Observers Design For Discrete Fuzzy Singular Time-delayed Systems And Solving Fuzzy Relations

This paper focuses on observers design for discrete fuzzy singular sys-tems and solving problems of fuzzy relations. For discrete fuzzy singular time-delayed systems, this paper gives concrete methods of designing ob-servers. About existence of observers, both delay-depend and delay-independ conditions are obtained for discrete fuzzy singular systems with constant time delays, while two delay-depend conditions are derived for rectangular discrete fuzzy systems with multiple time-varying delays using another ap-proach. In addition, based on the semi-tensor product (STP) of matrices, this paper considers solving the general decomposition problem of fuzzy rela-tions. Furthermore, solutions to a class of fuzzy relational inequalities (FRI) are investigated. First, equivalent forms of FRI are obtained via equivalent transformations. Then, efficient algebraic methods are given for solving the new forms.In Chapter 1, the background and some existed results of fuzzy ob-servers are introduced first. Then, for fuzzy relation we give a part of intro-duction, and show some results on the fuzzy relational calculations.Chapter 2 discusses methods of researching time-delayed systems, and introduces the backgrounds of singular systems and T-S fuzzy models. Fur-thermore, the definition of STP and some corresponding basic properties are reviewed.Chapter 3 investigates observers design for discrete fuzzy singular sys-tems with constant time-delays. Firstly, via equivalent transformations, the considered systems become standard discrete systems with constant time-delays. Then, via constructing common Lyapunoy functions, both delay de-pended and delay independed conditions on observers existence are obtained. Although delay depended condition is stronger, the delay independed one is also useful, especially when appropriate observers can not be designed with delay depended condition. However, for some fuzzy systems, methods of this chapter are inapplicable. Because appropriate common Lyapunov functions can not be found and there exist observers for those systems in fact. Hence, another design approach is discussed.Chapter 4 considers piecewise observers design for rectangular discrete fuzzy systems with multiple time-varying delays. First, we try to equivalently convert the interesting rectangular systems to square ones under the assump-tion of observability. Then, the final system models become standard discrete systems with multiple time-varying delays. Next, via choosing appropriate piecewise Lyapunov functions, the existence of observers is discussed. Two different delay depended conditions are obtained and the formulations of de-riving observer gain matrices are given. Compared with common Lyapunov functions, conservatism can be deduced via piecewise Lyapunov functions. We verify this point through examples.Chapter 5 introduces methods of solving general decomposition prob-lems of fuzzy relations (X o Y= R). Firstly, we give problem descriptions and some preliminaries. Then, the structure of solutions is researched and give necessary and sufficient conditions for the existence of solutions. Based on vector expression of multi-valued logic, several important equalities are proved. Taking use of them, we can obtain the algebraic form of considered decomposition problem. For corresponding algebraic form, the equivalent algebraic equations can be derived using proved formulations and STP of matrices. Finally, we show the concrete algorithm of solving corresponding algebraic equations. Then, the concrete algorithm is shown using STP. And two numerical examples are introduced to demonstrate effectiveness. It is noted that the methods of this chapter can also be used to deal with special decomposition problems of fuzzy relations (X o X= R) and solving fuzzy relational equations (X o X o X= R).Chapter 6 discusses how to solve a class of FRIs. In order to analyze considered FRIs more conveniently, we use an operator-Boolean Kronecker product, and prove a series of important transformations, which is related with the new operator. Taking use of them, more simple forms can be easily derived from considered FRIs. Then, for new forms of FRIs, the structure of solutions is studied. Necessary and sufficient conditions are obtained based on STP. Also, the concrete solving algorithm is introduced. Finally, effec-tiveness of solving methods is depicted by numerical examples.