Rochester Institute of Technology
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1981
Stability of time-delay systems
T. Lee
Sohail Dianat
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IEEE Transactions on Automatic Control 26N4 (1981) 951-953
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95 1
IEEE TRANSACTIONS ON AUTOWTIC CONTROL, VOL. AC-26,NO. 4, AUGUST 1981
(21 M Buslowi~,“On thedetermination of a transfer function ma&from the given
state equations for the hear multivariable time-lags system,’’Confr. Cybemef.,vol. 9,
no. 3, pp. 121- 129. 1980.
[3] F. R Gantmacher, Matrix Theory, vol. I . New Yo&: Chelsea, 1959.
[4] A Manitius and R Triggiani, “Sufficient conditions for function space controllability
and feedback stabilizability of linear retarded systems,”IEEE Trnns. Aufomf. Confr.,
vol. AC-23, pp. 659-6455. Aug. 1978.
[5] A Mostowskiand M. Stark, Inrroducrion 10 Higher Algebrn: Internationd Series oj
Monogrophs on Pure and Applied Mathematics, vol. 37. Warsaw, Poland: PWN, 1964.
[6] Proe. I€€€, Special Issue on MultidimensionalSystems, vol. 65, June 1977.
+
ihihx*(
t - S ) P , * ( S ) P o P l ( ~ ) x ( 1 - 7 ) dS d T .
Therefore, its derivative with respect to t is
Stability of Time-Delay Systems
T. N. LEE AND S. DIANAT
Then some simple manipulations and integrationby parts lead to
Ahstract-Thos paper gives necessary and sufficient conditions for the
stability of time-delaysystems of the form i ( t ) = A , x ( r ) + A , x ( r - h ) .
These new conditions arederived by Lyapunov’s direct method through
systematicconstruction of the conaponding “energy” function. Thos
function is known to exist, if a solution P,(O)of the algebraic nonlinear
can be determined.
matrix equation A , =e[AI+P~(o)lhPl(0)
I.
INTRODUCTION
Sets of differential-diFference equations havereceived considerable
attention recently dueto the many applications in the modeling of
physical systems [ I]-[3]. A study of the stability of such a system, because
of the time delay factor turns out to be very interesting. In this paper, new
conditions, derived from Lyapunov‘s second method, for the stabilityof a
class of timedelay system of the form f ( r ) = A , x ( t ) + A , x ( r - h ) , are
established. It is foundthat the existence of the corresponding useful
Lyapunov function is closely related to matrices A I and A,.
Substituting (2) and (3) into the foregoing equation results in
E. STABILITV ANALYSIS ANDPRELlMiNARIES
Consider a Hermitian matrix Po. It iswell known for any vector z ,
V= z*POz>O, if and only if Po is positive definite @.d.). By redefining
z(t)=x(r)+P,(r)*x(r),
where * denotes the convolution operator, the
same conclusion for any vector x is carried over, provided the linear
transformation PI(t ) e x( t ) exists. In what follows, a lemma needed in
constructing Lyapunovfunction for the time-delay system is developed
Lemma: Let the system be
(1)
f(r)=A,x(r)+A,x(r-h),
let P l ( r ) , a characteristic matrix of dimension ( n x n ) , be continuous and
differentiable in[0, h ] and 0 elsewhere, and set
v(xI,h)=(x(r)+Pl(r)*x(r))*~o(x(r)+Pl(r)*x(r))
Since ( - Q ) is negative definite (or n.d.), the conclusion of the lemma
Q.E.D.
follows.
The theorems and corollaries below present the stability results for the
system described by (l), resulting from using the Lyapunov‘ssecond
method.
7’heorem I : Let the system be described by (1). If for any given p.d.
Hermitian matrix Q, there exists a p.d. Hermitian matrix Po such that
Po[Al+P1(0)]+[AI+P1(0)]*PO=
-Q where for T E [ O , ~ ] ,P1(7) satisfies
P 1 ( ~ ) = [ A +P,(O)]P,(T)
1
with boundary value, P 1 ( h ) = A 2
and PI(7) =O elsewhere,
where Po is Hermitian and
x(r+e),
BE[-~,o].
(4)
then this system is asymptotically stable.
P m u t The conclusion of the theorem follows immediately by defining
If
P,(AI+PI(O))+(A~+PI(O))*PO=-Q
(2)
P[=(AI+PI(O))P~(T)
O<T<h
(3)
where P , ( h ) = A , and Q is p.d. Hermitian, then f ( x t , h ) = d / d r V ( x , , h )
(0.
Pruu) It follows from the definition of V ( x z ,h ) ,
Manuscript received February 11, 1981.
The authors are with the Department of Electrical Engineering and Computer Science,
The George Washington University, Washington, D C 20052.
as a Lyapunov function for the system (1). Since x ( r ) + P , ( t ) * x ( r ) = O if
and only if x(r)=O, it follows then from the Lemma, V ( x , ,h)>O and
f ( X 1 , h)<O.
By stabilitytheoremduetoLyapunov
[7], [9], [IO] this system is
asymptotically stable.
Q.E.D.
It is noted from the sufficient condition (4) of the above theorem; the
key to the success in the construction of a Lyapunov functioncorresponding to the system (1) is on the existence of the solution of P ~ ( T i.e.,
) , the
0018-9286/81/0800-0951$00.75
01981IEEE
952
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-26, NO. 4, AUGUST
solution of P , ( T ) = ~ ~ ~ ~ + ~ ~evaluated
~ ~ ) ) ' at
P ~T=h.
( O )This requires that
the nonlinear algebraic matrix equation
eAl +Pl(O)hP
0) = A 2
has asolutionfor P,(O).
(5)
Theorem 2: Let the system be described by (I), and furthermore, let ( 5 )
have a solution, P,(O), which is nonsingular. Then, the system isasymptotically stable, only if (4) of Theorem 1 is satisfied.
ProoJ Sincethesystem is asymptoticallystable,thecharacteristic
roots of I d - A , -A2e-hSI=0, have negative real parts. Substituting ( 5 )
into theleft-handside
of theforegoing equation resultsin I d - A , e l a , + P , ~ O ) - r l l h p'(0)l. Powerseriesexpensionoftheexponentialterm
yields
i(r)=A,x(r-h).
The system is asymptotically stable if and only if
a) - s/2 h <Re X i<O. &,hereX , are the eigenvalues of A , ,
b) h I A, I <ao. where wo E(0, s / 2 ) and satisfies
sinwo=-hReh,/oo
The above theorem is powerful in a sense that it is a necessary and
sufficient condition and it does not require any a priori condition. For
similar result, consult [8].
N.
APPLICATIONS AND EXAMPLES
The resultsestablishedaboveareillustratedbythefollowingsimple
examples.
Example I : Investigate the stability of the following system:
IM+Pl(0)-[I-Mh+(Mh)2/2!-(Mh)3/3!+
.-.]P,(O)J
=~M[Z+Pl(0)h-(Mh2)Pl(0)/2!+(M2k3)Pl(0)/3!-..]I
=(MI(I+P,(0)h-(Mh2)P,(0)/2!+
...I.
1981
x(r)=-3x(t)+0.135x(r-I).
whereM=sZ-AI-P,(0).
Solution: Since A , = -3, A2=0.135, it follows from (5) that P1(0)=
Since I+P,(O)h-(Mh2)P,(0)/2+ . . . converges uniformly to an entire
function, therefore the eigenvalues, X,, of matrix A , +P,(O) are a portion 2.7 12 e -'do) which yields P,(O)= 1.
Therefore, for any Q>O, we obtain a Po >O; in particular, for Q = 4 , we
of the discrete spectrum, y,. of the characteristic equation of ( S I - A , compute Po = 1. Now the system is stable by Theorem 1 and the correA2e-"'1=0, (h,}~{y,},foralliandsomej.
Sincesystem (1) is asymptoticallystable. Re y, ( 0 , for all j . This sponding Lyapunov function is
implies Re X, ( 0 , for all i.
It follows by a Theoremdue to Kalman [4],for every p.d. Hermitian Q .
there exists a p.d. Hermitian Po, such that Po[A,+P,(O)]+[A,+P,(0)]*Po
= -Q.
Its time derivative is
Sincetheexistence of P,(O) is quarenteedbythehypothesis
of this
$eorem, P , ( T ) , for TE[O,h ] satisfiestrivially the differential equation
P1(7)=[AI+ P I ( 0 ) ] P I ( with
~ ) , b o u n d q condition A , =e[al+Pl(o)lhPl(0).
This completes
proof
the
of the theorem.
Q.E.D.
Theorem 3: Let the hypothesisbe the same as stated inTheorem 2.
Example 2: Givenasystem of theform x ( t ) = x ( r ) + A , x ( r - I ) , inThe systemdescribedby
(I) is asymptoticallystable, if and only if vestigate the stability of the following cases.
la,!<1. for all i, where aiare the distinct eigenvalues of A , P , ( O ) - ' .
Case I : A , = - 2 e - ' . Since the given constants satisfy conditions a)
Proof: It follows from ( 5 ) that
and b) of Corollary 1, simple computations yield in connection with (4)
and ( 5 ) P,(O)= -2 and P'(T)= -2e-'.
A,PI(0)-'=e[~I+Pl'O)lh.
Hence by Theorem 1 the system is stable. In particular, for Q=2(1Therefore a,=ey,. where they, are the eigenvaluesof A I + P,(O). 1 a,1 < 1 e - I ) , we have Po = 1; the corresponding Lyapunov function is
implies Re y, <O.
Hence, the real parts of eigenvalues of A , +P,(O) are negative.
Therefore, the sufficiency part of the theorem follows immediately by
Q.E.D.
Theorem 1, and thenecessity part is theresult of Theorem 2.
Corollariesgivenbeloware
the simpleresultsin
the searchfora
solution of P,(O) of ( 5 ) .
Corolla? I : There exists at least one P,(O) satisfying (5). if
a) the eigenvalues, B, of (e-"1'A2h). are real and distinct, and
b) /3,2
- e - ' , for all i.
Proof: It follows from ( 9 ,
eP~(0)'P,(0)h=e-A~hA2k.
and its time derivative is
Case 2: A, = e 2 . This set of given constants also satisfies conditions a)
and b) of Corollary 1 and the Lyapunov function can be constructed as
before. Thus, P,(O)= 1 and Pl(T)=e2'.
Therefore. with Q=4, we have Po = - I. This results in
This, since by a) the right-hand side can be diagonalized, that
e4al=pi>-e-I
for all i
(6)
where the 6, are the eigenvalues of P,(O).
Since the nonlinear algebraic equation(6) has solution, therefore we are
Q.E.D.
able to compute F, and hence P,(O).
Corollay 2: There exists a unique P,(O) satisfying ( 5 ) . if
a) p, €real and distinct and
b) p, 2 0 , for all i.
Proof: The conclusion of this corollary follows from ( 6 ) . since e"F, is
negativefor 6, <O and isa continuousand monotonicallyincreasing
function of 6, for 6, a 0 ,
A specialization of the original result when the system (1) is such that
A , =O, is stated here without proof. Interested readers should refer to the
authors 151.
Theorem 4: Let the time-delay system be
Butsince V ( x , l ) = - [ x ( r ) + ~ ~ . x ( r - ~ ) e ~ ' d r it
] ~ follows
> O bythe
instability theorem due to Lyapunov. the system is unstable [7].
Example 3: Investigate the stability of the following system:
[=I: -: ~ ~ ] [ ~ ~ ] + [ ~ : ~ ~
-0.697
-Yl(r- 1)
-0.33 ][.x,(t-l)]'
Solurion: Soldng P , ( 0 ) = e - ( " l i P ~ ( o ) )by
A ,numerical method. After 10
iterations. we obtain
Hence,
IEE TRANSACTIONS ON AUTOMATIC
CONTROL, VOL.
AC-26,NO. 4, AUGUST 1981
V. CONCLUSION
In this paper we have established new results for the stability
of a linear
time-delay system. By defining the characteristic matrix function P 1 ( 7 )on
[O,h], we are able to derive necessary and sufficient conditions for the
time-delay system to bestable, with apn'ori condition of the existence of a
nonsingular solution P,(O). We believe similar results can be obtained for
the casewhere A , and A , are timevarying. The sufficient condition
follows trivially. The necessity part, as usual, is the heart of the problem;
research is in progress. Literature on the existence and uniqueness of the
solution of the Lyapunov matrix equation [4], [6] relates intimately to the
problem concerned.
REFERENCES
Equations.
New York:
R. Bellman and K. L. Cooke, Diflerenrial-Diflerence
Academic. 1963.
A. Halanay, Differentiul Equations, Stubiliry, L)scillations and Time-Lap. New Yo&:
Academic, 1966.
G . S. Ladde, "Stability of model ecosystems with time-delay," J . Themy Biol., vol. 61,
pp. 1-13, 1976.
R E.Kalman and J. E Bertram, "Control system analysis a i d design vie the 'second
method' of Lyapunov," J. Bmie Eng.. pp. 371-391, June 1960.
S . Dianat,and T.Lee. "Stability and optimization of a class of time-delay system,"
to be published.
J. P. LaSalle and S. Lefscheiz, Srabiliry by Lyapunm's Direcr Method with Application. New York: Academic, 1961.
M. Vidyasagar. Nonlinear SysfemAnabsis. Rentice-Hall. 1978,pp. 53-55.160-163.
V. K. Bamvell, "Numerical solution of differential-difference equation&" C.S. Dep.
Rep. CS-76-04, Univ. Waterloo, Waterloo, Ont. Crinada.
R.D. Driver. Ordinaru and Delm
, Differential Eauations. New York: SDrineer-Verlae.
-.
1977, pp. 354-355.
J. Hale, 77wor). of Funcrionul DifferentiulEquations. New York Springer-Verlag,
1977, pp. 105- 126.
. -
I
,
953
same time, functional analysis and operator theory have made a significant impact on modern system theory. These structures have potential for
even more extensive contributions.
The present study is a connective one in that we start with the operator
format and develop the state realization. The first comprehensive study of
this type is due to Saeks [I], [7]. Although Saeks used a Hilbert resolution
space format, most of hisdevelopmentused only algebraicproperties.
References [I] and [7] also left unclear the issue of state trajectories and
did not consider the nonlinear case.
In [2] and [3] Schnure introduced the concepts of state trajectory space
and used the geometric structure of Hilbert spacefully to establish
various controllability, observability, and minimality concepts. Independently, Steinberger [4] developedan alternative approach to statetrajectories and thenproceeded to solvea basic problem of optimalcontrol
introduced earlier by Porter [17] using hisstructure [5]. Further details on
these results are available in Feintuch [6, ch. 21.
The literature cited above deals only with linear operators. The present
studydeparts from thispatterninthat
we take upaversion
of the
nonlinear problem. In this regard, it is important to note the pioneering
work of DeSantis [20] who developed the state structure requirements for
nonlinear maps.
11.
POLYNOMIC OPERATORS ON HILBERT RESOLUTION SPACES
For brevity, our discussion here presumes some a priori knowledge of
Hilbert resolution spaces. The reader is referred to [I], [6]-[ IO] for details.
Someproperties of tensorproductsand polynomic operatorsarealso
introduced. A more complete discussion is available in [8], [ IO]-[15].
To simplify the discussion, attention is focused on the quadratic case.
The development and maintheorems,however,holdmoregenerally.
These extensions are noted in the closing section.
Let H denote a Hilbert space over the field F and P= {P': t Ev} denote
a resolution of the identity. For x, y E H the element x@y is the tensor
product. Similarly, for x EH we define
T(X)=(l,X,X@x),
xEH.
The space
H=closed S P ~ { T ( X x) :E H }
is a Hilbert space with inner product
((~94)>=x a i B j ( l + ( X i * y / ) + xi,^,)')
ij
The State Representationof Polynomic Maps
W. A. PORTER
where @=Zai(l,x i , xi@xl),#=Z/3,(1, y j , yj@yl), and the closure of H is
with respect to the n o m induced by this inner product.
The space H can be equipped with a resolution of the identity and this
is defined in the following specific way:
4 ' ~ = ~ a i ( l , P ' x i , P ' x i ~ P ' x r ) ,fEV.
Abstract-kt f be an arbitrary strictly causal polynomic map between
I
Hilbert resolutionspaces.Then
f hasthe form ~ = T Twhere T is a
4'=I ,
predetermined causal map rind T is linear and strictly causal. The maps f The properties Tu@ =?'Tu =q' for all b r a hold and
f apparent. Since lim,,,9'+=Z,ai(l,o,o) we augment
and T have state representations. The present paper develops the interrela- &d ( T r ) * = T are
the 9 " ' s with the zero operator to close the resolutionfrombelow;
tionship between these.
formally
I. INTRODUCTION
s={sr:rEv}u{O}.
The notion of "state" has played a highly visible rolein modem system
n e complementary projections Tf= I - 4'are also available. and aptheory. This role has been both theoretical and as a realization mechanism
for suchproblems as filtering, estiuiation, and optimal control. At the parently
lim+=
Yr@=x a i ( o , P,Xi, P f x , @ P ' x r C P ' x i @ 3 P r x i + P r x i @ P f x i ) . (1)
Manuscript received February I . 1980; revised February 11, 1981 and April 3, 1981. This
work Was supported io part by the Air Force Office of Scientific Research under Grant
AFOSR 78-3500.
The author is witb the department of Electrical Engineering, Louisiana State university,
Baton Rouge, LA 70803.
r
Here Pf =I - P' is used. We need also the incremental projections T ( k ) =
T k - T k - ' . Letting A ( k ) = P k - P k - ' where P k is an abbreviation for
PI*, it can be verified that
0018-9286/81/0800-0953$00.75 0 1981 IEEE