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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 5, NO. 4, DECEMBER 2007
Robust H∞ Static Output Feedback Control Design for Discrete-time Fuzzy Systems with Disk Closed-loop Pole Constraints Mourad Kchaou, Mansour Souissi and Ahmed Toumi
Abstract— This paper addresses the design of robust H∞ controller for an uncertain discrete fuzzy system. A static output feedback controller is designed such the prescribed H∞ performance condition is satisfied and the closed-loop poles are placed in specified disk. It is shown that the controller parameters can be obtained by solving a set of linear matrix inequalities. An illustrative example is provided to demonstrate the effectiveness c 2007 Yang’s of the approach proposed in this paper. Copyright ° Scientific Research Institute, LLC. All rights reserved. Index Terms— TS fuzzy model, Fuzzy static output, D-stability, H∞ control
I. I NTRODUCTION
I
N THE past few years, fuzzy logic has been successfully used in nonlinear system modeling and control. The TakagiSugeno (TS) model has become a quite popular and powerful tool to represent or approximate nonlinear system. However, there have been significant research efforts on the analysis and synthesis of (TS) fuzzy system where various techniques have been developed. When the stability and performance are the most important requirement for a control system, many works on (TS) fuzzy-model-based on model control for nonlinear system were developed to stabilize the (TS) fuzzy model with guaranteed H2 performance [6], H∞ performance [3], [4], [7], [8], [11] or mixed H2 / H∞ performance [16]. Furthermore, robust fuzzy control problem have been also considered for uncertain nonlinear systems, based on fuzzy model, since uncertainties often degrade systems performance and may lead to instability [5], [6], [10]. One can cite Tanaka et al [19] studied the robust stabilization problem for a class of uncertain nonlinear system via linear matrix inequalities ; Yong et al [17] presented the robust H∞ disturbance attenuation for a class of uncertain discrete-time fuzzy system . In practical application, it is also desirable to design a control system, which does not only satisfy an H∞ performance specification but also achieves satisfactory transient behavior by placing the closed-loop poles in a suitable region. This leads to the study of the problem of the robust H∞ control with pole placement constraints [1], [2], Manuscript received February 01, 2007; revised November 06, 2007. Mourad Kchaou and Ahmed Toumi, National School of Engineers of Sfax, BP: W, 3038 Sfax, Tunisia. Mansour Souissi, Preparatory Institute of Studies Engineers of Sfax, Tunisia. Email:
[email protected](M. Kchaou),
[email protected](M. Souissi),
[email protected](A. Toumi). Publisher Item Identifier S 1542-5908(07)10405-X/$20.00 c Copyright °2007 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on June 19, 2008 at http://www.YangSky.com/ijcc/ijcc54.htm
[4], [12] The aim of this paper is to design a static output feedback controller for a class of fuzzy uncertain system which guarantees the robust H∞ performance with pole placement in specified disk. Being based on Lyapunov function and an LMI approach, some sufficient conditions are derived in terms of a family of linear matrix inequalities to ensure the last specifications. It should be mentioned that, the sufficient conditions for the existence of admissible controllers are expressed as a non convex feasibility problem. To overcome such difficulty, we use the cone complementarity linearization method. [9], [3], [13] This paper is organized as follows. Problem formulation is presented in section 2. The analysis results for fuzzy controller, which guarantees the H∞ performance and the D-stability is developed in section 3. Section 4 interests of the synthesis of the controller. In section 5, a numerical example is given to illustrate the application of the proposed method. II. P ROBLEM F ORMULATION We consider a class of uncertain discrete-time nonlinear systems which can be described by the following discrete-time fuzzy model with parameter uncertainties: ½ r X x(k + 1) = hi (θ(k)) (Ai + ∆Ai )x(k) + (B1i + ∆B1i )u(k) i=1
¾ + (B2i + ∆B2i )w(k) (1) ½ ¾ r X z(k) = hi (θ(k)) C1i x(k) + D1i u(k) + D2i w(k) i=1
(2) (3)
y(k) = C2 x(k)
where θ(k) = [θ1 (k), θ2 (k) . . . θp (k)] is the premise variable vector, r is the number of model rules. hi (θ(k)) is the normalized weight for each rule, that is: hi (θ(k)) ≥ 0 and
r X i=1
hi (θ(k)) = 1
for
i = 1, · · · , r
(4) for all k, x(k) ∈ IRn is the system state, u(k) ∈ IRm is the control input, y(k) ∈ IRp is the output and z(k) ∈ IRq is the control output and w(k) ∈ IRl is the exogenous disturbance input with {w(k)} ∈ L2 [0, ∞).
KCHAOU, SOUISSI & TOUMI, ROBUST H∞ STATIC OUTPUT FEEDBACK CONTROL DESIGN FOR DISCRETE-TIME FUZZY SYSTEMS
∆Ai , ∆B1i and ∆B2i represent the time-varying parametric uncertainties having the following structure: £ ¤ £ ¤ ∆Ai ∆B1i ∆B2i = HF (k) Ei E1i E2i for
i = 1, . . . , r
(5)
where H, Ei , E1i and E2i are known constant matrices in appropriate dimensions and F (k) is an unknown real time varying matrix with Lebesgue-measurable elements satisfying F T (k)F (k) ≤ I. Based on the fuzzy models (1)-(2), fuzzy static output feedback controller is proposed to deal with the H∞ control problem. r X u(k) = hj (θ(k))Kj y(k) (6) j=1
The resulting closed-loop system is described by: x(k + 1) =
r X r X
˜ij w(k)} hi (θ(k))hj (θ(k)){A˜ij x(k) + B
i=1 j=1
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£ T ¤ x (k) wT (k) , hwe have z(k) = = i e e Cij Di . i=1 j=1 hi hj C ij ξ(k), where C ij = Since (14) and (4), we obtain the following X 11 ··· X 1r ∗ .. .. .. . . . ∗ (16) 0 for any uncertainties.
III. O UTPUT F EEDBACK F UZZY C ONTROLLER D ESIGN WITH D- STABILITY C ONSTRAINTS In this section, we develop the controller for uncertain fuzzy model (1)-(3) regarding the problem formulated in the previous section. The following theorem investigates the existence of sufficient conditions for the system (7) to be D-stable with H∞ norm bound. (Theorem 3.1 was moved to the top of this next page.) Lemma 3.1: [10] Given constant matrices D and E and a symmetric constant matrix P of appropriate dimensions and a scalar ε > 0, the following inequality holds: P + DF E + E T F T DT < 0 where F satisfies F T F ≤ I, if and only if P + ε−1 E T E + εDDT < 0
i=1 j=1 u=1 v=1 T
= −z (k)z(k)
T
hi hj hu hv ξ T (k)C ij C ij ξ(k) (18)
On the other hand, from (10) and (12) we obtain for all i and j 11 −P − Xij ∗ ∗ ∗ r X r 21 22 X −Xij −γ 2 I − Xij ∗ ∗ hi hj Aij B 2i −P −1 + εHH T ∗ i=1 j=1 Eij E2i 0 −εI < 0. (19) By Schur complement, one has ( −P − X 11 r X r ∗ ∗ ij X 21 22 ∗ −γ 2 I − Xij hi hj −Xij i=1 j=1 Aij B 2i −P −1 + εHH T T T T ) Eij Eij −1 T T +ε E2i E 2i 0 0 ( −P − X 11 r r ∗ ∗ ij XX 21 22 −γ 2 I − Xij ∗ = hi hj −Xij i=1 j=1 Aij B2i −P −1 T T T T ) Eij Eij 0 0 + ε 0 0 + ε−1 E2Ti E2Ti < 0. (20) H H 0 0 By Lemma (3.1) and (9) the above inequality is equivalent to 11 −P − Xij ∗ ∗ r X r X 21 22 −γ 2 I − Xij ∗ < 0 (21) hi hj −Xij eij ei i=1 j=1 A B −P −1
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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 5, NO. 4, DECEMBER 2007
Theorem 3.1: For a prescribed H∞ performance , the system (7) is D-stable in the disk D(C0 , r0 ) with H∞ norm bound γ if T T there exist a matrices P > 0, Ki , X ii > 0, Y ii > 0, X ij = X ji , Y ij = Y ji and some positives scalars ε and ε0 such that satisfying the following matrix inequality: −P − Xii11 ∗ ∗ ∗ −Xii21 −γ 2 I − Xii22 ∗ ∗
