Frequency Domain Stability Condition and Existence of Lyapunov Function of Composite Systems

The composite-system type analysis is generally acknowledged to be useful for a stability study of large-scale systems. However, in many cases it is highly difficult to construct Lyapunov functions for subsystems with best parameters. In this paper a method to construct Lyapunov functions of the subsystems suited for the composite-system analysis based on the frequency-domain stability condition is given. For that purpose we first extend the Lyapunov stability theorem of composite systems and then show that Lyapunov functions of subsystems which satisfy all the conditions of the above extended theorem exist if the L2-stability condition given in the frequency domain is satisfied. In the proof of the last statement, a method to construct the Lyapunov functions of subsystems is included. Using this result it is easy to construct a Lyapunov function of the whole system.