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Induced ℓ∞- Optimal Gain-Scheduled INDUCED ℓ∞ OPTIMAL GAIN-SCHEDULED FILTERING OF T−AKAGI-SUGENO FUZZY SYSTEMS Isaac Yaesh Control Department, IMI Advanced Systems Div., P.O.B. 1044/77, Ramat–Hasharon, 47100, Israel Uri Shaked School of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel Keywords: Takagi-Sugeno fuzzy systems, Polytopic uncertainties, H∞-optimization, ℓ∞-optimization, S -procedure, Lin- ear matrix inequalities. Abstract: The problem of designing gain-scheduled filters with guaranteed induced ℓ∞ norm for the estimation of the state-vector of finite dimensional discrete-time parameter-dependent Takagi-Sugeno Fuzzy Systems systems is considered. The design process applies a lemma which was recently derived by the authors of this paper, characterizing the induced ℓ∞ norm by Linear Matrix Inequalities. The suggested filter has been successfully applied to a guidance motivated estimation problem, where it has been compared to an Extended Kalman Filter. 1 INTRODUCTION response and an upper-bound on the ℓ1-norm of its transfer function (see (Dahleh and Pearson, 1987)). The theory of optimal design of estimators for linear In the present paper, the problem of discrete- discrete-time systems in a state-space formulation has time optimal state-estimation in the minimum in- duced ℓ∞ norm sense is considered for a class of been first established in (Kalman, 1960). The original − problem formulation assumed Gaussian white noise Takagi-Sugeno fuzzy systems. The plant model for models for both the measurement noise and the ex- the systems considered, is described by a collection ogenous driving process. For this case, the results of ’sample’ finite-dimensional linear-time-invariant of (Kalman, 1960) provided the Minimum-Mean- plants which possess the same structure but differ Square Estimator (MMSE). The Kalman filter has in their parameters. All possible plant models are found since then many applications (see e.g. (Soren- then assumed to be convex combinations of these spe- son, 1985) and the references therein). Following the cific plant models (namely a polytopic system where the ’sample’ plant models are denoted as its vertices introduction of H∞ control theory in (Zames, 1981), (Boyd et al., 1994)). The solution of the estima- a method for designing discrete-time H∞ optimal es- timators within a deterministic framework has been tion problem is characterized by LMIs (Linear Ma- developed in (Yaesh and Shaked, 1991), where the ex- trix Inequalities) based on the quadratic stability as- ogenous signals are of finite energy. The case where sumption. We note that more recent developments of the driving signal is of finite energy (e.g. piecewise (Geromel et al., 2000) include gain scheduled filter constant for a finite time) ,whereas the measurement synthesis for the cases of linear (Tuan et al., 2001) noise is white has been recently considered in (Yaesh and nonlinear (Hoang et al., 2003) dependence on the and Shaked, 2006). However, in some cases the min- parameters with parameter dependent Lyapunov func- tions. We also note that in (Salcedo and Martinez, imization of the maximum absolute value of the es- 2008) related results appear where the continuous- timation error (namely the ℓ∞ norm) rather than the error energy is required where− the exogenous signals time fuzzy output feedback and filtering were con- sidered in parallel to the discrete-time results of the are also of finite ℓ∞ norm. In such cases, an induced − present paper. ℓ∞ norm is obtained which is often referred to as an − ℓ1 problem due to the fact that the induced-ℓ∞ norm The paper is organized as follows. In Section 2, − for a linear system is just the ℓ1-norm of its impulse the problem is formulated and a key lemma character- 67 Yaesh I. and Shaked U. INDUCED âDˇ ¸Sâ´Ld¯ â´LŠ OPTIMAL GAIN-SCHEDULED FILTERING OF TAKAGI-SUGENO FUZZY SYSTEMS. DOI: 10.5220/0002169500670073 In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page ISBN: 978-989-674-001-6 Copyright c 2009 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics izing the induced ℓ∞ norm in terms of LMIs is pre- external disturbances and/or time (Tanaka and Wang, sented. In Section 3− the filter design inequalities are 2001) and rewrite (1) as : obtained. Section 4 considers a numerical example IF s1 is Mi1 and s2 is Mi2 and ... sp is Mip THEN dealing with a robust gain scheduled tracking prob- x k 1 A k x k B w k x 0 x lem. Finally, Section 5 brings some concluding re- ( + ) = i( ) ( ) + i ( ), ( ) = 0 y(k) = C (k)x(k) + D w(k) marks. i i (6) z k L k x k Notation: Throughout the note the superscript ( ) = i( ) ( ) i 1 2 N ‘T ’ stands for matrix transposition, R n denotes the = , ,..., n m n dimensional Euclidean space, R × is the set of where Mi j is the fuzzy set and N is the all n m real matrices, and the notation P > 0, for number of model rules. Defining s(t) = ×n n P R × means that P is symmetric and positive def- col s1(t),s2(t),...,sp(t) , ∈ { } inite. The space of square summable functions over ω Πp i(s(t)) = j 1Mi j(s j(t)) [0 ∞] is denoted by l2[0 ∞], and . 2 stands for the = Σ∞ T || ||1/2 standard l2-norm, u 2 = ( k=0uk uk) . We also use and || || 2 T ωi(t) . ∞ for the l∞-norm namely, u ∞ = supk(u uk). f (s(t)) = || || || || k i ΣN ω The convex hull of a and b is denoted by C o a, b , i=1 i(s(t)) { } In is the unit matrix of order n, and 0n,m is the n m we readily get the representation of (1). We, there- × zero matrix and Im,n is a version of In with last n m fore, assume indeed that the p premise scalar vari- − rows omitted. ables si(t),i = 1,2,..., p, and, consequently fi are ex- actly known and consider the following filter: xˆ(k + 1) = Axˆ(k) + K(k)(y Cxˆ), zˆ(k) = Lxˆ(k) 2 PROBLEM FORMULATION − (7) AND PRELIMINARIES where the filter gain is given by the following: N We consider the following linear system: K = ∑ Ki fi (8) i=1 x(k + 1) = A(k)x(k) + Bw(k), x(0) = x0 y(k) = C(k)x(k) + Dw(k) (1) N N and where A = ∑ Ai fi and L = ∑ Li fi. We will dif- z(k) = L(k)x(k) i=1 i=1 ferently treat, in the sequel, the case where C is con- n r where x R is the system states, y R is the mea- N surement,∈ w R q includes the driving∈ process and stant and the case where C = ∑ Ci fi. the measurement∈ noise signals and it is assumed to i=1 Our aim is to find the filter parameters Ki so that ∞ R m have bounded ℓ norm. The sequence z is the the following induced ℓ∞ norm condition is satisfied. state combination− to be estimated and A, B∈, C, D and − γ L are matrices of the appropriate dimensions. supw ℓ∞ z zˆ ∞/ w ∞ < (9) ∈ || − || || || We assume that the system parameters lie within To solve this problem we will first define another the following polytope polytopic system : Ω := ABCDL (2) Ω¯ := A¯ B¯ C¯ D¯ (10) which is described by its vertices. That is, for which is described by the vertices: Ω i := Ai Bi Ci Di Li (3) Ω¯ i := A¯i B¯i C¯i D¯ i , i = 1,...,N (11) we have The system of (10)-(11) will represent, in the sequel, Ω = C o Ω1,Ω2,...,ΩN (4) the dynamics of the estimation error for the system { } where N is the number of vertices. In other words: (1). The following technical lemma will be needed N N in order to provide convex characterization of the in- duced ∞ norm of the estimation error system: Ω = ∑ Ωi fi , ∑ fi = 1 , fi 0. (5) ℓ ≥ − i=1 i=1 Lemma 1. The system Assuming that f are exactly known, the above sys- i x¯(k + 1) = A¯(k)x¯(k) + Bw¯ (k), x(0) = x tem is just a Tagaki-Sugeno fuzzy system. To see this, 0 (12) z(k) = C¯(k)x¯(k) + D¯ w¯(k) one may introduce new parameters si(t),i = 1,2,..., p (so called premise variables, see (Tanaka and Wang, satisfies γ 2001)) possibly depending on the state-vector x(t), supw ℓ∞ z ∞/ w ∞ < (13) ∈ || || || || 68 INDUCED - OPTIMAL GAIN-SCHEDULED FILTERING OF TAKAGI-SUGENO FUZZY SYSTEMS if the following matrix inequalities are satisfied for and i = 1,2,...,N: λ T P 0 Li ¯T ¯ λ ¯T ¯ 0 (γ µ)I 0 > 0, λ < 1 (22) Ai PAi + P P Ai PBi − T − T < 0 (14) Li 0 γI B¯ PA¯i µI + B¯ PB¯i i − i We, therefore, obtain the following result: and λ ¯T P 0 Ci Theorem 1. Consider the estimator of (12) for the 0 (γ µ)I D¯ T > 0 (15) system of (1) with C = C,D = D,i = 1,2,...,N. The − i i i C¯i D¯ i γI estimation error satisfies (9) if (21) and (22) are satis- λ so that P > 0, µ > 0 and λ < 1. fied for i = 1,2,...,N so that P > 0, µ > 0 and < 1. The proof of this lemma is given in (Shaked and We next address the problem where C and D are Yaesh, 2007) and is also provided, for the sake of vertex dependent.
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