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Robustness of luenberger observers: Linear systems stabilized via non-linear control

Authors
Journal
Automatica
0005-1098
Publisher
Elsevier
Publication Date
Volume
22
Issue
4
Identifiers
DOI: 10.1016/0005-1098(86)90046-4
Keywords
  • Robust Stabilization
  • Observers
  • State Estimation
  • Uncertain Systems
  • Lyapunov Functions
  • Non-Linear Control
Disciplines
  • Design
  • Mathematics

Abstract

Abstract 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 non-linear. 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 non-linear. It is also of interest to note that the approach given here guarantees stability for some non-zero 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 combined plant-observer-controller system.

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