Stability of nonlinear systems with worst case power gain disturbances

When applying nonlinear ℋ_∞ analysis, an important constraint on the system is that all trajectories decay to the origin in the absence of disturbances. In this paper, we introduce a performance measure which does not need this constraint. This allows us to analyse the likes of limit cycle systems. Unlike nonlinear ℋ_∞ analysis, we find that the dynamics in the presence of the worst case disturbance can result in a sizeable limit set S_w. Often S_w bears little relation to the corresponding disturbance free limit set. We propose a Lyapunov like function W whose minimal set is precisely S_w. The set S_w is invariant in that trajectories due to the worst case disturbances in forward and reverse time tend to S_w