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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEPTEMBER 1994 1845

A Supervisory C ontroller for 'FuzzyControl Systems that Guarantees StabilityLi-Xin Wang

A6stract- A supervisory controller is a controller which operatesonly when some undesirable phenomena occur, e.g., when the state hitsthe boundary of constraint set. In this note, we develop a supervisorycontroller for nonlinear fuzzy control systems. The supervisory controllerworks in the following way: if the fuzzy control system (without thesupervisory controller) is stable in the sense that the state is inside theconstraint set, the supervisory control is idle; if the state hits the boundaryof the constraint set, the supervisory controller begins operation to forcethe state back to the constraint set. We prove that the fuzzy controlsystem equipped with this supervisory controller is globally stable inthe sense that the state is guaranteed to be within the constraint setspecified by the system designer. We also propose schemes by continuouslyswitching between supervisory and nonsupervisory modes. Finally, weapply a fuzzy controller with the supervisory controller to the invertedpendulum balancing problem where we require that the state variablesmust be within a fixed bound.

I. INTRODUCTIONConceptually, there are at least two different approaches to guar-antee the stability of a fuzzy control system. The first approach is

to specify the structure and parameters of the fuzzy controller suchthat the closed-loop system with this fuzzy controller is stable. Anexample of this approach is [l]. This approach often requires thefuzzy controller to satisfy some strong sufficient conditions whichgreatly limit the design flexibility and, therefore, the performance ofthe fuzzy controller. In the second approach, the fuzzy controlleris designed first without any stability consideration, then anothercontroller is appended to the fuzzy controller to take care of thestability requirement. Because there is much flexibility in designingthe fuzzy controller in this second approach, the resulting fuzzycontrol system is expected to show high performance. In this note,we will detail the second approach.The key is how to design the appended controller to guaranteestability. Because we want the fuzzy controller to perform the maincontrol action, the appended controller would be better a safeguardrather than a main controller. Therefore, we choose the appendedcontroller to work in the following supervisory fashion: if the fuzzycontroller works well, the appended controller is idle; if the purefuzzy control system tends to be unstable, the appended controllerbegins operation to guarantee stability. Thus, we call the appendedcontroller a supervisory controller. In this note, we say a system isstable if its state variables are uniformly bounded.In Section 11, we show the details of how to construct a supervisorycontroller for a nonlinear fuzzy controller system where the fuzzycontroller already exists and propo se mo difications of the supervisorycontrol which switch to the supervisory mode gradually. In Section111, we apply the supervisory controller to the inverted pendulumcontrol problem. Section IV concludes this note.

Manuscript received June 4, 1993; revised September 15, 1993.The author is with the Department of Electrical and Electronic Engineering,Hong Kong University of Science and Technology , Clear Water Bay, HongKong.IEEE Log Number 94027 10.

11. DESIGNOF THE SUPERVISORY CONTROLLERConsider the non linear system governed by the differential equation

where x E R is the output of the system, U E R is the control,-x = (2 ,k , . . .,z ( ~ - ' ) ) ~is the state vector which is assumed tobe measurable or computable, and f an d g are unknown nonlinearfunctions. We assume that g > 0. From nonlinear control theory[2] we know that this system is in normal form, and many generalnonlinear systems can be transformed into this form. The mainrestriction is that the control U is required to appear linearly in theequation.Now suppose that we have already designed a fuzzy controller

21 = U f k ) (2)for the system. This can be done by synthesizing fuzzy control rulesfrom human experts andor by trial and error using designing tools.Ou r task is to guarantee the stability of the closed-loop system and,at the same time, without changing the existing design of the fuzzycontroller u f .More specifically, we are required to design a con trollerwhose main control action is the fuzzy control us and that the closed-loop system with this controller is globally stable in the sense thatthe state g is uniformly bounded, i.e., Ig(t)l 5 M,, W > 0, whereM , is a constant given by the designer.For this task, we append th e fuzzy controller u f with a supervisorycontroller u3 which is nonzero only when the state 4hits the boundaryof the constraint set {g:121 5 M,}, i.e., the control now is

U = U&) +I*U&) (3)where the indicator function I* = 1 if 2 M , an d I* = 0if < M,. Therefore, the main control action is still the fuzzycontrol u f . Our task now is to design the us such that we alwayshave Ig(t)l 5 M , for all t > 0.Le t us first examine whether it is possible to design such a super-visory controller without any additional assumption. Substituting (3)into ( 1 ) we have that the closed-loop system satisfies

(4)Now suppose = M , and thus I" = 1.Because we assume thatf(g)an d g (g ) are totally unknown and can be arbitrary nonlinearfunctions, for any we can always find f ( g ) and g(g) suchthat the right-hand side of (4)is positive, and therefore we will have1g1 > M,. Thus, we must make some additional assumptions onf(g)an d g (g ) for such us design possible. We need the followingassumption.

Assumption: We can determine functions fU (g)an d gL (g)suchthat lf(g)I 5 f ' (g) an d 0 < g L ( g ) 5 g (g ) , i.e., we assume thatwe know the upper bound of I f(g)l and the lower bound of g ( g ) .In practice, the bounds f U ( g )and g L ( c ) are usually not difficultto find because we only require to know the loose bounds, i.e., f U ( g )can be very large and gL(z) can be very small. Also, we require tohave state-dependent bounds, which is weaker than requiring fixedbounds.Before we design the supervisory controller us ,we need to writethe closed-loop system equation into a vector form. First, define

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1846 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 9, SEPIZMBER 1994

where & = ( k , , . . . , k ~ ) ~E R is such that all roots of thepolynomial S + k1sn- + . . .+ k , are in the left-half complexplane. Using this U* , we can rewrite (4) asDefine

A, =

34 ) = -&Tg + g [ U f - U* + I*U,].

0 1

- 0 1 0 0 . 0 0 1 0 . . .0 0 0 0 .. .... .. . . .. .. . .. . . . . . . .

--kn -knpl . . . . .. . . . . . .0

b , = [j.]then (6) can be written into the vector form

Now we design the supervisory controller us such that 121 5 M,.Define the Lyapunov function candidatev = r2 4Tp 4 (10)

where P is a symmetric positive definite matrix satisfying theLyapunov equation(1 1)ATP+ PA, = -Q

where Q > 0 is specified by the designer. Because A, is stable, suchP always exists [2]. Using (9) and (11) and considering the case141 2 M ,, we have

V = -$gTQg+gTPb,[uf - U* + u.]5 14TPb,l(lufI+b*l)+zTPb,us. (12)

Our goal now is to design us such that V 5 0, i.e., the right-hand sideof (12) is nonpositive. Observing (12) and (5 ) , we choose the U, as

Substituting (13) into (12) we see that we have V 5 0. Therefore,the supervisory controller us of (13) guarantees that 141 is decreasingif Igl 2 M ,, therefore if we choose the initial Ig(0)l 5 M,, wealways have 121 5 M, . Because g > 0 an d 4 an d P are available,sign (gTPb,) in (13) can be determined. Also, all other terms in (13)are available, thus the U, of (13) can be implemented on-line.Because the I* in (3) is a step function, th e supervisory controllerbegins operation suddenly as g hits the boundary 141= M , and isidle as soon as the 4 is back to the interior of the constraint set141 5 M ,, therefore the system may oscillate across the boundaryline 141= M,. One way to overcome this chattering problem is tole t I* continuously change from zero to one. Specifically, we maychoose the I* as

0, 141< a1, 1412 M z (14)I*

- M - aMzpa a 5 141