Decentralized Control of Linear Switched Nested Systems with l₂-Induced Norm Performance

This paper considers a decentralized switched control problem where exact conditions for controller synthesis are obtained in the form of semidefinite programming (SDP). The formulation involves a discrete-time switched linear plant that has a nested structure, and whose system matrices switch between a finite number of values according to finite-state automation. The goal of this paper is to synthesize a commensurately nested switched controller to achieve a desired level of l₂-induced norm performance. The nested structures of both plant and controller are characterized by block lower-triangular system matrices. For this setup, exact conditions are provided for the existence of a finite path-dependent synthesis. These include conditions for the completion of scaling matrices obtained through an extended matrix completion lemma. When individual controller dimensions are chosen at least as large as the plant, these conditions reduce to a set of linear matrix inequalities. The completion lemma also provides an algorithm to complete closed-loop scaling matrices, leading to inequalities for controller synthesis that are solvable either algebraically or numerically through SDP.