Dynamic high-gain scaling: State and output feedback with application to systems with ISS appended dynamics driven by all States

We propose a dynamic high-gain scaling technique and solutions to coupled Lyapunov equations leading to results on state-feedback, output-feedback, and input-to-state stable (ISS) appended dynamics with nonzero gains from all states and input. The observer and controller designs have a dual architecture and utilize a single dynamic scaling. A novel procedure for designing the dynamics of the high-gain parameter is introduced based on choosing a Lyapunov function whose derivative is negative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). The system is allowed to contain uncertain terms dependent on all states and uncertain appended ISS dynamics with nonlinear gains from all system states and input. In contrast, previous results require uncertainties to be bounded by a function of the output and require the appended dynamics to be ISS with respect to the output, i.e., require the gains from other states and the input to be zero. The generated control laws have an algebraically simple structure and the associated Lyapunov functions have a simple quadratic form with a scaling. The design is based on the solution of two pairs of coupled Lyapunov equations for which a constructive procedure is provided. The proposed observer/controller structure provides a globally asymptotically stabilizing output-feedback solution for the benchmark open problem proposed in our earlier work with the provision that a magnitude bound on the unknown parameter be given.