An extended integral method to derive Lyapunov functions for nonlinear systems

Abstract

In this paper, Chin's (1987 b) integral method of deriving Lyapunov functions for the investigation of nonlinear system stability is extended. Whereas in Chin's method, each weighting function includes only one state variable, in the extended integral method the weighting functions can be any continuous functions of time, some or all state variables and their time derivatives. As a result of the less restrictive weighting functions, the sensitivity of the method to the form of the state-space model is reduced, which gives the user more freedom in deriving the Lyapunov functions. The method also extends the integral method in the sense that it can be applied to both autonomous and non-autonomous systems. Furthermore, the intrinsic method of Chin (1988) is shown to be a special case of the extended integral method. Many examples from the literature and from engineering applications are provided to show that the extended method is surprisingly easy to apply, and results in Lyapunov functions for more general systems.