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In this paper we give a general formulation of a number of control problems. This formulation considers a wide class of systems and any globally bounded state feedback controller that renders a certain compact set positively invariant and asymptotically attractive. We develop a converse Lyapunov theorem, and we prove that, by implementing the control law using a high-gain observer, we can recover asymptotic stability of the attractive set, its region of attraction, and trajectories.

In this note, we extend the separation results of a previous work to a case where a globally bounded state feedback controller renders a certain compact set positively invariant and asymptotically attractive. The extension covers a wide range of control tasks that arise in adaptive control, servomechanisms, and practical stabilization. It is shown that by implementing the control law using a high-gain observer, we can recover the performance of the state feedback controller.