Stable Adaptive Fuzzy Controllers with Application to ... · that controller designers can...

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART B: CYBERNETICS, VOL. 26, NO. 5, OCTOBER 1996 611

Stable Adaptive Fuzzy Controllers with Application to Inverted Pendulum Tracking:

Li-Xin Wang,

U

Member, IEEE

Abstruct- An adaptive fuzzy controller is constructed from a set of fuzzy IF-TH[EN rules whose parameters are adjusted on-line according to some adaptation law for the purpose of controlling the plant to track a given trajectory. In this paper, two adaptive fuzzy controllers are designed based on the Lyapunov synthesis approach. We require that the final closed-loop system must be globally staible in the sense that all signals involved (states, controls, pariuneters, etc.) must be uniformly bounded. Roughly speaking, the adaptive fuzzy controllers are designed through the following steps: first, construct an initial controller based on linguistic descriptions (in the form of fuzzy IF-THEN rules) about the unknown plant from human experts; then, develop an adaptation law to adjust the parameters of the fuzzy controller on-line. Wie prove, for both adaptive fuzzy controllers, that: 1) all signals in the closed-loop systems are uniformly bounded; and 2) the tracking errors converge to zero under mild conditions. We proviide the specific formulas of the bounds so that controller designers can determine the bounds based on their requirements. Finally, the adaptive fuzzy controllers are used to control the inverted pendulum to track a given trajectory, and the simulation results show that: 1) the adaptive fuzzy controllers can perform successful tracking without using any linguistic information; and 2) after incorporating some linguistic fuzzy rules into the controllers, the adaptation speed becomes faster and the tracking error becomes smaller.

I. INTRODUCTION UZZY control has been successfully applied to many F commercial products and industrial systems, where: 1)

no accurate mathematical models of the systems under control are available; and 2) human experts are available to provide linguistic fuzzy control rules or linguistic fuzzy descriptions about the systems. Conventional nonadaptive control methods require that the mathematical model of the system is known, while most of the existing adaptive control approaches deal only with linear systems. Recently, there have been some researches which use artificial neural networks as building blocks of adaptive controllers for unknown nonlinear systems [lo], [12]-[14]. However, these neural network adaptive controllers cannot incorporate linguistic control rules or linguistic system descriptionis directly into the controllers. Because so much human knowledge is represented in linguistic terms, incorporating it into controllers in a systematic way is very important. Although existing fuzzy controllers are capable of incorporating linguistic information, they are heuristic in nature in the sensa that there are no general design methods which guarantee the very basic requirements like stability,

Manuscript received June 11, 1993; revised May 14, 1995. The author is with th,? Department of Electrical and Electronic Engineering,

The Hong Kong University of Science and Technology, Kowloon, Hong Kong (e-mail: [email protected]).

Publisher Item Identifier S 1083-44 19(96)05357-5.

robustness, etc. The present practical successes of fuzzy control are mainly due to its low developing cost, high-speed implementation, and capability of incorporating linguistic information from human experts. As the application realm of fuzzy control expands from simpler problems (in the sense that there are only a few key variables, and that large number of trial-and-error experiments are permitted, like in washing machines, air conditioners, rice cookers, etc.) to more complex problems (in the sense that there are many key variables, and that unsuccessful trials are not permitted, like in chemical processes, power plants, aircrafts, etc.), there is a urgent need for systematic design methods of fuzzy controllers which: 1) assume no mathematical model of the system; 2) can incorporate linguistic information from human experts directly into the controllers; and 3) guarantee the global stability of the resulting closed-loop system.

The adaptive fuzzy controllers in this paper try to address this kind of problems. Roughly speaking, we use the following ideas. For the first requirement, we use nonlinear adaptive control concepts, where nonlinearity is required in order to cover a wider spectrum of practical systems, and adaptivity is required because the mathematical model is unknown. For the second requirement, we use fuzzy systems as basic building blocks of the adaptive fuzzy controllers so that linguistic fuzzy IF-THEN rules can be directly incorporated into the controllers. Finally, for the third requirement, we use some techniques in conventional adaptive control theory [ 11 J, [ 151, e.g., Lyapunov synthesis, parameter projection, and in conventional nonlinear control theory [5], [ 171, e.g., sliding control. More specifically, we use the Lyapunov synthesis approach to construct the basic adaptive fuzzy controllers, and use the sliding control and parameter projection to guarantee the boundedness of all signals.

In the conventional adaptive control literature, adaptive controllers are classified into two categories [12]: direct and indirect adaptive controllers. In direct adaptive control, the parameters of the controller are directly adjusted to reduce some norm of the output error between the plant and the reference model. In indirect adaptive control, the parameters of the plant are estimated and the controller is chosen assuming that the estimated parameters represent the true values of the plant parameters.

In fuzzy control, linguistic information from human experts can also be classified into two categories: 1) fuzzy control rules which state in what situations what control actions should be taken; and 2) fuzzy IF-THEN rules which describe the behavior of the unknown plant (e.g., we can describe the behavior of a car using the fuzzy IF-THEN rule: “IF

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678 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART B: CYBERNETICS, VOL. 26, NO. 5, OCTOBER 1996

I Fuzzy RuleBase I I

Fuzzifier I' Defuuifier y in V

X ' n u I + fuzzy

in U in V

Fig. 1. Basic configuration of fuzzy systems.

apply more force to the accelerator, THEN the speed of the car will increase," where the "more" and "increase" are characterized by fuzzy membership functions). Interestingly enough, adaptive fuzzy controllers which make use of these two classes of linguistic information just correspond to the direct and indirect adaptive control schemes, respectively. More specifically, direct adaptive fuzzy controllers use fuzzy systems as controllers, therefore linguistic fuzzy control rules can be directly incorporated into the controllers. On the other hand, indirect adaptive fuzzy controllers use fuzzy systems to model the plant and construct the controllers assuming that the fuzzy systems represent (approximately) the true plant, therefore fuzzy IF-THEN rules describing the plant can be directly incorporated into the indirect adaptive fuzzy controller. In this paper, we develop two indirect adaptive fuzzy controllers. A direct adaptive fuzzy controller was developed in [20].

In Section 11, we present a detailed description of fuzzy systems. In Section 111, we show the basic ideas, in a constructive manner, of how to construct indirect adaptive fuzzy controllers based on the fuzzy systems, and how to use the projection algorithm and sliding control to meet the control objectives. In Sections IV and V, the detailed design steps of two indirect adaptive fuzzy controllers are presented, and the properties (boundedness of the variables, convergence of the tracking error, etc.) of them are analyzed, respectively. In Section VI, the two adaptive fuzzy controllers are used to control the inverted pendulum to track a trajectory. Section VI1 concludes the paper.

11. DESCRIPTION OF FUZZY SYSTEMS

Fig. 1 shows the basic configuration of the fuzzy systems considered in this paper. The fuzzy system performs a mapping from U c R" to R. We assume that U = U1 x . . . x U,, where U, c R, i = 1 , 2 , . . . , n. We now present a detailed description of each of the four blocks in the fuzzy system in Fig. 1.

A. Fuzzy Rule Base

THEN rules 'The fuzzy rule base consists of a collection of fuzzy IF-

E('): IF ( 2 1 is F; and . . . and 5 , is FA)

THEN y is G1 (1)

where x = (51,. . . , x , ) ~ E U and y E R are the inputs and output the fuzzy system, respectively, F: and G' are labels of fuzzy sets in U, and R, respectively, and 1 = 1 , 2 , . . . , M . Each fuzzy IF-THEN rule of (1) defines a fuzzy implication 181 F; x e . . x FA i G' which is a fuzzy set defined in the product space U x R. Based on generalizations of implications in multivalue logic, many fuzzy implication rules were proposed in the fuzzy logic literature. Here, we quote four commonly used fuzzy implication rules [8]:

Min-operation rule of fuzzy implication

b F j x xF;-G2 (x , = min[pF,lx . . X F A ( ~ ) , P G ~ ( Y ) ~

Product-operation rule of fuzzy implication

PF," x xFi+Gl ( x ~ Y) = pFf x ... xFA ( x ) p G 1 (Y)

* Arithmetric rule of fuzzy implication

P F ; ~ X F ; - G ~ ( X , Y )

min[l, - pP,1 x .xFA t PGL ( y ) ]

* Maximum rule of fuzzy implication

p F , " x X F L + G I ( ~ , Y )

max[min(pFf x x F!, (XI, PGl

- PFj x xF;

where p F i X x F ; ( ~ ) is defined by

P F j x X F A ( ~ ) = P F j ( 5 1 ) * " . * P F ~ ( 2 n )

where "*" denotes the t-norm [8] which corresponds to the conjunction "and" in (1); the most commonly used operations for the t-norm are

min(u, v) fuzzy intersection

max(0, U + U - 1) U * v = uv algebric product (7)

bounded product.

B. Fuzzy Inference Engine

The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in R, based upon the fuzzy IF-THEN rules in the fuzy rule base and the compositional rule of inference [25]. Let A, be an arbitrary fuzzy set in U ; then, each R(l) of (1) determines a fuzzy set, A, o E(') , in R based on the following sup-star composional rule of inference:

PA, oR(2 ) (Y> = [ P A , * PF; x . x FA--.Gl (x, Y)1 (8) XEU

where * is the t-norm (7), and p F j X x F ; + G ~ ( ~ , ~ ) is determined by the fuzzy implication rules of (2)-(5). The final fuzzy set A, o (I?('), . . . , R(")) determined by all the M rules in the fuzzy rule base is obtained by combining pAZoR(2) ( y ) of (8) for 2 = 1 , 2 , . . . , M using fuzzy disjunction

pA,o(R( l ) , ,R(M))(Y) pA,oR(l) ( Y l + . ' ' + p A , o R ( M ) (Y)

(9)


WANG: STABLE ADAPTIVE FUZZY CONTROLLERS 619

where 4 denotes the t-cononn [8]; the most commonly used operations for i are

rrtax(u, w ) fuzzy union

min( 1, U + U) U + w = ci + 'U - U'U algebraic sum (10)

bounded sum.

C. Fuzzifier Thefuzzijier maps a crisp point x = ( 2 1 , . . . , z ~ ) ~ E U into

a fuzzy set A, in U . There are at least two possible choices of this mapping:

(i) A, is a fuzzy singleton with support x, i.e., PA, (x') = 1 for x' = x and p ~ = ( x ' ) = 0 for all other x' E U with x1 # x;

(ii) p ~ , ( x ) = 1 and p~ , (x ' ) decreases from one as x' moves away from x, e.g., pA,(x') = e x p [ - y w ] . where U' is a parameter characterizing the shape of PA, (x').

In the literature, it seems that only the singleton fuzzifier (i) has been used. We think that the nonsingleton fuzzifier (ii) may be useful if the inputs are corrupted by noise.

D. Defuzzifier The defuzzifier rnaps fuzzy sets in R to a crisp point in R.

There i)

i i)

iii)

are (at least) three possible choices of this mapping: maximum defuzzijier, defined as

where CLA,o(R(I), ..., I Z ( M ) ) ( Y ' ) is given by (9); center-average defuzziJer (which is the most commonly used defuzzifier in the literature), defined as

where j j ' is the point in R at which p G ( ~ ) (y) achieves its maximum value, and pAz0R(l) (y) is given by (8); and, modified center-average defuzziJer, defined as

where U' is a parameter characterizing the shape of p ~ i (y) such that the narrower the shape of pGi (y); the smaller is U'; for example, if pGi (y) = exp[-( v)'], then U: is such a parameter.

The modified center-average defuzzifier is justified as follows. Common sense indicates that the sharper the shape of ~ G i ( y ) , the stronger is our belief that the output y should be nearer to y' = arg s u p g r E R ( p ~ ~ (9')) [according to the rule R(') of (l)]. The standard center-average defuzzifier, (12), is a weighted average of the g"s, and the weight pA,oR(i)(yyl) determined by (8) do not take the shape of pGi(y) into consideration. This is clearly not satisfactory based on our common sense. An obvious improvement is the modified center-average defuzzifier (1 3).

Note that if we use the center-average or modified center-average defuzzifiers, we do not need to calculate the ~ A , ~ ( R ( I ) , , . , , R ( M ) ) ( ; Y ) of (9); we only need to calculate the p A c o R ( ~ ) (y) of (8) in the fuzzy inference engine.

E. Two Subclasses of Fuzzy Systems From Sections 11-A to 11-D, we see that the fuzzy systems

of Fig. 1 comprise a very rich class of static systems mapping from U c Rn to R, because within each block there are many different choices, and many combinations of these choices can result in useful subclasses of fuzzy systems. We now consider two subclasses of fuzzy systems which will be used as building blocks of our adaptive fuzzy controllers.

The set of fuzzy systems with singleton fuzzifier, center- average defuzzijier, and product inference is all functions f : U c R" -+ R of the following form:

where x = (XI, . . . , z,)~ E U , j j l is the point at which pGl (y) achieves its maximum value (without loss of generality, we assume that pGt(g') = l), and F," and G' are the fuzzy sets in (1).

The fuzzy system (14) is obtained by substituting (8) into (12) (center-average defuzzifier), substituting (6) into (3) and (3) into (8) and replacing * with algebraic product (product inference), and noticing that if A, is a fuzzy singleton with support x (singleton fuzzifier), then p A z o R ( ~ ) ( y ) = S U P ~ ! ~ U [ p A , nr==, (s ' , )pGi ($')I = H,"=I PF; ( X t ) .

If we fix the p F ; ( z Z ) ' s and view the jj"s as adjustable parameters, then (14) can be written as

Y(X) = e T w (15)

where 8 = ( j j l , . . . , y")* is a parameter vector, and ((x) = (<'(x), . . . , <"(x))~ is a regressive vector with the regressor <'(x) (which is called fuzzy basis function in [23]) defined as

In Section IV, we will use the fuzzy systems (15) as building blocks of our first adaptive fuzzy controller. The advantage of using the fuzzy system (15) is that although the y(x) is a nonlinear function of x, it is linear in its parameter 8 ; therefore, the adaptive fuzzy controller based on it is relatively easier to construct and analyze. The disadvantage of using the fuzzy system (15) is that since we cannot adjust the membership functions pF; ( T ~ ) during the adaptation procedure, the adaptive fuzzy controller is not efficient in utilizing its adjustable parameters (we will make this clear in Section IV). To overcome this disadvantage, we introduce the following fuzzy system.

The set of fuzzy systems with singleton fuzzijier, ceizter- average defuzzijier, product inference, and Gaussian membership ,function consists of all functions f : U C R" -+ R of



the following form: y to follow a given bounded reference signal ym(t), under the constraint that all signal involved must be bounded. More specifically, we have the following.

Y(X> = (17) Control Objectives: Determine a feedback control U = u(x 1 0) and an adaptation law for adjusting the parameter vector 0 such that:

E E ~ g l [ nzl exp (- f (9) ')I CE, [ rIL exp (- ; ( 9) ')I

which is obtained by substituting pF;(z,) = exp(--i

(A )2) (Gaussian membership function) into (14). 6:

We view g', 2: and af > 0 in (17) as adjustable parameters. We see that the fuzzy system (17) is not only a nonlinear function of x, but also nonlinear in its parameters. In Section V, we will use this fuzzy system as building blocks of our second adaptive fuzzy controller.

There are two main reasons for using the fuzzy systems (14) and (17) as basic building blocks of adaptive fuzzy controllers. First, it was proven in [ 191 that the fuzzy systems (14) and (17) are universal approximators, i.e., for any given real continuous function g on the compact set U , there exist fuzzy systems in the form of (14) and (17) such that these fuzzy systems can uniformly approximate g over U to arbitrary accuracy. Therefore, the fuzzy systems (14) and (17) are qualified as building blocks of adaptive controllers for nonlinear systems. Second, the fuzzy systems (14) and (17) are constructed from the fuzzy IF-THEN rules of (1) using some speciAc fuzzy inference, fuzzification, and defuzzification strategies; therefore, linguistic information from human experts [in the form of the fuzzy IF-THEN rules of (l)] can be directly incorporated into the controllers.

111. A CONSTRUCTIVE LYAPUNOV SYNTHESIS APPROACH

In this section, we first set up the control objectives, and then show, in a constructive manner, how to develop adaptive controllers based on the fuzzy systems to achieve these control objectives.

TO ADAPTIVE FUZZY CONTROLLER DESIGN

Consider the nth-order nonlinear systems of the form

x, = 5 2

2 2 = 23

i n = f(X.1, . . . , Z") + g(z1,. . . >%)U

y = 21 (1 8)

or equivalently, of the form

x(n) = f(x,k, . . . ,x(n-l)) + g(z,k, . . . ,z("-l) )U, y = 2 (IS')

where f and g are unknown continuous functions, U E R and y E R are the input and output of the system, respectively, and x = (z1,x2,. . . , = (z,i,. . . , z ( " - ~ ) ) ~ E R" is the state vector of the system which is assumed to be available for measurement. In order for (1 8) to be controllable, we require that g(x) # 0 for x in certain controllability region U, c R"; since g(x) is continuous, without loss of generality we assume that g(x) > 0 for x E U,. In spirit of the nonlinear control literature [5 ] , [17], these systems are in normal form and havle the relative degree equal to n. The control objective is to force

i) the closed-loop system must be globally stable and robust in the sense that all variables, x(t),O(t) and u(x 1 e), must be uniformly bounded, i.e., Ix(t)l 5 Mx < 00, l Q ( t ) l 5 MO < 00 and Iu(xl0)l 5 Mu < 00

for all t 2 0, where Mx, MO, and Mu are design parameters specified by the designer;

ii) the tracking error, e = ym - y, should be as small as possible under the constraints in (i).

In the rest of this section, we shall show the basic ideas of how to construct adaptive fuzzy controllers to achieve these control objectives.

To begin, let e = ( e ,& , . . . ,e(n- l ) )T and k = ( k n , . . . , k l )T E R" be such that all roots of the polynomial h(s) = sn + klsn-' + . + kn are in the open left-half plane. If the functions f and g are known, then the control law

applied to (19) results in

e(") + kle(n-l) + . . . + kne = 0 (21)

which implies that limt,oo e ( t ) = 0-a main objective of control.

Since f and g are unknown, we replace them by fuzzy systems f(x I 0,) and ij(x 1 Og), respectively, which are in the form of either (14) or (17). The resulting control law

1 U, = ~ [ - $(x 1 0,) + y?) + kTe] (22)

G(x I 0,) is the so-called certainty equivalent controller 1151 in the adaptive control literature. Applying (22) to (19) and after some straightforward manipulation, we obtain the error equation

e(") = -kTe + [f(x 1 8,) - f(x)] + [4(x]0> - g(x)]uc (23)

or equivalently

e = Ace + bc[($(x I 0,) - f b ) ) + (jr(x I 0,) - g(x>>ucl (24)

where

0 1 A, = . _ . . . . . . . . . . . . . . . . . . .

(25)

0 1 0 0 0 1 0 . . .

0 0 0 * * * I: - kn -kn- . . . . . . . . . . . . r o i

Since A, is a stable matrix (Is1 - A,/ = dn) + kls(n-l) + . * * + kn which is stable), we know that there exists a unique



positive definite symmetric n x n matrix P which satisfies the Lyapunov equation [ 171

R:P+ PA, = -Q (26)

where Q is an arbitrary n x n positive definite matrix. Let V, = $eTPe, then using (24) and (26) we have

In order for x, = p k l ) - e('-') to be bounded, we require that V, must be bounded, which means we require that V, 5 0 when V, is greater than a large constant P. However, from (27) we see that it is very difficult to design the U , such that the last term of (27) is less than zero. To solve this problem, we append another control term, U,, to the U,, i.e., the final control is

U = U , + U s . (28)

This additional control term U, is called a supervisory control. We now show how to determine the U , such that V, 5 0 when V, > V. Substituting (28) into (19) and using the same manipulation for obtaining (24), we have the new error equation

= L e + blC[(f(x 1 0,) - f(x)) + (ixx I 0,) - d X ) ) U C - g(x)usI. (29)

Using (29) and (26), we have 1 2

V, = --eTQe + eTPb,

x [CfCX I e,) - f(x)) + (i(x I 47) - s(x)>'1Lc - S(X)U.I

1 2

5 --.eTQe + JeTPb,l

x rlfcx I 0,)l+ If (41

- eTPb,g(x)u,. (30) 4- IS(X I &h4 + IdX)UCll

In order to design the U , such that the last term of (30) is nonpositive, we need to know the bounds of f and g, i.e., we have to make the following assumption-the only restrictive assumption for our adaptive fuzzy controllers.

Assumption 1: We can determine functions f U (x), g' (x) and gL(x) such chat If(x)l 5 fU(x) and gL(x) 5 g(x) 5 g'(x) for x E U,, where fU(x) < m,gU(x) < 00 and gL(x) > 0 for x: E U,.

Based on fU, gu and g~ and by observing (30), we choose the supervisory control U, as

1 U , = 1: sgn( eT Pb,) -

QL (XI

x nfcx I 0,)l+ f U W + I9(x I8 , )UCl+ lgU(x)Ucll (31)

where 1; = 1 if V, > V (which is a constant specified by the designer), 1; = 0 if V, 5 V , and sgn(y) = 1(-1) if y 2 0 (<O). Substituting (31) into (30) and considering the case V, > V , we have

1 V e L -ZeTQe + leTPb,I

In summary, using the control (28) with U , given by (22) and U , given by (31), we can guarantee that V, F: V < 00. Since P is positive definite, the boundedness of V, implies the boundedness of e, which in turn implies the boundedness of x. Note that all the quantities in the right hand sides of (22) and (3 1) are known or available for measurement, therefore the control law (28) can be implemented.

Our next task, in this constructive route, is to replace f and ij by specific formula of fuzzy systems of (14) or (17) and to develop an adaptation law for adjusting the parameters in the fuzzy systems for the purpose of forcing the tracking error converge to zero. First, define

0; = argmin sup Ij(x I 0,) - g(x)1 e , m , le,

where Rf and R, are constraint sets for 8 f and d,, respectively, specified by the designer. For R f , we require that Of is bounded, and for the fuzzy system (17), that the of's are positive, i.e.,

Rf = {of: l0fl 5 M,,of 2 a} (35)

where M f and o are positive constants specified by the designer. If we use the fuzzy system (14), ignore the af 2 o in (35). For R,, in addition to the constraints similar to (35), we also require that @(x I 0,) must be positive (since g(x) is positive). Observing (14) and (17), we have

R, = {%,: (%,I 5 M,,jjl 2 E , O ; 2 a} (36)

where M,, E , (r are positive constants specified by the designer. Since both fuzzy systems (14) and (17) are weighted averages of yl's, y' 2 E > 0 implies that the corresponding fuzzy systems are positive. If we use the fuzzy system (14), ignore the of 2 a constraint. Define the minimum approximation error

(37) w = ( f ( X I 0;) - m) + ( j (x I 0;) - g(x))Uc

then the error equation [see (29)] can be rewritten as

e = Ace - bcg(x)us + b,

x [(fb I 0,) - i ( x I 0;)) + (ijb I 6,) - 9(x 1 e;)),, + 4. (38)


682 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART B: CYBERNETICS, VOL. 26, NO. 5 , OCTOBER 1996

If we choose of (14) [or equivalently, (15)], then (38) can be rewritten as

e = Ace - bcg(x)us + bcw + bc [$TE(x) + $:E(x)u~]

and g to be the fuzzy systems in the form

(39)

where 4f = 6'f - e;, 4, = 8, - 6';, and [(x) is the fuzzy basis function (1 6). Now consider the Lyapunov function candidate

where y1 and 7 2 are positive constants. The time derivative of V along the trajectory of (39) is

. 1 V = --eTQe - g(x)eTPb,u, + eTPbcw 2

+ ,4 f I f 4 +yleTPbc<(x)]

where we used (26) and $f = ef, 4, = e,. From (31) and g(x) > 0 we have that g(x)eTPbcu, 2 0. If we choose the adaptation law

4, = -71eTPbc((x) (42) 8, = -y2eTPbc<(x)uc (43)

then from (41) we have 1 2

V 5 --eTQe + eTPbcw. (44)

This is the best we can hope to get because the term eT Pbcw is of the order of the minimum approximation error. If w = 0, i.e., the searching spaces for f and g are so big that the f and g are included in them, then we have V 5 0. Because the fuzzy systems in Definitions 1 and 2 are universal approximators, we can hope that the w should be small, if not equal to zero, provided that we use sufficiently complex (in terms of number of adjustable parameters) f̂ and 6.

If we choose f and 4 to be the fuzzy systems in the form of (17), then in orderto use the same strategy as above, we have to approximate f and g using Taylor series expnansions. Specifically, taking the Taylor series expansions of f (x 1 6';) and g(x I 6';) around Of and Os, we have

(45)

(46)

where O(14flz) and O ( ~ $ , ~ z ) are the higher order terms. Substituting (45) and (46) into (38), we have

e = Ace - bcg(x)us + b,w + b,

(47)

if1,*=0

0 J

+ l

e

Plant x(") = f(x) + g(x) U

6, = hf (0f,e,x)

ig = hg (Og,e,x) -I

determined by linguistic information

Supervisory Control

us = sgn(eTPbc) [Ifl+~+l~ucl+lguuc I]/gL

Fig. 2. The overall scheme of adaptive fuzzy control systems

where

= w + O(l 4#) + o( 14,12)>.c. (48)

The rest of the development is the same as (40)-(44) (we omit the details.)

The final problem is how to constrain the 8 f and 8, within the sets Rf (35) and R, (36), respectively. If we can keep Of E Rf and 8, E R,, then U, (22) and U, (31) will be bounded because in this case f is bounded, g > 0, and recall that e is bounded due to the supervisory control U,.

Clearly, the adaptation law (42) and (43) cannot guarantee that 8, E 02s and 0, E R,. To solve this problem, we use the parameter projection algorithm [4], [9]: if the parameter vectors 6'f and 8, are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets, then use the simple adaptation law (42) and (43); otherwise, i.e., if the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, then use the projection algorithm to modifiy the adaptation law (42) and (43) such that the parameter vectors will remain inside the constraint sets. We shall show the details in Sections IV and V. The overall control scheme is shown in Fig. 2.

We have shown all the basic ideas of constructing stable adaptive fuzzy controllers in a constructive manner. In Sec- tions IV and V, we will inverse the procedure by first showing the detailed design steps of the adaptive fuzzy controllers, and then proving that the adaptive fuzzy controllers so designed have the desired properties. We think that the way of presentation in this section should make it easier to understand how the adaptive fuzzy controllers are obtained, whereas the way of presentation in Sections IV and V should make it easier to use them.

Before we conclude this section, we make an assumption.



Assumption 2: There are the following linguistic descriptions about the unknown functions f(x) and g(x) (from human experts):

RS,'": IF x1 is A; and . . ' and x, is A;

THEN f(x) is C' (49)

and

Rf): IF x1 is Bf and . and z, is I?: THlEN g(x) is D s (50)

respectively, where AT, B,", C', and D" are fuzzy sets in R, r = 1 , 2 ,..., L f and s = 1,2 ,..., L,.

We allow L f = L, = 0, which means that there are no linguistic descriptions (49) and (50) about f(x) and g(x); therefore, Assumption 2 is not a necessity. We make this assumption for the purpose of emphasizing that our adaptive fuzzy controllers (in Sections IV and V) can directly incorporate these linguisric descriptions (if there is any) into their designs.

Iv . DESIGN AND STABILITY ANALYSIS OF FIRST ADAPTIVE FUZZY CONTROLLER

In this section, we choose f(x 1 0,) and j(x I 0,) to be the fuzzy systems in (14) or (15). We first present the detailed design steps of the adaptive fuzzy controller, and then study its properties.

Design of First Adaptive Fuzzy Controller Step I : Off-Line Preprocessing

Specify the k1 , . . . , k, such that all roots of sn+klsn-'+ . + k, = 0 are in the open left-half plane. Specify a positive definite n x n matrix Q. Solve the Lyapunov equation [see (26)], e.g., using the method in [24], to obtain a symmetric P > 0. Specify the design parameters M f , Ad,, E and v based on practical constraints (see Remark 4.1 for further discus- sion).

Step 2: Initial Controller Construction Define m, fuiczy sets F,' whose membership functions pF;% uniformly cover U,% which is the projection of U, onto the i'th coordinate, where 1, = 1 , 2 , . . . , ma and i = 1 , 2 , . . . , n. We require that the F,l.'s include the A;'S and Bf's in (49) and (50). Construct the fuzzy rule bases for the fuzzy systems f (x 1 19,) and g(x I e,), each of which consists of ml x m2 x ... x m, rules whose IF parts comprise all the possible combinations of the F;"s for i = 1 , 2 , . . . , n. Specifically, the fuzzy rule bases of f(x I 0,) and j(x I 6,) consist of rules

R("' f

Rkl>

'lZn): IF 2 1 is F;' and . . I and x, is F$

: IF x1 is F;' and ... and x, is F k THEN f(x 1 e,) is G ( ~ I > .

THEN j ( ~ 1 e,) is A ) (52)

(5 1)

respectively, where I , = 1 , 2 , . . . , ma, i = 1 , 2 , . . . , n, and G(''> .An) and H ( ' l ~ ~ . ~ ~ ' - ) are fuzzy sets in R which are

specified as follows: if the IF part of (51) or (52) agrees with the IF part of (49) or (50), set G(l1). or H(' l> A) equal to the corresponding C' or D", respectively; otherwise, set G(ll> A) and - J n ) arbitrarily with the constraint that the centers of G(''> . A) and H("> A) (which correspond to the j j l parameters) are inside the constraint sets Rf and R,, respectively. Therefore, the initial adaptive fuzzy controller is constructed from the linguistic rules (49) and (50). Construct the fuzzy basis functions

and collect them into a na, mi-dimensional vector ('(x) in a natural ordering for E1 = 1 , 2 , . . . ,ml, . . . ,1, = 1 , 2 , . . . ,m,, Collect the points at which pG(l1, . I = ) and p H ( l l , , I n ) achieve their maximum values, in the same ordering as t (x) , into vectors 19,(0) and 8,(0), respectively. The f"(x 1 6,) and j ( x I 19,) are constructed as

f(x I O f ) = e:5cx, i ( x 10,) = Q,Tl(x). (55)

(54)

Step 3: On-Line Adaptation Apply the feedback control (28) to the plant (18), where U, is given by (22), U, is given by (31), and f(x I 0,) and g(x I 8,) are given by (54) and (554, respectively. Use the following adaptation law to adjust the parameter vector: 8,

-yleTPb,<(x) if (let1 < M f ) or

and eTPbceF<(x) 2 0) (16, I = Mf

P{-yleTPbct(x)) if (IQ, I = M f Of =

andeTPb,B:('(x) < 0) (56)

where the projection operator P{*} is defined as [4], [9]

P{ -71 eT PbCE(4 1

Use the following adaptation law to adjust the parameter vector: 8,: 0 Whenever an element B,i of 0, = E , use

-y2eTPb,&(x)u, if eTPb,&(x)u, < 0 8 ' -

0 sa -

if eTPb,&(x)u, 2 0

where &(x) is the i 'th component of E(x).


684 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART B CYBERNETICS, VOL 26, NO 5, OCTOBER 1996

0 Otherwise, use

-y2eTPb&(x)uC if (lQ,i < Ad,) or (141 = Ad, and eTPb,8,T((x)uc 2 0)

e , = P{ -y2eTPbcJ(x)uc} if (IQ, I = Mg

and eTPb,8,T((x)u, < 0) (59)

where the projection operator P{*} is defined as

The following theorem shows the properties of this adaptive fuzzy controller.

Theorem I : Consider the plant (18) with the controln(28), where U , is given by (22), us is given by (31), and f and g are given by (54) and (55), respectively. Let the parameter vectors Bf and 8, be adjusted by the adaptation law (56)-(60), and let Assumptions 1 and 2 be true. Then, the overall control scheme guarantees the following properties:

i) lOf(t)l 5 M f , lO,(t)l 5 M,, all elements of 8, 2 E

and

I - M f i x

for all t 2 0, where Amin is the minimum eigenvalue of ( 4 ) T P , and Y m = (ym, Ym, . . . , Ym

ii)

(63)

for all t 2 0, where a and b are constants, and w is minimum approximation error defined by (37).

iii) If w is squared integrable, i.e., som lw(t)I2dt < 00, then limt+m le(t)l = 0.

Proof of this theorem is given in the Appendix. We now make a few remarks on this adaptive fuzzy controller.

Remark 4.1: For many practical control problems, the state x and control U are required to be constrained within certain regions. For given constraints, we can specify the design parameters k , M f , M , , e , and V , based on (61) and (62), such that the state x and control U are within the constraint sets. To do this, we need to know some fixed bounds of JymJ, Iyg’I, IfU(x)I,gU(x) and gL(x). Since these functions are known to the designer, it should not be difficult to determine these bounds. After these bounds are determined, we can specify the values of the right hand sides of (61) and (62) by properly choosing the design parameters. Note from (26) and (25) that A,,, is determined by k, therefore we can specify the k to achieve a required A,,,. In Section VI, we will show an example (the inverted pendulum tracking control) of how to specify the design parameters such that the state and control are within given constraint sets.

Ronark 4.2: From the definition of IT in (31) we see that the supervisory control U, is nonzero only when V, > v. Since v is usually a large number, the U , is more of a safeguard rather than an active control. If x is well-behaved in the sense that V, is not very large, then the control U of (28) becomes the fuzzy controller uC of (22).

Remark 4.3: From Step 2 we see that the linguistic information (49) and (50) is incorporated into the adaptive fuzzy controller by constructing the initial controller based on (49) and (50). If the linguistic rules (49) and (50) provide good descriptions about f ( x ) and g(x), then the initial f^ and 4 should be close to the f ( x ) and g(x), respectively; as a result, we can hope that the closed-loop system behaves approximately like (21). If no linguistic information is available, our adaptive fuzzy controller is still a well-performing nonlinear adaptive controller, in the sense of having the properties (i)-(iii) of Theorem 1. In summary, good linguistic information can help us to construct a good initial controller so that we can have a fast adaptation; we will show an example in Section VI to illustrate this point.

Remark 4.4: From (iii) of Theorem 1 we see that in order for the tracking error e ( t ) converge to zero, we require that the “minimum approximation error w” defined by (37) is small (in the sense of squared integrable). So a natural question is whether the fuzzy systems f and have the capability of accurately approximating the nonlinear functions f and g. In [19], we proved that fuzzy systems in the form of (14) or (17) are universal approximators, i.e., they are capable of uniformly approximating any real continuous function over a compact set to arbitrary accuracy. ATherefore, if we use sufficient number of rules to construct f and g, the w should be small.

Remark4.5: The basic idea of the projection algorithm in (56)-(60) is as follows: if the parameter vector is inside the constraint set or on the boundary of the constraint set but moving toward the inside of the constraint set [which corresponds to the cases of the first lines of (56), (58) and (59)1, then use the simple adaptation law based on the Lyapunov synthesis approach [see (42) and (43)]; if the parameter vector is on the boundary of the constraint set but moving toward the outside of the constraint set [which corresponds to the cases of the second lines of (56), (58) and (59)], then project the gradient vector ef or 8, onto the



supporting hyperplane to a, at eg t eg2

-- E eg,=E

Fig. 3. Illustration of the projection algorithm.

supporting hyperplane [9] at Of or 8, to the convex set Rf or a,. Fig. 3 shows a two-dimensional example for 8,.

Remark 4.6: In this adaptive fuzzy controller, we fix the fuzzy sets in the IF parts of the rules for the fuzzy systems f” and ij. An advantage of doing so is that the fuzzy systems f ^ and ij are linear in the parameter; therefore, we were able to use a relatively simpler adaptation law to adjust the parameters and convergence of the adaptation procedure is expected to be faster because we are not concerned with complicated nonlinear search problems. A disadvantage is that we have to consider all the possible combinations of the fuzzy sets in U,, because these fuzzy sets cannot change so that we should have rules to cover every region of U,, where by “cover” we mean that for each x E U, thFre should be at least one rule in the fuzzy rule bases of f and 6 whose “strength” (p+ (q) . . p F b (zn)) is not very small. Since in general the real trajectory of x(t) is only in a certain small region of U,, many rules in f and g are not used for an implementation of the adaptive fuzzy controller, i.e., this adaptive fuzzy controller is not efficient in utilizing its adjustable parameters. To overcome this disadvantage, we develop another adaptive fuzzy controller next in which the fuzzy sets in the IF parts are also adjustable during the adaptation procedure.

v. DESIGN AND STABILITY ANALYSIS OF SECOND ADAPTIVE FUZZY CONTROLLER

From Remark 4.6 we see that the adaptive fuzzy controller in Section IV may require a large number of rules for higher dimensional systems. A way to overcome this rule explosion problem is to allow the fuzzy sets in the IF parts of the rules also to change during the adaptation procedure so that in principle any rule can cover any region of Uc; as a result, we onlyAneed a small number of rules. Specifically, we will choose f and ij to be the fuzzy systems in the form of (17), and develop an adaptation law to adjust all the parameters $, 5; and gf. The price paid for this additional freedom is that the fuzzy systems f” and ij become nonlinear in the parameter, so that we have to use a more complicated adaptation law.

Design of Second Adaptive Fuzzy Controller Step 1: OfS-Line Preprocessing The same as the first adap-

tive fuzzy controller, except that we need to specify one more design parameter a.

Step 2: Initial Controller Construction Choose f(x I 0,) and ij(x I 8,) to be the fuzzy systems, i.e.,

where Of (e,) is the collection of the adjustable parameters $, 3;; and aii (&, Ztilfi and aii)). Clearly, f and fi in (64) and (65) are constructed based on M rules whose IF parts are characterized by the Gaussian membership functions

pF; (xi) = exp (- (q) ’) (66) Ofi

respectively, where I = 1 , 2 , . . . , M and i = 1,2,. . ., n. For this adaptive fuzzy controller, we assume that the A: and B,“ in (49) and (50) are also characterized by the Gaussian membership functions in the form of (66) and (67), and that L f I M and L, I M . Determine the initial ef(0) and 8,(0) as follows: for the L f (L,) pF;’s ( ~ Q ’ s ) that are the same as the ~ A ; ’ s

(plgf’s) in (49) [(50)], determine the 31fi(0) and aii(0) (3:,(0) and aL,(O)) based on the PAT’S ( p ~ t ’ ~ ) , and choose the $(O)’s ( j&(O)’s) to be the centers of the corresponding pc.’s ( p p ’s); the remaining parameters are chosen arbitrarily in the constraint sets (35) and (36).

Step 3: On-Line Adaptation

Compute ds(x’of) using the following algorithm: a*,



where 0 Otherwise, use

Equations (68)-(70) are obtained by taking differentials of f in (64) with respect to the corresponding parameters. Compute using the same algorithm (68)-(70), replacing f by g.

* Apply the feedback control (28) to the plant (18), where U , is given by (22), U , is given by (31), and f(x I O f ) and S(x 1 0,) are given by (64) and (65), respectively.

* Use the following adaptation law to adjust the parameter vector O f : 0 Whenever r fz = 0, use

iTbz = -y2eT Pb, % U ,

e eT 8 - +72eTPb,*&u, if (log1 = Mg

and e T P b C H ~ ~ u , < 0).

(76)

Properties of this second adaptive fuzzy controller are

Theorem 2: The adaptive fuzzy controller designed through

(i) I B f l 5 M f , the o i z ' s in 6'f 2 CT, l0,l 5 Mg, the 4 ' s in 8, 2 o, the vi 's in Os 2 E , and x and U satisfy (61)

summarized in the following theorem.

the above three steps guarantees the following properties:

and (62), respectively. -T1eTPbc$ if e ' P b , g < 0 (ii)

0 Otherwise, use

* Use the following adaptation law to adjust the parameter vector 6,: 0 Whenever j jb = 6 , use

(77)

for all t 2 0, where a and b are constants, and w is defined by (48).

(iiij If w is squared integrable, then limtim / e@)] = 0. Proof of this theorem is given in the Appendix. Remark 5.1: Remarks 4.14.5 apply to this adaptive fuzzy

controller. Remark 5.2: From (48) we see that in addition to the

minimum approximation error w we have another error term O(lOf - 8;12) + O(l0, - 6';12)u, in this second adaptive fuzzy controller. Because in this controller we have a larger searching space for f (x I 6';) and g(x I 0,) than in the first controller, the minimum approximation error w here should be smaller than the w in the first controller. Therefore, the performance of this controller should be more sensitive to the initial Of (0) and 8, (0) than that of the first controller. That is, if the initial 8,(0) and Q,(O) are close to the optimal 6'; and O g , respectively, then the total error w may be smaller than the w in the first controller; on the other hand, if the initial Of(0) and 8,(0) are far away from the optimal values, the w will be large. Because the initial controllers are constructed from the linguistic rules (49) and (50), these rules are more important for the second adaptive fuzzy controller than for the first one.

- y zeTPb ,3u , if eTPbc$$u, < 0

0 if eTPb,$$u, 2 0.

VI. APPLICATION TO INVERTED fjf = PENDULUM TRACKING CONTROL

In this section, we use our adaptive fuzzy controllers to control the inverted pendulum to track a sinewave trajectory. Fig. 4 shows the inverted pendulum system (or the cart-pole system). Let 21 = 6' and x2 = 8, the dynamic equations of the inverted pendulum system are [17]

(74)

0 Whenever aiz = r, use

-72eTPb,$uc if eTPb,$u, < 0 XI = 2 2



Fig. 4. The inverted pendulum system.

where g = 9.8 m/s2 is the acceleration due to gravity, m, is the mass of cart, m is the mass of pole, 1 is the half length of pole, and U is the applied force (control). We chose m, = 1 kg, m = 0.1 kg, and I = 0.5 m in the following simulations. Clearly, (78) is in the form of (18), thus our adaptive fuzzy controllers apply to this system. We chose the reference signal ym(t) = & sin(t) in the following simulations (other choices are certainly possible).

To apply the adaptive fuzzy controllers to this system, we first need to determine the bounds f U , g U , and gL. For this system, we have

0.025 2 9.8 + 1 . 1 2 2 5 - 2 0.05

3 1.1

= 15.78 + 0.03662; = f " ( ~ 1 , ZZ), (79)

= 1.46 = gu(z l ,x2 ) . (80) 1

5 1.1($ - E) If we require that 1x11 5 7r/6 (we will specify the design parameters such that this requirement is satisfied), then

Now suppose we require that

Since 1x11 5 (1z1I2 + 1x212)1/2 = 1x1, if we can make 1x1 5 7r/6, then lzll 5 7r/6. In this case we also have 1x2 1 5 7r/6. Our first task is to determine the design parameters V , k l , k 2 , E , M ~ and Mg, according to (61) and (62), such that the constraint (82) is satisfied. Since Iyml 5 7r/30, if we determine v and Amin such that (%)'/' 5 27r/15, then according to (61) we have 1x1 5 ~ / 3 0 + 2n-/15 = 7r/6. Since the number of design parameters is larger than the

0.15

0 2 4 6 8 10 12 14 16 18 20 -0.15

Fig. 5. (dashed line) for initial condition x(0) = (- 6, O ) T in Example 1.

The state z i ( t ) (solid line) and its desired value ylm(t) = & sin(t)

number of constraints, we have freedom in choosing the design parameters. We simply choose k1 = 2 and k2 = 1 (so that s2 + k l s + k2 is stable) and Q = diag( 10,lO). Then, we solve (26) and obtain

P = ['," 3. This P is positive definite with A,,, = 2.93. To satisfy the constraint for 1x1, we choose v = *(E) = 0.267. Finally, we determine M f and t such that I u I 5 180, according to (62). Again, we have additional freedom in choosing the M f and E. After some trial and error, we choose M f = 16, Mg = 1.6, and t = 0.7. It is straightforward to verify from (61) and (62) that the above choice of design parameters guarantees the state and control satisfy (82). For the second adaptive fuzzy controller, we choose n = lo-'.

Now we have finished the off-line preprocessing, i.e., Step 1 of the designs of both adaptive fuzzy controllers. Next, we simulate the two adaptive fuzzy controllers for this inverted pendulum tracking control problem, each for two cases: without any linguistic rules (Examples 1 and 3), and with some linguistic rules (Examples 2 and 4).

Example 1: In this example we used the first adaptive fuzzy controller, assuming that there are no linguistic rules (49) and (50). We chose r n l = m2 = 5 . Since I I C , ~ 5 7r/6 for both i = 1,2, we chose ~ F ; ( x , ) = e ~ p [ - ( w ) ~ ] , ~ F : ( x , ) =

2

e x p [ - ( w ) 2 ] i P F : ( X % ) = exp[-(&)2], PF;(%%) = 2 -9rj12 2 x - T / S 2

exp[-(-L;;724) 1, and PF;(G) = exp[-(*) 1, which clearly cover the interval [-7r/6,n-/6]. From the bounds (79)-(81) of f ( x 1 , x z ) and g (z l , zz ) we see that the range of f (x l ,x2) is much larger than that of g ( q , z z ) , therefore we chose y1 = 50 and y2 = 1. We use the same y1 and y2 in Examples 2-4. Fig. 5 shows the simulation results [x l ( t ) (solid line) and its desired value ym(t) = $ sin(t) (dashed line)] for initial condition x(0) = ( - $ , O ) T .

Example 2: Here we considered the same situation as in Example 1 except that there are some linguistic rules about f ( z1 , zz ) and g ( x 1 , x ~ ) based on the following physical intuition. First, suppose that there is no control, i.e., U = 0. In this case the acceleration of the angle 6' = 2 1 equals f(q, 2 2 ) .


N

IEEE TRANSACTIONS ON SYSTEMS, MAA AND CYBERNETICS-PART B: CYBERNETICS, VOL. 26, NO. 5, OCTOBER 1996

0 W12 W6 X1

Fig. 6. Linguistic fuzzy rules for f(z1, z2).

Based on physical intuition we have the following observation:

The bigger the 21, the larger the f ( s 1 , ~ 2 ) . (84)

Our task now is to transform this fuzzy information into fuzzy rules about f(xl, ~ 2 ) . Since (XI, x2) = (0,O) is an (unstable) equilibrium point of the system, we have the first rule

RF): IF XI is F; and x2 is F i , THEN $(XI, x2)is near zero (85 )

where F,” (i = 1 , 2 , j = 1 , 2 , . . . , 5 ) are the fuzzy sets defined in Example 1, and “near zero” is a fuzzy set with center zero (since only the centers of the THEN part fuzzy sets are used in the fuzzy systems, we do not need to specify the detailed membership functions of these THEN part fuzzy sets; that is, knowing their centers is sufficient). From Fig. 4 we see that the acceleration of z1 is proportional to the gravity mgsin(xl) , i.e., we have, approximately, that f(x1,x2) = as in(x l ) , where a is a constant. Clearly, f(q, Q) achieves its maximum at 5 1 = n/2; thus, based on (79) we approximately have a = 16. Therefore, we have the following fuzzy rules for f(z1,32)

RY): IF XI is F: and 5 2 is F;

~ j j j ) : IF x1 is F; and x2 is F;

RY): IF x1 is Ff and x2 is F;

RY): IF x1 is F; and xz is Fi

THEN f(x1,x2)is near - 8

THEN f(xl, Q) is near - 4

THEN f(x1,z2) is near 4

THEN f(xl, 2 2 ) is near 8

(86)

(87)

(88)

(89)

1’: n

I Fi’ Fi2 FI3 FI4 Fig

Fig. 7. Linguistic fuzzy rules for g(zl,zz)

where F,” (i = 1 , 2 , j = 1 , 2 , . . . ,5 ) are the fuzzy sets in Example 1, and the values * in “near *” are determined according to 16sin(n/6) = 8 and 8sin(n/12) = 4. Also based on physical intuition we have that the f ( z l , z z ) is more sensitive to x1 than to x2, we therefore extend the rules (85)-(89) to the rules where 5 2 is any F; for j = 1 , 2 , . . . ,5. In summary, the final rules characterizing XI, ~ 2 ) are shown in Fig. 6, which comprises 25 rules.

Next, we determine fuzzy rules for g(z1,x2) based on physical intuition. Since g(zl,52) determines the strength of the control U on the system and clearly this strength is maximized at z1 = 0, we have the following observation:

The smaller the 2 1 , the larger the g(x1 ,x~) . (90)

Similar to the way of obtaining the rules for f(x1,zz) and based on the bounds (80) and (81), we quantify the observation (90) into 25 fuzzy rules for g(zl,zz), which are shown in Fig. 7.

Fig. 8 shows the simulation results of the first adaptive fuzzy controller for initial condition x(0) = ( - & , O ) T , after the fuzzy rules in Figs. 6 and 7 are incorporated. Comparing Fig. 8 with Fig. 5 we see that the initial parts of control are apparently improved after incorporating these fuzzy rules.

Example 3 In this example we simulated the second adaptive fuzzy controller without using any linguistic rules. We chose: M = 15, the initial % ) , ( O ) and % i , ( O ) randomly in the interval [- g, E], the initial jjlf(0) randomly in [-3,3] , the initial fj:(O) randomly in [l, 1.31, and all the initial cr;, and cr’ equal to [E - (-$)]/15 = n/45. Fig. 9 shows the simulation results [x l ( t ) (solid line) and its desired value y m ( t ) = $ sin(t) (dashed line)] for initial condition x(0) =

94

(-&, qT.



0.151 I

I 0 2 4 6 8 10 12 14 16 18 20

-0.15

Fig. 8. (dashed line) for initial condition x(0) = ( -5 .

The state 51 ( t ) (solid line) and its desired value ym ( t ) = & sin@) in Example 2.

I V 0 2 4 6 8 10 12 14 16 18 20

-0.15

Fig. 9. (dashed line) for initial condition x(0) = (-&. O ) T in Example 3 .

The state z1 ( t ) (solid line) and its desired value y m ( t ) = 5 sin@)

Example 4: Here we considered the same situation as in Example 3 except that we used the linguistic rules in Figs. 6 and 7. Since Figs. 6 and 7 each has 25 rules while the fuzzy systems in the controller are constructed each by 15 rules, we only used the two sets of 15 rules corresponding to the middle three columns of Figs. 6 and 7, i.e., we used these 15f15 rules to construct the initial controller. Fig. 10 shows the simulation results in this case for initial conditions x(0) = (- $, O ) T and x(0) = ( & , O ) T , respectively. Comparing Fig. 10 with Fig. 9 we see that the control performance is apparently improved after incorporating these fuzzy rules.

VII. CONCLUSION

In this paper, we developed two adaptive fuzzy controllers which: 1) do not require an accurate mathematical model of the system under control, 2) are capable of incorporating fuzzy IF-THEN rules describing the system directly into the controllers, and 3) guarantee the global stability of the resulting closed-loop systems in the sense that all signals involved are uniformly bounded. We used the adaptive fuzzy controllers to control the inverted pendulum to track a sinewave trajectory, and the simulation results show that: 1 ) the adaptive fuzzy controllers could perform successful tracking without using any linguistic information; and 2 ) after incorporating some linguistic fuzzy rules into the controllers, the adaptation speed became faster and the tracking error became smaller.

Despite the practical successes, fuzzy control has still gained criticism from some more conventional control researchers.

0.15

-0.15 0 2 4 6 8 10 I 2 14 16 18 20

Fig. 10. (dashed line) for initial condition x(0) = (-&, O ) T in Example 4.

The state 2 1 ( t ) (solid line) and its desired value y m ( t ) = $ sin(t)

From this paper we see that a constructive combination of fuzzy control and conventional control ideas (rather than criticizing with each other) could produce useful results. In fact, we did not invent any new fuzzy systems, i.e., the fuzzy systems used in our adaptive fuzzy controllers are widely used in the fuzzy control literature; similarly, the ideas of Lyapunov synthesis, sliding control, and parameter projection are very common in conventional control. Using these common ideas in conventional control, we could study the long-standing problem of fuzzy control-lack of stability guarantee, and at the same time preserve the most important advantages of fuzzy control-model-free and the capability of incorporating linguistic and fuzzy information. From this paper we see that fuzzy control researchers and conventional control researchers should learn from each other: on one hand, fuzzy control researchers should ackonwledge that the mathematical rigor- ous approaches in conventional control have produced many powerful ideas and methods, and the applications of which to fuzzy control could make fuzzy control more a science rather than a technology; on the other hand, conventional control researchers should acknowledge the outstanding feature of fuzzy control-the capability of incorporating linguistic and fuzzy information in a systematic and efficient manner. Because SO

much human knowledge is represented in natural languages, incorporating it into controllers should be a common goal of all researchers on control.

APPENDIX

Proof of Theorem 1 i) To prove [Of\ 5 M f , let Vf = i O F O f . If the first line

of (56) is true, we have either \Of\ < M f or Vf = -yleTPb,07J(x) 5 0 when \Of1 = M f , i.e., we al- ways have lO f l 5 M f ; if the second line of (56) is true, we have = M f and Vf = -yleTPb,OFJ(x) + yleTPb, 'B ' ' ;~f~~(X) = 0, i.e., \Of\ 5 M f . Therefore, we have lOf(t)l M f , M 2 0. Using the same method, we can prove that lO,(t)l 5 M g , V t 2 0. From (58) we see that if O,, = E , then Q,, 2 0, i.e., we have O,, 2 E

for all elements O,, of O s . In Section I11 we proved that V, 5 V ; therefore, iXminle12 5 i eTPe 5 V , i.e., le1 5 (z)'l2. Since e = ym - x. we have 1x1 5 Iyml + \el 5


690 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART B: CYBERNETICS, VOL. 26, NO. 5 , OCTOBER 1996

( y m ( + which is (61). Finally, we prove (62). Since f(x I 0,) and g(x I e,) are weighted averages of the elements of Of and O s , respectively, we have If(x I d,)l I l O f l 5 My and ij(xl0,) 2 E (since all elements of 8, 2 t). Therefore, from (22) we have

(since Q is determined by the designer, we can choose such a QL we have

[ l4T)l2dr I [IV(O) I + IV(t) I1 2 XQ min - 1

From (31) we have

1 Define a = ---”--[lV(O)l + sup,20 IV(t)I] and b = A IPb,I2, (A.6) becomes (63) (note that sup,>o IV(t)I is finite because e, $f and 4, are all

(‘4.2) bounded). iii) If w E La, then from (63) we have e E L2. Because

we have proven that all the variables in the right hand

A Q m 1 n - 1

A Q r n i n - 1 1 % I ~ [Mf + IfU(x)/ f (Mf + IgU(x)l)l~cll. I g L (x) I Combining (A.l) and (A.2) we obtain (62).

ii) From (41) and (56)-(60), we have

1 2

V = --eTQe - g(x)e*Pb,u, + eTPbcw

+ 134;€eTPbcS€(X)u, (A.3)

where Il = O ( 1 ) if the first (second) line of (56) is true, 1 2 = O(1) if the first (second) line of (59) is true, 13 = O ( 1 ) if the first (second) line of (58) is true, 8,+ denotes the collection of QYz’s > E , O,, denotes the collection of Oyz’s = E, $g+ = Os+ - $+,

= O,, - OBE, and [+(x) (&(x)) is the collection of the corresponding elements of [(x) with respect to ds+ (OgE). Now we show that the last three terms of (A.3) are nonpositive. First, the term with 1,. If Il = 0, the conclusion is trivial. Let 11 = 1, which means that lQ f l = M f and eTPb OT[(x) < 0, we have

since = M f 2 16’;l. Therefore, the term with Il is nonpositive. Similarly, we can prove that the term with I2 is nonpositive. Finally, from (58) and the fact that $,, = Oyz - e$, = t - O g , 5 0, we have that the term with I3 is also nonpositive. Therefore, we have

V I --eTQe - g(x)eTPb,u, + eTPb,w.

4fQf = ( B , - 8 ; ) T O f = +[le, 1 i f -1Q;12+pf-O;/2] L 0,

1 2 (A.4)

From (31) and g(x) > 0, we have g(x)eTPbcu, 2 0 ; therefore, (A.4) can be further simplified to

1 2

V 5 --eTQe + eTPbcw

XQ min - 1 1 leI2 - 5 2 I-

1 2

x [lei2 - 2eTPb,w + J P ~ , W / ~ ] + -(Pb,w12

1 (‘4.5)

XQ min - 1 < - lei2 + 51Pb,w12 2 -

where XQ mln is the minimum eigenvalue of Q. Integrat- ing both sides of (A.5) and assuming that X Q ~ ~ , , > 1

side of (47) are bounded, we have e E L,. Using the Barbalat’s Lemma [17] (if e E L2 n L, and e E L,, then limtt, le(t)l = 0), we have limt,, Ie(t)i = 0.

0 Proof of Theorem 2 Comparing the error equation (39) of the first adaptive fuzzy

controller with the error equation (47) of the second adaptive fuzzy controller, we see that they are the same if we replace w by v and c(x) by and 8 [as in (47)]. Also, same as (SX), the adaptation law (72), (74) and (75) guarantees that

same procedure as in the proof of Theorem 1, we can prove 0

80,

a f i 1 2 0, jjh 2 t and ahn 2 a. Therefore, using exactly the

this theorem (we omit the details.)

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S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989. A. L. Schwartz, “Comments on “Fuzzy logic for control of roll and moment for a flexible wing aircraft,” IEEE Contr. Syst. Mag., Feb. 1992. J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. R. M. Tong, “Some properties of fuzzy feedback systems,’’ IEEE Trans. Syst., Man, Cybern., vol. SMC-10, no. 6 , pp. 327-330, 1980. L. X. Wang, “Fuzzy systems are universal approximators,” in Proc. IEEE Int. Con$ Fuzzy Systems, San Diego, CA, 1992, pp. 1163-1 170. -, “Stable adaptive fuzzy control of nonlinear systems,’’ IEEE Trans. Fuzzy Syst., vol. I , no. 2, pp. 146-155, 1993. L. X. Wang and J. M. Mendel, “Generating fuzzy rules by learning from numerical examples,” IEEE Trans. Syst., Man, Cybern. vol. 22, no. 6, pp. 1414-1427, 1992. -, “Back-propagation fuzzy systems as nonlinear dynamic system identifiers,” IEEE Int. Con$ Fuzzy Syst., San Diego, CA, 1992, pp. 1409-14 18. -, “Fuzzy basis functions, universal approximation, and orthogonal least squares learning,” IEEE Trans. Neural Networks, vol. 3 , no. 5 , pp. 807-814, 1992. -, “Three-dimensional structured networks for matrix equation solving,” IEEE Trans. Computers, vol. 40, no. 12, pp. 1337-1346, 1991. L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes,” IEEE Trans. Syst., Man, Cybern., vol. SMC-3, no. 1, pp. 28-44, 1973.

Li-Xin Wang (S’90-M’93) received the B S. and M.S. degrees from Northwestern Polytechnics Uni- versity, Xian, China, in 1984 and 1987, respectively, and the Ph D degree from the University of South- ern California, Los Angeles, in 1992, all in electrical engineering.

From 1987 to 1989, he was with the Department of Computer Science and Engineering, Northwest- ern Polytechnical University. From the fall of 1989 to the spring of 1992, he was a ResearcNTeaching Assistant with the Department of Electrical Engi-

neering-Systems, University of Southern California, where he worked toward the P h D degree From the summer of 1992 to the summer of 1993, he was a Postdoctoral Fellow in the Department of Electrical Engineering and Computer Science, University of California at Berkeley. Since the fall of 1993, he has been an Assistant Professor with the Department of Electrical and Electronic Engineering, the Hang Kong University of Science and Technology, Hang Kong He is author of Adaptive Fuzzy Systems and Control. Design and Stability Analysis (Englewood Cliffs, NJ: Prentice-Hall, 1994) and A Course In Fuzzy Systems and Control (Englewood Cliffs, NJ: Prentice-Hall, 1996). His research interests include intelligent control, fuzzy systems, and process control.

Dr. Wang received a Phi Kappa Phi Student Recognition Award in 1992 for his work on fuzzy systems.


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