Time-Varying Optimal Disturbance Rejection in Presence of Plant Uncertainty Seddik M. Djouadi, Charalambos D. Charalambous Abstract— The optimal robust disturbance rejection problem plays an important role in feedback control theory. Here its time-varying version is solved explicitly in terms of duality and operator theory. In particular, the optimum is shown to satisfy a time-varying allpass property. Moreover, optimal performance is given in terms of the norm of a bilinear form. The latter depends on a lower triangular projection and a multiplication operator defined on special versions of spaces of compact operators. DEFINITIONS AND NOTATION • B(E,F ) denotes the space of bounded linear operators from a Banach space E to a Banach space F , endowed with the operator norm ‖A‖ := sup x∈E, ‖x‖≤1 ‖Ax‖, A ∈ B(E,F ) • ℓ 2 denotes the usual Hilbert space of square summable sequences with the standard norm ‖x‖ 2 2 := ∞  j =0 |x j | 2 , x := ( x 0 ,x 1 ,x 2 , ··· ) ∈ ℓ 2 • P k the usual truncation operator for some integer k, which sets all outputs after time k to zero. • An operator A ∈ B(E,F ) is said to be causal if it satisfies the operator equation: P k AP k = P k A, ∀k positive integers The subscript “ c ” denotes the restriction of a subspace of operators to its intersection with causal operators, that is B c (E, F ) (see [3], [2] for the definition.) Bounded and causal linear operators can be represented by lower triangular “infinite” matrices, with respect to the canonical basis, {e i } ∞ 1 of ℓ 2 , where the entries S.M. Djouadi is with the Department of Electrical and Com- puter Engineering at The University of Tennessee, Knoxville. djouadi@ece.utk.edu C.D. Charalambous is with the Electrical & Computer Engi- neering Department, University of Cyprus, Nicosia, 1678, Cyprus. chadcha@ucy.ac.cy of {e i } are all zero except that the entry at the i-th position is 1. The symbol “⊕” denotes the direct sum of two spaces. “ ⋆ ” stands for the adjoint of an operator or the dual space of a Banach space depending on the context [10], [14]. I. I NTRODUCTION The optimal robust disturbance attenuation problem plays a fundamental role in feedback optimization [26], [19]. In particular, it has been shown in [26], for linear time-invariant (LTI) systems, using a counter example based on a ”two-arc” result, that approximate solutions employing state space robust control theory may result in arbitrary poor solutions. An exact solution based on operator theory and duality theory for LTI systems has been proposed in [16], [15]. In this paper, we consider the optimal disturbance rejection problem is considered for time-varying systems generalizing certain results which hold in the LTI case. Characterization of the optimal solution in part by duality theory has been proposed in [17], albeit for continuous time systems. It was also shown there that for time-invariant nominal plants and weighting functions, time-varying control laws offer no improvement over time-invariant feedback control laws. Analysis of time-varying control strategies for optimal disturbance rejection for known time-invariant plants has been studied in [24], [5]. A robust version of these problems were considered in [23], [12], [13] in different induced norm topologies. They showed that for time-invariant nominal plants, time-varying control laws offer no advantage over time-invariant ones. The Optimal Robust Disturbance Attenuation Problem (ORDAP) was formulated by Zames [25], and considered in [4], [11], [26], [19]. In ORDAP a stable uncertain linear time-varying plant P is subject Proceedings of the European Control Conference 2007 Kos, Greece, July 2-5, 2007 TuD14.1 ISBN: 978-960-89028-5-5 2395