A Research On Optimization Theory Of Mamdani Fuzzy Systems

Known as a kind of widely applied fuzzy systems, Mamdani fuzzy system has been assigned great research value for its three outstanding characteristics.Taking complex industry process control as background, this research focuses on optimization problems of fuzzy systems. The objective of it is to find co-relationships among varieties of parameters and factors of Mamdani fuzzy system and to establish better optimization mechanism. Theory analysis supplemented with computer imitations is adopted as the primary method to advance this study.This study focuses on parameter optimization problem and rule base optimization problem of Mamdani fuzzy system. The main works of this paper can be summarized as the following two sections:In aspect of parameter optimization, the focus is how to enhance the effect and the converging speed of the membership function optimization in which multiple kinds of parameters are involved and the sample set scale of it is relatively small. The study is started from extracting key factors and related key properties that can play a significant role in Miso-Mamdani fuzzy system optimization. Three key factors of Mamdani fuzzy systems which are defined as L-parameter, Q-characteristic function and β-characteristic function are extracted. critical properties such as the sample error is delivered from the board to the center area by the proportion (1+β)~-1, the Q-characteristic function has a special relationship with a designed vector graph and a method to derive the Q-characteristic function from the sample data.Based on the obtained factors and properties, new parameter optimization mechanisms are established. Based on the new representative form of the membership function, a membership function local optimization mechanism for Mamdani fuzzy systems is designed. Since the optimization is approximately converted into a group of special programming problems, Its significant advantages are time saving and better quality of the solution. By applying these key properties about β-characteristic function, a new Mamdani fuzzy system parameter globe optimization mechanism is established. The initial condition of the optimization are an input space partition and a sample set; the optimizing process is decomposed into some typical and relatively simple sub-processes that can be converted approximately into solving optimization problems with linear constraints and with double quadric multi-variable polynomial objective functions and into solving optimization problems of eliminating direction loops from designed vector graphs. These advantages theoretically enable the mechanism to be simpler in form, to be more stable in quality and to be faster in converging speed comparing to the correspondent mechanisms based on evolution. So, it is suitable for application objects that need online optimizing and that the scales of their sample sets available are relatively small. Furthermore, according to thisoptimization principle, more efficient optimization mechanisms for two special kinds of 2-dimensional Mamdani fuzzy systems are erected which convert the parameter optimization problem into sub-problems such as optimization problems with linear constraints and double quadric multi-variable polynomial object functions, clustering problems, problems of eliminating direction loops from designed vector graphs and quadric programming problems. Meanwhile, a set of imitation experiments are carried out to study the mentioned optimization mechanisms. From the results, the functions of the parameter local optimization mechanism are verified. Furthermore, a comparing experiment between the triangle fuzzy set based fast optimization method and the globe optimization method for the 2-dimensional Mamdani fuzzy systems has been carried out, whose result shows the optimization mechanism established in the paper is better.In aspect of fuzzy rule base optimization, the focuses are how to reduce the quantity of rules of Mamdani fuzzy system and how to optimize the distribution of weight degree of each rule. At first, new mechanisms to decompose and to merge Mamdani fuzzy rules are studied. A rule fusing mechanism, which can fuse k+l Mamdani fuzzy rules into k Mamdani fuzzy rules with better performance if the weight degree of the k+l rules are linear related, is presented. On the other hand, the method of rule classing based on constant factor is extended to rule classing based on 1-dimensional function. Based on this idea, a new Mamdani fuzzy rule classing mechanism is established. This new mechanism reduces the rule base scale by replacing the rule quantity with the quantity of rule classes.Since T-S fuzzy control system is viewed as a special kind of Mamdani fuzzy systems, The functions of the rule base optimization mechanism to improve T-S fuzzy control system synthesis are studied. Two major functions are proved theoretically: the rule whose weight degree function are linear related with the weight degree function of the rest can be merged into the rest rules. In this way, the fuzzy system's input/output function doesn't change but the quantity of rule decreases essentially; in turn, the quantity of subsystems is reduced and the controller design is simplified; For a given subsystem, if the weight degree function of a non-dominated rule is larger than a positive number in the subspace, the weight degree of the dominated rule can increase by the rule fusing mechanism. Therefore, the estimated up-board of the uncertain term AAh(/d),ABh(n)declines significantly. In turn, the effect of the piecewise linear synthesis method for T-S fuzzy control systems can be improved. A comparing imitation experiment is carried out to demonstrate these effects.Through this research, the form of couple mechanism between the parameters in rule conclusion part and the membership functions is acquired. And a new parameter optimization decomposing model is established in which the optimization of the two kind parameters can be combined moreefficiently. This optimization mechanism has features of good stability in quality, fast in converging speed and low cost in computation. Under the condition of small sample set, this work can alleviate the degree of experience dependence of the current membership function quick optimization methods.Through this research, a new fuzzy rule fussing mechanism is established. It can be implemented to restrain the scale of Mamdani rule bases and to redistribute weight factors of rules to more appropriate status. This rule fussing mechanism can be applied to improving the effects of typical T-S fuzzy control systems synthesis method.Through this research, three significant factors (L-parameters, ^-characteristics function , ^-characteristics function) and a series of related properties of Mamdani fuzzy systems are extracted. These new theoretical factors and properties can provide grounds for further studying and designing fuzzy system optimization mechanism.