Decentralized Stabilization of Interconnected Systems With Time-Varying Delays

Abstract

This technical note establishes decentralized delay-dependent stability and stabilization methods for two classes of interconnected continuous-time systems. The two classes cover the linear case and the Lipschitz-type nonlinear case. In both cases, the subsystems are subjected to convex-bounded parametric uncertainties and time-varying delays within the local subsystems and across the interconnections. An appropriate Lyapunov functional is constructed to exhibit the delay-dependent dynamics at the subsystem level. In both cases, decentralized delay-dependent stability analysis is performed to characterize linear matrix inequalities (LMIs)-based conditions under which every local subsystem of the linear interconnected delay system is robustly asymptotically stable with an gamma -level <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gain</i> . Then we design a decentralized state-feedback stabilization scheme such that the family of closed-loop feedback subsystems enjoys the delay-dependent asymptotic stability with a prescribed gamma-level <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on a representative example.