$H_{2}$ State-Feedback Control for Continuous-Time Systems under Positivity Constraint

Abstract

This study is concerned with the $H_{2}$ state-feedback controller synthesis problem under positivity constraint on the closed-loop system. This problem is believed to be a nonconvex problem in both continuous- and discrete-time system settings and hence remains to be the most challenging issue in positive system theory. For this hard problem, in the discrete-time system setting, the authors recently proposed a technique for the lower bound computation of the best achievable $H_{2}$ performance by a specific treatment of finite impulse responses (FIRs) of the closed-loop systems. The goal of this paper is to extend this idea to the continuous-time system setting. Even though there is no notion of FIR in (finite-dimensional) continuous-time system impulse responses, the truncation of the Taylor series expansion of the matrix exponential function in the impulse response paves the way for obtaining a semidefinite programming problem (SDP) for the lower bound computation. We show that, by increasing the truncation degree, we can construct a sequence of SDPs that generates a monotonically non-decreasing sequence of the lower bounds. By combining this lower bound computation technique with heuristic upper bound and suboptimal gain computation techniques, it becomes possible to draw definite conclusion on the quality of the computed suboptimal gains.