Adaptive fuzzy logic control

ADAPTIVE FUZZY LOGIC CONTROL Hoon Kang and George Vachtsevanos SCHOOL OF ELECTRICAL ENGINEERING GEORGIAINSTITUTE OF TECHNOLOGY ATLANTA, GEORGIA 30332-0250

Abstract The objective of this paper is to introduce an adaptive fuzzy logic control system based on Lyapunov stability criteria. We consider the feedback control system in the crisp domain, and then, obtain the fuzzy control laws under the identification-control principle. The design is based on stability and hierarchy of identification and control. The fuzzy rulebase is stored in a fuzzy hypercube and the fuszy control action is computed via a fuzzy inference mechanism. lnitial conditions for the elements of a fuzzy hypercube are obtained by an off-line fussy clustering mechanism with large-grain uncertainty. T w o fussy algorithms are developed: The first one is called a f u t z y identification-learning algorithm and the second is a furzy control-inferencing algorithm. The fuzzy identification-learning algorithm updates the membership functions on the action side of the rules and the fuzzy control-inferencing algorithm calculates fuzzy control data. This approach guarantees stability, convergence and robustness of the closed-loop feedback system.

1

Introduction: Fuzzy Logic Control

Fuzzy logic control has been cast by many investigators as an expert system paradigm where human-like intelligence may be adopted and applied to some complex systems. This fuzzy logic artificial intelligence setting adjusts controller parameters or membership functions based on specified performance characteristics. We introduce in this paper a systematic design procedure for fuzzy linguistic controllers with adaptive or learning capability. This method can be applied to a class of nonlinear systems which suffer from vagueness uncertainty and dynamic/parametric changes. One way to address these difficulties is to divide the uncertain objects or domains into a finite number of manageable quantities and to assign fuzzy sets to each linguistic represention; then, the relations that gov'ern the control objectives may be obtained [1,2]. Thus, we are dealing with quantitative objects of an infinite point space in terms of qualitative reasoning of finite rules. In this paper, the objects are the vector fields of the invariant manifolds or the switching manifolds in a hyperspace. Since fuzzy set theory includes binary set theory [3], it is more convenient to design and analyze fuzzy logic control systems on the basis of classical control theoretic techniques, and to map the appropriate crisp domains into corresponding fuzzy sets 141. However, heuristics still take part in the design of fuzzy logic controllers, if a priori knowledge of the process is incomplete [5]. For example, the shape and locations of membership functions for each linguistic variable are arbitrarily chosen but the membership function of a linguistic term can be of a particular type rather than a simple triangular function once the possibility distribution function of the linguistic term is found. Moreover, the size of a quantized subspace is another heuristic design parameter. The main advantage of fuzzy linguistic control is the relaxation of certain conditions or assumptions imposed on the process, so that we may address a black box approach by suggesting that the system poleiero orders, linear or time-invariant conditions need not be specified. Other advantages include flexibility that endows one with a linguistic rulebase that matches with a particular application; parallelism that accelerates the time-requirement significantly so that a real-time application is possible by sacrificing the hardware complexity; robustness that allows large-grain uncertainty or changes in the process behavior; and finally, combination of control theoretic techniques and possibility approaches that completes the requirement of stability, convergence, and feedback compensation of structural complexity. The fuzzy semantics of linguistic variables relate to complexity and uncertainty. We also consider fumy optimality as an equivalent concept of classical optimality, as the limiting case of infinity resolution in the universe of discourse where the same performance characteristics are obtained. In 'This research has been partially supported by the MARC/FORD Motor Company under Contract No. E21-693 and the ONR under Contract No. NOOO14-89-J-3113.

0-7803-0236-2 192 $3.00 Q 1992 IEEE

407

fact, there are several rules fired as a result of fuzzy inferencing and these control rules contribute to decision making which may not be optimal. For this reason, a fuzzy logic controller is of a suboptimal type in general. However, we have flexibility in choosing a fuzzy control input which is optimal especially if the control action minimizes an energy criterion 16). Fuzzy logic control is similar to adaptive gain scheduling control, but the regions in the look-up table are fuzzified with grades of membership. In this way, the fuzzy semantics are established, connected, and updated according to some performance index. The resolution, on which the quantitative effects depend, must be predetermined; it will then be increased or decreased via the trade-off between the performance criterion and the implementation cost. Our approach incorporates (i) learning for identification and (ii) inferencing for control so that the fuzzy logic system can be adapted to the process while the fuzzy logic controller simultaneously calculates and applies the control inputs to the process. In Figure 1, the inner-loop performs fuzzy identification (i) while the outer-loop achieves fuzzy control (ii). The objective of fuzzy identification here is to obtain the input/output relation of the dynamic behavior of the process. The fuzzy controller is a set of ‘if-then’ rules consisting of premise fuzzy propositions and consequence fuzzy propositions. We will use fuzzy hyper-cells (the elements of a fuzzy hyper-subspace) for the premise part which has fixed membership functions assigned to each fuzzy cell, and will define fuzzy window operators for the consequence part which has time-varying membership functions as adaptively inferenced control actions. (ii) control loop

1 Desired Commands

FUZZY CONTROLLER

c

PROCESS

Measurements

P

c

b

(i) identification loop

-

-

Figure 1: Block Diagram of Adaptive Fuzzy Logic Control

2 2.1

Main Results: Adaptive Fuzzy Logic Control Mathematical Notations and Definitions

As a preliminary excursion to the adaptive paradigm for fuzzy logic control, we consider some mathematical notations and definitions relating to fuzzy sets and systems. Let X be the universe of discourse for a fuzzy variable z whose membership function is denoted by p a ( z ! . For example, z may be the speed of a car or a steering angle of a truck-trailer. Every membership function has the properties of normality and convexity. Q is a fuzzy set characterizing an element of the linguistic set f ( z ) for the premise part (for example, a =‘z is positive large’), or a real value (an open, closed interval) in 3 for the consequence part (e.g., Q = 5.75). A linguistic variable is defined as a collection of sets V = { z , p a ( z ) ,~(z),S,,,S,,) where f ( z ) is a set of the linguistic terms describing z, S,, is the syntactic rule, and Se, is the semantic rule governing z [7]. It is noted that f ( z ) has a finite number of linguistic terms. However, the following definition of fuzzification has an infinite number of linguistic elements for f ( z ) in the continuous case and this concept is applied to the membership functions for the consequence part of fuzzy rules.

Definition 1: We define a ‘fuzzy window operator’ TA,, as a mapping from X to X x (0,1] with a unique inverse mapping DAa. The fuzzy window operator is a sort of fuzzifier defined as follows:

408

0

3 ~ in , Continuous Universe of Discourse ( X = 92)

(i) Singleton case: A singleton a E X can be represented by the characteristic function c , ( t )

{ F(z-cz)+~ + +

The fuzzy window operator is defined as

7~~: a E X

=

--‘pu(z)

:,(a

(~-AcY