ANALYTICAL STRUCTURES, ANALYSIS AND APPROXIMATIONS OF FUZZY CONTROLLERS

Abstract

The great majority of the industrial plants are controlled by means of simple Proportional-Integral-Derivative (PID) controllers due to their simplicity in structure and assure acceptable performance for a wide range of industrial plants. These plants often present characteristics such as high order, time delays, nonlinearities and so on. Many innovative methodologies have been devised in the past 50 years to handle these. Intelligent Control is one of the powerful, robust and popular methods, developing since past 20 years to handle such characteristics. Fuzzy Logic is one of the alternative approaches to intelligent control, which has been used since last four decade due to its various advantages. The various design parameters of fuzzy controllers include, i) number and types of membership functions for input and output variables, ii) t-norm and t-conorm, iii) reasoning method, iv) defuzzification method, and v) normalization and denormalization factors. The analytical studies of fuzzy controllers provide the mathematical foundations for its change. Such study includes the derivation of analytical structure, behavior of derived structure with variation of input variables as well as the change of structural parameters, stability conditions, relations (similarities and differences) with conventional controllers, appropriateness and applicability for control purposes. The available literature on analytical structures are limited to the following parameters: odd numbers of symmetrical triangular and nonlinear membership functions for input variables, symmetrical triangular and singleton (crisp) membership functions for output variables, Intersection and Algebraic Product t-norms, Zadeh-OR and Lukasiewicz-OR t-conorms, different (twelve) methods of reasonings, Center of Area, Center of sum, Height and linear defuzzifier. In fuzzy control scheme, use of trapezoidal membership function is also a common practice. Again if one wants to compare the results of singleton and triangular output membership functions, he has to derive two structures individually. Also, during last two decades, studies have been reported on tuning of membership functions using genetic algorithm, neural network and optimization methods for optimum performance of fuzzy controllers. Hence there is a need of a unique analytical structure which is valid for trapezoidal, triangular and singleton membership functions. This should also give an analytical solution of parameter tuning of membership functions for optimum performance, which are generally done using various tools ofartificial intelligence. In view of above, analytical structures for simplest as well as multi-fuzzy sets FLC are studied in the present work for the following parameters: i) fixed symmetrical triangular membership functions for input variables, ii) intersection and algebraic product t-norm, iii) Zadeh-OR and Lukasiwicz-OR t-conorm, iv) Mamdani, Larsen Product, Drastic product and FIi (proposed by author) reasonings, v) Center of Sum, Center of area, Mean of Maxima, Height, First of Maxima, Last of Maxima, Middle of Maxima defuzzification methods with asymmetrical /symmetrical, trapezoidal/triangular membership functions for output variable. These structures are derived using new formulations for COA, COS, MOMx,Height, FOM, LOM, MOM defuzzification methods. Following are the results obtained, i)unique analytical structures for simplest and multi-fuzzy sets fuzzy controller are derived, which is valid for linear, nonlinear control rules, parameters mentioned in last paragraph, ii) justification of why FOM, LOM, MOM, MOMx defuzzification methods are not appropriate for control purposes, iii) role of overlapping area between two consecutive fuzzy sets on the performance of systems, iv) a proposal of fuzzy implication and analytical study of simplest fuzzy controller using proposed implication is done and found that the proposed implication is appropriate for control purposes, v) comparative performance of derived structures with conventional algorithm of fuzzy control produces same results, vi) stability conditions are derived using well known small gain theorem, and vii) proposed analytical structure of simplest-FLC using drastic product reasoning are derived and analyzed. On the basis of concept of nonlinearity variation of fuzzy controller, it is found that, Drastic Product is not an appropriate reasoning for control purpose for Simplest Fuzzy Controller using output membership function other than singleton. The analytical study of role of each parameter of fuzzy controller over performance of systems are studied and the following results are obtained i) Lukasiewicz-OR t-conorm produces better control action as compared to Zadeh-OR t-conorm, ii) Mamdani Minimum reasoning produces better control action as compared to Larsen Product and Proposed reasoning, and if Larsen Product and proposed reasoning are compared then, proposed reasoning produces better control action, iii) trapezoidal membership function produces better control action as compared to triangular membership function, and iv) on the basis of ISE minimization, the ranking of defuzzification methods is COA>COS>Height for producing the control action, while LOM, FOM, MOM, MOMx are not appropriate for control purposes. Once a unique analytical structure is obtained for symmetrical /asymmetrical, trapezoidal /triangular output membership functions, the analytical study is performed for the following regarding output membership functions: shape, fixed symmetrical and asymmetrical, online regulation of parameters of membership functions. The following results are obtained: i) the analytical structure with asymmetrical fuzzy sets produce better control action as compared to symmetrical. The performance of fuzzy controllers can be improved by changing the shape of output membership functions of negative universe of discourse, ii) the performance of simplest Fuzzy controller can also be improved by online regulation of parameters of output membership functions. It is well known that, the amount of information stored in knowledge-base of fuzzy controller is mainly depends upon the number of membership functions used, for input and output variables. Further, from available literature it is known that, i) the number of membership function is an important design factor for producing effective control, ii) as the number of membership functions increases, the defuzzified output comes closer to a linear function of input as a result the nonlinearity of the fuzzy controllers minimizes. Consequently, as the number of membership function for input and output variables increases, the systems take more time to produce the output, due to the large number of computational steps involved in complex algorithm of approximate reasoning. Hence, if one can approximate the output of MFS-FLC without increasing the membership functions of fuzzy controllers, then the nonlinearity of fuzzy controllers can in be minimized within less computational steps and low computational memory as a result the performance of simplest-FLC is enhanced. In view of above discussions, three methods are developed for approximation of multi-fuzzy sets-FLC with simplest-FLC using the derived analytical structures. These are named as, Approximation Method-1, Modified Approximation Method-I, and Approximation of TKS-FLC. The comparative study of each method with conventional multi-fuzzy sets-FLC with application to control systems has been done and found that, the proposed approximation methods produces nearly same results. The common features of all methods are i) minimization of computational memory, ii) minimization of computational steps and iii) freedom to mould the control surface of fuzzy controller according to our need via changing the compensating factors. A significant result has been obtained by changing the compensating factors. It has been found that, the scheme is able to handle wide range of characteristics like time delays, high order, nonlinearities, which are difficult to control by conventional and existing Intelligent Control schemes. The tuning criterion for compensating factors has also been proposed. The work undertaken in this dissertation provides the analytical solutions of many new aspects of Fuzzy Controllers, few of which are i) unique analytical structures for trapezoidal, triangular and singleton output membership functions, ii) use of asymmetrical output membership functions for enhancing the performance of Fuzzy Controllers, iii) approximations of large rule Fuzzy Controllers by four rule Fuzzy Controller, iv) a proposal of new Fuzzy implication which is appropriate for control purpose, v) role of overlapping area between two consecutive Fuzzy sets on the performances of systems and vi) a new way of implementing Fuzzy Controller for handling systems with nonlinearities, large dead time and higher order.