A Lyapunov approach to frequency analysis

This paper proposes a Lyapunov approach to frequency analysis for general systems. The notion of frequency response is extended to general systems through a connection in linear systems. Lyapunov approaches to the characterization of frequency response are established for linear systems, homogeneous systems and nonlinear systems, respectively. In particular, we show that for linear systems, quadratic Lyapunov functions are sufficient for the characterization; for homogeneous systems, homogeneous Lyapunov functions are sufficient; and for general nonlinear systems, locally Lipschitz Lyapunov functions will be used. We also develop a Lyapunov approach for the characterization of the peak of the output. This approach is demonstrated to be effective on linear systems. An LMI based method for performing frequency analysis on linear differential inclusions is developed. Through a numerical example, an interesting phenomenon is observed about the relation between the frequency response and the L/sub 2/ gain of linear differential inclusions.