ANALYSIS, TUNING AND DEVELOPMENT OF FUZZY PID CONTROL SYSTEMS

Abstract

Although fuzzy control has been in existence for over two decades, the complexities of fuzzy control from a conventional theoretical point of view has hindered its wider acceptance by the larger control communities. One clear difficulty is that one cannot guarantee control performances in terms of conventional mathematics. The problem arises because fuzzy logic mathematics deals largely with fuzzy mathematics while conventional control analysis deals with differential equations, time domain and frequency domain analysis. Hence, any research effort which sets out to analyze and design a fuzzy controller in a manner that meets the acceptance of both fuzzy and conventional control communities could not be an easy task. However, a bridge to link up the two diverse communities is sorely needed if fuzzy control is to expand and grow rapidly to complement conventional control. The objective of this present work is to seek a systematic approach to (1) analyze the internal structures of fuzzy controllers, (2) design and tune the fuzzy controllers, and (3) develop and synthesize fuzzy control systems to achieve specified control performance specifications. This thesis also presents a methodology that translates fuzzy control systems into its equivalent conventional control system by visually mapping out the internal structures of fuzzy controllers. This mapping allows fuzzy control systems to be simplified and transformed into the conventional control domain space. It therefore enables conventional analysis to be performed on a fuzzy control system for the purpose of evaluating its stability, transient and steady-state performances. It also leads to the development of simplified analytic models of fuzzy controllers which behaves dynamically like a fuzzy controller but does not depend on fuzzy mathematics in real-time computations. The author contributed to the objective of theoretical development and understanding of the input-output structure of fuzzy controllers by introducing the novel concept of fuzzy space. Using a visualization technique, the author successfully translates the fuzzy control domain into the conventional control domain. In general, a fuzzy controller is nonlinear. Its nonlinearity usually exists in two main forms, one introduced by the designer arising from the use of heuristic rules. The other arises from the definition of the input fuzzy sets. The former results in various degrees of non-linearity. The latter is usually a severe form of non-linearity. The net result of these two types of non-linearity is the peculiar behavior of fuzzy control that is very different from conventional proportional-integral-derivative (PID) control. The author contributed towards the understanding of these two main forms of nonlinearity by introducing the novel notion of fuzzy variable structure control (FVSC). The variable structure is uncovered when the fuzzy space is transformed into a conventional state space and the output contour mapping obtained from the visualization technique is examined in the state space, The superposition of the fuzzy space onto the state space results in a combined space or a common platform. Analysis using the combined space shows that a fuzzy controller in many situations operates as a variable structure controller that partitions the state spaces into different operating regions. The manner in which the state space is being partitioned is entirely unique to fuzzy control systems. A fuzzy control system is usually confined to two or three inputs, since a human being is unlikely to factor in too many inputs anyway. Consequently, the combined space in 2D or 3D is usually adequate to address most of the practical design issues, The common platform also links together fuzzy mathematics with conventional analog or discrete control mathematics so that analysis of control performances can now be done using the same control terminology. For instance, if a linearly structured fuzzy controller operates in its linear region in a linear process, the conventional frequency domain analysis, such as gain margin, phase margin, steady-state error analysis, etc., can be readily applied. Linear analysis is also a good approximation in linear processes controlled by a mildly nonlinear-structured fuzzy controller. The distinction between conventional and fuzzy control is not significant in such a situation. A case in point is that arising from its mild form of nonlinearity, a fuzzy PI controller can be tuned using the approach applicable to a conventional PI controller. On the other hand, a fuzzy controller outperforms a conventional PID controller when its operating structure becomes highly nonlinear, as in the case of a fuzzy PD controller. Here, linear analysis no longer applies and nonlinear control analysis takes over. The author contributed towards the nonlinear analysis of the nonlinear fuzzy PD controller by using Lyapunov stability analysis to examine each region in the nonlinear mapping. By ensuring that stability conditions are satisfied in each region, the fuzzy controller can be designed to be globally stable. Since the analysis is done in the conventional state space using the conventional nonlinear control analysis technique, it is acceptable to both communities. This thesis is also devoted to the theoretical analysis of control performances, simulations and experimental verification of the methodology and design applications. The technique discovered by the author also implies that conventional computerized control design software can be readily extended to fuzzy control systems with little additional effort. A control designer can therefore choose to apply a conventional design approach or a fuzzy design approach as he deems fit for his application. Fuzzy control is therefore completely integrated into the toolbox of the control community at large.