Robust [formula omitted] and [formula omitted] control for positive continuous-time uncertain linear systems

Abstract

This paper addresses the problems of H2 and H∞ control design for positive continuous-time linear systems with uncertain parameters belonging to a polytope. Although it is well known that the optimal H∞ state-feedback gain can be obtained from the bounded real lemma condition with a diagonal Lyapunov matrix without any conservatism for precisely known positive linear systems, the same is not true for the H2 norm, neither for output-feedback control. When uncertainties affect the plant, a fixed diagonal structure for the Lyapunov or any other matrix variable involved in the conditions can produce poor results in terms of H2 and H∞ guaranteed costs, specially if the matrix constrained to be diagonal is involved in the computation of the control gain. In this sense, sufficient linear matrix inequalities (LMIs) for the existence of a stabilizing gain and an H2 (or H∞) guaranteed cost are proposed. As main characteristic, the gain and the Lyapunov matrix can be treated as independent optimization variables in the conditions. Using relaxations in the stability of the system, an algorithm is constructed to iteratively solve the LMIs. Therefore, no change of variable is necessary and, differently from other LMI-based methods for positive systems from the literature, there is no need to impose a diagonal structure on the matrix used to recover the gain to enforce closed-loop positivity. Moreover, static output- or state-feedback controllers, including bounded entries or decentralized structures, can be computed in a straightforward way. With an appropriate initialization, the iterative procedure, implemented in two phases, is assured to end up, at the first step, with a finite bound for the real part of the eigenvalues of the closed-loop positive system. Whenever a stabilizing gain is obtained, the second step provides non-increasing H2 (or H∞) guaranteed costs along the iterations. Numerical examples show that the proposed technique provides good estimates for the H2 and H∞ norms when compared with other methods available in the literature.