Stability analysis for time-dependent nonlinear systems. An interval approach

One of the most important issues in control is determining the stability of a system. Since the 1960’s, Lyapunov-based methods have been developed to determine the stability of linear and nonlinear systems. However, when the system is nonlinear, time-dependent and uncertain, in a set-membership context, stability analysis is challenging and no reliable methods have been developed. This paper proposes an original set-membership based approach for establishing the stability of non-linear, uncertain, time-dependent systems. Two new concepts G-Stability (which is the stability of nonlinear time-dependent uncertain trajectories) and capture tubes (which is an invariant stability region for time-dependent systems) are introduced and illustrated for an autonomous, uncertain, robotic sailboat. Then, G-Stability is used to formulate and prove the safety for a set of capture tubes. This result is then used to analyse the safety of a squad of uncertain, robotic sailboat moving in their environment (no collision among the robots).