Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions

In this paper, we present several conditions which are both necessary and sufficient for quadratic stability of an uncertain/nonlinear system. These conditions involve multiplier matrices which characterize the uncertain/nonlinear terms in the system description. It is known that some of these conditions are sufficient for quadratic stability. One of the main contributions of this paper is to demonstrate that these conditions are also necessary conditions, hence, they are not conservative conditions for quadratic stability. By presenting multiplier matrices for many common types of uncertain/nonlinear terms, the paper also demonstrates the usefulness of the multiplier matrix approach in the analysis and control of nonlinear/time-varying/uncertain systems.