A revisit to inverse optimality of linear systems

Abstract

In this article, we revisit the problem of inverse optimality for linear systems. By applying certain explicit formulae for coprime matrix fraction descriptions (CMFD) of linear systems, we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index. Compared to existing results in the literature, the proposed condition is simpler to check and interpret. Moreover, it reduces the redundancy in the solutions of the associated algebraic Riccati equation (ARE). As a direct application of the proposed results, we consider the problem of inverse optimality of observer-based state feedback. To be specific, for the case where the state is not fully known, we consider the inverse optimality problem of an observer-based state feedback for the closed-loop system augmented by an observer. More precisely, it is shown that the observer-based state feedback is inverse optimal for the closed-loop system with some general forms of cost functions, only if the original state feedback is inverse optimal for the original system with certain cost functions, irrespective of the choice of the observer. This coincides with existing results in the literature. Some other applications of the proposed results are also discussed. We also illustrate the proposed results through an example.

Keywords:

• inverse optimality
• coprime matrix fraction description
• algebraic Riccati equation
• observer

Acknowledgements

The work of the authors was partially supported by University of Newcastle and also by the Australian Research Council through Discovery Projects. The authors would like to thank the anonymous reviewers for their helpful comments and suggestions which have helped us to improve the quality of this article.