Global exponential stabilization for a class of uncertain nonlinear systems with time-varying delay arguments and input deadzone nonlinearities

Global Exponential Stabilization for a Class of Uncertain Nonlinear Systems with Time-vaving Delay Arguments and Input Deadcone Nonlinearities hy YEONG-JEU

Institute Taiwan

SUN and

of Electrical 80424.

JER-GUANG

Engineering,

HSIEH

National

Sun Yut-Sen

University,

Kaohsiung,

R.O.C.

In this paper, theglobal e.xponentialstabilization,for a class of uncertain nonlineur ABSTRACT : systems with time-varying delay arguments and input deadzone nonlinearities is considered. A composite,feedback control is proposed such that the jeedback-controlled system is globally exponentially stable if a simple sufjcient condition is met. Furthermore, ,for slightly more restricted systems, we show that the global exponential stability can always be achieved with any pre-specified convergence rate. A numerical example is provided to illustrate the use of our main results.

I. Introduction Time delay is encountered often in various areas, such as chemical engineering systems, the AIDS epidemic, aircraft stabilization, ship stabilization, manual control, the turbojet engine, the nuclear reactor, the microwave oscillator, the rolling mill and systems with lossless transmission lines. It is frequently a source of instability and a source of generation of oscillation in many control systems. The feedback control of time-delay systems, with or without uncertainties, has been extensively studied in recent years; see, for example. (l)-(9) and references therein. In particular, global exponential stabilization for time-delay systems, with or without uncertainties, constitutes an important area for practical control design. In general, the global exponential control for time-delay systems is not as easy. even in the linear case, as without time delays. In the past, there has been a number of interesting developments in control design for time-delay control systems, but mostly were restricted to either linear cases or local results. On the other hand, nonlinearities often appear in practical control systems. For decades, nonlinear control engineers have been concerned with nonsmooth nonlinearities common in physical systems, such as saturation, relays, hysteresis and deadzones (10). These nonlinearities are frequently due to friction, which may vary with temperature and wear. Also, these nonlinearities may appear in mass-

,I‘The Franklm lnst~tute 001~~0032/96$9.50+0.00

@

Pergamon

00M-llO32(95)00065-8

619

Y.-J. Sun and J.-G. Hsieh produced components, such as valves and gears, which can vary from one component to another (11). In particular, deadzone nonlinearity is important in itself, but other nonsmooth nonlinearities, such as hysteresis, can also be modeled using deadzones (11). It is the purpose of this paper to investigate the global exponential stabilization for a class of uncertain nonlinear systems with time-varying delay arguments and input deadzone nonlinearities. This paper is organized as follows. In Section II, the problem formulation is presented. A composite feedback control is proposed such that the feedbackcontrolled system is globally exponentially stable if a simple sufficient condition is met. Furthermore, for slightly more restricted systems, we show that global exponential stability can always be achieved with any pre-specified convergence rate. An example is given in Section III to illustrate the main results. Finally, a conclusion is drawn in Section IV.

II. Problem Formulation For convenience, paper as follows:

and Main Results

we define

1-164 Q>O P
some notation

that will be used throughout

this

set of real numbers, n-dimensional real space, set of all real m by n matrices, absolute value of the real number a, unit matrix, transpose of the matrix A, conjugate transpose of the matrix A, induced Euclidean norm of the matrix A, the maximum (resp. minimum) eigenvalue of the symmetric matrix Q, matrix measure of A; p(A) = ilman[A* +A], symmetric matrix Q is positive definite, symmetric matrix Q - P is positive semidefinite, { 4: [t, -H, t,] -+ 2: c$ is continuousj, sup II W(t) II, with w(t) E C(f, W, ,-H<.t
Definition 1 The deadzone nonlinearities D(u, d,, d2), with d, > 0 and d2 > 0, as shown in Fig. 1, is defined to be the collection of all functions 4. 99 -+ 9 satisfying Journal

620

of the Frankhn lnst~tutc Elsevm Saence Ltd

Global Exponential

Stabilization

FIG I. The deadzone nonlinearities

dz(u-i=),

4(u) =

I

0,

dz(u+~),

for Nonlinear Systems

D(u, u,. LA).

r: < u,

if if

-r

if

< 24d r, u<

-u,

for any P and r with 0 d r, P < d,. From the preceding definition, it is clear that d2u E D(u, d,. d2) for any d,. Before presenting the problem formulation, let us introduce two lemmas which will be used in the proof of our main theorem. Lemma 1 (12) For any matrices

X and Y with appropriate

dimensions,

XT Y+ YTX < rXTX+

we have

k YT Y,

for any r > 0 Lemma 2 Suppose (A,, B) is a completely controllable pair, z, r, and U; are positive constants, Q is a positive definite symmetric matrix, and the pair (A, E) is completely observable for any matrix such that Q = EET. Then the Riccati equation (A,+cxZ)~P+P(A,+~~~)--~~,~~PBB~P+Q has a unique positive definite symmetric solution Proqf: Equation (1) can be rewritten as dTp+

PA-

PBBTP+Q

= 0

(1)

P.

= 0,

(2)

where

A= A,+d, Vol. 3328, No. 5. pp. 619 631, 1995 Prmted in Great Bntain. All nghtb reserved

I?=(2r,dZ)“2B. 621

Y.-J. Sun and J.-G. Hsieh It can easily be shown that the pair (A, @ is completely controllable and the pair (A, E) is completely observable. Consequently, we conclude that Eq. (2) has a unique positive definite symmetric solution Pin view of the lemma on p. 47 in (13). This completes our proof. n In this paper, we consider a class of uncertain nonlinear systems with timevarying delay arguments and deadzone nonlinearities described as a(t) = &x(t)+

c A,x(t-hh,(t))+A~,(z(t), i= 1

t)+AO~(z(t),

t > 0,

t)+B$(u),

0) x(t) = q(t),

t~[-H,O],

(3b)

where A -E Px ‘, ViE r~, x E 92’”is the state vector, (A,, B) is a completely controllable are bounded by some pair, the time delays hi(t), hi 3 0, and all delay arguments constant H 2 0, y(t) is a given continuous vector-valued initial function, z(t) = [x(t),x(t-h,(t)), . . ,x(t-h,,,(t))], u = [~,,a~, . . , u,,]~E&?~ is the input, $J(u) = [4,(u,), . , I,]’ and $,(uJ ED(u, d,, d2) represents the input deadzone nonlinearity for each iEp. For the existence of solutions of Eq. (3), we assume that the unknown terms A@,;, ie 2 are all continuous vector-valued functions. It is our goal to find a feedback control law u such that the global exponential stability of the uncertain feedback-controlled system can be guaranteed. Before presenting our first main result, i.e. Theorem I, we make an assumption as follows. (Al) There exist a,s, ai 3 0, Vie rii, a continuous vector-valued function A$(z, and a continuous real-valued functionf(z, t) such that for all their arguments

t)

IId@,(2,t)ll d aolIx(f)lI+ i 4l~~(~-M~>)ll, A%(z, t) = BAQj(z, t), ,= I

with

IIA%k t)Il
(Al)

t).

term A@, is usually called the mismatch uncercalled the match uncertainty. result for the global exponential stability of the system (3) under the feedback control proposed

is globally

exponentially

stable under

u(t) = UI (t>+u2(t),

the com-

(4)

where

622

Journalofthe

Frankhn Institute Elsevier Sc~cnce Ltd

Global Exponential

Stabilization

for Nonlinear

Systems

u,(t) = -r,BTPx(t), P is the unique positive definite symmetric parameters c( > 0, r, > 0 and Q > 0,

(5) solution

of Eq.

(1) with

h2(z, t)/2d2BTPx(t)

242(t) = -

(6)

h(z, t)112d2BTP.u(t)ll +&e-l”

wz,t) = Pd, +(d2)-‘f(z,

design

0,

(7)

with E> 0 provided

and

fl>

2c(*,

(8)

that

aa >

a, + f (IlAJ +a,). ,= 1

Ln,, (P)

In this case, the guaranteed

convergence

rate c(* is given by

I

I,,

x* =

~-[E.,,,(P)/~~mln(P)l

a,+

[ Proof: The system (3) can be rewritten k(t) = Aox(

(9)

C(II4lfaJ i=

1 .

(10)

as

C A,x(t-h,(t))+A@,(z(t), ,=I

t)+BA@3(z(t),

t)

n, =

&x(t)+

C A,x(t-h;(t))+A$(z(t),t)+d>BU, i= I

+d~Bu?

+BA~,,(z(t),t)+B[~(u)-d,u]. Furthermore,

one has I@j(u&dzuiI

6 d,d,,

Vi~p,

u,E%‘,

which implies

Let V(x(t)) The time derivative given by

of V(x(t))

Vol. 3328, No. 5, pp. 619-631. 1995 Printed in Great Britam. All nghts reserved

= x’(t)Px(t).

along the trajectories

(11) of the closed-loop

system is

623

Y.-J. Sun and J.-G. Hsieh p(‘(x(r)) = -2aV(x(r))-sT(t)Qx(r)+

6 -2aV(.y(t))+

c 2xT(t-h,)A;rPx(t) /= I

112d2.vT(t)PBll’h”

~2ll.4--h,)/I llA,ll IlPll II-Qf)IlI=I lIl/2d2xT(f)PBll +cemP’

+ 1/2dzxT(t)PBIlpd, + l12dgT(t)PB~I(dz)

‘,f’+2a,~IPll ll_v(t)ll’

+ C 2a,I/.~(~-h)l/IIPII Il.~(N. ,=I Applying preceding

Lemma 1 to the second inequality, we have ,>,

P(X(t))

d

-2aV(.\-(r))+

C

,=I

term and the last term of right hand

of the

1?1

IIAJ llpll lI-“(t--lzi)ll’+ 1 IlAd II’ll lIx(z)ll~ i= I

+2a llpll Il,y~t~l12 + _(I12dzxT(t>P~llh>(~e~“‘> 0 (I12dz.u’(t)P~Ilh)+(&e~‘~‘) ,,I

,12

+ 1 aillpllIl.y(r-h,)lI’+ 1 a~ll~llIldf)ll’~ ,=I i= I By the inequality 06--it can be deduced

ab a+b


Vu,b>O

A(a+b)

>O,

that

p(X(t)) d -2aV(X(t))+

,,1 1 ,=I

,I,

llA,ll IIPII lI~x(t-~hi)ll’ + C IlA,ll lIPI Ilx(~)ll’+~ec~’ ,= I

111

1?1

+24lIPIl ll-$f)l12+C a,llPII lIx(f-hII’+ ,=I

1 a,llPlIIlx(M2 ,=I

d - 2@W40) + 20, [~,,,,(~)/L(~)l &(O)

111 +2[~~,,,(P)/i.,,,(P)]C(IlAjjl+ai)V(x(t-hh,(t>>)+Ee~“‘. ,=I

(12)

Let us define W(r) = V, (t)+aeC”‘,

Vr > 0,

(13)

where V,(t) = V(x(t)),

Qt > 0,

(14)

and

624

lournal of the Frankhn lnsl~lulr Elsevier Sc~cnce Ltd

Global Exponential Stabilization for Nonlinear Systems a=&>O. W(t) is positive for all t 3 0, in view of V,(t) being nonnegative for all from Eqs (12), (13), (9) and (lo), it can be readily obtained that

Clearly,

t 3 0. Thus,

I@(t)= V, (t)-afieCv’ < -2a v, (t) + 2a,, [LaxVY/L(~)I

V/I(0 ,,, +2[i.,,,(P)l~,,,(P)l C (iI4 +a,)& (t ,=I

6 - 21W(t)+ 24 Lx(~>/~mlnWI w(t) ,?I +2[i,,,(P)ljt,,,(P)1 C (llA,ll+a,) Wt I=1

h,(t))+(2ax*-afi+c)eC”’

= - 2~W(t)+ 24 [iax VYL,, WI w(t) ,,,

I,,

=

-go W(t) +

c gfw(t-h,(t)),

v’t 3 0,

,=I

where, in view of Eqs (9) and (IO), go =

2c(-2a,,[/1,,,(P)/3.,,,(P)l= 2cr*+2[~,,,(P)lj.,,,(P)l 1 (II4 +a,) > 0, /=I

g, = 2[3~,,,(P)/i~,,,~(P)l(lIA,Il +a,) 2 0, Vienl. It follows that

which implies 11 W, I/ < epY~‘(ey~~f’ll W(0) II) +

’

IOe--Y~~(‘-‘) (;+Ws;

‘f’t>O,

or equivalently, I/ W, Ile”l)’ < (eBoHl( W(0)

11)+ j-”’ t+0’ (&+‘lldi;

V’taO.

(15)

Define W(t) Vol. 3328. No 5, pp. hlY-631. lYY5 Prmted in Great Brimn All rights reserved

= 11W,lleBo’,

Vt 3 0.

625

Y.-J. Sun and J.-G. Hsieh It is obvious one has

that W(t) > 0, V’t > 0. Thus, from Eq. (15) and the definition

P(t)

d eUoHljW(0) lj +

d max{e”oHIl W(O)ll, I} + Applying that

the generalized

Bellman

Vt 3 0.

l?‘(s) ds,

inequality

(16)

(14) to Eq. (16), it can be deduced

,

IV(t) < rnax{e”OHII W(O)ll, l} exp Consequently,

of m(t),

Vt b 0.

in view of Eq. (IO), we have

W(r)I/ W(t) d // W,ll = W(t)eP”(~’ < max{e80HII W(O)ll, 13 exp = max{egnHl/ W(O)//,1) exp(-2a*t), Similarly,

according

V’t 3 0.

to Eqs (1 l), (13) and (14), it is readily deduced

Ilx(t>II2 d Gn,,(~>)-’ W(O) = (ATl,,(f9-’ v, (0 d (L”(f?-’ ~(i”mln(P))-‘max{egoHliw(0)II,

ljexp(-2c(*t)

that

W(t)

= k’exp(-2a*t),

Vt 3 0,

where k = Consequently,

(~“min(P))~‘max{eYoHIl W(O)l/, 1).

we conclude

that

lI_u(t)ll < kexp(-a*r), This completes It is noted parameters satisfied. To tion is made

Vt 3 0.

n

our proof.

that in the design process we may appropriately select the free design a > 0, r, > 0 and Q > 0 in Eq. (1) such that the inequality (9) is guarantee the satisfaction of the inequality (9), an additional assumpin the following.

(A2) Now we present

rank(B)

= n.

our second main result.

Theorem II The system (3) satisfying (Al) and (A2) is globally exponentially stable with any pre-specified convergence rate z* > 0 under the composite control u defined by 626

Global Exponential Stabilization for Nonlinear Systems

u(t) = ~l(o+~,(o, where

u,(t) = -rlBTx(t),

U2(t) = -

h2(z,t)2d2BTx(t) h(z, t)I12d2BTx(t)Il +&eCp”

h(z, t) =pd, +(W'f(z, with

E >

0, fl>

0,

2cr* and f$(llAill+ai) i= 1

&%J-a*+aO+ rI >

d2Anin(BBT)

’

Proof! From (A2), we have 3,,i”(BBT) > 0. It can be readily deduced that, for all XEZ. .X=(4, -AT)x

< 2&4,)x=.x

and

-2r,dzxTBBTx Consequently,

< -2r,dz&,,n(BBT).\-T.~.

we have, for any x E R’\{O},

.~~[(A,~-ctZ)~+((A,-zl)-2r,d~BB~]x = xT[.4T + A0 + 2crZ- 2r, d2BBT]x < 2[p(A,) + a - r, dzAmln(BBT)]xTx < 0, if we let ct = a*+a,+ In that case, the matrix

f (lIA,Il +a,) > 0.

I=I

defined by

Q = -[(A,+uZ)T+(Ao-aZ)-2r,d2BBT] is positive definite. Thus, Eqs (1) and (9) are evidently P = I. This completes our proof in view of Theorem

satisfied if we let, in addition, I. n

III. Example We provide an example in the following to illustrate our main results. Consider the following uncertain nonlinear system with time-varying delay arguments and deadzone nonlinearities described as k,(t)

= -4.2x,(t)+0.1x2(t)+0.9x2(t-3+2cos(t))+axz(t-I-

+bx:(t)+c~,(t)x~(t-lVol. 332B, No. 5, pp. 619-631, 1995 Prmfed in Great Britam All nghts reserved

sin(t)) sin(t))-4(u),

(17a)

627

Y.-J. Sun and J.-G. Hsieh

i,(t)

= O.lx,(t)-4.lx~(t)+0.5x,(t-3+2cos(t))+O.lx,(t-3+2cos(t)) +O.k,(t-l-

sin(t))-hx:(t)-cx,(t)x?(f-l-

sin(t))-4(u),

(17b)

where Ial d 2, In comparison

Ihl B 2,

ICI f 2

and

$(u)ED(u,O.~,

with Eqs (3) and (17), it can be obtained

n=m=2,

p=dZ=l,

h,(t)

d, = 0.1,

1).

that

= 3-2cos(t),

h2(t) = 1 + sin(t),

Ao=[-;I; _::;I.A, =[K,;:;I>Az=[~;~ ;], I?=[-;]. do

0 1’

ax*(t-

It can be readily

1 - sin(t))

obtained

A@,, = -hx:(t)-cx,(r)x,(t-l-

that //A, I/ = 0.9079,

/IA@, II d

2/lx(t-

sin(t))

1 - sin(t>>ll,

JIA, 11= 0.5,

/IA@, ll < 2l.~~(t>l+2lx, (t>x2(t- 1 - sin(t))l,

which implies a, = a, = 0,

a, = 2,

,f=

2~x~(t)l+2~x,

(t)xz(t-

in view of (Al). By selecting the parameters

c( = 4, it can be verified,

YI = 1,

0.11 0

0 0.1

Hence Eq. (9) is evidently

Aim(PI

C(P L,,(P)

1 ’

satisfied,

_o.043

0.04

I,,,,(P)

= 0.11,

1>0,

&+“(P) = 0.1.

for in this case ,,I

= 3.6363 > 3.4079 = a,+

~(IIA,J+a,). i= I

from Eq. (lo), we have x* = a-(3.,,x(P)/~~,,,(P))

a,+

From Eq. (5), we have uI = 0.11x,(t)-0.1x,(t). which Eq. (8) is satisfied, then we may obtain 628

-0.043

from Eq. (I), that

P= [

Furthermore,

0.0682

Q=

1 - sin(t))],

i i= I

ll4l +a, = 0.25131. In addition,

let E = /I = 1, under

Journal of the Frankhn lnst~tute ElsevierScmce Ltd

Global Exponential Stabilization for Nonlinear Systems x2 (1) 1 5

70

5

0

-5

-

1 0

-

1 5

FIG 2. Typical phase trajectories

system for Eq. (17).

of the uncontrolled

2u, h2

u2= hl2u,

I + exp(-

t)

with

h = 0.1 +21x:(t)/ +21x, (t)xz(t-

1- sin(t))l,

in view of Eqs (6) and (7). Consequently, by Theorem I, we conclude that the system (17) with u = U, + u2 is globally exponentially stable with the guaranteed convergence rate c(* = 0.25 13 1. With

a = b = c = 2,

4(u) =

u-0.1,

if

0,

if

1u-0.1,

if

0.1 bu -0.1


some typical phase portraits of the uncontrolled system trolled system are depicted in Figs 2 and 3, respectively.

and the feedback-con-

Z V. Conclusion The global exponential with time-varying delay

stabilization for a class of uncertain nonlinear systems arguments and input deadzone nonlinearities has been

Vol. 332B, No. 5, pp. 619431, 1995 Printed m Great Bntain. All nghts reserved

629

Y.-J. Sun and J.-G. Hsieh

-6 -6

-4

-2

cl

2

4

6

x,(0 FIG 3. Typical phase trajectories

of the feedback-controlled

system for Eq. (17).

in this paper. A composite feedback control has been proposed such that the feedback-controlled system is globally exponentially stable if a simple sufficient condition is met. Furthermore, for slightly more restricted systems, we have shown that the global exponential stability can always be achieved with any pre-specified convergence rate. A numerical example has also been provided to illustrate the use of our main results. It is of interest to consider the global exponential stabilization problem for more general uncertain nonlinear time-delay systems and more general input nonlinearities, such as saturation, relay with dead zone and quantization. considered

References (1) T. A. Burton, (2) (3)

(4) (5)

630

“Stability and Periodic Solutions of Ordinary and Functional Differential Equations”, Academic Press, Orlando, FL, 1985. R. D. Driver, “Ordinary and Delay Differential Equations”, Springer, New York, 1975. F. H. Hsiao, J. G. Hsieh and M. S. Wu, “Determination of the tolerable sector of series nonlinearities in uncertain time-delay systems under dynamical output feedback”, ASME J. Dyn. Syst. Meas. Control, Vol. 113. pp. 5255531, 1991. V. B. Kolmanovskii and V. R. Nosov, “Stability of Functional Differential Equations”, Academic Press, London, 1986. W. H. Kwon and S. J. Lee, “LQG/LTR methods for linear systems with delay in state”, IEEE Trans. Autom. Control, Vol. AC-33, pp. 681-687, 1988. Journal

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Global Exponential Stabilization for Nonlinear Systems (6) T. T. Lee and C. L. Shih, “Discrete-time

(7)

(8) (9)

(10) (11)

(12) (13) (14)

optimal control for linear time-delay systems with deterministic disturbances”, IEEE Proc. D, Control Theory Applic., Vol. 138, pp. 573-578, 1991. T. H. Li, C. H. Lee and F. C. Kung, “A new robust stability test for a class of linear systems with uncertainties and multiple time delays”, J. Franklin Inst., Vol. 33lB, pp. 115-124, 1994. E. Tissir and A. Hmamed, “Further results on the stabilization of time delay systems containing saturating actuators”, Int. J. Sci. Syst., Vol. 23, pp. 615-622, 1992. K. Watanabe, E. Nobuyama and M. Ito, “A new algorithm for finite spectrum assignment of single-input systems with time delay”, IEEE Trnns. Autom. Control, Vol. AC-37, pp. 1377-1383, 1992. G. J. Thaler and M. P. Pastel, “Analysis and Design of Nonlinear Feedback Control Systems”, McGraw-Hill, New York, 1962. D. A. Reeker, P. V. Kokotovic, D. Rhode and J. Winkelman, “Adaptive nonlinear control of systems containing a deadzone”, Proc. 30th Conjbence on Decision and Control, Brighton, England, pp. 211 l-21 15, 1991. K. Zhou and P. P. Khargonekar, “Robust stabilization of linear systems with normbounded time-varying uncertainty”, Syst. Control Lett., Vol. 10, pp. 17-20, 1988. B. D. 0. Anderson and J. B. Moore, “Optimal Control: Linear Quadratic Methods”, Prentice-Hall, Englewood Cliffs, NJ, 1989. E. H. Yang, “Perturbations of nonlinear systems of ordinary differential equations”, J. Math. Anal_vsis Applic., Vol. 103, pp. l-15, 1984.

Vol. 3328, No. 5, pp. 619431, 1995 Printed in Great Brmin. All rights reserved

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