Observer-based Robust Control Synthesis for Polytopic Uncertain Systems

Abstract

It is important to consider model uncertainties of systems in practical and realistic engineering applications. Model uncertainties are usually expressed as norm-bounded type or polytopic type. Especially, polytopic uncertain systems are a significant class of control systems from both theoretical and practical standpoints, because many control systems are identified as polytopic uncertain systems. In addition, polytopic uncertainty systems usually describe parametric uncertainty in engineering practice more precisely than the norm-bounded uncertainty. While there have been many research works on the observer-based controller design, most of them dealt with norm bounded uncertain systems. Due to the difficulties for selecting a nominal observer model, the observer-based controller design scheme for polytopic uncertain systems has not been reported yet.

With this background, this thesis contributes to design an observer-based controller not only for regulation problems but also for reference tracking problems for discrete-time systems with the polytopic uncertainties. To this end, the robust controller and observer gains are computed such that all possible controller and observer poles of the uncertain systems are located in the open unit disk of the complex plane. Finally, the observer model is selected in a way that the whole closed-loop system is stable. For the sake of selecting such an observer model, it is first expressed as a convex combination of known models and then the convex combination coefficients are computed by solving bilinear matrix inequalities (BMIs). Unlike the linear matrix ineqaulity (LMI), BMI formulation has some drawbacks such as insufficient computer solvers and difficulties when handling numerical errors. Therefore, we have also presented an alternative algorithm which exploits LMI solver iteratively instead of BMI solver. This algorithm is called iterative linear matrix inequality (ILMI).

We can extend the results into the stability problem of switched systems. Hence, the other topic of the thesis is to analyze the stability condition of switched polytopic uncertain systems. With this analysis, the control synthesis and switching rule for stabilization of switched polytopic uncertain systems are discussed.