Hierarchical approach to fuzzy logic control

This paper proposes a framework for hierarchically structured control systems that may incorporate fuzzy as well as conventional control strategies in an analytically tractable form. This framework is further used to develop a formalism whereby certain fundamental system theoretic properties of the resulting closed loop system, such as stability and robustness to modeling uncertainty, can be analytically verified. The proposed approach is based on large scale systems theory and incorporates nonlinear singular perturbation in the study of hierarchicaUy structured control systems. I N T R O D U C T I O N Fuzzy logic based control is fundamentally based on representing the knowledge of operation of a given system in terms described by humans and thus must meet the constraints we often attribute to such knowledge. In particular, the so called fuzzy rule set, representing this knowledge, must be structured in a manner that streamlines the inferetw, e process and preserves the integrity of the resulting knowledge base. In this respect a hierarchical scheme can constitute a suitable structure wherein the fuzzy rule set in conjunction with other forms of structured knowledge carry a sense of hierarchy, or abstraction based semantic structure. The diagram in Figure 1 depicts this idea. Note that the diagram may be extended to mul"Permission to make digital/hard copy of all or part of this material without fee is granted provided that copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery, Inc.(ACM). To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee." © 1996 ACM 0-89791-820-7 96 0002 3.50 605 Figure 1 Hierarchical architecture for fuzzy control. tiple levels. Further, the low level rule set or low level processor is bracketed in the dashed block implies that its inputs come from both the set of observables, that is x , as well as from the high level processor or rule set. Usually the consequent clauses of the high level rule set determines the settings for variables used in the low level rule set. Observables may be directly measureables such as process outputs, or indirectly estimated ] . With this in mind we consider the application of fuzzy logic in complex multi-layer systems where both implementation and analysis do not immediately follow the streamlined approach that has been proposed in connection with single layer fuzzy controllers. We do not suggest, however, that some esoteric techniques need to be employed. Rather, a careful appfication of established tools may very well lead to useful results. S I N G U L A R P E R T U R B A T I O N M O D E L O F F U Z Z Y C O N T R O L S Y S T E M S A proper framework for analytical study of fuzzy control systems must take into account the approximate nature of any functional model. Further, in our view, such l . W e will s ee e x a m p l e s o f this shor t ly . a framework must deal with the issue raised earlier namely that fuzzy control is essentially meant to be used in complex systems. One possible approach here is based on considering the plant model as f ( x ) + g ( x ) u with f ( . ) and g(.) each made up of segments that correspond to the various time scales of the plant behavior, that is V/1(x) f(x) = :2(x>l . :3(x)/ The resulting system would look like (1) = fI(Xl, X2, X3)+ ~ gl, i(XI'X2'X3)Ui i= I,m (2) similarly for x 2 or x 3 as for instance in eg 2 = f 2 ( X l , X2, X3) + ~_~ g2, i(Xl, X2, X3)Ui i= 1,m (3) Note that we have made the dependence of f l ( . ) and f l (-) on segments of x explicit for clarity. Also note that we need not be limited to three stages or time scales; in some cases, we may have two, and in others, there may by more than three segments to the state vector. Three seems natural, however. In process control, for instance, valves and servos are the fastest subsystems, the local control systems are the medium, and the slowest time scale represents the overall behavior of the system. The fact that is important here is that the medium and fast subsystems are describable using standard dynamical system models; they arc relatively simple and can generally be stabilized using conventional control techniquesin particular linear control laws. That would mean we can have stable fast and medium time scale subsystems, and in particular, they can be sufficiently stable as to provide the basis for a sensible use of the singular perturbation framework in the present context[12,8,6]. Based on this idea, we can have C 1 functions[lO] h I and h 2 such that x 2 = h2(Xl) x 3 -h3(Xl) , (4)