Neurocomputing 74 (2010) 481–486

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

� Corr

210096

E-m

journal homepage: www.elsevier.com/locate/neucom

Letters

Multilayer neural networks-based direct adaptive control for switchednonlinear systems

Lei Yu a,b,�, Shumin Fei a,b, Fei Long c, Maoqing Zhang d, Jiangbo Yu a,b

a Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 210096, PR Chinab School of Automation, Southeast University,Nanjing 210096, PR Chinac Institute of Intelligent Information Processing, Guizhou University, 550025 Guiyang, PR Chinad Institute of Mechanical and Electrical Engineering, Soochow University, 215021 Suzhou, PR China

a r t i c l e i n f o

Article history:

Received 14 January 2010

Received in revised form

31 July 2010

Accepted 19 August 2010

Communicated by M.-J. Ertime technique is given. It is proved that the resulting closed-loop system is asymptotically Lyapunov

Available online 19 October 2010

Keywords:

Switched nonlinear systems

Multilayer neural networks

Average dwell-time

Tracking error performance

12/$ - see front matter Crown Copyright & 20

016/j.neucom.2010.08.024

esponding author at: School of Automation, S

, PR China. Tel.: +86 13851900139.

ail address: slender2008@gmail.com (L. Yu).

a b s t r a c t

This paper is concerned to present a direct adaptive neural control scheme for switched nonlinear systems

with unknown constant control gain. Multilayer neural networks (MNNs) are used as a tool for modeling

nonlinear functions up to a small error tolerance. The adaptive updated laws have been derived from the

switched multiple Lyapunov function method, also an admissible switching signal with average dwell-

stable such that the output tracking error performance is well obtained. Finally, a simulation example of

two Duffing forced-oscillation systems is given to illustrate the effectiveness of the proposed control

scheme.

Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction

Recently, switched systems have received a great deal ofattention, mainly because many real-world systems, such asmechanical systems, transportation systems and power systemscan be modeled as switched systems. At present, there are manyresults on the control analysis and synthesis of switched linear(nonlinear) systems, see [1–6,12–15] and references therein. Themain efforts mainly focus on the analysis of dynamic behaviors,such as stability, controllability, reachability, and observability,and aim to design controllers for assuring stability and optimizingperformance.

It is well-known that adaptive control design remains open fornonlinear systems using an approximator tool. Typically, the tooluses either neural networks (NNs) or fuzzy logic systems toparametrize the unknown nonlinearities [7–13,18,20]. However,the researches on adaptive neural control for switched nonlinearsystems have been less investigated. Few attempts have beenmade (and are being made) to pursue this novel area. In [12], anadaptive NN feedback control scheme and an impulsive controllerare given under all admissible switched strategy for output

10 Published by Elsevier B.V. All r

outheast University, Nanjing

tracking error disturbance attenuation of nonlinear switchedimpulsive systems. Han et al. [13] have presented an adaptiveneural control scheme for a class of switched nonlinear systemswith switching jumps and uncertainties in both system models andswitching signals . So far, theoretical results and constructiveprocedures for designing satisfactory adaptive neural controllersare really very sparse.

In this paper, the motivation of our work is to introduce a directadaptive neural control scheme for a class of switched nonlinearsystems with unknown constant control gain, and give an illus-trative example for practical application. The main feature of thiswork include: (i) MNNs are used as an approximator tool formodeling nonlinear unknown functions; (ii) By designing a directadaptive neural controller with average dwell-time technique[14,15], a novel-type multiple switched Lyapunov function isdeveloped to construct an stable adaptive controller and effec-tive adaptation laws, which can be proved that the resultingclosed-loop switched system is asymptotically Lyapunov stablesuch that the actual output follows the any given bounded desiredoutput.

The outline of this paper is as follows. In Section 2, the directadaptive control problem for switched nonlinear systems isintroduced. In Section 3, a direct adaptive tracking controller isdesigned based on MNNs. A numerical example is treated toillustrate the effectiveness of the design approach in Section 4.Finally, we give a conclusion in Section 5.

ights reserved.

L. Yu et al. / Neurocomputing 74 (2010) 481–486482

2. Problem formulation

Consider a class of switched nonlinear systems as follows:

_xi ðtÞ ¼ xiþ1ðtÞ ð1r irn�1Þ

_xn ðtÞ ¼ fsðtÞðxðtÞÞþbsðtÞðxðtÞÞuðtÞþdsðtÞðx,tÞ

yðtÞ ¼ x1ðtÞ

8><>: ð1Þ

where tZ0, x¼ ðx1,x2, . . . ,xnÞT ARn denotes the state vector of the

systems, which is available. uAR,yAR are the control input and thecontrol output, respectively, and dAL2½0,1Þ is the external dis-turbance. sðtÞ : ½0,1Þ-X :¼ f1,2, . . . ,Ng is a piecewise constantfunction called switching signal (or law), which takes values inthe compact set X. If sðtÞ ¼ i, then we say the i-th subsystem isactive on and the remaining subsystems are inactive. The functionsfiðxðtÞÞðiAXÞ are unknown nonlinear smooth functions, and bi

denote the unknown constant control gain. For conciseness, weuse x, b, u, d instead of x(t ), b(x(t)), u(t), d(x,t) for short, respectively.

The aim of this paper is to develop the adaptive tracking controlscheme using MNNs for a class of switched nonlinear systems withunknown constant control gain such that the actual output y ofsystem (1) follows the any given bounded desired output signal yd.To achieve the proposed control objective, we have the followingassumptions.

Assumption 2.1. The desired output signal yd and its time derivativesup to the n-th order are continuous and bounded. Define the desiredtrajectory Yd ¼ ðyd, _yd , . . . ,yðn�1Þ

d ÞT which is continuous and available,

and YdAOdARn withOd known compact set, then the output trackingerror is e¼ Yd�x¼ ðe1, . . . ,en�1,enÞ

T¼ ðe, _e, . . . ,eðn�1ÞÞ

T , i.e., e1 ¼

yd�x1,e2 ¼ _e ¼ _yd�x2, . . . ,en ¼ eðn�1Þ ¼ yðn�1Þd �xn, also eðnÞ ¼ yðnÞd �

_xn .Define K¼(k1, k2,y,kn), where K is Hurwitz vector.

Assumption 2.2. For tZ0,8xARn, the nonlinear functions fi(x)satisfied jfiðxÞjrFiðxÞ , i.e., fi(x) can be either positive or negative,without loss of generality, we shall assume fiðxÞ40. In addition,di(x,t) are bounded, assuming that the upper bound of thedisturbance di(x,t) are Di(x), i.e., jdiðx,tÞjrDiðxÞ. Where, Fi(x) andDi(x) are both known positive continuous functions.

Assumption 2.3. The constant control gain bi(x) satisfiedbiðxÞZbL,i40, where bL,i are the lower bounded of bi(x) , andthey are given positive constants.

If fi(x), bi are both known and disturbance vector di¼0, accordingto the feedback linearizable techniques [16,20], we choose thecontrol law as follows:

u� ¼1

bsðtÞyðnÞd �fsðtÞðxÞþ

Xn

i ¼ 1

kieðn�iþ1Þ

!ð2Þ

Substituting (2) into (1) we have

_xn ¼ yðnÞd þXn

i ¼ 1

kieðn�iþ1Þ ) eðnÞ þXn

i ¼ 1

kieðn�iþ1Þ ¼ 0 ð3Þ

Then (3) can be written as: en1þk1eðn�1Þ

1 þk2eðn�2Þ1 þ � � � þkn�1e1 ¼ 0.

So limt-1e1ðtÞ ¼ 0, the control objective is achieved.However, fi(x), bi are both unknown and disturbance vector

dia0 , so the control objective cannot be achieved. As we know,MNNs are used as a tool for modeling nonlinear functions up to asmall error tolerance. In comparison to RBF NNs which are alsocapable of approximating continuous functions [11,12], MNNshave the advantage that the basis function set as well as thecenters and variations of radial-basis type of activation functionsare estimated online and hence, they need not to be specified apriori. This means that we need not to fix a priori the compact setover which the neural networks approximations are employed. Inthis paper, we use MNNs to approximate the function u* over a

compact set Oz � Rp as follows:

u�ðz,W�,V�Þ ¼ u1ðz,W ,VÞþeðzÞ ¼WT SðVT zÞþeðzÞ zAOz � Rp ð4Þ

where W* and V* are ideal MNNs weights, and eðzÞ is the MNNsapproximation error such that jeðzÞjrDe with the constant De40for all zAOz; z¼(z1, z2, y, zn +2)T, z ¼ ðzT ,1ÞT ; V ¼ ðv1,v2, . . . ,vlÞARðpþ1Þ�l, W ¼ ðw1,w2, . . . ,wlÞARl are the first-to-second layer andthe second-to-third layer adjustable weights; SðVT zÞ ¼

ðSðvT1zÞ,SðvT

2zÞ, . . . ,SðvTl�1zÞ,1ÞT with Sðz‘Þ ¼ 1=ð1þeð�rz‘ÞÞ, r¼ yðnÞr þ

KT e ; p¼n+2; and the number of hidden units l41.According to the discussion in [7,11], W* and V* are defined as

follows:

ðW�,V�Þ ¼ argminW ARl ; V ARðpþ 1Þ�l

miniAX

supzAOz

ju1�u�j

( )( )ð5Þ

Also, we define ðW ,V Þ as the estimate value of (W,V) , respectively.So the estimate value errors can be given by

~W ¼ W�W�

~V ¼ V�V�

(ð6Þ

Thus,

u�1ðz,W�,V�Þ�u1 ðz,W ,V Þ ¼� ~WTðS�SuV

Tz�W

TSu ~V

Tzþdu ð7Þ

where S ¼ SðVTzÞ; Su ¼ diagð ^S1u, ^S2u, . . . ,SluÞ, ^Smu ¼ Suðv

TmzÞT ¼ fdðSðz‘ÞÞ=

dz‘jz‘ ¼ vTmz,m¼ 1,2, . . . ,lg,

du ¼�~W

TðS�SuV�T zÞþW�T

ðSðV�T zÞ�S�Suð� ~VTzÞÞ.

The residual term du is bounded by [11]

jdujrJV�JF � JzWTSuJFþJW�J � JSuV

TzJþJW�J1 ð8Þ

From (4) and (8), we have

jdujþjeðzÞjrJV�JF � JzWTSuJFþJW�J � JSuV

TzJþJW�J1þDe ¼HTjðz,tÞ

ð9Þ

where H¼ ðJV�JF ,JW�J,JW�J1þDeÞT ,jðz,tÞ ¼ ðJzW

TSuJF ,JSuV

TzJ,1Þ.

3. Direct adaptive tracking controller design

Define K¼(k1, k2,y,kn) is a constant matrix, satisfying thatCðsÞ ¼ k1þk2sþ � � � þknsn�1 ¼

Pni ¼ 1 kis

i�1 ¼ ðsþlÞn,l40, i.e.,ki ¼ Cn�i

n liðiAXÞ. We can define the filtered tracking error x as

x¼ enþXn�1

i ¼ 1

kn�ieiþkn

Ze1 dt

� �¼

d

dtþl

� �n�1 Ze1 dt

� �ð10Þ

The time derivative of (10) is

_x ¼ _enþXn

i ¼ 1

kieðn�iþ1Þ ¼Xn

i ¼ 1

kieðn�iþ1Þ þyðnÞd �fsðtÞðxÞ�bsðtÞu�dsðtÞ

ð11Þ

Remark 3.1 (Leu et al. [11]). It has been shown that definition (11)has the following performances:

(i)

if x¼ 0, then limt-1

Re1ðtÞ dt

� �¼ 0;

(ii)

if e(0)¼0 and jxjoM, 8t40 with constant M40, theneðtÞAOc , Oc ¼ fejjeðjÞjr2j�1lj�nM,jAXg;

(iii)

if eð0Þa0 and jxjoM, 8t40 with constant M40, theneðtÞAOc within a time-constant n=l.

For the switching signal sðtÞ , a switching sequence is given by

S :¼ fði0,t0Þ,ði1,t1Þ, . . . ,ðik,tkÞ, . . . ,jikAX,kANg ð12Þ

L. Yu et al. / Neurocomputing 74 (2010) 481–486 483

where (ik, tk) denotes that the ik-th subsystem is switched on at tk ,and the ik +1-th subsystem is switched off at tk +1. Where t0 is theinitial time, tk40 is the k-th switching time. When tA ½tk,tkþ1Þ, thetrajectory of the switched nonlinear system (1) is produced by theik + 1-th subsystem, defining tP ¼ tk�tk�1 as dwell-time of the ik-thsubsystem .

Assumption 3.1. For tAðtk�1,tk�AOmðmAXÞ and tA ½tk,tkþ1Þ

AOmþ1ðmAXÞ, there is a constant mZ0 such that

jxðtkþ1ÞjrmjxðtkÞj ð13Þ

where in this paper we assume m¼ 1 .

Definition 3.1 (Han et al. [13], Hespanha and Morse [14], and Persis

and Satis [15]). Given some family of switching signalsY, for eachsand each tZt0Z0, let Nsðt0,tÞ denotes the number of disconti-nuities of s in the interval [t0,t). For given tD,N040, we denote byYave½tD,N0� the set of all switching signals for which

Nsðt0,tÞrN0þt�t0

tDð14Þ

The constant tD is called the average dwell-time and the chatterbound N0. So the switching signal Yave½tD,N0� �Y consisting of allswitching signals having the same persistent dwell-time tD40 andthe same persistent chatter bound N040.

Assumption 3.2. For each piecewise constant switching signal sthe switched nonlinear system can be defined as follows:

_x ¼ fsðx,u,dÞ ð15Þ

For each sAY , it is shown that (15) is uniformly asymptoticallystable over Y such that

JxJrððcjxðt0ÞjÞ1=de�ðt�t0ÞÞ

dþg0

Z t

t0

juðtÞj2 dtþCðtÞ 8tZt040

ð16Þ

where c,d,g0 are suitable positive real numbers, and CðtÞ is thecontinuous nonnegative function.

Lemma 3.1 (Hespanha and Morse [14], and Persis and Satis

[15]). Given a set Of9ffiðx,u,dÞ : iAXg of nonlinear maps from Rn

to itself for which Assumption 3.2 holds. For any average dwell-time

tDZt�D which t�D is a finite constant and any chatter bound N040, the

switching system (15) is uniformly asymptotically stable over

Yave½tD,N0� such that there exist continuously differentiable functions

Vi : R-Rn,iAX, positive constants a1,a2 and functions b1,b2 of class

K1 satisfying that

a1ðJxJÞrViðxÞra2ðJxJÞ, ð17Þ

@Vi

@xfiðx,u,dÞr�2b1ViðxÞ, ð18Þ

ViðxÞrb2VjðxÞ, ð19Þ

for each xARn and i,jAX.

With the function approximation (4), we present the adaptivecontrol law as follows:

u¼ u1ðz,W ,V Þþu2þu3

u2 ¼ ZðtÞsgnðxÞ u1þ1

bL,iFiðxÞþjy

ðnÞd jþ

Xn

i ¼ 1

kieðn�iþ1Þ

����������

!" #

u3 ¼ txþðHTjðz,tÞþDiðxÞ=bL,iÞsgnðxÞ

8>>>>><>>>>>:

ð20Þ

where ZðtÞ ¼ f10Vx4V0

VxrV0, Vx ¼ x2; t,V0 are design positive para-

meters; H denotes the estimate value of H at time t.Then, the adaptive update laws which are both deduced by the

switched Lyapunov function candidate (described later) can be

expressed by [17,18]

_W ¼GwðS�Su ~V T zÞx_V ¼GvzW T Sux _H ¼Ghjðz,tÞjxj:

8<: ð21Þ

where Gw,Gv and Gh which are both positive constant in generaldenote the adaptive gains.

From (3) and (11), the output tracking error dynamic equation ofswitched nonlinear system (1) is described by

_en ¼ yðnÞd �_xn ¼ yðnÞd �fsðtÞðxÞ�bsðtÞu�dsðtÞ

¼ �Xn

i ¼ 1

kieðn�iþ1Þ þbsðtÞ½u��u1�u2�u3�b�1

sðtÞdsðtÞ� ð22Þ

We can also conclude that

_x ¼ _enþXn

i ¼ 1

kieðn�iþ1Þ ¼ bsðtÞ½u��u1�u2�u3�b�1

sðtÞdsðtÞ� ð23Þ

Theorem 3.1. Consider the system (1), satisfying Assumptions 3.1and 3.2, Definition1 and Lemma1. The proposed direct adaptive

control law (20), adaptive update laws (21), together with the

switching signals with average dwell-time method (14) guarantee

that the resulting closed-loop switched system is asymptotically

Lyapunov stable such that the output tracking error convergence to0.

Proof. Consider the switched multiple Lyapunov function candi-date:

V ¼Xn

i ¼ 1

yiðtÞVxþVG ¼Xn

i ¼ 1

yiðtÞx2þbsðtÞ½G�1

w~W

T ~W

þtrðG�1v~V

T ~V ÞþG�1h~H

T ~H � ð24Þ

where the characteristic function:

yiðtÞ ¼1 tAOi

0 t=2Oi,

(

Oi ¼ ftj the ith system is active at time instant tg

For tAðtk�1,tk�AOmðmAXÞ and tAðtk,tkþ1�AOmþ1, from (13) and(24), we have

DVðtÞ ¼ Vxðtkþ1Þ�VxðtkÞ ¼ x2ðtkþ1Þ�x

2ðtkÞo0 ð25Þ

From the switching signal (14), using the tracking error dynamicEq. (22) and the weights update laws (21), and taking the timederivative of V, also from _~W ¼

_W , _~V ¼

_V , and _~H ¼

_H , we obtain

_V ¼ _Vxþ_VG ¼ 2x _xþ2bsðtÞ½G�1

w~W

T _W þtrðG�1

v~V

T _V ÞþG�1

h~H

T _H �

¼ 2bsðtÞx½u��u1�u2�u3�b�1sðtÞdsðtÞ�

þ2bsðtÞ½G�1w

~WT _W þtrðG�1

v~V

T _V ÞþG�1

h~H

T _H �

¼ �2bsðtÞxu3þ2bsðtÞx½u��u1�u2�b�1sðtÞdsðtÞ�

þ2bsðtÞ½G�1w

~WT _W þtrðG�1

v~V

T _V ÞþG�1

h~H

T _H �

r�2bsðtÞtx2þ2bsðtÞx½� ~W

TðS�SuV

Tz�W

TSu ~V

TzþduþeðzÞ�u2

�HTjðz,tÞsgnðxÞ�þ2bsðtÞ½G�1

w~W

T _W þtrðG�1

v~V

T _V ÞþG�1

h~H

T _H �

r2bsðtÞf�tx2þG�1

w~W

T½_

W�GwðS�Su ~VTzÞx�

þtr½G�1v~V

Tð_V�GvzW

TSuxÞ�þG�1

h~H

T½_H�Ghjðz,tÞjxj�g

r�2tbsðtÞx2r0 ð26Þ

Because V(t) is the monotone non-increasing nonnegative function,it is clear from VðtÞAL1 that there is the existence of limt-1VðtÞ.From _V r�2tbsðtÞx

2r�gx2r0 (where, g40), integrating it, we

0 2 4 6 8 10 12 14 16 18 20 22 2424−1.6

−1.2

−0.8

−0.4

0

0.4

0.8

1.2

1.6

time (sec)

actual output y = x2desired output yd

Fig. 2

0 2 4 6 8 10 12 14 16 18 20 22 2424−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

tracking error e=yd − y

L. Yu et al. / Neurocomputing 74 (2010) 481–486484

obtainR1

0 gx2ðtÞ dtrVð0Þ�Vð1Þoþ1 and hence, xAL2. Using the

conditions of x,yd,xAL1, and the smoothness of system functions,it can be seen from (23) that _xAL1. According to Barbalat’s lemma[3,19], xAL2 and _xAL1, we can conclude that x-0 as t-1.Therefore we have limt-1e1ðtÞ ¼ 0, also limt-1eiðtÞ ¼ 0ðiAXÞ. Thusthe proof is complete. &

Remark 3.2. In the direct adaptive neural controller designed bythis paper, u2, u3 has to be discontinues, so this will lead tochattering. Generally, chattering must be eliminated for the con-troller to perform properly. It can be achieved by smoothing out thecontrol discontinuity for u2, u3 as follows [20]:

u2 ¼ ZðtÞsatxl0

� �u1þ

1

bL,iFiðxÞþjy

ðnÞd jþ

Xn

i ¼ 1

kieðn�iþ1Þ

����������

!" #ð27Þ

u3 ¼ txþðHTjðz,tÞþDiðxÞ=bL,iÞsatxl0

� �ð28Þ

where l0 is a design parameter, and satð�Þ is the saturation function.

4. Simulation results

Consider the switched nonlinear system (1) of two Duffingforced-oscillation systems with constant control gain [21]:

S1 :_x1 ¼ x2

_x2 ¼�x31�0:1x2þð1:5þ0:5sinð5tÞÞuþ10cosðtÞ

(

S2 :_x1 ¼ x2

_x2 ¼�x21þ0:5x2þð1:0þ0:2sinð2tÞÞuþ12cosðtÞ

(

The design of control objective is that the actual output y¼x1 and _y ¼ x2

follows the desired output signal yd¼0.5(sin(t)+0.3 sin(3t)) and_yd ¼ 0:5cosðtÞþ0:45cosð3tÞ, respectively. In the simulation, accordingto the design procedures in Section 3, we choose the parametersas follows: FðxÞ ¼ ðF1ðxÞ,F2ðxÞÞ ¼ ðjx1j

3þ0:1jx2j,jx1j2þ0:5jx2jÞ, D¼

ðD1,D2Þ ¼ ð10,12Þ, bL ¼ ðbL1,bL2Þ ¼ ð1,0:8Þ, K ¼ ðk1,k2Þ ¼ ð2,1Þ, Gw ¼

2,Gv ¼ 4,Gh ¼ 2,t¼ 0:5, V0¼2,tD ¼ 2. The number of hidden units forthe MNNs is taken as 40. The initial values of state vectors is x(0)¼(0.2 ,0.1)T, and the initial weights values of MNNs are chosen randomly inthe interval [�1 , 1].

The simulation results are shown in Figs. 1–6. From Figs. 1 and 2,we can observe that the actual output and its derivative follow the

0 2 4 6 8 10 12 14 16 18 20 22 2424−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

actual output ydesired output yd

Fig. 1

time (sec)

Fig. 3

desired output signal and its derivative, respectively. So thesatisfactory tracking performance is obtained, and the trackingerror performance is well-achieved in Fig. 3. Also, Fig. 4 denotes theswitching signal using average dwell-time method. In addition, thebounded value H and V are shown in Figs. 5 and 6. It is shown thatthe signals are bounded while the switching signal exhibits fastswitching using average dwell-time technique.

Remark 4.1. During the simulation procedure, it is seen that thevalue of the system tracking error is related to controller parametert and V0 , which can be chosen arbitrarily. The smaller the t, thebigger the V0, and the smaller the tracking error will be. However,on the other hand, when V0 is smaller and t is bigger, theperformance of the control system will be affected, such as thechatting phenomena will happen, the value of control input may beincreased, etc. Therefore, it is important to select an appropriate tand V0 according to the controlled system.

0 2 4 6 8 10 12 14 16 18 20 22 24240

1

2

33

time (sec)

σ (t)

Fig. 4

0 2 4 6 8 10 12 14 16 18 20 22 24240.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

time (sec)

The

valu

e: H

<

Fig. 5

0 2 4 6 8 10 12 14 16 18 20 22 24−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

time (sec)

The

valu

e: V<

Fig. 6

L. Yu et al. / Neurocomputing 74 (2010) 481–486 485

Remark 4.2. As compared with the method in [11–13], we haveaddressed a direct adaptive neural controller to enhance robust-ness and maintain boundedness. By means of Lyapunov stabilitytheory, we have designed a novel-type multiple switched Lyapunovfunction consisting two parts Vx and VG to guarantee the asymp-totic stability of the closed-loop switched nonlinear system. As thesimulation results show, the direct adaptive neural controller couldtrack and stabilize the desired trajectory within a short time notonly to a fixed point, but also to an arbitrary orbit. Therefore, fromthe control application point of view, the control scheme proposedin this paper has a good satisfactory control performance.

5. Conclusion

In this paper, we present the direct adaptive tracking controlscheme using for a class of switched nonlinear systems withunknown constant control gain. We employ MNNs to approximateunknown functions and design a direct adaptive tracking controllerto enhance system robustness and stabilization. The proposed

control scheme can guarantee that the resulting closed-loopsystem is asymptotically Lyapunov stable such that the actualoutput of system (1) follows the any given bounded desired outputsignal. Simulation results show the satisfactory tracking perfor-mance. Generally, the number of RBF NNs contained in the neuralnetwork system heavily influences the performance of the con-troller and the complexity of a neural network system, with regardto the conventional control scheme. Namely, the larger the numberof the nodes is, the more complexities the controller will contain.Therefore, how to select the optimal number of hidden units on thistopic remains an open research problem for the switched nonlinearsystems. The results in [22] which the number of the nodes andparameters of RBF NNs are determined online by the generalizedgrowing and pruning algorithm could be very useful for thisproblem for further research.

Acknowledgments

This work is supported by the National Natural Science Founda-tion of China (Nos. 60835001, 60804017, 60764001), and the WestTalent Project of Chinese Academy of Science (2007414). Theauthor would like to thank the associate editor and the reviewersfor their constructive comments and criticisms which were veryhelpful in the improvement of the manuscript.

References

[1] Z.D. Sun, S.S. Ge, Analysis and synthesis of switched linear control systems,Automatica 41 (2005) 181–195.

[2] Z.D. Sun, S.S. Ge, T.H. Lee, Controllability and reachability criteria for switchedlinear systems, Automatica 38 (5) (2002) 775–786.

[3] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003.[4] M. Branicky, Multiple lyapunov functions and other analysis tools for switched

and hybrid systems, IEEE Transactions on Automatic Control 43 (4) (1998)475–482.

[5] G.M. Xie, L. Wang, Controllability and stabilizability of switched linear-systems, Systems and Control Letters 48 (2) (2003) 135–155.

[6] H. Yang, V. Cocquempot, B. Jiang, On stabilization of switched nonlinearsystems with unstable modes, Systems and Control Letters 58 (2009) 703–708.

[7] F.L. Lewis, A. Yesildirek, K. Liu, Multilayer neural network robot controller withguaranteed tracking performance, IEEE Transactions on Neural Networks 7 (2)(1996) 388–398.

[8] C.F. Hsu, K.H. Cheng, T.T. Lee, Robust wavelet-based adaptive neural controllerdesign with a fuzzy compensator, Neurocomputing 73 (1–3) (2009) 423–431.

[9] Y.J. Liu, S.C. Tong, W. Wang, Adaptive fuzzy output tracking control for a class ofuncertain nonlinear systems, Fuzzy Sets and Systems 160 (19) (2009)2727–2754.

L. Yu et al. / Neurocomputing 74 (2010) 481–486486

[10] M. Krstic, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive ControlDesign, Wiley, New York, 1995.

[11] Y.G. Leu, W.Y. Wang, I.H. Li, RGA-based on-line tuning of BMF fuzzy-neuralnetworks for adaptive control of uncertain nonlinear systems, Neurocomput-ing 72 (10–12) (2009) 2636–2642.

[12] F. Long, S.M. Fei, Neural networks stabilization and disturbance attenuation fornonlinear switched impulsive systems, Neurocomputing 71 (2008)1741–1747.

[13] T.T. Han, S.S. Ge, T.T. Lee, Adaptive neural control for a class of switchednonlinear systems, Systems and Control Letters 58 (2009) 109–118.

[14] J. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time,in: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix,USA, 1999, pp. 2655–2660.

[15] C.D. Persis, R.D. Santis, A.S. Morse, Switched nonlinear systems with state-dependent dwell-time, Systems and Control Letters 50 (2003) 291–302.

[16] I. Kanellakopoulos, P.V. Kokotovic, A.S. Morse, Systematic design of adaptivecontrollers for feedback linearizable systems, IEEE Transactions on AutomaticControl 36 (1991) 1241–1253.

[17] K. Indrani, B. Laxmidhar, Direct adaptive neural control for affine nonlinearsystems, Applied Soft Computing 9 (2009) 756–764.

[18] M. Hernandez, Y. Tang, Adaptive output-feedback decentralized control of aclass of second order nonlinear systems using recurrent fuzzy neural networks,Neurocomputing 73 (1–3) (2009) 461–467.

[19] J.J. Slotine, W.P. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs,New Jersey, 1991.

[20] L. Yu, S.M. Fei, X. Li, Robust adaptive neural tracking control for a class ofswitched affine nonlinear systems, Neurocomputing 73 (10–12) (2010)2274–2279.

[21] J.G. Lin, A new approach to duffing equation with strong and high ordernonlinearity, Communications in Nonlinear Science and Numerical Simulation4 (16) (1999) 132–135.

[22] G.B. Huang, P. Saratchandran, N. Sundararajan, A generalized growing andpruning RBF (GGAP-RBF) neural network, IEEE Transactions on Neural Net-works 16 (2005) 57–67.

Lei Yu was born in 1983. He received his M.S. degree inControl Theory and Control Engineering from HefeiUniversity of Technology, Hefei, PR China, 2007. He iscurrently pursuing the Ph.D. degree at School of Auto-mation, Southeast University, Nanjing, PR China. Themain research interests include switched nonlinearsystems, robust adaptive control, neural networkcontrol, etc.

Shumin Fei was born in 1961. He received the Ph.D.degree from Beihang University, Beijing, PR China, 1995.From 1995 to 1997, he did postdoctoral research in theResearch Institute of Automation at Southeast Univer-sity. He is now a professor in the Research Institute ofAutomation at Southeast University, Nanjing, PR China.His research interests include analysis and synthesis ofnonlinear systems, robust control, adaptive control andanalysis and synthesis of time-delay systems, etc.

Fei Long was born in 1973, received the bachelor degreein basic mathematics from Guizhou University,Guiyang, PR China, 2002, Ph.D. degree in Control Theoryand Control Engineering from Southeast University,Nanjing, PR China, 2006. Since October 2006, he hadbeen with Guizhou University, Guiyang, PR China. Hewas a professor in the college of Computer Science andInformation Since 2008. He was also head of Institute ofIntelligent Information Processing, Guizhou University.His research interests include switched systems, net-work control systems, nonlinear systems, H1 control,neural network control, and adaptive control, etc.

Maoqing Zhang was born in 1954. He received his M.S.degree from Yanshan University, Qinghuangdao, PRChina, 1983. He is now a professor in the ResearchInstitute of Mechanical and Electrical Engineering atSoochow University, Suzhou, PR China. His researchinterests include control theory and control engineer-ing, and fuzzy control, etc.

Jiangbo Yu was born in 1983. He is currently pursuingthe Ph.D. degree at School of Automation, SoutheastUniversity, Nanjing, PR China. The main research inter-ests include nonlinear systems, robust control andartificial intelligence, etc.