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

TL;DR

This paper introduces a novel Lyapunov inequality allowing asymmetric matrices, enabling new stability analysis and suboptimal control design methods for linear systems, with practical applications demonstrated through numerical examples.

Contribution

It proposes a new Lyapunov inequality with asymmetric matrices and applies it to suboptimal LQ control, providing stability conditions and cost bounds.

Findings

Asymmetric Lyapunov matrices can be used for stability analysis.

Derived sufficient conditions for suboptimal control with asymmetric Lyapunov matrices.

Numerical examples demonstrate the practical applicability of the proposed methods.

Abstract

The Lyapunov inequality is an indispensable tool for stability analysis in linear control theory. It provides a necessary and sufficient condition for the stability of an autonomous linear-time invariant system in terms of the existence of a symmetric positive-definite Lyapunov matrix. This work proposes a new variant of this inequality in which the constituent Lyapunov matrix is allowed to be asymmetric. After analysing the properties of the proposed inequality for a class of matrices, we derive new results for the stabilisation of linear systems. Subsequently, we utilize the developed results to obtain sufficient conditions for the suboptimal linear quadratic control design problem wherein addition to having an asymmetric Lyapunov matrix, which serves as a design matrix for this problem, we provide a characterization of the cost associated with the computed stabilizing suboptimal control laws. This characterization is done by deriving an expression for the upper bound on cost in terms of the initial conditions of the system. We demonstrate the applicability of the proposed results using two numerical examples -- one for suboptimal control design for a linear time-invariant system and another for the consensus (state-agreement) protocol design for a multi-agent system, wherein we see how the asymmetry of the Lyapunov (design) matrix emerges as an inherent requirement for the problem.