Analysis And Design For Fuzzy Systems Via The Semi-tensor Product Method

Compared with conventional control systems, fuzzy control systems have the fol-lowing two unmatched advantages. First, it can realize easily and effectively human control strategies and experience in many real applications. Second, it can achieve bet-ter control performance independent of the mathematic model of the controlled system. During the past few decades, increasing attention has been devoted to fuzzy logic and fuzzy logic systems, and many nice results have been proposed for the analysis and synthesis of fuzzy control systems promoting the development of fuzzy system theory and its application. The fuzzy logic control has been proved to be a successful control approach to many complex nonlinear systems or even nonanalytic systems.Many of industrial processes and systems have multiple input and output variables. Whereas, in a multi-variables fuzzy systems, the number of rules increases exponen-tially with the number of variable involved. To deal with the rule-explosion problem, a lot of processing methods have been generated. But inference error of many of these methods is inevitable. Moreover, there are two different kinds of uncertainty-the ran-domness and fuzziness often coexist in many physical processes and systems in the real-world, such as robot control systems, power systems and signal processing, etc. As probabilistic methods and fuzzy techniques are effective approaches to deal with the randomness and fuzziness respectively, it is a worthwhile and meaningful job to bridge the gap between fuzziness and probability. Recently, the semi-tensor product (STP) of matrices was proposed, and up to now, it has been widely applied in many fields and lots of fundamental results have been presented. It is noted that fuzzy logic can be consid-ered as an extended mix-valued logic, and by the STP method, the complex reasoning process can be converted into a problem of solving a set of algebraic equations, which greatly simplifies the process of logical reasoning. We hope that this thesis can provide some new ways to study fuzzy systems and further enrich the theory and application of fuzzy systems.In this paper, we investigate the analysis and control design for multi-variable fuzzy systems, hierarchical fuzzy systems and stochastic fuzzy system. The main con-tents of this thesis are composed of the following parts:The first part investigate the fuzzy logic controller(FLC) analysis and design for multi-input multi-output fuzzy systems based on the semi-tensor product of matrices. Firstly, we give a new expression for the fuzzy control rules via expressing the input and output variables with the sector form. Based on this form, we convert the fuzzy reasoning into an algebraic form by constructing structural matrices of the FLC. Then a new framework is established to study multi-variable FLC. According to the proposed approach, a simulation example is given to demonstrate its effectiveness. A set of rea-sonable least in-degree controls are obtained through the analysis of structural matrices. when the control rules are incomplete, the algorithm of least in-degree controls is given. Moreover, the consistency of fuzzy control rules is introduced, and some principles are proposed for dealing with the inconsistency of fuzzy controls. Finally, the proposed ap-proach is applied to design of fuzzy controllers for the energy management and control strategy of parallel hybrid vehicles (PHV).The second part studies the semi-tensor product decomposition of mixed-valued logical functions and the modeling of hierarchical fuzzy systems. Firstly, the approach to the semi-tensor product decomposition of mixed-valued logical functions is pro-posed, and the serial decomposition, parallel disjoint and parallel non-disjoint decom-position have been realized. Then the sufficient and necessary condition is given for the semi-tensor product decomposition of mixed-valued logical functions, and the al-gorithm of the other decomposition function and all its possible solutions are devel-oped when one decomposition function is known. This method is also applicable to the decomposition of k-valued logic function and Boolean logic function. Based on the semi-tensor product decomposition of mixed-valued logical functions, a new kind of scheme is proposed to get the structural matrices of the hierarchical fuzzy systems including serial hierarchical structure, parallel hierarchical structure and hybrid hierar-chical structure. The algorithm of this scheme is developed such that one can easily design the involved structural matrices and fuzzy rules in the muddle layers of the hi-erarchical structure. It is well worth pointing out that, using this method, the same input-output model can be get as the conventional layer fuzzy logic system, and the total number of the rules can be greatly reduced.The third part considers the stochastic fuzzy logic and stochastic fuzzy systems based on one new method, that is, the semi-tensor product of matrices, and obtain some new results about stochastic fuzzy systems. Firstly, some concepts and properties on stochastic fuzzy logic are given. Then, the design of stochastic fuzzy controller is studied based on the semi-tensor product, and the algebraic equation of fuzzy reasoning is obtained. Moreover, the structural matrix and the probability transition matrix are obtained. Finally, a numerical example is provided to demonstrate our new results.Innovations of the thesis mainly include the following aspects:●The algebraic form of fuzzy reasoning of multi-variable fuzzy systems is ob-tained based on semi-tensor product of matrices. when the control rules are incomplete, the algorithm of least in-degree controls has been given. Some principles are proposed for dealing with the inconsistency of fuzzy controls.●The semi-tensor product decomposition of mixed-valued logical functions is pro-posed, and the serial decomposition, parallel disjoint and parallel non-disjoint decom-position have been realized. Based on the semi-tensor product decomposition of mixed-valued logical functions, a new kind of scheme is proposed to get the structural matrices of the hierarchical fuzzy systems including serial hierarchical structure, parallel hierar-chical structure and hybrid hierarchical structure. It is noted that the results obtained in this paper not only are simple, but also have more advantages than the existing ones in some cases.●The stochastic fuzzy logic and stochastic fuzzy systems are discussed based on the semi-tensor product of matrices. By vector representation of stochastic fuzzy rules, the structural matrix and the probability transition matrix are constructed. Then, the algebraic equation of stochastic fuzzy reasoning is obtained.