On the Design of Robust H ∞ Filter-Based Tracking Controller for a Class of Linear Time Delay Systems with Parametric Uncertainties A. Alif, M. Darouach and M. Boutayeb Abstract— : This paper deals, for the first time, with the design problem of linear robust H∞ filter-based tracking controller for a class of linear time delay systems subjected to parametric uncertainties. Sweet method is proposed to guarantee the filtering process of the system output at the same time as its robust H∞ tracking and output model following. Finally, the viability and the efficiency of the proposed method are clearly proved by numerical simulations. Index Terms— model following, Robust H ∞ tracking control, Filtering process, time-delay systems, paramet- ric uncertainties. I. I NTRODUCTION A lot of attention has been devoted to the problem of robust tracking and model following for uncertain linear time-delay systems in the last decade, see for instance [1], [2] and [3] and the reference therein. However, all these works suppose the entire knowledge of the system state for the stability purpose. Also, they all consider the particular class of delayed systems subjected to matched uncertainties. Indeed, based on the matched structure, and by means of the norm tools, all these works consider the boundedness of the further terms corresponding to the un- certainties, perturbations and the delayed terms. Thereafter, the knowledge of these bounds, or just their estimations, as has been made in [3], is exploited in the construction of some types of stabilizing state feedback controllers that may achieve the asymptotic tracking purpose. However, the structure of these controllers which is nonlinear, and often discontinuous, may lead to several problems concerning the implementation purpose. Moreover, all these works suppose the entire knowledge of the system state to ensure the stabilization purpose. However, this is not the case in several practical situations for technical and economical reasons. On the other hand, in the few works where a linear tracking controller has been provided, see for instance [4], only the practical tracking purpose can be achieved, which means that only the boundedness of the tracking error can be guaranteed, and not its asymptotic stability. In this note, for the first time, the problem of robust H ∞ filter-based track- ing and model following control for a class of linear time A. Alif and M. Darouach are with UHP-CRAN-CNRS-UMR 7039, IUT de Longwy, 186 rue de Lorraine, 54400 Cosnes et Romain, France Adil.Alif@iut-longwy.uhp-nancy.fr darouach@iut-longwy.uhp-nancy.fr M. Boutayeb is with LSIIT-CNRS-UMR 7005,University of Louis Pasteur Strasbourg, Pole API, Bu.S.Braut, 67400 ILLKIRCH Mohamed.Boutayeb@ipst-ulp.u-strasbg.fr varying delay systems subjected to parametric uncertainties is tackled. Aware of all the aforementioned limitations in the existing works within this framework of study, our major goal is to ensure in the same time the robust H ∞ tracking and the filtering processes, (RHTFP), for the more general class of linear time varying delay systems subjected to parametric uncertainties. To the author’s best knowledge, this problem has never been done before. Sweet sufficient conditions are provided to ensure the existence and the design of linear and easily implementable robust H ∞ filter- based tracking controller that achieves the (RHTFP) goal. Furthermore, a constructive procedure is provided to com- plete the design purpose. Finally, the validity and efficiency of the proposed approach are clearly proved through some numerical simulations. II. PROBLEM STATEMENT AND PRELIMINARIES We consider a class of uncertain linear time varying delay systems described by the following differential-difference equations: ⎧ ⎨ ⎩ ˙ x(t)= Ax(t)+ A d x(t − τ (t))+ Bu(t)+ B w w(t) x(t)=0, t ∈ [−τ (t), 0] y(t)= Cx(t)+ C d x(t − τ (t))+ Du(t)+ D w w(t) (1) where x(t) ∈ IR n is the state vector, u(t) ∈ IR r is the input vector, y(t) ∈ IR m is the output vector, w(t) ∈L q 2 [0, ∞) the external disturbances. τ (t) represents the time varying lag which is supposed to be a differential equation satisfying ˙ τ (t) < l< 1, ∀t ≥ 0. Furthermore, assume that the coefficient matrices are decomposed as follows: A = A +ΔA(t), A d = A d +ΔA d (t) (2) B = B +ΔB(t), C = C +ΔC(t) (3) C d = C d +ΔC d (t), D = D +ΔD(t) (4) B w = B w +ΔB w (t), D w = D w +ΔD w (t) (5) A, A d , B, B w , C, C d , D and D w are constant matrices with appropriate dimensions, and the uncertainty parts have the following structure:  ΔA(t) ΔA d (t) ΔB(t) ΔB w (t) ΔC(t) ΔC d (t) ΔD(t) ΔD w (t)  =  M 1 M 2  Δ(t)  N 1 N 2 N 3 N 4  (6) where N 1 ∈ IR d×n , N 2 ∈ IR d×n , N 3 ∈ IR d×r and N 4 ∈ IR d×q are constant known matrices. Δ(t) ∈ IR s×d is an Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 ThB06.5 0-7803-9568-9/05/$20.00 ©2005 IEEE 7181