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sets and systems Fuzzy Sets and Systems 85 (1997) 4961
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Fuzzy adaptive control of a certain class of SISO discretetime processes 1 JeanMichel Renders a, Marco Saerens b, Hugues Bersini b'* "Laboratoire d'Automatique, cp. 165, Universit~ Libre de Bruxelles, 50 av F. Roosevelt, 1050 Bruxelles, Belgium b lnstitut de Recherches lnterdisciplinaires en Intelligence Arttficielle, cp. 194/6, Universitk Libre de Bruxelles, Avenue Franklin Roosevelt, 1050 Bruxelles, Belgium
Received February 1995; revised July 1995
Abstract In this manuscript, we address the problem of the stability of a certain class of SISO discretetime processes controlled by an adaptive fuzzy controller, by using Lyapunov stability theory. These results were recently obtained for adaptive neural controllers, and are extended here to adaptive fuzzy controllers of Sugeno's type. In order to achieve tracking of a reference signal with this kind of fuzzy system, we allow both the membership functions and the consequent part of the rules to be adjusted by a parameter adaptation law. We first present the gradientbased (steepest descent) adaptation law, and we argue that this gradientbased adaptation law can be simplified dramatically. Thereafter, we show the asymptotic stability of the overall system (the convergence of the tracking error to zero) when using this simplified parameters adjustment law. Unfortunately, this result can only be proved when the outputs of the fuzzy controller can be expanded to the first order around the optimal parameter values that allow perfect tracking; that is, when the parameters are initialized not too far from their optimal values (local stability). However, when the set of tunable parameters is restricted to the set appearing in the linear consequent part of the rules (i.e. the membership functions of the premises are not modified) and when the reference signal is the delayed desired output, the stability result is strictly valid: the parameters do not have to be initialized around the perfectly tuned values. In this case, the algorithm can be simplified further by only considering the sign of the derivative of the output of the process in terms of its last influential input. Keywords: Fuzzy control; Adaptive control; SISO processes
1. Introduction O u r aim in this paper is to discuss some parameter adjustment algorithms that can be used in the framework of adaptive control with fuzzy sys* Corresponding author. Email:
[email protected] 1Part of this work has been presented to the second European Congress on Intelligent Techniques and Soft Computing (EUFIT '94). Aachen.
terns (for motivations a b o u t using fuzzy adaptive controllers, see 147]). It has been argued and illustrated in previous works [2,3,48] the ease to import methods and ideas emerging in the connectionist c o m m u n i t y for control applications as soon as the fuzzy controller is supplied with a gradient m e t h o d for the automatic tuning of its parameters. The same authors [2, 3] have derived such a gradient m e t h o d for one Sugeno's type of fuzzy controllers and p r o p o s e d a D A F C (Direct Adaptive F u z z y
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Controller) methodology very close to a methodology based on neural nets instead [36,37]. An adaptive fuzzy PID has equally been proposed by adding the automatic and online tuning method on a classical fuzzy PID structure [2]. However, most of the algorithms developed until now are gradientbased [2, 8, 26], and do not provide any information on the stability of the overall process. Stability is a fundamental requirement in adaptive control theory [1, 10, 18,21,39], and certainly a crucial issue for the future development and application of fuzzy adaptive controllers. Only recently, researchers tried to design adaptation laws that ensure the stability of the closedloop process (for adaptive fuzzy controllers: see [14,47]; see [9, 16,29] in the context of robotics applications; see [1114, 19, 20, 27, 30, 35, 36, 38, 44] in the context of adaptive control with neural nets). On the other hand, some strong connections between neural network architectures and fuzzy systems have been enlightened [34]. Indeed, radialbasis neural networks (based on local nonmonotonous activation function like Gaussian or trapezoid, rather than global monotonous ones like the sigmoid) are nearly equivalent to Sugeno's type of fuzzy controllers with consequence to make any fuzzyneural comparison or even any neurofuzzy merging often very confusing. In this work, we propose to use Sugeno's type of fuzzy systems for the adaptive control of a certain class of singleinput singleoutput (SISO) discrete processes. In order to achieve adaptive tracking of a reference signal with this kind of fuzzy system, we allow both the membership functions and the consequent part of the rules to be adjusted by applying a parameter adaptation law. Inspired by the work already done in neurocontrol [22], we first present the gradientbased (steepest descent) adaptation law, and we argue that this gradientbased adaptation law can be simplified dramatically. We then apply the Lyapunov stability theory in order to prove the asymptotic stability of the overall system (the convergence of the tracking error to zero) when using this simplified adaptation law [28,34], provided that the initial parameters of the fuzzy system are not too far from their optimal values (local stability). This is different from the work of Wang
[47] and Johansen [13, 14], where affine processes, linear in the input signal, were considered. When restricting the parameter tuning to the consequent part of the rules, linear with respect to the parameters to adapt, the stability results are strictly valid; that is, the parameter values do not have to be initialized around their perfectly tuned values. Notice that a similar result can also be proved for the control of continuoustime processes [30, 36]. These proposed algorithms have been studied experimentally in the framework of neurocontrol [30, 31, 36]. Indeed, for radial basis networks which are a particular case of Sugeno's type of fuzzy systems considered here [34], the proposed parameter adjustment law has been applied to the control of various discretetime processes. We showed that there is no perceptible difference between (1) the gradientdescent algorithm, (2) the algorithm designed by Lyapunov stability theory and (3) the further simplified signbased algorithm also based on Lyapunov theory, either in asymptotic performance, or in convergence speed [30, 31, 36].
2. Description of the fuzzy control system The output of the fuzzy control system (Fig. 1) provides at each time step k the input u ( k ) to the plant. For a singleinput singleoutput process of order p, the fuzzy system is given the input of the reference model r ( k ) , as well as the p last outputs of the plant ( y ( k ) , y ( k  1). . . . , y ( k  p + 1)), and the (p  1) last controller outputs ( u ( k  1),u(k  2), . . . , u ( k  p + 1)), in order to be able to reconstruct implicitly the state of the plant. The fuzzy controller must provide the series of control actions u ( k ) that minimize the difference between the current output and the output of the reference model: e ( k ) = (y(k)

ym(k)).
2.1. P r o c e s s d e s c r i p t i o n
For the sake of simplicity, the algorithm will be described and analyzed in the case of a singleinput singleoutput (SISO) system; for multiple inputs and multiple outputs systems (MIMO systems), see [33]. The process is described in terms of its
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r
rule /~i, and d~, ( m = 0 , 1 , . . . , p  1 ) , ~ (n = 1, ... ,p  1), ~ i are fuzzy sets whose membership functions din(X,&), i • ~ ~ , (~x , • &.), ~ and ~ (x,• &.), are denoted by the same symbols. The vectors i &., i &.i contain the adjustable parameters of the J,,., membership functions; for instance, for Gaussian membership functions, the centers and the variances of the densities. By using the definition of 4~k, we can rewrite (2a) as
Ym
) REFERENCE MODEL
I
~i: IF 4~k is (gi AND r(k) is ~ i Fig. 1. Overall architecture used for fuzzy adaptive control of a discretetime SISO process.
inputoutput representation: y ( k + 1) = F [ y ( k ) . . . . , y ( k  p + 1), u(k  d + 1). . . . , u ( k  p + 1)],
(1)
where d ~> 1 is the time delay or relative degree of the process, and p is the order of the process. F is a continuously differentiable deterministic function. Let us also define ~bk = [ y ( k ) , y ( k 1),..., y ( k  p + 1 ) , u ( k  1 ) , u ( k  2)..... u ( k  p + 1)] T.
In the paper, we deal with Sugeno's type of fuzzy model [41,42]; namely fuzzy models of the form: ~i: IF y ( k ) is ~¢~ AND y ( k  1) is s¢'[ A N D . . . AND y ( k  p + 1) is ~¢pi 1
(3)
where u~(k) is computed by the consequent equation of the ith rule (Eqs. (2a) and (2b)), and the weight w i represents the overall truth value of the ith rule for the input: w i = ago(y(k), i .jj.o).agi(y(k
AND u(k  1) is ~
(2b)
where cg~ is the cartesian product of ~¢o,i d l , . . . , 1. .