Co-design of linear low-and-high gain feedback and high gain observer for suppression of effects of peaking on semi-global stabilization

Abstract

This paper presents a co-design of linear low-and-high gain state feedback and linear high gain observer to suppress the adverse effects of peaking and thus to enlarge the class of nonlinear systems that can be semi-globally asymptotically stabilized by output feedback. It is known that a globally asymptotically stable nonlinear system cascaded by a stabilizable linear system through its output can be semi-globally asymptotically stabilized by a combination of linear low gain and high gain feedback of the state of the linear subsystem as long as the linear subsystem is right invertible and has all its invariant zeros located on the closed left-half plane. Such low-and-high state feedback can be implemented by an observer. To retain a domain of attraction arbitrarily close to that under a given state feedback law, a high gain observer is employed to achieve arbitrarily fast decay to zero 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 the desired domain of attraction under the state feedback law. Existing high gain observer designs thus require the linear subsystem to be observable. The proposed co-design does not resort to making all state observation errors to decay to zero arbitrarily fast and thus allows the linear subsystem to be detectable but not observable. Moreover, the proposed design does not resort to saturating the control input and results in linear feedback laws that are parameterized in a single parameter and easy to tune.