IDENTIFICATION AND CONTROL BY FUZZY NEURAL MODELS : THEORY, DESIGN, AND STABILITY ANALYSIS

Abstract

This thesis develops the learning algorithms for min-max neural networks, the general linear-in-the-parameters (LIP) neural models, and the generalized fuzzy radial basis function (f-RBF) networks, and applies them in direct fuzzy control of nonlinear systems and the adaptive minimum variance control for dynamic linear systems. The first task of this thesis is to develop a rigorous differentiation theory for minmax functions to deal with the two important operations min and max that are often involved in fuzzy neural networks, and based on it, further explores the ?-rule and its learning for min-max neural networks. In this thesis, we prove that under certain conditions any min-max function is continuously differentiable almost everywhere in the real number field R, and derive the explicit formula for the differentials of min-max functions. These results are then applied to develop the ?-rule suitable for the training of min-max neural networks. The differentiation theory of min-max functions actually provides a theoretical basis and computation approach for any problem involving min and/or max operations. Especially, it provides a powerful analytic tool to formulate worst-case estimations for nonlinear systems directly in the time domain. The conventional least squares algorithms are often used in dealing with the identification of linear-in-the-parameters (LIP) models. However, they are applied with the pre-condition that their regression matrix should be of full rank. Otherwise, windup and burst phenomena may happen during training process. The second task of this thesis is to develop a new learning algorithm for general LIP models, and further to investigate its application in adaptive minimum variance (MV) control. This thesis first proposes the off-line form of the new learning algorithm, named as minimum quadratic (MQ) learning algorithm, with the combined cost function, and then studies its robust version. The general recursive form and various specialized recursive forms of the new learning algorithm are also further developed in this thesis. The combined cost function, which is a convex combination of the cost function of weighted sum of squared errors and the cost function of maximum of squared errors, is used since it is better than any one of them alone in the sense that it can strike the right balance between the two cost functions, and at the same time, with it we can well estimate the maximum value of errors which is important in robust control. The MQ learning algorithm, derived with the combined cost function, can be extended to a large class of time-variant quadratic forms of cost functions. When choosing different time-variant quadratic forms of cost functions, the MQ learning algorithm is specialized to many interesting and useful off-line or on-line forms for different purposes. Comparing the new MQ algorithms with other training methods, we find that the MQ learning method has several distinct features. It incorporates both the weighted least squares and the worst-case parameter estimations as a special case. It does not assume the regression matrix to be of full rank. What is more, the recursive MQ learning algorithm is free of windup and burst phenomena. The application of the recursive MQ learning algorithms in adaptive MV control is also investigated in this thesis. For a given dynamic linear system, we design many forms of adaptive MV controllers using various specialized recursive MQ learning algorithms. We also provide a general proof on the stability of the closed-loop system with the adaptive MV controller designed by the general form of recursive MQ algorithm. As a direct consequence, all the closed-loop systems are also stable when the adaptive MV controllers are designed by various specialized recursive MQ algorithms. System models for identification and control can be classified into fuzzy model and non-fuzzy model according to whether they process fuzzy data set (knowledge base) or non-fuzzy data set ( data base). The identification and control problems based on non-fuzzy data set have been intensely studied using multilayer neural networks and RBF networks. The identification and control problems based on fuzzy data set have also been studied using the truth value flow inference,· the fuzzy relation equation, the fuzzy graph method, and the Sugeno-Takagi inference method. The third task of this thesis is to build a new kind of fuzzy neural networks, termed as Generalized Fuzzy Radial Basis Function (f-RBF) networks, and then develop its training rule and training strategies based on the differentiation theory of min-max function and fuzzy set theory. Its application in direct fuzzy control for a type of nonlinear systems is also further discussed. The f-RBF networks can process mixed data set integrating both a fuzzy data set (rule data) and a non-fuzzy data set (numerical data). It is a type of fuzzy neural networks which integrates the fuzzifying process, defuzzifying processes, and rule inference process into a united network structure by merging RBF networks and fuzzy systems. When used in fuzzy control systems, we can design a direct fuzzy controller for a given nonlinear system by a group of relatively minimum number of fuzzy rules which are obtained by training an f-RBF network with a combined data set. Using the f-RBF networks and sliding control technique, we prove that any nonlinear stabilizing control can be decomposed into the sum of a fuzzy control, a linear control, and an error compensation. In case the error is very small, the nonlinear controller can be approximately decomposed into the addition of a fuzzy controller and a linear controller.