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Applied Soft Computing 11 (2011) 5715–5723

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l ho mepage: www.elsev ier .com/ locate /asoc

MAC-based previous step supervisory control schemes for relaxing bound indaptive fuzzy control

ed Taoa,∗, Shun-Feng Sub

Department of Computer and Communication Engineering, Technology and Science Institute of Northern Taiwan, No. 2, Xue Yuan Street, Beitou, Taipei 112, TaiwanDepartment of Electrical Engineering, National Taiwan University of Science and Technology

r t i c l e i n f o

rticle history:eceived 25 March 2009eceived in revised form 8 February 2011ccepted 20 March 2011vailable online 12 April 2011

eywords:MAC

a b s t r a c t

In this paper, a novel scheme of incorporating a learning mechanism into previous step supervisorycontrollers for adaptive fuzzy control is proposed to relax bounds required in the control process. Intraditional supervisory adaptive fuzzy control approaches, the use of fuzzy estimators for approximatingsystem functions and a robust supervisory control law are necessary to deal with any possible uncertain-ties caused in the system. This kind of supervisory controller depends on the robust bounds of systemfunctions so that it can ensure the Lyapunov stability of controlled systems. However, in those approaches,the output may not be able to follow the reference trajectory well if the robust bounds are predicted

upervisory controlobust bounduzzy controlyapunov stability

improperly. In our implementation, CMAC (Cerebellar Model Articulation Controllers) is used as thelearning mechanism because of its quick learning capability. Under the Lyapunov stable criterion, theproposed CMAC learning mechanism can improve the output performance and can relax the robustbound limitation so that practical systems can easily be realized. In summary, the proposed approach notonly can relax bounds for previous step supervisory controllers in adaptive fuzzy control, but also cansignificantly improve the control performance of the system.

© 2011 Elsevier B.V. All rights reserved.

. Introduction

Soft computing techniques enable handling of imprecision andncertainty often encountered in solving practical problems requir-

ng reasoning and learning. Soft computing techniques involveomputations related to neural network (NN), fuzzy logic techniqueFL), genetic algorithm (GA), and others. Over the years, researchersave found suitability in using hybrid techniques involving thebove three methods, such as GA-NN, FL-NN, FLGA and GA-FL-N [1]. Some of above researchers are off-line analyses, whilee need on-line hybrid techniques for adaptive fuzzy control in

he proposed method. Fuzzy logic controllers [2–6] or in shortuzzy controllers were proposed to control plants that are poorlynderstood in mathematic models but are familiar to professionalperators. In recent developments, fuzzy controllers have excel-ent performances in situations where the plant parameters and

tructures have some uncertainties or unknown variations [7–10].daptive fuzzy control [9,11–17] is an important methodology forealing with system uncertainties in fuzzy control. We found that

∗ Corresponding author. Tel.: +886 2 8943356 421; fax: +886 2 28943357; mobile:886 0910313047.

E-mail addresses: tedtao@ms15.hinet.net (T. Tao),u@orion.ee.ntust.edu.tw (S.-F. Su).

568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.asoc.2011.03.019

even though the Lyapunov theorem has guaranteed the stability ofthe fuzzy control system [18–20], if certain conditions are not satis-fied, the system may still be not able to achieve acceptable controlperformance in our study of fuzzy adaptive controllers [21].

Recently, the robust control methodologies have been consid-ered in ensuring the stability of the controlled systems and canalso sometimes provide the theoretical stability for fuzzy systems[11,18], especially for adaptive fuzzy controllers. The objective ofrobust control is to maintain certain desired performance of a sys-tem despite of the existence of parameter drift or disturbance [22].In robust control schemes, only the nominal plant is considered andparameter drifts and possible disturbances will be viewed as uncer-tainties of the nominal plant. Those uncertainties can be expressedas a single disturbance force injection of the input of the system inrobust control [23]. It can be expected that robust controllers areeasier to realize while compared to adaptive controllers [24,25].Nevertheless, robust controllers are hard to realize when systemparameters are unknown in practical systems.

Considerable research results have been reported for the appli-cation of adaptive fuzzy control techniques based on supervisorycontrol [26,27]. In fact, those supervisory control approaches can

also be said to be a robust control scheme in that they are to drivethe system into a reasonable region when the current control per-formance is not acceptable. The first supervisory controller (SC)is addressed in [28] and many similar SCs are proposed to guar-

5 Comp

aoaCPadtittanw

rsriwtcitoatstb

om[isuftcmaottthaHfi

atSut

2

wa

716 T. Tao, S.-F. Su / Applied Soft

ntee the initial control performance [27,29–35] under differentriginal control mechanisms (the so-called main control [36]), suchs fuzzy systems, neural networks, Cerebellar Model Articulationontrollers (CMAC) [37–40], genetic algorithms [30], self-tuningID controller [33], etc. These supervisory control schemes have

promising advantage of requiring no prior knowledge of systemynamics. Nevertheless, a serious drawback of theses approaches ishat the robust bounds of some system parameters must be antic-patable when implementing supervisory controllers in ensuringhe stability of the system. In fact, those bounds will be difficulto obtain while the system functions are unknown in practicalpplications. Thus, the supervisory adaptive fuzzy controllers can-ot follow reference trajectories very well when the bounds arerongly predicted.

In this paper, we propose to use the previous control action toeplace the unknown perfect control in the robust control structureo that the bound required in the supervisory controller will beelaxed. We further proposed to incorporate a leaning mechanismnto the supervisory controller to learn all possible errors. In other

ords, in our approach, the learning mechanism not only is used inhe adaptive fuzzy control, but also is employed in the supervisoryontrol to improve the control performance. As a consequence, its not necessary to estimate the bound of the system function andhe system can still have robust stability as required. Although somether bounds are still required, their values are usually very smallnd can be eliminated by the learning mechanism. It is easy to seehat the proposed approach can easily be realized for any practicalystems. In this paper, based on Lyapunov theory and the use ofhe sliding control structure, the proposed approach is proved toe stable in the Lyapunov sense.

It should be noted that the learning mechanism used is fornline learning. Thus, it is important that the learning mechanismust have a quick learning property. In our implementation, CMAC

37,38] acts as the learning mechanism because of its quick learn-ng capability. In the literature, there are also some CMAC-basedupervisory control approaches [26,27,40]. Most of them are tose CMAC for taking the place of the fuzzy estimators in adaptiveuzzy control. In Ref. [26], CMAC acts as a traditional adaptive con-roller in their control law and a sliding control based supervisoryontroller is also proposed. The authors further proposed a fuzzyechanism to ease the transition between the adaptive controller

nd the supervisory controller. It can be found that the problemf model errors and the moderation of the requirement of the sys-em function bounds are not handled in the approach. In Ref. [27],he authors added a compensated controller in the control struc-ure to reduce possible model errors. Their approaches indeed canave effects when there are model errors in estimators, but theirpproaches still need to estimate the bound of the sliding controller.owever, with our proposed approach, the supervisory adaptive

uzzy control can work well even the bound of the system functions unknown or does not exist.

This paper is organized as follows. The supervisory control for nonlinear system is considered in Section 2. Adaptive fuzzy con-roller and bounds required in supervisory control are discussed inection 3. The proposed approach will be derived in Section 4. Sim-lation examples are provided to demonstrate the performance ofhe proposed method in Section 5. Section 6 concludes this paper.

. Supervisory control for nonlinear systems

Consider an nth-order nonlinear system of the form

(n) ′ (n−1) ′ (n−1)

x = f (x, x , . . . , x ) + b(x, x , . . . , x )u + dy = x

, (1)

here f and b are unknown but bounded continuous functions, d isn external bounded disturbance, and u and y ∈ R are the input and

uting 11 (2011) 5715–5723

the output of the system, respectively. Let x = (x, x′, . . . , x(n−1)) ∈ Rn

be the state vector of the system. The control objective is to force thesystem output to follow a given bounded reference signal r underthe constraints that all signals involved must be bounded. Now,the task is to design a robust controller for an uncertain system andsuch a robust controller is designed to ensure the stability of thesystem in a Lyapunov sense.

First, consider that there is no disturbance in the system (i.e.d = 0). It can be found that the system output can be ensured toapproach to the reference signal r, if the perfect control law isdescribed in Eq. (2),

u∗ = 1b

[r(n) − f (x, x′, . . . , x(n−1)) + kT e], (2)

where k and e are defined as k = [k0, k1, k2, . . . , kn−1]T and e =[e, e′, e′′, . . . , e(n−1)]

T, and e(t) is the tracking error and is defined

as e(t) = r(t) − x(t). The controller in Eq. (2) is usually referred to asthe perfect control law. Now, apply the perfect control law to Eq.(1) with d = 0, and then we have

e(n) + kT e = 0. (3)

It is easy to verify that if k is selected such that the roots of thecharacteristic equation as Eq. (3) are all in the open left-half plane,then the system will asymptotically track the reference input r.

Secondly, with the use of the Lyapunov stability theorem [18],Theorem 1 [9] is introduced to ensure the stability of supervisorycontrol systems.

Theorem 1. Consider the existence of uncertainties and externalbounded disturbances (i.e. d /= 0) in a control system. If a robust con-troller is considered as:

u = u∗ + us, (4)

where the supervisory control us is defined as

us = sat(S/Dmax) × Dmax ={

Dmax, for S/Dmax ≥ 1S, for −1 < S/Dmax < 1−Dmax, for S/Dmax ≤ −1

,

Dmax =∥∥d/b

∥∥∞ = sup |d/b| = dU

bL, (5)

the lower bound of b is bL (0 < bL ≤ |b|) and the upper bound of d is dU

(|d| ≤ dU), then the Lyapunov stability theorem will be satisfied in thenth-order nonlinear system (1).

Proof. Consider the Lyapunov function as

V1 = 12

S2, (6)

where the sliding surface S(t) is defined as the integral of the char-acteristic polynomial,

S(t) =∫

S(t) dt, where S(t) = e(n) + kT e. (7)

If u = u * + us, then

V1 = S(e(n) + kT e) = S[r(n) − f (x, x′, . . . , x(n−1)) − b(u ∗ +us) − d + kT e]. (8)

By introducing the perfect control law as Eq. (2) into Eq. (8), Eq.(9) is gotten.

V1 = −S(bus + d). (9)

It can be directly verified that if the supervisory control us isselected as Eq. (5), Eq. (9) is always negative. In other words, thesystem is stable in the sense of Lyapunov.�

T. Tao, S.-F. Su / Applied Soft Computing 11 (2011) 5715–5723 5717

ed pre

ffe

3s

mfttutI

R )

wafr

ic

p

c

b

[t

Fig. 1. The block diagram of the proposed bound relaxing CMAC-bas

However, the perfect control law needs to know the functions and b. If they are unknown, there is no way of forming the per-ect control law. Thus, the adaptive fuzzy control schemes will bemployed to resolve this problem in the next section.

. Adaptive fuzzy controller and bounds required inupervisory control

Fuzzy set theory, introduced by Zadeh in 1965 [41], has foundany applications in a variety of fields in recent years. When those

unctions f and b are unknown in a practical system, the fuzzy con-rol schemes as proposed in [42,43] can be employed to approachhose mentioned variables in the control system. In this paper, thesed fuzzy system is to perform a mapping from the current stateso the estimated f and b. The mapping consists of a set of fuzzyF–THEN rules in which the lth rule is of the form

(l) : IF x1 is F (l)1 , and . . . , and xn is F (l)

n THEN p = �(l),(10

here x1,. . ., xn are the state variables defined in Eq. (1), F (l)1 , . . . , F (l)

n

re the corresponding fuzzy labels, p is the output variable for theuzzy system, and �(l) is the corresponding output value for the lthule.

By using the product operations for the conjunction relationsn the premise parts of fuzzy rules, the output of a fuzzy systemonsisting of N rules is obtained as follows.

=

∑Nl=1

(∏ni=1�

F(l)i

(xi))

�(l)

∑Nl=1

(∏ni=1�

F(l)i

(xi)) = εT (x)�. (11)

The term ε(l) =(∏n

i=1�F(l)

i

(xi)/(∑N

l=1

(∏ni=1�

F(l)i

(xi))))

is

alled the fuzzy basis function [44], where � (l) (xi) is the mem-

F

i

ership degree of xi belonging to the fuzzy label F (l)i

. � =�(1), �(2), . . . , �(N)]

Tand ε = [ε(1), ε(2), . . . , ε(N)]

Tare referred to as

he regressive vectors.

vious step supervisory control for an adaptive fuzzy control system.

Thus, the fuzzy system has two corresponding output values �fand �b for f and b, respectively. Based on Eq. (11), f and b then canbe approximated as

f (x) = pf =

∑Nl=1

(∏ni=1�

F(l)i

(xi))

�(l)f∑N

l=1

(∏ni=1�

F(l)i

(xi)) = εT �f , (12)

b(x) = pb =

∑Nl=1

(∏ni=1�

F(l)i

(xi))

�(l)b∑N

l=1

(∏ni=1�

F(l)i

(xi)) = εT �b. (13)

Now, by using Eqs. (12) and (13) to replace f and b respectively inEq. (2), we have the fuzzy perfect control variable as the followingequation.

u∗ = 1

b[r(n) − f (x, x′, . . . , x(n−1)) + kT e]. (14)

With the use of supervisory control as Eq. (5), the fuzzy perfectcontrol variable becomes u = u∗ + us. According to the Lyapunovtheorems, the adaptive update law can be obtained as follows

�b = −rbSεu∗ and �f = −rf Sε, (15)

where rb and rf are adjusting ratio constants. Here we introduceTheorem 2 to ensure the stability of adaptive fuzzy laws in Eq. (15),which has been proved in Refs. [42,43].

Theorem 2. If a robust adaptive fuzzy controller is consideredas: u = u∗ + us, where the supervisory control us is defined asus = sat(S/Dmax) × Dmax, the adaptive update laws are �b = −rbSεu∗

and �f = −rf Sε, and the bound Dmax is defined as

Dmax =∥∥∥∥ıf + ıbu∗ + d

b

∥∥∥∥∞

= sup

∣∣∣∣ıf + ıbu∗ + d

b

∣∣∣∣ (16)

then the Lyapunov stability theorems will be satisfied and the controlsystem (1) will be stable in the sense of Lyapunov.

Now, the supervisory bound condition Dmax is considered. In

order to find Dmax, it is easy to see that

Dmax ≤ 1bL

sup |(ıf + d)| + |ıb|bL

sup |u∗|, (17)

5718 T. Tao, S.-F. Su / Applied Soft Computing 11 (2011) 5715–5723

F l: (a)

d

wb

s

wstvtoprtmpisi

4c

piglaisch

ig. 2. The performance of using the traditional supervisory adaptive fuzzy controisturbance d, and (c) the system function f and the estimated system function Pf .

here bL is the lower bound of b (0 < bL ≤ |b|). Furthermore, it cane found that

up |u∗| ≤ 1bL

sup(|r(n)| + f U + |kT e|

), (18)

here fU is the upper bound of f (|f| ≤ fU). The above bounds are theame as those defined in [11,18,26,27]. Thus, it is necessary to knowhe upper bound of the system function f in estimating the supremealue of the fuzzy perfect control variable u∗ as Eq. (18). However,he upper bound of the system function f is sometimes difficult tobtain in practical systems, or it even does not exist. Thus, in thisaper, we propose to use a CMAC-based learning mechanism foresolving this problem. The idea is to replace perfect control withhe previous step control value in Eq. (14) and the proposed learning

echanism is employed to learn the different values between theerfect controller and the previous step control value. As a result,

t is possible to predict the difference and to compensate it in theupervisory control system. The proposed method will be discussedn the next section.

. Bound relaxing CMAC-based previous step supervisoryontrol

The Cerebellar Model Articulation Controllers (CMAC) firstlyroposed by Albus in the literature [37,38], have several advantages

ncluding local generalization [45,46] and rapid learning conver-ence [47,48]. CMAC seems to be a good candidate for on-lineearning control [25] and can be thought of as a learning mech-nism that imitates the cerebellum of a human being. The CMAC

s often referred to an associative neural network, where only amall subset memory cells mapped by the input vector or so-alled state instantaneously determines the output. In fact, CMACas been regarded as a look-up table neuron computing tech-

the reference signal r and the output y, (b) the control variable u and the external

nique and has the property of fast learning convergence. Whena datum is recorded into the CMAC table, it is distributed intoL cells. L is also called the floor number of a variable and indi-cates the layer number each variable has. Detailed descriptioncan be found in the literature [25,37,38,49]. A significant prop-erty of CMAC is that its learning has effects on the neighborhoodinstead of a certain spot. Thus, similar inputs even for untrainedinputs still lead to similar outputs. This property is called thegeneralization capability [46,49]. The generalization property ofCMAC can be successfully used for control [25] and image appli-cations [49]. The CMAC uses a set of association cell indexes Cvk

=[c1(vk), c2(vk), . . . , cj(vk), . . . , cNm (vk)]T as address indexes via then-dimension input vector vk = (x1, x2, . . . , xn) to extract the Lstored weights W from memory cells, and only those activated cellindexes cj(vk) will be set to be 1 which are mapped by input vec-tor, and the others are set to be 0. Thus, the out put of CMAC canbe described as uCMAC = CT

vkW . In our approach, the CMAC learn-

ing mechanism will be employed into the supervisory controller tolearn all possible errors.

The block diagram of the bound relaxing CMAC-based previ-ous step supervisory control is illustrated in Fig. 1. The proposedcontroller can be written as

u = u(k − 1) + M0(e(n) + εTf ��f ) + uCMAC + us, (19)

where uCMAC = CTvk

W is the proposed CMAC learning controller, andεT

f��f is outputs of the fuzzy estimator. According to robust meth-

ods, it is assumed that b−1 can be divided into a nominal part M0and an varying part �M (i.e. b−1 = M0 + �M). Here the nominal value

M0 is usually the mean value of b−1 and is a constant. In our imple-mentation, the input variables for the CMAC learning mechanismconsidered in this paper are the tracking error and the error deriva-tive. In order to ensure the stability of the proposed controller, the

T. Tao, S.-F. Su / Applied Soft Computing 11 (2011) 5715–5723 5719

F ve fuzt functi

stf

abmut

wobta

u

wmbatls

ig. 3. The performance of using the CMAC-based previous-step supervisory adaptihe external disturbance d, and (c) the system function f and the estimated system

upervisory control is employed to guarantee that the control sys-em will be stable in the sense of Lyapunov, which is proved in theollowing.

As mentioned earlier, the idea is to use previous controlction so that the bounded for the perfect control law cane reduced and the supervisory control can easily be imple-ented. Let u∗ = u(k) = (1/b)[r(n)(k) − f (k) + kT e(k)] as Eq. (2) and

(k − 1) = (1/b)[x(n)(k − 1) − f(k − 1) − d(k − 1)] from Eq. (1). Owingo b−1 = M0 + �M, we can deduce Eq. (20).

u∗ = u(k − 1) + 1b

[r(n)(k) − x(n)(k − 1) − f (k) + f (k − 1) + kT e(k) + d(k − 1)]

= u(k − 1) + M0[r(n)(k) − x(n)(k − 1) − f (k) + f (k − 1) + kT e(k) + errb]

+ 1b

d(k − 1),

(20)

here errb is the error caused by the variation of b and is a functionf �M/M0. Since the control task is to track r, the term x(n)(k − 1) cane written as x(n)(k) − errtracking. Also, the transition error kT e(k) isermed as errtransition. With the definitions of e(n)(k) = r(n)(k) − x(n)(k)nd �f = f(k) − f(k − 1), Eq. (20) can be rewritten as

∗ = u(k − 1) + M0(e(n) − �f ) + M0E + 1b

d(k − 1), (21)

here E = errb + errtracking + errtransition. It is easy to know that E isostly a function of tracking errors. It is supposed that −�f can

e approximated by the fuzzy estimator in adaptive fuzzy control

nd the supervisory controller us is employed to overcome the dis-urbances and the others errors. If the system can find a way ofearning an approximation of E, it will largely reduce the use of theupervisory control. Thus, in this paper, we propose to learn such a

zy control: (a) the reference signal r and the output y, (b) the control variable u andon Pf .

relationship from the supervisory control. In fact, as shown in latersimulation, it can be found that such an approach has excellenttracking performance.

Theorem 3. If a robust controller is considered as Eq. (19), where thesupervisory control is defined as us = sat(S/Dmax) × Dmax with Dmax as

Dmax = 1bL

sup∣∣�d − ı�f − M−1

0 ıW − ıM

∣∣ ∼= 1bL

sup |�d|, (22)

the adaptive fuzzy law as

��f = r�f Sεf , (23)

and the CMAC updating law as

W = M0−1rmSCvk

, (24)

then the Lyapunov stability theorems will be satisfied in the nth-ordernonlinear system as described in Eq. (1).

Proof. First, define the Lyapunov function as

V = 12

(S2 + 1

r�f�T

�f ��f + 1rm

WT W

). (25)

The error vectors ��f and W are defined as ��f = �∗�f

− ��f and

W = W∗ − W , where the optimal output for the fuzzy estimator is�∗

�fand the optimal weight for the CMAC is W*. Similarly, their

model errors are ı�f and ıW, respectively. So, the following equa-tions can be defined.

−�f = εTf �∗

�f + ı�f = εTf ��f + εT

f ��f + ı�f , (26)

5720 T. Tao, S.-F. Su / Applied Soft Comp

M

c

u

wt

S

bOaT

S

wwlih

S

Wb

V

M

V

WL

aTacCc

Fig. 4. The control output of the CMAC mechanism.

0E = CTvk

W∗ + ıW = CTvk

W + CTvk

W + ıw . (27)

With the above notation and Eq. (21), the proposed controlleran be rewritten as follows.

= u(k − 1) + M0(e(n) + εTf ��f ) + uCMAC + us = u∗ + us

− (M0ı�f + ıW + M0εTf ��f + CT

vkW) + 1

bd(k − 1), (28)

here u* is the perfect control as Eq. (2). Now, take Eq. (28) intohe sliding surface as defined in Eq. (7) and we have

˙ = e(n) + kT e = −�d − bus + bM0ı�f + bıW + bM0εTf��f + bCT

vkW . (29)

It is known that �d = d(k) − d(k − 1) and = (M0 + �M)−1 = M0

−1(1 − (�M/M0) + O((�M/M0)2)), where((�M/M0)2) is the complexity upper bound notation indicatingll terms that have powers equal to or higher than (�M/M0)2.hus, Eq. (29) can be written as

˙ = −�d − bus + ı�f + M−1

0 ıW + M0−1(M0εT

f��f + CT

vkW) + ıM, (30)

here ıM is the error term caused by approximating b with M−10 and

ill also be included into the total error E. In other words, the CMACearning mechanism will also take this error effects into accountn learning. Let ıall = (1/b)(�d − ı�f − M−1

0 ıW − ıM) and then weave

˙ = −b(us + ıall) + (εT

f ��f + M−10 CT

vkW). (31)

ith Eq. (31), the time derivative of the Lyapunov function (25)ecomes

˙ = −bS(us + ıall) −

(1

r�f�T

�f��f − εT

f��f S

)−(

1rm

WT W − M−10 CT

vkWS

). (32)

It is easy to verify that if Eq. (23) ��f = r�f Sεf and (24) W =−10 rmSCvk

are satisfied then Eq. (32) becomes

˙ = −bS(us + ıall). (33)

hen the bound of supervisory control is selected as Eq. (22), theyapunov condition V < 0 will be assured.�

Note that Eq. (23) is the adaptive rule of the fuzzy estimatornd Eq. (24) is the update rule of the CMAC learning mechanism.he implementation of Eq. (23) is the same as that in the original

daptive fuzzy control. For the realization of Eq. (24), the adjustableonstant rm is set as 1/L, where L is the floor number used in theMAC mechanism. This selection is equivalently to set the learningonstant as 1 for fast learning [25,48,50]. From the definition of

uting 11 (2011) 5715–5723

saturation function, it can be found that us is a bounded version ofS. Thus, in our implementation, us takes the place of S in Eq. (24).Thus the updating rule for the CMAC learning mechanism becomes

Wnew = Wold + 1LM0

usCvk. (34)

In the above proposed controller as shown in Fig. 1, it can befound that the system function f is no longer required in the robustbound condition. In fact, in Eq. (19), it can be found that the previousstep of the control u(k − 1) takes the place of the fuzzy perfect con-trol variable u∗. Thus, it is not necessary to figure out the supremevalue of the system function f while calculating the bound of therobust control in Eq. (22). However, the bounds of model errorsı�f, ıW, ıM and disturbance �d are still needed while calculatingthe supreme value of the supervisory control. Fortunately, the dis-turbance �d is supposed to be bounded, and the modeling errorsof the fuzzy estimator and the CMAC compensator are supposed tobe small under the universal approximator theorems. Hence, thebound limitation can be replaced by Dmax ∼= (1/bL)sup|�d|. In latersimulation it can be seen that even though f is a divergent func-tion, the proposed approach still can work well. In other words, therobust bound becomes relaxing with the CMAC learning mecha-nism, so that the previous step supervisory adaptive robust fuzzycontroller can easily be realized.

5. Simulations

To illustrate the effectiveness of the proposed schemes, twoexamples are considered in this paper. An inverted pendulumsystem as used in [11,43] is employed as the first example. Thedynamics of this nonlinear system is

x1 = x2

x2 = f + bu + d; y = x1; f = g sin x1 − (mlx2

2 sin x1 cos x1)/(mc + m)

l[4/3 − m cos2 x1/(mc + m)];

b = cos x1/(mc + m)l[4/3 − cos2 x1/(mc + m)]

. (35)

where x1 is the angle of the pole with the range of initial anglesin 0 ∼ ±0.2 (rad), x2 is the angular velocity of the pole, g is thegravity (9.8 m/s2), mc is the mass of the cart (1.0 kg), m is themass of the pole (0.1 kg), u is the force applied to the cart, d isthe external disturbance (−5Nt ≤ d ≤5Nt), and the length of thepole l is 0.5 m. Let the reference signal be r(t) and the trackingerror be e(t) = r(t) − y(t). The error derivative is simply calculatedas e(t) = [e(k) − e(k − 1)]/ts, where ts is the sampling time.

First, the traditional supervisory adaptive fuzzy controller isadaptively tuned according to Theorem 2 and also is applied in [43].There are 25 rules utilized in this two input and one output adaptivefuzzy controller. The membership functions of fuzzy sets used aretriangular functions in this paper. The bound values of fuzzy sets areset to ±1 for the input and ±5 for the output, and the initial centralvalues of the membership functions used are set to be [− 1, − 0.05,0, 0.05, 1]T for the input e, [− 5, − 0.625, 0, 0.625, 5]T for the output�f and [0, 0.875, 1, 1.125, 2]T for the output �b. The reference signalis r(t) = (�/10)[sin(t) + 0.3 sin(3t)] (rad), the initial value of y(0) is 0.2(rad) and the external disturbance is a square wave with its ampli-tude being 5(Nt). From the above conditions and Eq. (35), the rangeof the parameter b is 1.38 < b < 1.48. The mean value of b is 1.43 andthus M−1

0 is also chosen to be 1.43 (i.e. M0 = 1.43−1 = 0.7). Accordingto Eq. (5) (Dmax = sup|d/b|), the robust control bound Dmax is set tobe 4. In this example, rb = 0.01 and rf = 1 are used for adaptive fuzzyrules in Eq. (15).

The simulation result of the traditional supervisory adaptivefuzzy control system is shown in Fig. 2. In those figures, the (a)part shows the reference signal r and the output y, the (b) partshows the control variable u and the external disturbance d, and

T. Tao, S.-F. Su / Applied Soft Computing 11 (2011) 5715–5723 5721

F for f =e tion P

ttutdbwirbtbts

tiuiontsaEfode

ig. 5. The performance of using the traditional supervisory adaptive fuzzy controlxternal disturbance d, and (c) the system function f and the estimated system func

he (c) part shows the system function f and the estimated sys-em function Pf. Note that the disturbance d also can be seen as thencertainties (�f) of the system function f. Thus the system func-ion f includes the uncertainties (�f = d) in our simulation so as toemonstrate the control effects. From the simulation results, it cane observed that the outputs cannot follow the reference signal veryell with these fuzzy control schemes when the reference signal

s time-varying and with external disturbances. It is obvious thatoot mean square error (RMSE) still has 0.017 after 2 s. In fact, theound of the supreme values of the perfect control law u∗ is difficulto obtain from a practical system f in Eq. (16). In this simulation, theound of the robust control is set as Dmax = 4 by Eq. (5), which takeshe place of Eq. (16) in this example, so that the tracking errors willtill exist.

Now, the proposed CMAC-based previous step supervisory con-roller as Eq. (19) is employed. According to Eq. (22), the bound Dmax

s still set as 4. The related values for the adaptive fuzzy controllersed are the same as that used in above. The control performance

s shown in Fig. 3. The part (a) shows the reference signal r and theutput y, the part (b) shows the control variable u and the exter-al disturbance d, and the (c) part shows the system function f andhe estimated system function Pf. In order to deduce the estimatedystem function Pf from the proposed method, we let control vari-ble u in Eq. (19) be equivalent to u∗ + us, where u∗ is described inq. (14). After learning, the regressive vector of the membership

unction �f becomes [− 4.98, − 0.59, 0, 0.645]T. The control outputf the CMAC learning mechanism (uCMAC) is shown in Fig. 4, whichemonstrates that the CMAC is mostly depends on the derivativerror e. From the simulation results, it can be found that although

ex − 1: (a) the reference signal r and the output y, (b) the control variable u and thef .

the reference signal is time-varying and with external disturbances,the output can follow reference signal very well after 2 s with theproposed control structure. It can be found that the RMSE is muchsmaller than that in the traditional supervisory adaptive fuzzy sim-ulation. Besides, it can be observed in the part (c) that the estimatedsystem function Pf is closer to the system function f in Fig. 4 thanthat in Fig. 3. It is obvious that the performance of the previous-stepCMAC-based supervisory adaptive fuzzy control is better than thatof the traditional supervisory fuzzy controller.

For a better demonstration, the second example considered isa first-order unstable system, which is considered in Ref. [21]. Theconsidered system is

x = f + u + d; f = ex − 1y = x

. (36)

It can be found that this system possesses drastically unstablebehavior than that of Example 1. In this simulation, the referencesignal r, the external disturbance d, the initial values for x, and themax bound values (bmax) of fuzzy sets, the robust bound Dmax andthe membership functions are set to same as those in the first exam-ple. The initial central values of the membership functions are setto be [− 1, − 0.05, 0, 0.05, 1]T for the input and [− 5, − 1.25, 0, 1.25,5]T for the output.

First, the simulation of using the traditional supervisory adap-tive fuzzy control according to Theorem 2 is shown in Fig. 5.

Although Eq. (15) is used to adaptively achieve the supervisorycontrol, it still can be found that the system output cannot followthe reference signal well when the disturbance exists. The mainreason is that the supreme value of the fuzzy perfect control u∗

5722 T. Tao, S.-F. Su / Applied Soft Computing 11 (2011) 5715–5723

F ive fuv mated

ifttpiTstbrape

6

sLfcvias

to

ig. 6. The performance of using the CMAC-based previous-step supervisory adaptariable u and the external disturbance d, and (c) the system function f and the esti

n Eq. (18) is difficult to obtain from the increasing exponentialunction f = ex − 1. Another reason is that the controller will changehose parameters according to the derived adaptive laws whenhe errors persistently exist [21]. The performance of using theroposed approach is shown in Fig. 6. The nominal value M0 = 1

s chosen for control, and the robust bound Dmax is set to be 4.he control performance is very good as expected. In fact, in ourtudy, other values for the robust bound were also considered, andheir performances and RMSEs are almost the same. Thus, it cane concluded that the proposed approach can indeed relax boundsequired in a supervisory controller for adaptive fuzzy control andlso can improve the control performance of the system. Thus, theroposed controller can easily be realized for any practical systemsven if the bound of the function f is unpredictable.

. Conclusions

The supervisory control is employed to adaptive fuzzy controlystem such that the control system will be sable in the sense ofyapunov. In order to avoid chattering phenomenon, a saturationunction is introduced in our approach. However, the output stillannot follow the reference command well in the traditional super-isory fuzzy control systems if the bound of the system function fs unpredictable. In our implementation, the previous step controlction and a CMAC learning mechanism are employed to deal with

uch problems.

In this paper, the proposed controller can significantly improvehe control performance of adaptive fuzzy control even if the boundf the system function is unknown or does not exist. The idea

[

[

zzy control for f = ex − 1: (a) the reference signal r and the output y, (b) the control system function Pf .

is simple, but some tedious derivations are conducted to ensurethe system to be Lyapunov stable. This approach can free therobust bound limitation so that the practical system can easily berealized. It indeed can improve the performance of robust fuzzycontrol systems. Finally, simulation results demonstrate that pro-posed schemes can make the plant output track the time-varyingtrajectories quickly and accurately.

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