H<inf>2</inf> State-Feedback Synthesis under Positivity Constraint: Upper and Lower Bounds Computation of the Achievable Performance

Abstract

This paper is concerned with the $H_{2}$ state- feedback synthesis problem under positivity constraint on the closed-loop system. This problem is believed to be a non-convex problem and hence exact treatment is not known to this date. With this difficulty in mind, in this paper, we first derive several semidefinite programs (SDPs) for the computation of the upper bounds of the achievable performance as well as suboptimal gains. However, if we rely only on the upper bound computation, we cannot say anything quantitatively on the quality of the computed suboptimal gains. For such quantitative evaluation, we next derive an SDP for the lower bound computation of the achievable performance. The key idea in deriving such an SDP is that, if the closed-loop system is positive, then the Lya- punov variable in the standard SDP for the $H_{2}$ state-feedback synthesis should be (elementwise) nonnegative. By numerical examples, we illustrate the effectiveness and limitation of the proposed strategy with upper and lower bounds computation. Keywords: $H_{2}$ state-Feedback synthesis, positivity constraint, upper and lower bound computation.