Fuzzy model-reference adaptive control

1606 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 25, NO. 12, DECEMBER 1995

ZZ odel-Reference Adaptive Tang-Kai Yin, Student Member, I E E , and C. S. George Lee, Fellow, IEEE

Abstract-This paper proposes a fuzzy model-reference adaptive control (Fuzzy-MRAC) to deal with controlling a plant with unknown parameters which are dependent on known variables. The proposed method uses the fuzzy basis function expansion (FBFE) to represent the unknown parameters and change the identification problem from identifying the original unknown parameters to identifying the coefficients of the FBFE. That is, the dependency property of unknown parameters is absorbed into the fuzzy basis functions and their linear combination coefficients. This data representation is substantiated by the Stone Weierstrass theorem which indicates that any continuous function can be represented by the FBFE. With the aid of the FBFE, the unknown parameters can be estimated more precisely and better performance can be expected from the fuzzy-MRAC than the traditional MRAC. Furthermore, the adaptation scheme of the proposed fuzzy-MRAC is based on both the tracking error and the prediction error. Combining these two sources of error information, the proposed fuzzy-MRAC will provide more adaptation power than a traditional adaptive control. Since the proposed fuzzy-MRAC can be considered as an extension of the traditional MRAC, its stability and convergence properties are preserved. Computer simulations were conducted to show the validity and the performance of the proposed fuzzy-MRAC and its improvements over the traditional MRAC.

I. INTRODUCTION HE objective of a controller is to drive the output of a plant to keep at a reference value or to follow another

signal. In many dynamic systems, the parameters of the systems usually change slowly and become unknown due to different operating environments. Thus, a controller with an adaptation scheme is usually designed to estimate the unknown parameters to control the system.

For many decades, adaptive control has been applied to control systems with constant or slowly changing unknown parameters. It has found many applications in such areas as robot manipulators, ship steering, aircraft control and process control [I]. In these applications, an on-line adaptation scheme is usually used to estimate the unknown parameters of the system, and an appropriate controller is then designed to control the plant to satisfy a desired performance.

Many approaches have been proposed for different adaptive systems. Miller and Davison showed that an adaptive controller can provide arbitrarily good transient and steady-state response, although the controller uses a very high gain and is not practical for implementations [a]. Lozano and Brogliato devised an adaptive controller for a simple first-order nonlinear system in which the apriori knowledge of the plant parameters

Manuscript received April 1, 1994; revised October 7, 1994. This work was supported in part by a scholarship from the Ministry of Education, Taiwan, R.O.C.

The authors are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 USA.

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0018-9472/95$04

is not required [3]. Kanellakopoulos et al., and Marino and Tomei used an adaptive output-feedback approach to control systems with output nonlinearities [4-61, and Messner et al. proposed a repetitive learning scheme to perform disturbance estimation and cancellation [7]. This repetitive method used predefined functions to represent disturbance information which is similar to our proposed approach of using the FBFE to represent the parameter information.

Self-tuning regulators or adaptive controllers were originally used to control systems with stochastic properties. k r o m et al. made a expository reviews on the theory and applications of self-tuning regulators [SI. Borison proposed self-tuning regulators for a class of multivariable systems based on the minimum variance strategy [9]. Johnson et al. designed a self- tuning regulator for adaptive control of aircraft winghtore flutter [lo]. Compared with direct adaptive methods, the self- tuning method is indirect. Nevertheless, the self-tuning concept is useful and can be applied to design adaptive controllers.

Recently, due to the computational simplicity and the learning power of neural networks, many researchers attempted to use neural networks andor fuzzy set theory to represent complex plants and construct adaptive controllers. Narendra and Parthasarathy detailedly discussed identification and control of dynamical systems using neural networks [ 111. Jagannathan et al. showed good tracking performance through a Lyapunov’s stability approach in their model reference adaptive control using multilayer neural networks [ 121. Wang designed a stable adaptive fuzzy controller for nonlinear systems [ 131. These approaches dealt with complex plants which cannot be well represented by mathematical models.

This paper focuses on the plants with known structures but unknown parameters which are dependent on known variables. In many cases, a plant operates in a certain working region and the parameters change due to different states of the plant. The parameters are unknown and may be too complex to describe precisely in mathematical models; however, they can be expressed in terms of some known variables of the system. For example, in a spring-mass system, the unknown coefficient of the spring is mainly dependent on the known variable-the length of the spring. Hence, in these cases, instead of viewing the unknown parameters merely as constant or slowly varying variables, we can take advantage of this dependency property in formulating the estimatiodidentification scheme. In this paper, we use the fuzzy basis function expansion (FBFE) to represent and approximate the unknown parameters to take advantage of this dependency property.

The concepts of fuzzy sets and fuzzy inference systems, proposed by Zadeh [14], have been applied to many control and decision systems. Many complex models which lack

.OO 0 1995 IEEE

YIN AND LEE: FUZZY MODEL-REFERENCE ADAPTIVE CONTROL 1607

suitable mathematical equations to describe their dynamics can be well approximated by fuzzy sets and inference systems. For example, using fuzzy sets and inference systems, Sugeno and Yasukawa successfully modeled the trend of stock prices [15], which is a complex system that is difficult to describe mathematically.

With the FBFE data (i.e., parameter) representation, we need to update two groups of coefficients: the coefficients in the fuzzy sets and the coefficients in the linear combination of the fuzzy basis functions. The first group of coefficients will be updated by off-line learning while the second group of coefficients will be updated by on-line learning. The on-line adaptation scheme of the proposed fuzzy-MRAC is based on both the tracking error and the prediction error. Combining these two sources of error information, the proposed fuzzy- MRAC will provide more adaptation power than a traditional adaptive control which solely utilizes either tracking error or prediction error. The off-line parameter learning for fuzzy sets is developed to establish the fuzzy basis function. This learning scheme matches the FBFE more precisely to the unknown parameters. The insensitive property of the number of fuzzy sets used to model the unknown parameters is also investigated. Finally, since the proposed fuzzy-MRAC can be viewed as an extension of traditional MRAC's, the stability and convergence properties are preserved. The use of the FBFE to achieve efficient estimations of the unknown parameters results in better adaptations and performance for the proposed fuzzy-MRAC. The improved adaptations and performance can be seen in the computer simulations and will be discussed in Section 111.

This paper is organized as follow. Section I1 delineates the proposed fuzzy-MRAC with the FBFE. Representation of the unknown parameters of a plant by the FBFE will be briefly discussed. The on-line adaptation scheme based on both the tracking error and the prediction error will be derived. The off-line parameter learning scheme is described. Then, practical considerations for the implementation of the proposed fuzzy-MRAC will be presented. In Section 111, computer simulations will be conducted on two examples to verify the performance of the proposed fuzzy-MRAC. Finally, conclusion is summarized in Section IV.

11. FUZZY MODEL REFERENCE ADAPTIVE CONTROL

m is the number of these known variables, and the superscript "7"' denotes matridvector transpose.

Dejinition I : A fuzzy basis function is defined as

where p 2 j ( x c ) is the membership function of a fuzzy set, and M is the number of fuzzy rules [17].

We note that Cglp,(x) = 1. So p3(x) can be viewed as a weighting function. For each point x, p3(x) is the weighting for the j th fuzzy rule. Combining all these fuzzy rules, we have the FBFE as follows.

Dejinition 2: A fuzzy basis function expansion (FBFE) is

(3) ,=1

where 19, is the j th linear combination coefficient [17]. From Definition 2, it is clear that the FBFE is a linear combi-

nation of the fuzzy basis functions with unknown coefficients 0,'s. This linearity is needed to employ the adaptation law from a Lyapunov function for our proposed fuzzy-MRAC. With the use of the FBFE, the unknown parameters to be estimated are changed from the physical parameter vector a to 0 and the coefficients of the fuzzy sets in the FBFE. Because much of the varying property of a is absorbed into the fuzzy sets of the FBFE, a can be estimated more accurately through I9 and the coefficients of the fuzzy sets in the FBFE.

Next, the Stone-Weierstrass theorem will be discussed which substantiates that the FBFE can be used to represent any continuous functions. Let E be any set and F be a family of real-valued functions on E. F is an algebra if f , g E F, c E R, then f + g E F, fg E F and cf E F . That is, F is an algebra if it is closed under the operations of addition, multiplication and multiplication by constants [ 181. Also F separates points of E if whenever x,y are distinct points of E, there exists a function f E F such that f(x) # f(y) [18].

The following theorem is one version of the Stone- Weierstrass theorem and is very important for the FBFE to be useful to any continuous functions.

Theorem I : (Stone Weierstrass theorem [lS]). Let M be a compact metric space, C ( M ) be the set of all continuous real-valued functions on M , and F be a subset of C ( M ) . If

A. Fuzzy Basis Function Expansion

p ~ ( z ) in a universe of discourse U , where p ~ ( z ) E [0,1] for any x E U [16]. A fuzzy set introduces vagueness by eliminating the sharp boundary that divides members from nonmembers in the group. Consider a Gaussian membership function p(z) which can be expressed as

1) F is an algebra, 2) F separates points of M .

of F, is equal to C ( M ) .

A f u u Y set is by a membership function 3) F contains the constant functions, then F , the closure

That is, given E > 0 and a function g on M , we can find a function f E F such that

2; Ilf (XI - g(x)II < E .

(x - E ) * P ( Z ) = e.+.;.-) ('1

where 3 and are the mean and the variance of the Gaussian membership function, respectively. Let x = ( Z I , X Z , . . . , x , )~ , where Q , ~ z , . . . , z, are known variables,

It can be checked that the set

M

f(x) I f(x) = Cp,(x)B, is the FBFE 3=1

1608 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL 25, NO 12, DECEMBER 1995

satisfies the three conditions of the Stone Weierstrass theorem. A more detailed treatment of this theorem and the FBFE can be found in [17], [19]

B. Tracking-Error-Based Adaptive Control

The use of the FBFE to approximate the unknown parameters is based on the assumption that the unknown parameters in the plant can be expressed as a function of the known variables. Fortunately, this assumption is not very restrictive to many practical systems. For example, the equation for the free oscillation of a nonlinear mass-spring system can be expressed as [20]

(4) m, + f l i ; + kz + k'x3 = 0.

This equation can be rewritten as

mi + f i + cz = 0 (5)

where c = k + k'x2 is the unknown parameter of the known variable z. Another example is the dynamic motion equation of an n-link robot manipulator [21],

H(q)G + C(q, + g(q) = 7 (6)

where the unknown parameters H(q) (inertial matrix) and C(q),q) (coupled Coriolis and Centripetal terms) are functions of known variables q and q (generalized coordinates). Hence, for most industrial systems, the unknown parameters in a plant can be expressed as a function of known variables.

Consider the block diagram of the on-line adaptation for 6 as shown in Fig. 1. In this paper, we focus on the MRAC in which the dynamic equation of the plant can be expressed as

any(") + a,-ly("-l) + (7)

any?) + cy,-ly?-l) + ' . ' + cy0ym = T

and the dynamic equation of the reference model is specified as

(8)

where y is the plant output, ym is the output of the reference model, u is the input of the plant, T is the reference signal, and y(%) indicates the ith derivative of y(t). We assume that the sign of the high frequency gain & is known and does not change in the overall process. Without loss of generality, we assume a, > 0. In this plant structure, the plant to deal with is a nonlinear system because the parameters a, contains the known variables 21, z2, . . . , IC, which are elements from the states y , y('), . . . , y("-'). The off-line parameter learning takes care of the time-invariant part of the unknown parameters, whereas the on-line adaptation takes care of the slowly time- varying part. Thus, the plant is nonlinear and slowly time- varying but confined in a certain region such that the fuzzy sets can be applied. The discussions of this paper are for single-input-single-output systems only. To extend to multi- input-multi-output systems, more research efforts must be performed on the proposed fuzzy-MRAC.

In formulating our problem, let e A y - ym be the tracking error, a = (an, an-l , . . . , a ~ ) ~ be the actual parameter vector of the plant, cy = (cy,, an-l , . . . cy^)^ be the parameter vector of the reference model, ,Ll = (&1, & - 2 , . . . ,PO)'

reference model

h a

fuzzy prediction-error based estimation

and fuzzy trackingerror t based adaptation I

Fig. 1. The block diagram of the on-line adaptations for 6.

be the parameter vector of the tracking error equation, a ' = (iin,iin-l,. . . , & o ) ~ be the estimated parameter vector of a, 5 A a - a be the estimation error vector, x = ( z ~ , x z , . . . , x , ) ~ be the vector of known variables X I , X ~ , . . . ,xm, i9 = (QT, Qz-l,. . . , be the vector of the linear combination coefficients in the FBFE, a = ((x)Q be the FBFE for the parameter vector a, a = ('(x)% be the FBFE for the estimated a value due to the estimated %, and 8x 8 - 0 be the error vector in estimating 8. [(x) is the fuzzy basis functions and is an (n + 1) x (n + l)M matrix expressed in the form of

(9)

The first step in the controller design is to select &-I, ,&-2, . . . ,Po such that the error polynominal en + &-len-' + . . . + ,BO is a stable (Hunvitz) polynomial. Define z as

z 2 y t ) - pnPl e(n-1) - . . . - Doe. (10)

Adding -u,z to both sides of (7) and rearranging terms, we have

Let u be

YIN AND LEE FUZZY MODEL-REFERENCE ADAPTIVE CONTROL 1609

b =

where v = ( z , y("-l), . . . , y)' and a = t8. Then the right hand side of (11) is

C. Prediction-Error-Bused Estimation

We note that besides the tracking error e = y - ym, if we plug the estimated parameter vector a into the dynamic equation of the plant, the prediction error el can be obtained as follows.

u - [ z , y("-'), . . . , yl[a,, un-l , . . . , a0lT = u - vTa.

-0 - 0

. 0

Using (IO), a = <(x)6' and 8 = 8 - 8, we have

u - vTa = vT<8 - vT<0 = vTt8.

Using (lo), the left hand side of (1 1) is

where a, # 0; otherwise, the plant is not of order n.

a matrix form as Let e be [e, e , 6 , . . . , e("-')]'. Then (14) can be written in

where

A = A =

e = Ae + (L) v T t h , an

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

. . . . 0 0 0 . . . 1

-Po -P1 -p2 ... -pn-l

. . . . 0 0 0 . . . 1

--Po -P1 -p2 ... -pn-l

and

Consider a Lyapunov function candidate

where exp[- J," X(r)dr] is an exponential forgetting weighting. The reason to use the exponential forgetting weighting is to have parameter tracking ability. If the weighting is constant, then the old information is of the same importance as the new information. Therefore, if 6' is slowly time-varying, then P1 (t) is a bad form to deal with the information. Theoretically, the FBFE can approximate the parameter a(t) as precisely as possible. However, in reality approximation error may occur due to the limitation of the number of fuzzy sets and the computational complexity. We can relax the requirement that 6' is constant to the requirement that 6' is constant locally; that is, 0 can be globally slowly changing. Thus, with the exponential forgetting weighting, we can use a fewer number of fuzzy sets and reduce the computational complexity.

We note that the integral

is always positive semidefinite. Except in a rare case, w(t) will give nontrivial information and the integral is positive definite. So P l ( t ) exists.

Taking the derivative of P;'(t), we have

where P and PI are symmetric positive definite matrices. P

symmetric positive definite matrix. P1 will be given in the next subsection because the adaptation of 8 depends not only on the tracking error e, but also on the prediction error el. The proof of the convergence is provided in the next subsection.

satisfies PA + ATP = -Q, where Q = QT is a chosen X ( t ) = X o ( l - y) (21)

where XO and IC0 are constants, then this assignment of X ( t ) will give us an upper bound of Pl(t) and the corresponding lower bound of P,'(t).


Consider = (eTATPe + eTPAe) + ze"TETvbTPe

1 - 1 - 1 - - -eTQe - -Xe"TP;lQ - ---OTwTwj 5 0 (25)

because both PY' and [IIP1III - PI] are symmetric positive an an

where definite matrices. Then

e = -P1[ETvbTPe + wTwe"].

where I is an identity matrix of appropriate dimension.

we have The initial condition is to assign IIPl(0)II 5 ko. By (22), Because 6' is 10Cdb constant, e" = 8. Equation (26) is the

adaptation law for e. Therefore,

1 2 -1. k0

due to the negative derivative of (25). Hence, the stability of the proposed method is guaranteed. Next, the convergence is obtained by Barbalat's Lemma [l].

Lemma I : (Barbalat [l]): If the differentiable function f ( t ) has a finite limit as t 4 oo and if f is uniformly continuous, then f ( t ) -+ o as t 4 oo.

By the property of the Lyapunov function, V 2 0. So V is lower bounded. From (25), V is negative semi-definite. Therefore, V has a finite limit. To use Barbalat's Lemma, V has to be uniformly continuous in time. It usually is, except in special cases, for example, an improper system with an impulse input. Then by Barbalat's Lemma,

(23)

From (20) and (21),

d - [PT'] = -XoPFl+ (Xo/ko)llP1IIP;l+ WTW. dt

Solving this differential equation, we have

~ , l ( t ) = Prl(0)e-'ot

+ I' e-'o(t-T) [ 211p111p;1 + WTW 1 dr.

Using (22) and (23), we have

P,'(t) 2 P,l(0)e-'ot

So $1 is the lower bound of P,'(t), and koI is the upper bound of P l ( t ) . Now we are ready to derive the update equation for 4.

With the bounds of P1 given, let us reconsider the Lyapunov function candidate in (16):

v(e, S ) = eTPe + LH"TP;~s. an

Taking the derivative of V and using (20), we have

V(m) = 0. (28)

If the system is of persistent excitation, then X and wTw will be a positive number and a positive definite matrix, respectively. Then (24) indicates that P 1' has the lower bound of 11, and by (28) that @(a) will approach zero, e and e" will converge to 0. Thus, the convergence of the tracking error and the estimated parameters is obtained.

LO

D. Off-line Leaming of Parameters in Fuzzy Sets To use the FBFE to represent the unknown parameter vector

a = (4, we not only have to estimate e as discussed in the previous subsection, we also need to determine appropriate fuzzy basis function expansion E ; that is, we need to determine a) the means and the variances of the Gaussian membership functions (in (l)), and b) the number of fuzzy sets and the maximum number of fuzzy rules (in (3)). The flow chart in Fig. 2 shows the adaptation of E and 6 .

The estimated parameter vector can be written as

En 0 ' .

a =

- - E(n+l)x((n+l)lM)O((n+l)n/l)x1

where n is the order of the plant, and M is the num fuzzy rules for one parameter a k .

YIN AND LEE: FUZZY MODEL-REFERENCE ADAPTIVE CONTROL 1611

1 initial assignmend

I I I I

t or iterations >= N cq>

Fig. 2. The flow chart for the adaptation of 6 and 6.

Using the FBFE ((1)-(3)), a k can be expressed as

IC = 0 , . . . ,n.

Hence, iik is a function of d j k , O t j k and 3 z j k .

During the on-line adaptation of the fuzzy-MRAC, g Z j k and 2 z j k , which dictate the spread and the mean of the Gaussian membership function, respectively, are fixed while are estimated. After each on-line adaptation, we have a new data a k at each sampled time and a new estimated d j k . We note that is gradually approaching a better estimated value by the convergence property; therefore, we can let 8; be the on the final sampled point and form the function

G+ - + ^+ k - ak ( @ 3 k , c ~ j k 7 X ~ j k ) 7

where 8:k are fixed and O Z j k and Z Z j k are the parameters to be learned.

Let p be the number of sampled points in one on-line adaptation run. Define the error

P

E k = c ( & k ( S ) - i ? ~ t ( S ) ) ~ , k = 0 , . . . ,n. s=l

Then E k is a function of u t j k and ICzJk. We can use a gradient- descent method on Ek to update o z J k and x z j k . For simplicity,

we shall label ~ i j k and Zijk as cxk t , t = 1,. . . , (2(n + 1)M), then the amount of update for each parameter is

where q k is the learning rate which is usually decided based on empirical study. One way is to update q k according to the error E k on consecutive epochs during the off-line learning.

After each update for o z j k and &jk in the off-line learning, we can continue the next on-line adaptation to update 8. The alternating off-line and on-line learning processes are run until the error is less than the desired value or the number of iterations is reached.

E. Practical Considerations With the above off-line learning of the parameters in the

membership functions of the fuzzy sets in the FBFE, our next step is to decide the number of fuzzy sets in the corresponding universes of discourse and the number of fuzzy rules. Since the performance of the proposed fuzzy-MRAC depends on the approximation of the unknown parameters a = (8, intuitively more fuzzy sets are needed if a is a complex curve. Therefore, one can increase the number of fuzzy sets from an initial small number to a larger number until the performance is satisfactory. Unfortunately, this intuition to improve the performance cannot be guaranteed. Actually, we must realize that as more fuzzy sets are used, the more difficult the fuzzy-MRAC will be due to the increased number of parameters of the membership functions. That is, for fewer fuzzy sets, the FBFE is not very complex but d and the coefficients in the fuzzy sets are precise. On the contrary, if we have more fuzzy sets, the FBFE will be more complex but 8 and the coefficients in the fuzzy sets are not precise. This is illustrated in the simulations of the following two examples.

y + 6sin(10(y2 + y2))y + y = u y + sin(y2 + yz)y + y = u

system (1) system (2)

The performance index used is the integral of absolute error (IAE)

The reference signal is a sinusoidal signal and the integral is from t o = 0 to t f = 10. The simulation results are shown in Fig. 3. It can be seen in Fig. 3 that the performance of the fuzzy-MRAC is quite insensitive to the number of fuzzy sets from 2 to 7 for both y and y. So a better way to determine the number of fuzzy sets is to run the fuzzy-MRAC algorithm on many different numbers of fuzzy sets, and from the results to decide the number of fuzzy sets to use for any specific plant.

Without any special consideration, the number of fuzzy rules can be simply chosen as the maximal number due to the number of fuzzy sets. For example, if l i k consists of y and

and there are 5 fuzzy sets on y and y, respectively, then the number of fuzzy rules for l i k is 5 x 5 = 25.


(x 0.01) 15, I

I 4 5

Number of fuzzy sets for ( y, y )

fUZZy-MRAC

MRAC

Fig 3 1, and dashed line for Example 2 )

The IAE's for different numbers of fuzzy sets. (Solid line for example

Ym:*-) :. - _' o.:r-l 0.6 ~ ~ : , - , ~ ~ ~ i :._ . _I

0.4

0.2

OO 5 10 OO 5 t O t ( S E ) t (E)

Fig. 4. Step response of the first example system. (a) MRAC. (b) Com- posite-MRAC. (c) Fuzzy-MRAC, initial. (d) Fuzzy-MRAC, after 2 gradient-descent parameter learning. (Solid lines for desired reference outputs; dashed lines for the controlled system response.)

111. COMPUTER SIMULATIONS

Computer simulations were conducted on two example systems to verify the validity and performance of the proposed fuzzy-MRAC. Four adaptive control strategies are sim- ulated on the following two example systems: (a) standard MRAC [l], (b) composite-MRAC [1], (c) fuzzy-MRAC with the initial result without off-line parameter learning, and (d) fuzzy-MRAC with 2 iterations of off-line parameter learning. A composite-MRAC is a tracking-error-and-prediction-error- based MRAC without incorporating the fuzzy set concept. The reference input signals used to test the systems are a step input and a sinusoidal signal. A disturbance d = 0.2sin(20t) is

Fig. 5. Tracking of parameter a1 = 0 1 + 2exp-(Y2+(y)') of the first example system. (a) MRAC. (b) Composite-MRAC. (c) Fuzzy-MRAC, initial. (d) Fuzzy-MRAC, after 2 gradient-descent parameter learning. (Solid lines for actud values; dashed lines for the estimated values in the adaptations.)

added to both example systems to examine the robustness of these MRAC's.

The first example system considered is:

$ + (0.1 + 2e-(y'+(y)2))y + y = u + d system (3)

where y and u are the output and input of the system, respectively, d is an added disturbance to test the robustness of the MRAC's. The system is a nonlinear system with the unknown parameters, a2 = l , a l = 0.1 + 2e~(y'+(Y)~), and a0 = 1, which are functions of known variables y and 5. Although a2 and a0 are constants which are independent of g and y, we still denote them as unknown parameters and estimate them.

Fig. 4 shows the performance of the four adaptive control strategies when the reference signal T is a step input and Fig. 5 shows the tracking performance of the estimated a1 for each method. Fig. 6 shows the system response when the reference signal r is a sinusoidal signal and Fig. 7 again shows the tracking performance of the estimated 81 parameter. Finally, Table I shows the IAE errors for these four control strategies.

For the second example, we let the parameter a1 be the same as that in the first example while a2 and a0 are varying parameters. The second example system is

(1 + O.l(y2 + ($)")$ + (0.1 + 2 e - ( Y z + ( q y

+ (-2 + yy)y = u + d system (4)

This is a nonlinear system with 3 unknown parameters, a2, a1 and ao, which are functions of known variables y and y. Fig. 8 shows their response when r is a step input. Since the tracking of the estimated parameters (a2,al and ao) are similar, we want to see how the performance will change as the parameters a2 and a0 are varying instead of constant; thus we only show the estimated 61 in Fig. 9. Fig. 10 shows the system response

YIN AND LEE: FUZZY MODEL-EFERENCE ADAPTIVE CONTROL 1613

0.2 0.2

0.1 0.1

0 0

4.1

Y m:*-* Ym:'-*

y I_ ~ y:'.--'

0 5 10 0 5 10 4.1

0.2 0.2

0.1 0.1

0 0

-0.1

Ym:*-s Ym:'-,

y : '- ~ *' y : '. - -9

0 5 10 0 5 10 4.1

Fig. 6. Tracking of parameter al = 0.1 + 2 e x , - ( ~ 2 + ( Y ) 2 ) of the first example system. (a) MRAC. (b) Composite-MUC, (c) F ~ ~ ~ ~ . M R A c , initial, (d) Fuzzy-MRAC, after 2 gradient-descent learning. (Solid lines for actual values; dashed lines for the estimated values in the adaptations.)

Fig. 8. step response of the second example system. (a) MRAC. (b) Com- posite MRAC. (C) FuzzY-MRAC, initial. (d) Fuzzy-MRAC, after 2-gradient-descent parameter learning. (Solid lines for desired reference outputs; dashed lines for the controlled system response.)

2':m 1.5

-0 1 I

0.5

5 10 0 ' 0

Fig. 7. Tracking of parameter al = 0.1 + 2 e , p - ( ~ 2 + ( ~ ) 2 ) of the first example system. (a) MRAC. (b) Composite-MRAC. (c) Fuzzy-MRAC, initial, (d) FUZZY-MRAC, after 2 gradient-descent parameter learning. (Solid lines for actual values; dashed lines for the estimated values in the adaptations.)

Fig. 9. Tracking of parameter a1 = 0.1 + 2 ~ X , - ( Y ~ + ( $ ) ~ ) of the second example system. (a) M U C . (b) Composite-MRAC. (c) Fuzzy-MRAC, initial. ( 4 Fuzzy-MRAC, after 2 gradient-descent parameter learning. (Solid lines for actual values; dashed lines for the estimated values in the adaptations.)

when r is a sinusoidal input and Fig. 11 shows the tracking of the estimated 61. Table I1 shows the IAE errors.

From these simulation results, the fuzzy-MRAC with the aid of the FBFE is the best among the four schemes or as good as the composite-MRAC or the standard MRAC. The concept of combining both sources of error information will have better adaptation ability is dependent on the adaptation power of the individual method. When the complexity of the unknown parameters increases, more a priori knowledge is needed to ease the adaptation mechanism. It is shown that composite-MRAC is better than standard MRAC by Slotine and Li in [l]. However, this superiority is violated here due to the more complex unknown parameters. With the FBFE,

the proposed fuzzy-MRAC can take care of more complex unknown parameters and results in better performance. It is worth pointing out that the computational complexity of the proposed FBFE-based MRAC is much higher than the traditional MRAC. This is a trade-off between computations and performance specifications.

All these simulations were carried out using MATLAB (on a Sun Sparcstation) with A = 0.01 seconds sampling interval. The differential equations of the systems were calculated using the MATLAB subroutine rk23.m.

We have also conducted computer simulations on testing the proposed fuzzy MRAC for unmodeled dynamics. Reconsider the above two example systems to have unmodeled dynamics

1614 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL 25, NO 12, DECEMBER 1995

of MRAC, it should be designed with the same care as the other MRAC’s for unmodeled dynamics. Detailed discussions dealing with unmodeled dynamics can be found in [22]. The design process should be guided according to the guideline listed in [22] if there exists substantial unmodeled dynamics in the plant.

In our computer simulations, for the comparison purpose we let one of the reference signals be a step input for 10 seconds. However, persistency of excitation in MRAC systems is important as stated in [l], [22]. Thus, persistency of excitation should be maintained if the step reference signal continues past 10 seconds.

Finally, the approximation error of the FBFE will gradually approach a steady state in which the plant follows the reference model. By the Stone Weierstrass theorem, the approximation error should be asymptotically reduced to zero; however, in real applications the number of fuzzy basis functions is finite

0.2 Y m .*-’

y ’_ - :

Y m d-’

y .’_ . -‘ 0 1

0

-01

t(=)

Y m L’

:I_ - _I

lo t (sec) t (W

Fig 10 Sinusoidal response of the second example system. (a) MRAC. (b) Composite MRAC (c) Fuzzy-MRAC, initial. (d) Fuzzy-MRAC, after 2 gradient-descent parameter learning (Solid lines for desired reference outputs; dashed lines for the controlled system response)

and a enor will exist asymptotically,

IV. CONCLUSION

t (se) t (W

Fig. 11. Tracking of parameter a1 = 0.1 + 2 e ~ p - ( Y ~ + ( y ) ~ ) of the second example system. (a) MRAC. (b) Composite-MRAC. (c) Fuzzy-MRAC, initial. (d) Fuzzy-MRAC, after 2 gradient-descent parameter learning. (Solid lines for actual values; dashed lines for the estimated values in the adaptations.)

In this paper, a FBFE-based MRAC was proposed to deal with controlling a plant with unknown parameters which are dependent on known variables. The unknown parameters in the plant are represented by the FBFE and this representation is substantiated by the Stone-Weierstrass theorem. In order to improve the parameter estimation performance, both the tracking error and the prediction error are used in the on-line adaptation scheme. Considering the fuzzy-MRAC as an extension of the composite-MRAC, the convergence and stability properties of the fuzzy-MRAC are preserved. An off-line parameter learning scheme was proposed to determine the coefficients of the fuzzy sets. It was found, from computer simulations, that the number of fuzzy sets used to model the unknown parameters can be quite small (usually 2 to 5 fuzzy sets). Also from the computer simulations of the two example systems, the performance on parameter estimation and system response (output tracking error) were found to be better than traditional MRAC’s and composite-MRAC’s. Extension of the proposed FBFE-based MRAC to multi-input-multi-output systems will be considered in the future.

s2+j::++229 such that each of the plants becomes REFERENCES

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[2] D E Miller and E. J. Davison, “An adaptive controller which provides an arbitranly good transient and steady-state response,” ZEEE Trans. Automat Contr, vol 36, no 1, pp 68-81, Jan 1991.

[3] R Lozano and B. Broghato, “Adaptive control of a simple nonlinear system without a priori information on the plant parameters,” IEEE Trans Automat Contr., vol. 37, no 1, pp 30-37, Jan. 1992

[4] I. Kanellakopoulos, P. V KokotoviC and A. S. Morse, “Adaptive output- feedback control of systems with output nonlinearities,” IEEE Trans. Automat. Contr.. vol. 37, no. 11, pp 1666-1682, Nov. 1992.

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(30) Cliffs, NJ Prentice-Hall, 1991. 229

K y - 1 - -

u a2s2 + ais + uo + 30s + 229

where the first term represents example system (3) and example system (4) with unknown parameters a0 I a1 and a2 as stated before. Unfortunately, the standard MRAC, composite-

to perfom satisfactory control in both example systems. The problem probably comes from the complex parameter nature of the two example systems such that the bandwidths of the controllers cannot be derived. We do not recommend the use of the Automat. Contr., vol 38, no 1, pp 17-32, Jan 1993

proposed fuzzy-MRAC to deal with unmodeled dynamics. Since the proposed fuzzy MRAC is considered a member

and the proposed fuzzy-MRAC

YIN AND LEE FUZZY MODEL-REFERENCE ADAPTIVE CONTROL 1615

[7] W. Messner, R. Horowitz, W. W. Kao and M. Boals, “A new adaptive learning rule,” IEEE Trans. Automat. Contr., vol. 36, no. 2, pp. 188-197, Feb. 1991.

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[lo] T. L. Johnson, C. A. Harvey and G. Stein, “Self-tuning regulator design for adaptive control of aircraft wing/store flutter,” IEEE Trans. Automat. Contr., vol. 27, no. 5, pp. 101P1022, Oct. 1982.

[ l 11 K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” ZEEE Trans. Neural Networks, vol. 1, no. 1, pp. 4-27, Mar. 1990.

[12] S. Jagannathan, F. L. Lewis and 0. Pastravanu, “Model reference adaptive control of nonlinear dynamical systems using multilayer neural networks,” in Proc. of IEEE Int. Con$ on Neural Networks, vol. 7, pp. 4766-4771, 1994.

[13] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp. 146-155, May 1993.

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[17] L. X. Wang and J. M. Mendel, “Fuzzy basis functions, universal approximation, and orthogonal least-squares learning,” IEEE Trans. Neural Networks, vol. 3, no. 5, pp. 807-814, Sept. 1992.

[18] R. R. Goldberg, Methods of Real Analysis, John Wiley & Sons, 1976. [ 191 J. S. R. Jang, “ANFIS:Adaptive-network-based fuzzy inference system,”

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[20] K. Ogata, Modern Conrrol Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1970.

[21] K. S. Fu, R. C. Gonzalez and C. S. G. Lee, Robotics: Control, Sensing, Vision, and Intelligence.

[22] S. Sastry and M. Bodson, Adaptive Control-Stability, Convergence, and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989.

New York McGraw-Hill, 1986.

Tang-Kai Yin (S’93) received the B.S. degree in 1990 from the National Taiwan University, Taiwan, R.O.C., and the M.S.E.E. degree in 1994 from Pur- due University, West Lafayette, IN, both in electrical engineering. Currently, he is pursuing the Ph.D. degree at the School of Electrical and Computer Engineering, Purdue University.

He received a three-year scholarship from the Ministry of Education, Taiwan, R.O.C., for his doc- toral study. His research interests include adaptive control, fuzzy systems, neural networks and intelli-

’

gent control. Mr. Yin is a member of Tau Beta Pi.

C. S. George Lee (F’93) received the B.S. and M.S. degrees in electrical engineering from Wash- ington State University, Pullman, in 1973, and 1974, respectively, and the Ph.D. degree from Purdue University, West Lafayette, IN, in 1978.

In 1978 and 1979, he taught at Purdne Univer- sitv, and from 1979 to 1985, at the University of I .#’

...A Michigan, Ann Arbor. Since 1985, he has been kith the School of Electrical and Computer Engineering, Purdue University, where he is currently Professor of Electrical and Comouter Enemeerine. His current

I - research interests include neural-network-based fuzzy control systems, computational robotics, and intelligent robotic assembly systems.

Dr. Lee was an IEEE Computer Society Distinguished Visitor from 1983 to 1986, the Organizer and Chairman of the 1988 NATO Advanced Research Workshop on Sensor-Based Robots:Algorithms and Architectures, and the Secretary of the IEEE Robotics and Automation Society from 1988 to 1990. Currently, he is Vice-president for Technical Affairs of the IEEE Robotics and Automation Society, and an Associate Editor of the International Joumal of Robotics and Automation. He is a co-author of Robotics: Control, Sensing, Vzsion, and Intelligence (McGraw-Hill), and a co-editor of Tutorial on Robotics (Second Edition), (IEEE Computer Society Press). Dr. Lee is a member of Sigma Xi and Tau Beta Pi.

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Fuzzy model-reference adaptive control