Semi-Global Stabilization of a Single Input Single Output Nonlinear System by Linear Low-and-High Gain Output Feedback

A globally asymptotically stable nonlinear system cascaded by a single input single output linear system through its output has been shown to be semi-globally asymptotically stabilizable by low-and-high gain feedback of the state of the linear system if the linear system is controllable and with its invariant zeros located at the origin. Such low-and-high state feedback can be implemented by an observer. To retain a domain of attraction arbitrarily close to the domain of attraction under a given state feedback, high gain observer is employed to achieve arbitrarily fast decay of the observation errors of all states and the effect of the peaking phenomenon associated with the high observer gain is overcome by saturating the control input outside a region that contains the desired domain of attraction. In this paper, we propose a co-design of the linear low-and-high gain state feedback and the high gain observer for semi-global stabilization of such a cascaded system without resorting to saturating the control input. Moreover, our design does not rely on making all state observation errors decay to zero arbitrarily fast.