Dynamic Output-Feedback Control for Extended Singular Systems

Abstract

This thesis addresses the dynamic output-feedback control for extended singular systems in terms of linear matrix inequalities. First, this thesis provides some backgrounds about the dynamic output-feedback control and the extended singular systems including singular Markovian jump systems and singular fuzzy systems. Second, dynamic output-feedback control for singular Markovian jump systems is considered. Third, dynamic output-feedback control for singular fuzzy systems is studied. Fourth, dynamic output-feedback control for singular Markovian jump fuzzy systems is investigated.

Chapter 2 considers the dynamic output-feedback control for singular Markovian jump systems. In Section 2.1, for the dynamic output-feedback stabilization of continuous-time singular Markovian jump systems, this section introduces the necessary and sufficient condition, whereas the previous researches suggested the sufficient conditions. A special choice of the block entries of Lyapunov matrices leads to derive the necessary and sufficient condition in terms of linear matrix inequalities. Two numerical examples show the validity of the derived results. In Section 2.2, the problem of dynamic output-feedback stabilization for singular Markovian jump systems with partly unknown transition rates is considered. First of all, for the augmented systems, the stabilization conditions are formulated in terms of non-convex matrix inequalities. For these conditions, this section successfully derives new necessary and sufficient conditions in the form of linear matrix inequalities under partly unknown transition rates by using the variable elimination technique. Two numerical examples are provided to demonstrate the validity of the derived results. In Section 2.3, a dynamic output-feedback stabilization problem of descriptor Markovian jump systems with generally uncertain transition rates is considered. First, a new necessary and sufficient condition to relax inequalities including generally uncertain transition rates is introduced. For the closed-loop systems with a dynamic output-feedback controller, the stabilization criterion is achieved as non-convex matrix inequalities. For the obtained criterion, this section gives an improved necessary and sufficient condition in terms of linear matrix inequalities under completely known transition rates. Then, the proposed condition is extended for the descriptor Markovian jump systems with generally uncertain transition rates. To show the validity of the proposed control, a numerical example is given.

Chapter 3 considers the dynamic output-feedback control for singular fuzzy systems. In Section 3.1 a dynamic output-feedback control for singular T-S fuzzy systems is introduced in light of fuzzy-weighting dependent Lyapunov functions. Based on a set-equivalence technique and fuzzy-weighting dependent Lyapunov functions, this section derives a sufficient condition ensuring the singular T-S fuzzy systems to be admissible (regular, impulse-free and stable) in the form of strict parametric linear matrix inequalities (PLMIs). Then, for the closed-loop systems with a dynamic, rather than static, output-feedback controller, an admissibility criterion is given with strict PLMIs by specially representing the block entries of matrices in Lyapunov function. By appropriately choosing the structures of PLMI variables, the strict LMIs are obtained, where slack variables are included in the relaxation process fully exploiting the properties of the fuzzy weighting functions. Three numerical examples are provided to show the effectiveness of the introduced dynamic control. In Section 3.2, an admissibilization condition for singular interval-valued fuzzy systems with a dynamic output-feedback controller is introduced using a linear matrix inequality (LMI) approach. The derivation of the admissibility criterion (satisfying regularity, non-impulsiveness and stability) for the closed-loop system of the singular interval-valued fuzzy systems using the dynamic output-feedback controller is concerned. Here, the derived criterion is represented as the parameterized matrix inequalities depending on the membership functions of the system and the controller. To relax the derived parameterized matrix inequalities, this section proposes a relaxation lemma based on the properties of the membership functions and their relations. By using this lemma, the parameterized matrix inequalities are converted into the matrix inequalities independent of the membership functions but not convex. Therefore, by introducing the structures of the variables and the congruent transformation matrix, a sufficient condition for the admissibility criterion is successfully given in terms of strict LMIs. A numerical example is given to show the feasibility of the proposed control.