Stabilization and $\mathcal {H}_{2}$ Static Output-Feedback Control of Discrete-Time Positive Linear Systems

This article investigates the problem of designing $\mathcal {H}_{2}$ robust (or gain-scheduled) static output-feedback (or state-feedback) controllers for discrete-time positive linear systems affected by time-invariant (or time-varying) parameters belonging to a polytope. For this purpose, an iterative procedure based on robust (parameter-dependent) linear matrix inequalities (LMI) is proposed. Unlike most approaches, where the controller is obtained by means of a change of variables, the synthesis conditions deal with the control gain directly as an optimization variable, which is specially appealing to cope with closed-loop positivity or structural constraints. The existence of feasible initial conditions for the iterative procedure and some relaxation strategies adopted to reduce the conservativeness of the method are also discussed. Numerical examples borrowed from the literature and statistical comparisons show that the proposed technique is in general less conservative than other approaches, providing solutions and handling cases where traditional design techniques for positive systems cannot be applied.