Lyapunov-like Stability Inequality with an Asymmetric Matrix and Application to Suboptimal LQ Control Design

Abstract: The Lyapunov inequality is an indispensable tool for stability analysis in the linear control theory. This work proposes a new variant of this inequality where-in the constituent matrix is allowed to be asymmetric. After developing the stability conditions based on the proposed inequality for a class of linear systems, we utilize these conditions to derive new results for the suboptimal linear quadratic control problem where we characterize the cost of the stabilizing controllers. We also demonstrate, by a numerical example, that the proposed results can be easily molded for the structured suboptimal consensus protocol design for multi-agent system where we also see that the asymmetry condition of the design matrix turns up inherently.