COMMENT

ROBUST ADAPTIVE SLIDING MODE

CONTROL USING FUZZY MODELLING FOR

A CLASS OF UNCERTAIN MIMO NONLINEAR

SYSTEMS

1 Introduction

The controller proposed in ‘Robust adaptive sliding modecontrol using fuzzy modelling for a class of uncertainMIMO nonlinear systems’ is pointed out to be infeasible inpractice due to the unknown portion of the controlled plantthat exists in the proposed controller. A modified resultis presented for the problem under a relatively stringentassumption.

Sliding mode control (SMC) has been widely applied tothe robust control of nonlinear systems. However, SMC doeshave certain drawbacks including chattering caused by adiscontinuous control action. To overcome these difficulties,fuzzy sliding mode controllers (FSMCs) have been proposed.The main advantage of FSMCs is that the control methodachieves asymptotic stability of the closed-loop system.Although FSMCs have found practical applications in manyfields, several fundamental problems still exist in the controlof complex systems, for example the stability and robustnessanalysis of a FSMC system is usually very difficult.

Lin and Chen [1] addressed the problem of controlling anunknown MIMO nonlinear affined system, their goal was todevelop an adaptive MIMO FSMC to overcome theinteractions among the subsystems using a decouplingneural network and to facilitate robust properties by fine-tuning the consequent membership functions. The maincontribution of [1] was the implementation of an advancedadaptive SMC strategy within the framework of a fuzzymodel. Unfortunately, however, there is an error whichcannot be overlooked in their development of the controlscheme. The error is in (7) in [1], that is, u ¼ uueq þ uuh ¼uueq þ G�1uh; where we note that G is unknown, whichmeans that the proposed control law u cannot beimplemented in practical applications. Because [1] uses u ¼uueq þ G�1uh to obtain (23) (sliding manifold) and theirtheorem 2, these are also incorrect. Thus, the addressedcontrol problem has not been completely solved.

In the following, we present our modifications to thiscontrol problem under a relatively stringent assumption.

2 The modified results

Consider the following MIMO nonlinear affined system:

yðrÞ ¼ fðxÞ þ GðxÞu þ dðtÞ ð1Þ

The variables and functions of system (1) and theassumptions used to derive the control law for system (1)are the same as in Lin and Chen [1] and are omitted here forbrevity.

First, we make the following assumption on the basis ofLin and Chen [1].

Assumption 1: The bounds of the elements of GðxÞ; that is,the minima giimin of gii ðgii > 0Þ and the maxima jgijjmax ofjgijj; are all known. The matrix G1 is positive definite andinvertible and is defined as:

G1 ¼

g11min �jg12jmax � � � �jg1mjmax

�jg21jmax g22min � � � �jg2mjmax

..

. ... ..

. ...

�jgm1jmax �jgm2jmax � � � gmmmin

26664

37775

Define the sliding manifold as follows

s ¼ ½s1; s2; . . . ; smT

¼

eðr1�1Þ1 þ a11e

ðr1�2Þ1 þ � � � þ a1;ðr1�1Þe1

eðr2�1Þ2 þ a21e

ðr2�2Þ2 þ � � � þ a2;ðr2�1Þe2

..

.

eðrm�1Þm þ am1e

ðrm�2Þm þ � � � þ am;ðrm�1Þem

26666664

37777775

ð2Þ

We choose the control law as follows:

u ¼ uueq þ uuh ¼

uueq1 þ uuh1

uueq2 þ uuh2

..

.

uueqm þ uuhm

26664

37775 ð3Þ

From (2), the time derivative of s is:

_ss ¼ ½_ss1; _ss2; . . . ; _ssmT

¼

eðr1Þ1 þ a11e

ðr1�1Þ1 þ � � � þ a1;ðr1�1Þ _ee1

eðr2Þ2 þ a21e

ðr2�1Þ2 þ � � � þ a2;ðr2�1Þ _ee2

..

.

eðrmÞm þ am1e

ðrm�1Þm þ � � � þ am;ðrm�1Þ _eem

266666664

377777775

¼ yðrÞd � yðrÞ þ

Pr1�1

i¼1

a1ieðr1�iÞ1

Pr2�1

i¼1

a2ieðr2�iÞ2

..

.

Prm�1

i¼1

amieðrm�iÞm

2666666666664

3777777777775

¼ yðrÞd � fðxÞ � GðxÞu � dðtÞ þ

Pr1�1

i¼1

a1ieðr1�iÞ1

Pr2�1

i¼1

a2ieðr2�iÞ2

..

.

Prm�1

i¼1

amieðrm�iÞm

266666666666664

377777777777775

ð4Þ

From (4), by letting _ss ¼ 0, we obtain the equivalentcontrol as:

uueq ¼ GG�1

yðrÞd � ffðxÞ þ

Pr1�1

i¼1

a1ieðr1�iÞ1

Pr2�1

i¼1

a2ieðr2�iÞ2

..

.

Prm�1

i¼1

amieðrm�iÞm

266666666664

377777777775

266666666664

377777777775

ð5Þ

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004522

First, we will find a suitable hitting control uuh: Consider theLyapunov function candidate:

Va;i ¼1

2s2

i ð6Þ

From (2), taking the time derivative of (6), and using (3)and (4), we have:

_VVa;i¼si yðriÞdi �fi�

Xm

j¼1

gijuueq j�diðtÞþXri�1

j¼1

aijeðri�jÞi �

Xm

j¼1

gijuuhj

!

ð7Þ

Suppose that the hitting control uuhi can be expressed asfollows:

uuhi ¼ KhisgnðsiÞ ð8Þ

where Khi > 0; i ¼ 1; 2; . . . ;m; to be determined. Then:

_VVa;i ¼ si Pa;i �Xm

j¼1

j6¼i

gijKhjsgnðsjÞ

2664

3775� sigiiKhisgnðsiÞ

¼ si Pa;i �Xm

j¼1

j6¼i

gijKhjsgnðsjÞ

2664

3775� jsijgiiKhi ð9Þ

where

Pa;i ¼ yðriÞdi � fi �

Xm

j¼1

gijuueq j � diðtÞ þXri�1

j¼1

aijeðri�jÞi Þ ð10Þ

Denote

Fa;i ¼ j fijmaxþXm

j¼1

jgijjmaxjuueq jj þXri�1

j¼1

aijeðri�jÞi

��� ���þ yðriÞdi

��� ���þDi

ð11Þ

Since

si Pa;i �Xj¼1

m

j6¼i

gijKhjsgnðsjÞ

2664

3775 � jsij Fa;i þ

Xj¼1

m

j 6¼i

jgijjmaxKhj

0BB@

1CCA

ð12Þ

hence

_VVa;i � jsij Fa;i þXj¼1

m

j6¼i

jgijjmaxKhj

0BB@

1CCA� jsijgiiKhi

� jsij Fa;i þXj¼1

m

j6¼i

jgijjmaxKhj

0BB@

1CCA� jsijgiiminKhi ð13Þ

if we choose Khi>0; i ¼ 1; 2; . . . ;m; such that:

giiminKhi �Xj¼1

m

j 6¼i

jgijjmaxKhj >Fa;i ð14Þ

then, _VVa;i < 0:Equation (14) can be rewritten as follows:

g11min �jg12jmax � � � �jg1mjmax

�jg21jmax g22min � � � �jg2mjmax

..

. ... ..

. ...

�jgm1jmax �jgm2jmax � � � gmmmin

26664

37775

Kh1

Kh2

..

.

Khm

26664

37775>

Fa;1

Fa;2

..

.

Fa;m

26664

37775

ð15Þ

Using assumption 1, from (15), we have:

Kh1

Kh2

..

.

Khm

26664

37775> G�1

1

Fa;1

Fa;2

..

.

Fa;m

26664

37775

we should take

Kh1

Kh2

..

.

Khm

26664

37775 ¼ G�1

1

Fa;1

Fa;2

..

.

Fa;m

26664

37775þ

�h1

�h2

..

.

�hm

26664

37775¼D

Ka;h1

Ka;h2

..

.

Ka;hm

26664

37775 ð16Þ

where �hi>0; i ¼ 1; 2; . . . ;m; are some small constants, tobe chosen by the designer. Thus, we have determined thehitting control. In the following, we will further determinethe adaptive law.

Plugging u or (3) into (4), we have:

_ss ¼ yðrÞd � f ðxÞ � GðxÞðuueq þ uuhÞ � dðtÞ þ

Pr1�1

i¼1

a1ieðr1�iÞ1

Pr2�1

i¼1

a2ieðr2�iÞ2

..

.

Prm�1

i¼1

amieðrm�iÞm

2666666666664

3777777777775

þ GGðxÞuueq � GGðxÞuueq

ð17Þ

Plugging uueq or (5) into the last term of (17), we have:

_ss ¼ ½ ff ðxjuÞ � f ðxÞ þ ½GGðxjwÞ � GðxÞuueq � GðxÞuuh � dðtÞð18Þ

Plugging fðxÞ ¼ ff ðxju�Þ � zf ; GðxÞ ¼ GGðxjw�Þ � zG; into(18), we have

_ss ¼ ½ ff ðxjuÞ � ff ðxju�Þ þ ½GGðxjwÞ � GGðxjw�Þuueq

� GðxÞuuh � dðtÞ þ zf þ zGuueq ð19Þ

Define ~uui ¼ ui � u�i ; ~wwij ¼ wij � w�ij; then:

_ss ¼

~uuT1jf

~uuT2jf

..

.

~uuTmjf

2666664

3777775þ

~wwT11jg ~wwT

12jg � � � ~wwT1mjg

~wwT21jg ~wwT

22jg � � � ~wwT2mjg

..

. ... . .

. ...

~wwTm1jg ~wwT

m2jg � � � ~wwTmmjg

2666664

3777775

uueq1

uueq2

..

.

uueqm

2666664

3777775

�

d1ðtÞd2ðtÞ...

dmðtÞ

266664

377775� GðxÞuuh þ zf þ zGuueq ð20Þ

Now, consider the Lyapunov candidate:

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 523

Vb ¼ Vb;1 þ Vb;2 þ � � � þ Vb;m ð21Þ

where

Vb;i ¼1

2sT

i si þ1

gi

~uuTi~uui þ

Xm

j¼1

1

bij

~wwTij ~wwij

!

Noting the fact that _~uu~uui ¼ _uui; _~ww~wwij ¼ _wwij; and with (20), weobtain the derivative of V as:

_VVb ¼ _VVb;1 þ _VVb;2 þ � � � þ _VVb;m ð22Þ

where

_VVb;i ¼ sTi _ssi þ

1

gi

~uuTi_~uu~uui þ

Xm

j¼1

1

bij

~wwTij_~ww~wwij

¼ si~uu

Ti jf þ zfi þ

Xm

j¼1

~wwTijjg þ zGij

� �uueq j � diðtÞ

"

�Xm

j¼1

gijuuhj

#þ 1

gi

~uuTi_~uu~uui þ

Xm

j¼1

1

bij

~wwTij_~ww~wwij

¼ 1

gi

~uuTi ðsigijf þ _~uu~uuiÞ þ

Xm

j¼1

1

bij

~wwTijðsibijjguueq j þ _~ww~wwijÞ

þ si

Xm

j¼1

zGijuueq j � sidiðtÞ � si

Xm

j¼1

gijuuhj þ sizfi

If we choose the adaptive law as

_uui ¼ �sigijf ; _wwij ¼ �sibijjguueq j ð23Þ

where the following relations have been used

_~uu~uui ¼ _uui; _~ww~wwij ¼ _wwij

then

_VVb;i ¼ sizfiþsi

Xm

j¼1

zGijuueq j�sidiðtÞ�si

Xm

j¼1

gijuuhj

¼ si zfiþXm

j¼1

zGijuueq j�diðtÞ�Xj¼1

m

j 6¼i

gijKhjsgnðsiÞ

2664

3775�jsijgii

Denote Pb;i ¼ zfi þPmj¼1

zGijuueq j � diðtÞ; then:

_VVb;i < si Pb;i �Xj¼1

m

j 6¼i

gijKhjsgnðsiÞ

2664

3775� jsijgiiKhi ð24Þ

Denote Fb;i ¼ jzfijmax þPmj¼1

jzGijjmaxjuueq jj þ Di; because:

si Pb;i �Xj¼1

m

j6¼i

gijKhjsgnðsiÞ

2664

3775 � jsij Fb;i þ

Xj¼1

m

j6¼i

jgijjmaxKhj

2664

3775

hence

_VVb;i � jsij Fb;i þXj¼1

m

j6¼i

jgijjmaxKhj

2664

3775� jsijgiiKhi

Similarly, by using assumption 1, we should take:

Kh1

Kh2

..

.

Khm

26664

37775 ¼ G�1

1

Fb;1

Fb;2

..

.

Fb;m

26664

37775þ

�h1

�h2

..

.

�hm

26664

37775¼D

Kb;h1

Kb;h2

..

.

Kb;hm

26664

37775 ð25Þ

such that _VVb;i < 0:Combine (16) and (25), we should take:

Khi ¼ maxfKa;hi;Kb;hig; i ¼ 1; 2; . . . ;m ð26Þ

The above discussion can be summarised as thefollowing theorem.

Theorem 1: Consider a nonlinear plant (1) with a controller(3). The adaptive law (23) and hitting control (8) and (26)make the tracking error ultimately bounded.

3 Conclusions

The controller proposed by Lin and Chen [1] has beensuggested to be infeasible in practice due to the unknownportion of the controlled plant that exists in the controller.A modified result has been presented for the problem undera relatively stringent assumption. A detailed derivation hasbeen presented.

4 Acknowledgment

The authors thank the reviewers for their helpful sugges-tions on improving this comment.

15th July 2003

Y.A. Zhang, Y.A. Hu and F.L. Lu

The Naval Aeronautical Engineering Institute,264001, Yantai, Shandong,People’s Republic of China

q IEE, 2004

IEE Proceedings online no. 20040439

doi: 10.1049/ip-cta:20040439

5 Reference

1 Lin, W.S., and Chen, C.S.: ‘Robust adaptive sliding mode control usingfuzzy modeling for a class of uncertain MIMO nonlinear systems’,IEE Proc., Control Theory Appl., 2002, 149, (3), pp. 193–201

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004524