A general invariance principle for nonlinear time-varying systems with application to mobile robots

Abstract

A general invariance principle is proposed from the output-to-state view-point for general nonlinear time-varying systems. A simple and intuitive criterion is proposed using an integral inequality involving the output function and a modified detectability condition. When applied to systems having a Lyapunov function, the well-known LaSalle's invariance principle can be deduces based on our approach. A similar criterion, called the integral invariance principle, was proposed by Byrnes and Martin for nonlinear time-invariant systems. In general, it can not be directly applied to time-varying systems. Our proposed method can be viewed as an extension of the integral invariance principle for time-invariance systems to time-varying systems. The extension is not trivial and can be used in divergence research area, including the adaptive control, the tracking control and the control of driftless systems. In particular, a globally tracking controller for 4-wheeled mobile robots is proposed based on the invariance principle derived in this paper. From this example, it can be seen that as LaSalle invariance principle is used in study the stability of time-invariant systems, our results can be also used to study the stability of time-varying systems along a similar argument.