Delay-dependent exponential stability criteria for stochastic systems with polytopic-type uncertainties

J ControlTheory Appl 2009 7 (3) 291-296 DOI 10. 1007/sl 1768-009-8008-3

Delay-dependent

exponential

stability criteria for

stochastic

systems

with polytopic-type uncertainties

Yumei LI 1'2, Xinping GUAN 2, Dan PENG 2, Changchun IIUA 2, Xiaoyuan LIJO? (l.Institute of Mathematics and System Science, Xinjiang Universiry Urumuqi Xinjiang 830046, China;

2.Institute ofElectrical Engineering, Yanshan University, Qinhuangdao Hebei 066004, China)

Abstract: This paper considers the problem of delay-dependent exponential stability in mean square for stochas-tic systems with polytopic-type uncertainties and time-varying delay. Applying the descriptor model transformation and introducing free weighting matrices, a new type of Lyapunov-Krasovskii functional is constructed based on linear matrix inequalities (LMIs), and some new delay-dependent criteria are obtained. These criteria include the delay-independenvrate-dependent and delay-delay-independenvrate-dependent/rate-indelay-independenvrate-dependent exponential stability criteria. These new criteria are less conservative than :;t;.lr*

ones. Numerical examples demonstrate that these new criteria are effective and are an improvement over existing Kelrrords: Stochastic system; Exponential stability in mean square; Time-varying state delay; Delay-dependent criteria; Linear matrix inequality (LMI)

L Introduction

Recently, delay-dependent stability cirtellra for stochas-tic delay systems have attracted extensive attention (see t1-81). In view of the robustness of stochastic stability, the linear and semilinear systems were studied rn f1,21, re-spectively. [3] investigated the stability of linear and semi-linear stochastic differential equation by means of the ex-ponential stability. Verriest [4] presented stability of linear stochastic differential equation via Riccati equations. Based on the LMI approach, [5-8] gave the delay-dependent ro-bust stability criteria of uncertain stochastic systems, re-spectively. However, the criteria in [5] involved the param-etenzed model transformation. To determine the stability of system, [6] and [8] used some inequality consffaint. [7] used a descriptor integral inequality constraint, and the criteria in [7, 8] with matrix constraint P ( o1 (o > 0 is a scalar, P is the product of Lyapunov matrix). These results show considerable conservativeness.

This paper presents some new delay-dependent exponen-tial stability criteria for stochastic system with polytopic-type uncertainties and time-varying delay. First, apply-ing descriptor model transformation [9], we set de-scriptor stochastic system and construct a new type of Lyapunov-Krasovskii functional. Second, based on the idea of [10], some free weighting matrices are intro-duced to exclude constraint conditions in [6-8]. Finally, using LMI algorithm, we obtain delay-dependent and delay-independent exponential stability criteria for stochas-tic system with polytopic-type uncertainties and time-varying delay. These criteria include delay-dependenVrate-independent and delay-delay-dependenVrate-independent/rate-dependent expo-nential stability criteria. In contrast with the existing stabil-ity criteria, these new criteria are less conservative. Numeri-cal simulation examples show that these results are effective

and an improvement over existing ones.

For convenience, we adopt the following notations: tr(A)(,aT) denotes trace(transpose) of the matrix A; A 2 0 (A > 0) denotes positive semidefinite (positive definite) matrix A; L2r^(l-r,0];R") is the family of lR'-valued stochastic _{proiesses q(s), -r ( s ( jo such that 4(s) is} Fo- measurable for every second unu

J_" .E lla(s)ll2 ds < oo; and E{'} denotes mathematical expectation operator with respect to the given probability measure P.

2 Preliminaries

Consider the robust stability of system ( 1) with polytopic-type uncertainties, that is, assume that system (1) has the following form:

dr(t) : lAr(t) + Aar(t - h(r))ldt

+ fcn(t) r car(t - h(t)))dB(t), (1)

r(t) : 9(t), Yt e l-r, 0l ,

where z(t) € IR.' is the state vector, the system matrices A, Aa, C , and C a are assumed to be uncertain but belong to a known convex compact set of polytopic type, namely

( A , A a , C , C a )

e A ,

(2)

where J-l is a given convex bounded polyhedral domain de-scribed by q vertices as follows:

q

9 , : {(A, Aa, C, Ca) : D €n(Ax,

A p a ,

C p ,

C p a )

k : r

q

t n ) o ; D € r : 1 ) .

( 3 )

k : 1

The time delay h(t) is a time-varying continuous function that satisfies

0 ( h ( t ) ( r

Received 14 January 2008; revised 24 July 2008.

This work was supported by the National Natural Science Foundation of China (No .60525303,606M004, 60704009) and Natural Science Foundation of Hebei Province. China No.F2005000390. F2006000270).

Y. U et al. / J ControlTheory Appl 2009 7 (3) 291-296

h ( t )

< / - , < 1 ,

(s)

where r and p, are constants, cp(t) is a continuous vector-valued initial function, and g ': {p(") : -r < s ( 0} e L2so([-r,0] , R'). It is well known that (1) has a unique so-lution, denotedby n(t, rp), which is square integrable. So (1) admits a trivial solution r(t,O) : 0. Ap, Apy,C*,Cna arc known constant matrices with appropriate dimensions. The variables /ft) arc an rn-dimensional Brownian motion de-fined on a complete probability space (A , F , P) with a nat-ural filtration {F r}r>o (i.e., F t : o {o(s) : 0 ( s ( t}).

Definition L System (1) is said to be exponentially sta-ble in mean square if there exists a positive constant os such that

.lim sup

] roge ll*(t)ll' ( -ao.

f * m - t

2.1 Delay-dependent

robust exponential

stability

To discuss

the stability of system

(1), first, we introduce

the descriptor

system

approach,

set

s ( t ) : A r ( t ) * A 6 r ( t - h ( t ) ) ,

(6)

s(t) : Cr(t) + Car(t - h(t)).

(7)

Then system

(1) becomes

the following descriptor

stochas-tic system

dz(t) : q(t)dt + g(t)dl3(t).

(8)

Moreover, equations in (6) and (7) ensure the following zero equations

z lrr (t)Nt + nr (t - h(t))N2 + sr(r)Ab + er(r)N4]

x [An(t) * Aar(t - h(t)) - s(r)] : o,

(e)

z lrr (flrt + rr (t - h(t))72 + qr (t)rt + sr &)T4l

x lCn(t) * C6r(t - h(t)) - e(r)l : 0,

(10)

where

lI, andT, (r : 1,2,3,4) are appropriately

dimen-sioned

matrices.

On the other hand, the Newton-Leibniz

for-mula provides

r ( t \ - r ( t - h ( t \ \ : l L

e t

' t ' - r 1 ' ; i ( t ) d

t :

J ' - n'''o(s)ds

* s '

( 1

1 )

where

_{sr : I f" . g(s)dp(s)]r by 1l r), get}

' J t - h t t \ " '

J:-r,^q(s)ds

: r(t) - n(t - h(t)) - s,

G2)

then, we obtained the following Theorem.

Theorem 1 Consider system (1) with polytopic-type uncertainties (3) aad a time-varying delay satisfying (4) and (5). Given scalars r ) 0 and p, 1 1, system (1) is robust exponentially stable in mean square, if there exist symmet-ric positive definite matsymmet-rices Pp ) 0,Q* 2 0, Zx ) 0, and n[f _{{1,,i : 7,2,3) and appropriately dimensioned}

marri-ces

AI, andT, (r :1,2,3,4) suchthat

Rl? : R!!) u"a

the following LMIs hold for k : L,. . . , ei

l-

o\\' o\r,,

ol!,)

ofn)

a\!)1

| - o"fr'o52'o!ri^'

alr|)

|

o ( k )

_{: l * * o \ 2 ) o \ \ )}

_{o l < o}

( 1 3 )

l - . 7 " o f i '

o l

L * * * i - z r )

and

Ia{t)

n{f)

n{{)l

R G ) : l ) a t i t a l t ) l r o ,

L . .- a55'.1

where an asterisk

x denotes

a block induced

easily by

sym-metry and

al\) : NtAn

_{*aTlrfl i Ttcx + cT:t + Qn}

+ r Rl\)

+al!)+

R\Z)',

a\t) : NrAan

+ ATN; tT1c61"

+ cfrf + rn\!)

-nl!)+ Ry")',

@13)

: Pn

- Nr+ A;,^/f + CTr{,

an):A'*NI

-Tr*c[r[,

ilfr) : NzAan

+ ArdkNf

lT2C61"

- (1 - tDQx + CTrr[

+rR$) - Ryr)

-Rr?r)',

@fr)

_{: -Nz r A'o*N? + CIIT{,}

a*n)

_{: dkvf - Tz-r chT[ ,}

ar\!"r)

: -AIB

_{-.nrf +}

"a!!),

ilf) : -,^rf - Tr, af,n)

: Px - rf, - rn * r21".

Proof Choose a Lyapunov-Krasovskii functional for system (1) to be

v ( t ) : f U(t),

i : l in which q

v(t): I r(t)rP1-n(t),

It:l Q r t

v2ft)

_{: p, J-o,,,zr(s)Q6z(s)ds,}

t (

ftr

) ) ,

) 1

) t

t l g

d ' t ) t (cr',

Rli

Rti

r ) r(ct

Rli

Rti

v4(t)

:

fi,f,I,'*rt

v ( t \ :

i f ' r

f l t J O J a - h ( a where dT : lrT (a), rT (a - h

lali' I

o , * , :

|

* l

L x

w(t)

:

_{P, J:, Ji*,}

o' {,)

*!!)q(s)dsdd,

s)z

7,9(s)ldsdd,

drft6ddsdo,

853)

Pr,,Q*,zr,R[? (i,i : 7,2,3) are positive definite ma-trices with appropriate dimensions. Let L be the weak in-finitesimal operator of (8), then, by It6 differential formula,

Y. LI et al. / J Contol Theory Appl 2009 7 (3) 291-296

T

t + sT

Z*s.

which implies that system (1) is exponentially stable in mean square. The proof of Theorem 1 is completed.

Remark I By constructing an output feedback con-troller, we can obtain the stability criteria for system with output feedback in the same methods. So, it is without loss of generality for the discussion of system (1).

Note that a delay-dependent and rate-independent expo-nential stability criterion for system (1) with polytopic{ype uncertainties (3) and a delay satisfying (4) and (5) can be derived from Theorem 1 by choosing Q* : 0 as follows.

Corollary I Given scalar r- ) 0, system (1) with polytopic-type uncertainties (3) and a time-varying delay satisfying (4) is exponentially stable in mean square if there exist symmetric positive-definite P1,,Zk,R*) (k : \,-.. ,S) and appropriately dimensioned matrices N" and

T, (r :7,2,J) such

_{that Rl|) : a!? {t,,i : L,2,3)}

and

the following LMI holds for k : I,2,. - . , q:

6 @ :

p(r)

-where

6\\) : NtAx -raTlrf * Ttc* + cf rl

+rRl\)

+,q13)

+ R\!)',

6y) : NzAa*

+ ArdkN[

tTzCa* + CIIT;

+rnlrf;)

- Ryr)

-Ry]',

a t a O l l ( i : 7 , 2 , 3 ; j : 2 , 3 , 4 ) a r e d e f i n e d i n ( 1 3 ) . 2.2 Delay-independent rate-dependent robust

expo-nential stability

If we set the matrices 21" and Al? Q' : i : 1,2,3) to zere, then we can obtain a delay-independent and rate-dependent exponential stability criterion for system (1) with polytopic-type uncertainties (3) and a time-varying delay satisfying (4) and (5). In this case, Theorem 1 becomes the following corollary.

Corollary 2 Given scalar p < 1 and system (1) with polytopic-type uncertainties (3) and a time-varying delay satisfying (5) is robust exponentially stable in mean square, if there exist symmetric positive-definite matrices P1" and Qp and appropriately dimensioned matrices N" and T, (r : I,2,3,4) such that the following LMI holds for k : 1 , 2 , . . . , e 1

293

Lu:ovq<

_{f 1"r, lor(t)zno|)l}

b - 1

- f: -.nlor(s)zxs(s)l

as),

J t _ h ( t )

where

_{1 : l.' .. . qt(")n!3)q(s)ds.}

r t _ n \ t )

Substitute (12) inro (18), get Lo:oVs

.,*'l;iAl

_-;fii]

t+x-zt'

_l;fir],]

,,n,

By Lemma [11], for any matrix Zn ] 0,

-,,'

_[;i{]l

_{,. *}

Obviously,

(r7)

L.-ovs:

_{i {n(,)e'}

lfttfi|Eg,l,

*, Ii-^u,r'

_[;fi]]

o(")a"+x],

(18)

: -?

_{n,',*'

_lfifi

Efr)1,

[;fir]

','l;l,)l

6l\) o\2)

o$)

o;?^)

nl\)1

* ,6$t

o$t 6a) P{xt

₁

* * <

o $ ) E { n )

_{o l < o}

Q 2 )

* r < . " * f ^ ' 0 1

*

*

r< * -zr)

Ia!!)

al!)

a!!)l

| - n\\)

a{a\)

l,

L *

ft!:'j

(20)

Lu:oVs

. E {"' [#iJ,

fg,],-,,'

_[;ii]

ifill

+y*

sr

zxs-,'

[;ii]l,r,lti'i.]'*]

,

and

where {r : [zr(t), rr(t - h(t))]. Combining L,:sVi (i : I,2,3,4,5) and adding the terms on the left of (9)-(10) to Lr:sV, we can express Lo:oV as

q

Lu:sV < I {q"(r)qrotn)q1t1

k : 1

-

Ji-rurtt lsr (s)zns(s)]

ds + sr znsj'

where

_{4r (t) : lrr (t) , *r (t}

- h(t)), qr (t),.st (t) ] , or :

[(Rl3)

)', 1a!!)

;r, o,

01,

lo\\)

o\P

o\\)

o\?1

I r I f - | | - / r " \ - / a \ - r k ) I

4 i ( r . r

- |

x * i i * i , ? , , * i f , l * o z ; 1 o r .

I x * @ | . ] J ' Q \ \ ' l I " " y r . l L x * r< Ofn'J

Since

E (sr zxs) :

" fr_n,utr

lor (s)zt s(s)] ds, it

fol-lows that

EL,:ov(t)a i tn'(r1qro{x)nQ). Qr)

K : L

By Schur's complement, 6t@) I 0 is equivalent to LMI (13). From the proofofTheorem 1 [12], there exist a scalar o such that

,]jg'"p Tnsn ll"(t)ll'(

-o,

6l\t Eat

af,)

6li)l

:

'*' u_rff',-r\,',|

.,

(23)

* * ." agt)

294

_{Y. LI etal. /J ControlTheoryAppl 2009 7 (3) 291-296}

where

6l? : NrAn

r A'rNl

_{*Trcn + c[r[ + en,}

6ly : NtAna+

ATNI

tTycl,a+

qrf,

ot? : P* - Nt + Ark

N? + CTr?,

6\rn):aTw[

_{_ Tr*cfrf ,}

'6y)

_{: NzAxa}

_{+ ATdNI}

_*T2caa

_{- (r - der"}

+c[dT;,

619")

:-lh + AT'N?

+ CTdT{,

6n) : AT.NI

_{- Tz * c[oTf, 6tt) : -t/, - nJ,}

6yn)

_{: -,^/f - rr, 6f,) :2pn - TI - rn.}

In addition, if there is no stochastic

uncertainty il system

(l),.that is, B(t) is assumed

ro be zero, system

_{1i; degener_}

ate into system

_{[10]. We obtain the following corollary.}

Corollary 3 Given scalars

_{r ) 0, and p < 1,}

sys-pm (1) with l3(t) : 0 and with time-varying

delay

saiis-fying (a) and (5) is asymptorically

stable,

iithJre exiit

sym-metric positive-definite

_{matrix p1a ) O,ex 2 0, and Rff)}

(i, j : I,2,3) and appropriately

dimensioned

matrices

N'

(, : I,2,3,4) such thar RIP : A!!) anathe foltowing

L M I s

h o l d

f o r k : I , 2 , . . . , q :

T r'(ft) \-(k) \-(k) I | - 1 , 7 2 1 2 2 1 3 |

r ( A ) : l * z ! , 9 2 \ ? l < o

L - .' ,!'')

(24)

and

R&) :

) 0 ,

where

'{t) : NrA*

* eTul *Ttcn + c[r[,

i{!) : Pr"

- Nr+.4TNf

+ cTr{,

tll) : aTtt - rt * c[rf , i[t) : -r/, ; Nf ,

i[? :-.^rf - rr, D[1)

- Pk

- rf, - rn.

Remark 3 When p : 0, the delay is time invariant. From Theorem 1 and Corollary 2, we can easily obtain the delay-dependent and delay-independent robuit expo-nential stability criteria for continuous-time linear stochas-tic system with polytopic-type uncertainties and with time-invariant state delay, respectively.

3 Numerical simulation

In this section, for the purpose of illustrating the useful-ness and flexibility of the methods in this paper, we present some simulation examples.

Example 1 Consider the following time-varying delay llrteT 11 with polyropic-type uncerrainties ([10] Example 2), where

4: 19

-o^'?!"r1

. o,: f

-o't

-0'351

L 1 - 0 . 4 6 5 - p ) '

L 0

0 . s I

and

_llpll

_{< 0.035}

_{tt2l.I_,et}

_{p^:0.02b [10] and}

_ser

, . _ l o

- 0 . 1 2 + 1 2 p ^ j

" ' -

l t - 0 . 4 6 5

- p* l'

a , : f o

-

- o l 2 - t z P ^ f

L 1

_ 0 . 4 6 5

* p ^ ) ,

Ata

_{: Azd: A, : l-9j-9 itl}

_-

L o 0.3

I

!!en p : 0, the upper bound on the time delay obtained in [10] is 0.863. However, by Corollary 3, the syitem X1 is robustly stable for delay r : 0.8758, which is better than the values in [10]. Table I shows a comparison of the up-per bounds for p, I 0 obtained by Fridman and Shaked s method [13], He's merhod [10], and our methods (Corollary 3). It is clear that the upper bounds obtained by Corollary i are larger than those given in tl3l and t101.

Tbble 1 Calculation results for Example 1.

f nff)

nl!)

a{!).1

| - af,, nLE,

I

L * - ft53,-]

where

El?

: NrAr"

* ATNI

_{]- en-r "n\l +al3) + R\?' ,}

t!t) : NtAan

+ ATN;

_{+ "afl - Rl? + RtE)'}

_,

tff) : Pn

_{- Nt+ ATNI ,}

tt ) : NzAan

+ AT'NI

- (1 - u)e* + rR\9

-Ry') - Ryr)',

E ? :-A& + ATrN{, E[? : -AiB

-,^'f + 'n[!).

Remark 2 Corollary 3 is equivalenr ro Theorem 2 tl}l. Moreover, because the new Lyapunov-Krasovskii func-tional is different from that in [10], ihe results obtained from Corollary 3 are less conseryative than existing ones [10].

Besides, if we assume Aak : Can :0, system (l) turns into a stochastic system without time delay, the followins corollary can be acquired.

Corollary 4 Given scalars r ) 0 and tt I 1, system (1) with time-varying delay satisfying (4) and (5) ani with 4o* : _Cat" : 0 exponentially itable in mean square, if there exist symmetric positive-definite matrix pr" j O, ana appropriately dimensioned maffices AL and f, (, : 1,8,4) such that the following LMI holds for k : 1,2,. . . ,q:

0 . 1 0.9 any p Fridman _[13] 0.782 0.736 He [10] 0.863 0.786 Corollary 3 0.8758 0.8041 0.454 0.454 0.454 0.454 0.603 0.4548 T i( / ' ) i ( / c ) n ( k ) l | " ] t - 1 3 2 1 4 |

r ( f t )

_{: I * rjf)ijf) l<0.}

L . . tti))

Example 2 Consider the robust stabiliff of the uncer-tain stochastic delay system 12 with the foilbwing parame-ters:

a n d lpl ( 1, lal ( 1.

The parameter uncertainty can be represented by a four-vertex polytope and the upper bound of the time delay r, whichguarantees that the given system is exponentially

sta-': i;'-,1Tr"]

' Ad:

_{l;', -, fitr'] '}

": [%'

;:,'._'r:;y]

, cd:

l-3'0,1'f

r,] ,

Y. LI et aI. / I ControlTheory Appl 2009 7 (3) 291-296

295

t3l

t4l i5l

t61

vl

t8l li >l

ble in mean square, as given in Table 2. Set the initial con-ditions as z(0) : 10.04,0]r; Figs.l and 2 show the state response of polytopic model with four-vertex systems (each system has two states), as p - 0.5, z-* : 0.5598, and pl is any value, rmax : 0.3893, respectively.

Table 2 The upper bound of r for Example 2, as a function of the bound p.

1-r O 0.1 0.5 0.9 anY P

rnax 0.7498 0.7093 0.5598 0.4223 0.3893

References

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E. Verriest, P. Florchinger. Stability of stochastic systems with uncertain time delayspl. Systems & Control I'etters,1995,24(l): 41

^ 1

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tl0l Y. He, M. Wu, J. She, et al. Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertaintiesUl. IEEE Transactions on AutonThtic Control, 2O04,

49(s): 828 - 832.

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Yumei LI received the B.S. degree in Theoretical and Applied Mechanics from Lanzhou Universrty, Lanzhou, China, and the M.S. degree in Control Theory and Control Engineering from Xinjiang Uni-versity, Urumuqi, China, in 1998 and 2004, respec-tively. She is currently pursuing her Ph.D. degree in the Institute of Electrical Engineering, Yanshan University, Qinhuangdao, China. In July 1998, She joined the Faculty of the Department of Mathemat-ics and System Science at Xinjiang Universiry Urumuqi, Chha, where she is currently an instructor. Her research interests are in robust control and robust filtering of stochastic systems and in distributed control and cooper-ative control of multiagent system. E-mail: zwzlym@ 163.com'

Xinping GUAN received the B.S. degree in Math-ematics from Harbin Normal University, Harbin, China, fhe M.S. degree in Applied Mathematics, and the Ph.D. degree in Electrical Engineering, from Harbin Institute of Technology, in 1986, 1991, and 1999, respectively. He is currently a professor and Dean of the Institute of Electrical Engineering, Yan-shan University, Qinhuangdao, China. He is the (co)author of more than 100 papers in

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i- o.oz

I _o.o1 0 -0.02 0.05 0.04 0.03 x' o.o2 |i 0.01 0 -0.01 -0.4 0.05 0.04 - 0.03 rr 0.02

'i o.o1

0 -0.01 -0.02 0.05 0.04 0.03 \j" 0.02 tr 0.01 0 -0.01 -0.02 . - . . . r " 1 ',. _{- xz} 200400 600 8001000 t l s . . . v' - 1 - x " 2 0 200400 600 8001000 t l s , " " ' x r : - Y '. '-2 0 200400 600 8001000 t l s " - l : - Y | ' - 2 200400 600 8001000 t l s ' .--- _x. t . - ^ 2 0 200 400 600 8001000 t l s - - t - x " 2 0 200400 600 8001000 t l s ' . . . . t r ';. _{- x,} 0 200 400 600 8001000 t l s I - y 0 200 400 600 8001000 t l s

Fig. 1 The state response of system with 4 vertices (as p : 0.5, t.max : 0.5598). )l x )< X

:

x

Fig. 2 The state response of system with 4 vertices (as p is any value, Tmax : 0.3893)'

4 Conclusions

This paper presents some new stability criteria for stochastic time-varying delay systems with polytopic-type uncertainties. Based on the equivalent descriptor stochas-tic system and some free weighing matrices, a new type of Lyapunov-Krasovskii functional is constructed, and new techniques are developed to make the criteria less conserva-tive. Finally, numerical examples demonstrate that the cri-teria presented here perform much better than the existing

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Y. U et al. / J Contol Theory _{Appt 2009 7 (3) 291-296}

in time-delay systems, wireless sensor networks, congestion control ofnet-works, robust control and in intelligent control for nonlinear systems, dis-tributed control and cooperative control of multiagent system, chaos con-trol, and synchronization. He is serving as an associate editor of IEEE Ttansaction on Systems, Man and Cybernetics (part C) and a reviewer of Mathematic Review of America. He is National Science Fund for Distin. guished Young Scholars of China, 2005. E-mail: xpguan@ysu.edu.cn.

DanPENG received the B.S. degree in Applied Mathematics from Beihua University, Jilin, China, the M.S. degree in Operational Research and Cy-bemetics and the Ph.D. degree in Control Theory and Control Engineering from Yanshan University, Qinhuangdao, China, in 2O01,2OO3, and 2008 , re-spectively. Since June 2008, she is an instructor of the College of Science, Yanshan University, ein-W _{huangdao, China. Her research interests are tn} 2-D time-delay systems and in robust control and adaptive control. E-mail: dpeng1219@yahoo.com.cn.

ChangchunHUA received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from yan-shan University, Qinhuangdao, China, in 2000, 2O02, and 2005, respectiVely. He is currently an as-sociate professor in the Institute of Electrical Engi-neering, Yanshan University. His resemch interests are in nonlinear control systems, control systems de-sign over netwofk, teleoperation systems, and intel-ligent control. E-mail: cch@ysu.edu.cn.