Nonlinear Output Feedback Control: An Analysis of Performance and Robustness

Abstract

By considering a non-singular performance cost functional, observer backstepping designs and adaptive observer backstepping designs are compared to high-gain observer designs for an output feedback system and a parametric output feedback system. For the output feedback system, if the initial error between the initial condition of the state and the initial condition of the observer is large, the high-gain observer design has better performance than the observer backstepping design. Whilst, for the parametric output feedback system, if the a-priori estimate for the bound of the uncertain parameter is conservative, the adaptive observer backstepping design has better performance than the high-gain observer design. In the sense of gap metric robustness, by a backstepping procedure, a robust state feedback controller is developed for the nominal plant in strick-feedback form. For the closed-loop, the controller achieves gain-function stability, and stability if the initial condition is zero. By the gap metric robustness theory, the controller achieves robustness to plant perturbations which are small in gap sense. In this way, it is shown that for any perturbed plant the controller stabilizes the closed-loop in the presence of input and measurement disturbances if the gap metric distance between the nominal and perturbed plant is less than a computable constant. For output feedback control, a nominal plant in output-feedback form is considered, and the observer backstepping procedure is amended to design a robust controller and an observer in the presence of input and measurement disturbances. The closed-loop is shown to be gain-function stable, and stable if the initial condition is zero. If the nonlinearities are only locally Lipschitz continuous, the results are only local to input and measurement disturbances; if the nonlinearities are globally Lipschitz continuous, then results are global to input and measurement disturbances. By gap metric robustness theory, if the initial condition is zero the controller is shown to be robust to plant perturbations in a gap metric sense. As an application, the theory is applied to a system with time delay, and it is shown that if the time delay is suitably small, the controller is able to achieve stability of the closed-loop. To investigate the robustness of high-gain designs to loop disturbances and plant perturbations, a restricted class of nonlinear nominal plant in normal form are considered. An amended high-gain observer control design is shown to be robust to loop disturbances and has a non-zero plant perturbation margin, which is independent of the high-gain factor.

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