模糊控制器设计外文资料翻译

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 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 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 Introduction

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 design[1]-[11]. Most of the above results are designed based on either state feedback control or observer-based

control[1]-[7].Very few results deal with fuzzy output feedback[8]-[11]. The scheme of static output feedback control is very important and must be used when the system states are not completely available for feedback. The static output feedback control for fuzzy systems with time-delay was addressed [9][10] and a robust H∞ controller via static output feedback was designed[11]. But the derived conditions are not solvable by the convex programming technique since they are bilinear matrix inequality problems. 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].The research of bilinear systems has been paid a lot of attention and a series of results have been obtained[12][13].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 researchers[14]-[16]. The paper [14] studied the robust stabilization for the FBS, then the result was extended to the FBS with time-delay[15]. The problem of robust stabilization for discrete-time FBS (DFBS) was considered[16]. But all the above results are obtained via state feedback controller.

In this paper, 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. 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 P 0 denotes P being a positive definite matrix. In symmetric block matrices, an asterisk (*) is used to represent a symmetric term and diag{...}stands for a block-diagonal matrix. The notioni,j,l 1meansi 1j 1l 1. 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

Riif 1(t)isM1iand...and v(t)isMvithen x(t 1) Aix(t) Biu(t) Nix(t)u(t)

y(t) Cix(t)

i 1,2,...,s

s

sss

(1)

i

WhereRdenotes the fuzzy inference rule, sis the number of fuzzy rules.

Mji,j 1,2...v

is fuzzy set and

j(t)

n

x(t) Ris premise variable.Is the state

y(t) [y1(t),y2(t),..,yq(t)]T Rqu(t) Rvector,is the control input and is the system output. The matricesAi,Bi,Ni,Ci are known matrices with appropriate dimensions. Since the static output feedback control is considered in this paper, we simply setv qand

1(t) y1(t),..., v(t) yq(t)

.

By using singleton fuzzifier, product inference and center-average defuzzifier,

the fuzzy model

(1) Can be expressed by the following global model

x(t 1) i 1hi(y(t))[Aix(t) Biu(t) Nix(t)u(t)]y(t) i 1hi(y(t))Cix(t)(2)Wherethe

grade

s

s

q

hi(y(t)) i(y(t))/ i 1 i(y(t)), i(y(t)) j 1 ij(y(t))

s

.

ij(y(t))

is

of

that i(y(t)) 0and. Then we have the following

s

h(y(t)) 0, i 1hi(y(t)) 1

conditions:i.Based on parallel distribute compensation, the fuzzy controller shares the same premise parts with (1); that is, the static output controller for fuzzy rule i is written as

Membership of

s

i 1 i(y(t)) 0

yj(t)

in

Mji

. We assume

Riif y1(t)isM1iand... andyv(t)isMvi

(3)Hence, the overall fuzzy control law can be represented as

then u(t) Fiy(t)u(t) i 1hisin i

Where

s

i 1hi sin i i 1hi Ficos iy(t)

(4)

ss

i

i [ ,],i 1,2,...,s

22.

Fi R1 qis a vector to be determined and 0is a scalar to be assigned.

By substituting (4) into (2), the closed-loop fuzzy systems can be represented as

x(t 1) i,j,l 1hihjhl ijlx(t)y(t) i 1hiCix(t)

s

s

(5)

.

where

ijl Ai BiFjCl cos j Ni sin j

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 be used in our main results.

Lemma 1 Given any matricesM,NandP 0with appro …… 此处隐藏：17542字，全部文档内容请下载后查看。喜欢就下载吧 ……