inequality (6).
