Robustness of Luenberger Observers: Linear Systems Stabilized via Nonlinear Control

Given a dynamical system whose state equations include time-varying uncertain parameters, it is often desirable to design a state feedback controller leading to uniform asymptotic stability of a given equilibrium point. If, however, the controller operates on some estimate of the state, instead of the true state itself, it is of interest to know whether the desired stability will be preserved; e.g., suppose that the measured output is processed by a Luenberger observer. This paper concentrates on the scenario above and in addition, our analysis permits the controller to be nonlinear. As a first step, inequalities are developed which have implications on the system\´s robustness; that is, when the uncertain parameters satisfy these inequalities, it becomes possible to separately design controller and observer. This amounts to an extension of the classical separation theorem to the case when the controller is nonlinear. It is also of interest to note that the approach given here enables us to guarantee stability for some nonzero range of admissible parameter variations. This is achieved by introducing a certain "tuning parameter" into the Lyapunov function which is used to assure the stability of the cambined plant-observer-controller system.