A Liapunov-Krasovskii criterion for ISS of systems described by coupled delay differential and difference equations

Abstract

The input-to-state stability of time invariant systems described by coupled delay differential and difference equations is here investigated. A new Liapunov-Krasovskii theorem to check such type of stability is proved. An example taken from the literature, concerning an electrical circuit, is reported, showing the effectiveness of the methodology.

Full text

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Automatica 44 (2008) 2266–2273
Contents lists available at ScienceDirect
Automatica
journal homepage: www.elsevier.com/locate/automatica
On the Liapunov–Krasovskii methodology for the ISS of systems described by
coupled delay differential and difference equations$
P. Pepea,∗, I. Karafyllisb, Z.-P. Jiangc
aDipartimento di Ingegneria Elettrica e dell’Informazione, Università degli Studi dell’Aquila, Monteluco di Roio, 67040 L’Aquila, Italy
bDepartment of Environmental Engineering, Technical University of Crete, 73100 Chania, Greece
cDepartment of Electrical and Computer Engineering, Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA
a r t i c l e i n f o
Article history:
Received 4 April 2007
Received in revised form
26 October 2007
Accepted 21 January 2008
Available online 5 March 2008
Keywords:
Delay differential equations
Continuous time difference equations
Neutral systems
Input-to-state stability
Liapunov–Krasovskii functional
a b s t r a c t
The input-to-state stability of time-invariant systems described by coupled differential and difference
equationswithmultiplenoncommensurateanddistributedtimedelaysisinvestigatedinthispaper.Such
equations include neutral functional differential equations in Hale’s form (which model, for instance,
partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal,
hydraulic and electrical engineering. A general methodology for systematically studying the input-to-
state stability, by means of Liapunov–Krasovskii functionals, with respect to measurable and locally
essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the
functional evaluated at the solution has been studied and solved by introducing the hypothesis that the
functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case.
It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state
stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example
taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the
proposed methodology.
© 2008 Elsevier Ltd. All rights reserved.
1. Introduction
In this paper systems described by coupled delay differential
and difference equations forced by measurable, locally essentially
bounded inputs, are considered. The importance in engineering
applications of the systems here considered is well known (see
Niculescu (2001) and Rasvan and Niculescu (2002) and the
references therein). It is assumed that the functionals involved
in the dynamics and the input are such that the Carathéodory
conditions are verified. The notion of ISS, given in Sontag (1989),
has had a great impact on the study of nonlinear delay-free
systems and we are confident that this notion will have a great
impactalsofordelayedsystems.Forinstance,theconceptofinput-
to-state stability for coupled delay differential and difference
systems here introduced can be used when studying the internal
dynamics of recently studied nonlinear delay control systems (see
Germani, Manes, and Pepe (2000, 2003)), or when studying the
behavior of lossless transmission lines with forcing inputs (see
Rasvan and Niculescu (2002)). In the seminal paper Teel (1998),
$This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Associate Editor Antonio Loria
under the direction of Editor Hassan Khalil.
∗Corresponding author. Fax: +39 0862 434403.
E-mail addresses: pepe@ing.univaq.it (P. Pepe), ikarafyl@enveng.tuc.gr
(I. Karafyllis), zjiang@control.poly.edu (Z.-P. Jiang).
the notion of input-to-state stability has been generalized to
systems described by nonlinear retarded functional differential
equations and sufficient conditions are stated in the setting of
Liapunov–Razumikhin methodology. In the paper (Pepe & Jiang,
2006) the input-to-state stability and the integral input-to-state
stability (see Angeli, Sontag, and Wang (2000) and references
therein) from a perspective of Liapunov–Krasovskii functionals for
systemsdescribedbyretardedfunctionaldifferentialequationsare
addressed. In the paper (Pepe, 2007a) the input-to-state stability
of systems described by neutral functional differential equations
in Hale’s form, with linear difference operator, is studied and a
Liapunov–Krasovskiimethodologyispresented.Thepaper(Rasvan
& Niculescu, 2002) focuses on forced oscillations for lossless
propagation systems, described by linear coupled differential and
difference equations with scalar, globally Lipschitz, nonlinear
perturbations, depending on a linear combination of the unknown
variables which on the left-hand side of the mathematical
model appear differentiated. A single delay, not involving these
unknown variables, is considered. The input function, which
appears in both the differential and the difference equations of
the model, is supposed to be piece-wise continuous and bounded.
Computationally checkable LMI conditions, involving the Lipschitz
constant, which yield the exponential stability of the solution are
provided.
In this paper, we take a step further to study the input-
to-state stability property for a class of systems described by
0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.automatica.2008.01.010

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P. Pepe et al. / Automatica 44 (2008) 2266–2273
2267
coupled delay differential and difference equations. Particularly,
wepresentforthefirsttimetheLiapunov–Krasovskiimethodology
for systematically studying the input-to-state stability of general
nonlinear infinite-dimensional systems of this kind, with multiple
noncommensurate and distributed time delays which may affect
all variables in the model, and with the forcing input, measurable
and locally essentially bounded, appearing in the differential
equation of the model, see Section 4. The input here does not
appear in the difference equation, otherwise the very weak
hypothesis on such input, which may well describe unknown
disturbances, would not allow the continuity (not even piece-
wise) of the solution, which would be just measurable. This point
is very critical (see the technical problems for the correct use
of Liapunov–Krasovskii functionals addressed in Pepe (2007a,b))
and deserves further deep investigations, which are beyond the
aims of this paper. The studied class of systems includes general
neutral systems in Hale’s form with nonlinear difference operator
(for instance, systems described by Eq. (19), in the time-invariant
case, in Pepe (2007a)). We then show, in Section 5, that the
proposed methodology leads to computationally checkable LMI
conditions upon specification of our systems into the context of
linear systems. It is proved that a linear neutral system with
stable difference operator is input-to-state stable if and only if the
trivial solution of the unforced system is asymptotically stable.
In Section 6, a nonlinear example concerning an electrical device,
taken from the past literature, is worked out in detail to illustrate
the effectiveness of the approach advocated in the paper.
2. Preliminaries
The symbol ¯R indicates the extended real line [−∞,+∞].
The symbol | · | stands for the Euclidean norm of a real vector,
or the induced Euclidean norm of a matrix. Ij is the identity
matrix of dimension j, 0jis a zero square matrix of dimension j,
0i,j is a zero matrix in Ri×j, i,j positive integers. A function u :
R+→ Rm, m positive integer, is said to be essentially bounded if
esssupt≥0|u(t)| < ∞. We indicate the essential supremum norm
ofanessentiallyboundedfunctionwiththesymbol?·?∞.Forgiven
times 0 ≤ T1< T2, we indicate with u[T1,T2): [0,+∞) → Rmthe
function given by u[T1,T2)(t) = u(t) for all t ∈ [T1,T2) and equal to
0 elsewhere. An input u : R+→ Rmis said to be locally essentially
bounded if, for any T > 0, u[0,T)is essentially bounded. A function
w : [0,b) → R, 0 < b ≤ +∞, is said to be locally absolutely
continuous if it is absolutely continuous in any interval [0,c], 0 <
c < b. For a real ∆ > 0, a positive integer n, L2([−∆,0];Rn)
is the Hilbert space of square Lebesgue integrable functions
mapping [−∆,0] into Rn, C([−∆,0],Rn) is the space of continuous
functions mapping the interval [−∆,0] into Rn, endowed with
the supremum norm. C1([−∆,0],Rn) is the space of continuously
differentiable functions mapping the interval [−∆,0] into Rn.
W1,∞([−∆,0];Rn) is the space of absolutely continuous functions
mapping the interval [−∆,0] into Rnwith essentially bounded
derivative, endowed with the same norm of C([−∆,0],Rn). For a
given function s : [−∆,b) → Rn, 0 < b ≤ +∞, the function st :
[−∆,0] → Rn,t ∈ [0,b), is defined as st(τ) = s(t +τ),τ ∈ [−∆,0].
Let us here recall that a function γ : R+→ R+is: of class K if it
is zero at zero, continuous and strictly increasing; of class K∞if
it is of class K and it is unbounded; of class L if it is continuous,
nonincreasing and converges to zero as its argument tends to +∞.
A function β : R+× R+→ R+is of class KL if it is of class K in
the first argument and is of class L in the second argument. Here
an Nafunctional is any functional defined in the product space
C([−∆,0];Rd) × C([−∆,0];Rn), with d,n positive integers, and
taking values in R+, such that, for suitable functions γaand ¯ γaof
class K∞, the following inequalities hold for anyφ ∈ C([−∆,0];Rd)
and any ψ ∈ C([−∆,0];Rn):
γa(|φ(0)|) ≤ Na
ψ
For instance, the N2norm (see Pepe and Verriest (2003)) in the
product space C([−∆,0];Rd) × C([−∆,0];Rn) is an Nafunctional.
??φ
??
≤ ¯ γa(?φ?∞+ ?ψ?∞).
(1)
3. The system equations
Let us consider a system described by the following nonlinear
coupled delay differential and difference equations (see Fridman
(2002), Germani et al. (2003), Hale and Martinez Amores (1977),
Niculescu(2001),Pepe(2005,2007a),PepeandVerriest(2003)and
Rasvan and Niculescu (2002))
˙ξ(t) = f(ξt,xt,u(t)),
x(t) = g(ξt,xt),
t ≥ 0, a.e.,
t ≥ 0
(2)
ξ(τ) = ξ0(τ),
where: t ∈ [0,+∞); ξ(t) ∈ Rd; x(t) ∈ Rn; n,d are positive
integers; u(t) ∈ Rmis the measurable locally essentially bounded
input function, m is a positive integer; ξ0 and x0 are functions
in C([−∆,0];Rd) and C([−∆,0];Rn), respectively; ∆
the maximum involved delay; f is a locally Lipschitz continuous
functional mapping C([−∆,0];Rd) × C([−∆,0];Rn) × Rminto
Rd, independent of the second argument at zero (see Definition
5.1, p. 281 in Hale and Lunel (1993)); g is a locally Lipschitz
continuousfunctionalmappingC([−∆,0];Rd)×C([−∆,0];Rn)into
Rn, independent of the second argument at 0. We assume that
f(0,0,0) = 0, and g(0,0) = 0, thus ensuring that ξ(t) =
0,x(t) = 0, for every t ≥ 0, is the solution of system (2) and
(3) corresponding to zero initial conditions and zero input (i.e. the
trivial solution). Note that the difference equation in (2) holds at
0 too, that is the initial conditions satisfy the matching condition
x0(0) = g(ξ0,x0). Let us now introduce the following hypotheses
involving the functionals f,g,u.
(Hp1) For any (φ,ψ,v) in the space
W1,∞([−∆,0];Rd) × W1,∞([−∆,0];Rn) × Rm,
it happens that
x(τ) = x0(τ),τ ∈ [−∆,0],
(3)
>
0 is
limsup
h→0+
where, for 0 < h < ∆: φh∈ W1,∞([−∆,0];Rd) is given by
?φ(s + h),
ψh∈ W1,∞([−∆,0];Rn) is given by
?ψ(s + h),
moreover the functional
g(φh,ψh) − g(φ,ψ)
g(φh,ψh) − g(φ,ψ)
h
∈ Rn,
(4)
φh(s) =
s ∈ [−∆,−h],
s ∈ (−h,0];
φ(0) + v(s + h),
(5)
ψh(s) =
s ∈ [−∆,−h],
s ∈ (−h,0];
ψ(0),
(6)
(φ,ψ,v) → limsup
h→0+
h
(7)
is bounded on bounded sets U
W1,∞([−∆,0];Rn) × Rm;
(Hp2) for any continuous function s : [−∆,+∞) →
functional F : C([−∆,0];Rd) × R+→ Rd, defined as
F(φ,t) = f(φ,st,u(t)),
is bounded on any bounded set U ⊂ C([−∆,0];Rd) × R+, and
satisfies the modified Carathéodory conditions in C([−∆,0];Rd)×
R+
(see Section 2.4, p. 100 in Kolmanovskii and Myshkis
(1999)).
?
⊂
W1,∞([−∆,0];Rd) ×
Rn, the
(8)

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P. Pepe et al. / Automatica 44 (2008) 2266–2273
Remark 1. By applying the existence and uniqueness of solutions
theoremsfortimedelaysystemstosystem(2),fromthehypothesis
Hp2it follows that system (2) admits a unique solution on a right
maximal time interval [0,b), 0 < b ≤ +∞, with ξ(t) component-
wise locally absolutely continuous and x(t) continuous. Moreover,
if b < +∞, then ξ(t) is unbounded in [0,b) (see Section 2.6, p.
58 in Hale and Lunel (1993) and Sections 2.2 and 2.4, p. 96, 100 in
Kolmanovskii and Myshkis (1999)).
?
As is well known, given a continuous functional
V : C([−∆,0];Rd) × C([−∆,0];Rn) → R+,
the upper right-hand Dini derivative of the time function w(t) =
V
xtx(t)
??ξt+h
h
(9)
??ξt
??
, with
?ξ(t)
?
solution of (2), is given, for t ≥ 0, by
??
D+w(t) = limsup
h→0+
V
xt+h
− V
??ξt
xt
??
.
(10)
Definition 2 (See Driver (1962) and Pepe (2007a)). Let V
C([−∆,0];Rd) × C([−∆,0];Rn) → R+be a continuous functional.
The upper right-hand Dini derivative
:
D+V : C([−∆,0];Rd) × C([−∆,0];Rn) × Rm→¯R
of the functional V is defined, for φ
C([−∆,0];Rn), v ∈ Rm, as
D+V
ψ
?
where, for 0 < h < ∆, 0 < θ ≤ h: φ?
?φ(s + θ),
ψ??
(11)
∈
C([−∆,0];Rd), ψ
∈
??φ
?
,v
?
1
h
= limsup
h→0+
V
??φ?
h
ψ?
h
??
− V
??φ
ψ
???
θ∈ C([−∆,0];Rd) is given by
s ∈ [−∆,−θ],
s ∈ (−θ,0];
,
(12)
φ?
θ(s) =
φ(0) + f(φ,ψ,v)(s + θ),
θ∈ C([−∆,0];Rn) is given by
?ψ(s + θ),
ψ?
ψ??
θ(s) =
s ∈ [−∆,−θ],
s ∈ (−θ,0];
ψ(0),
(13)
h∈ C([−∆,0];Rn) is given by
?ψ(s + h),
ψ?
h(s) =
s ∈ [−∆,−h],
s ∈ (−h,0].
g(φ?
s+h,ψ??
s+h),
(14)
In the following, it will be useful to consider the input-to-
state stability of (only) the difference part of system (2), x(t) =
g(ξt,xt) (see Pepe, Jiang, and Fridman (2008)). A continuous-time
difference equation can be re-written, in many ways, as a discrete-
time equation on a suitably chosen Banach space (see Germani
et al. (2003) and Pepe (2003), for an explicit expression of the
discrete-time system in the case of discrete delays). For instance,
there exists a suitable functionˆG by which the difference equation
can be transformed into the discrete-time system in the Banach
space C([−∆,0];Rn)
ˆ χ(k + 1) =ˆG(ˆ χ(k),ˆζ(k)),
where ˆ χ(k)(τ) = x(kδ + τ),ˆζ(k)(τ) =
0 < δ ≤ a ([−a,0] being the interval of independence of the
functional g with respect to the second argument) is such that,
for a suitable integer r > 1, ∆ = rδ,
setting χ(k) = ˆ χ(rk), ζ(k)(τ) =
τ ∈ [−∆,0], a suitable functional G exists such that the following
k = 0,1,...,
(15)
?ξ((k + 1)δ + τ)
ξ(kδ + τ)
?
, τ ∈ [−∆,0],
∆
r≤ a <
∆
r−1. Moreover,
ξT(rδk + τ)
?
ξT(rδ(k + 1) + τ)
?T,
discrete-timesystemcanbeusedwithLiapunovmethodologiesfor
ISS problems (see Jiang and Wang (2001) and Karafyllis (2006a,b))
χ(k + 1) = G(χ(k),ζ(k)),
If one considers ζ(k) as an input, then by Theorem 3 in Pepe et al.
(2008), one can check if system (16) is ISS with respect to such an
input.Inthefollowingsections,theISSofsystem(16)willbemeant
with respect to ζ.
k = 0,1,....
(16)
4. The Liapunov–Krasovskii theorem for ISS
Definition 3 (See Sontag (1989)). The system described by the
equations (2) is said to be input-to-state stable (ISS) if there exist a
KLfunctionβandaK functionγ suchthat,foranycontinuousinitial
state
x0
locally essentially bounded input u, the solution exists for all t ≥ 0
and furthermore it satisfies
????
Let us here recall that a continuous functional V : C([−∆,0];Rd) ×
C([−∆,0];Rn) → R+islocallyLipschitzwithrespecttothenormof
the uniform topology if ∀
ψ
R+there exist a neighborhood Uφ,ψof
that the inequality |V(y1,z1)−V(y2,z2)| ≤ Lφ,ψ(?y1−y2?∞+?z1−
z2?∞) holds ∀
?ξ0
?
(satisfying the matching condition) and any measurable
?ξ(t)
x(t)
?????≤ β(?ξ0?∞+ ?x0?∞,t) + γ(?u[0,t)?∞).
(17)
?φ
?
∈ C([−∆,0];Rd) × C([−∆,0];Rn) →
?φ
∈ Uφ,ψ.
ψ
?
and a constant Lφ,ψsuch
?y1
z1
?
,
?y2
z2
?
Lemma 4 (SeePepe(2007a)).Let V : C([−∆,0];Rd)×C([−∆,0];Rn)
→ R+be a continuous functional, locally Lipschitz with respect to the
norm of the uniform topology. Let the function w : [0,b) → R+be
defined as w(t) = V(ξt,xt), where (ξt,xt) is the solution of system (2)
and (3) on a right maximal time interval [0,b),0 < b ≤ +∞.
Then D+w(t) = D+V(ξt,xt,u(t)),
t ∈ [0,b), a.e.
Next two lemmas are needed in order to solve the problems
concerning the absolute continuity on the time domain of a
continuous functional evaluated at the solution (see Pepe (2007b),
as far as systems described by retarded functional differential
equations are concerned).
Lemma 5. Let V
a continuous functional, locally Lipschitz with respect to the norm
of the uniform topology. Let the initial conditions in (3) belong
to W1,∞([−∆,0];Rd) × W1,∞([−∆,0];Rn). Let the function w :
[0,b) → R+be defined as w(t) = V(ξt,xt), where (ξt,xt) is the
solution of system (2) and (3) on a right maximal time interval
[0,b),0 < b ≤ +∞.
Then the function t → w(t) is locally absolutely continuous in [0,b).
Proof. We have to prove that, for any given c ∈ (0,b), the func-
tion w is absolutely continuous in [0,c]. Let c ∈ (0,b) be arbitrarily
given. Let [−a,0], 0 < a < ∆, be the interval of independence of
thefunctional g withrespecttothesecondargument.Astepproce-
dureishereused.Thefirststepisasfollows.Let? = min{a,c}.From
the hypotheses on the functional g in Hp1, and from Theorem 2 in
Pepe (2007a), it follows that, in[0,?], the part x(t) of the solution is
almosteverywheredifferentiablewithessentiallyboundedderiva-
tive.Therefore xt∈ W1,∞([−∆,0];Rn),t ∈ [0,?].Itfollowsthat,for
any t1< t2in [0,?] the following equalities–inequalities hold
:
C([−∆,0];Rd) × C([−∆,0];Rn) →
R+be

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P. Pepe et al. / Automatica 44 (2008) 2266–2273
2269
?xt2− xt1?∞≤
sup
τ∈[−∆,0]|x(t2+ τ) − x(t1+ τ)|
????x(−∆) +
˙ x(θ)dθ
=
sup
τ∈[−∆,0]
?t2+τ
−∆
˙ x(θ)dθ − x(−∆) −
????≤ ess
?t1+τ
−∆
˙ x(θ)dθ
????
=
sup
τ∈[−∆,0]
????
?t2+τ
t1+τ
sup
τ∈[−∆,?]|˙ x(τ)|(t2− t1).
(18)
Analogously, the part ξtof the solution belongs to W1,∞([−∆,0];
Rd),t ∈ [0,?] too, and the inequality holds
?ξt2− ξt1?∞≤ ess sup
τ∈[−∆,?]|˙ξ(τ)|(t2− t1).
(19)
It follows that the function t → (ξt,xt) is absolutely continuous in
[0,?]. Since the functional V is locally Lipschitz, it follows that it is
uniformly Lipschitz on the compact set S? = {(ξt,xt),t ∈ [0,?]}.
It follows that the function t → w(t) is absolutely continuous
in [0,?]. If c > ?, the reasoning is repeated in the same way in
[ka,min{(k + 1)a,c}], k = 1,2,..., until k is such that (k + 1)a >
c. Therefore, the function t → w(t) is absolutely continuous in
[0,c].
?
Lemma 6. The following statements are equivalent:
(1) The system described by (2) with continuous initial conditions is
ISS;
(2) The system described by (2) with initial conditions in W1,∞([−∆,
0];Rd) × W1,∞([−∆,0];Rn) is ISS.
Proof. The implication 1 =>
implication 2 => 1. So, let us hypothesize that for any initial
conditions in W1,∞([−∆,0];Rd) × W1,∞([−∆,0];Rn) and any
measurable locally essentially bounded input, the inequality (17)
holds for the corresponding solution (ξ(t),x(t)).
Claim: The inequality (17) holds also in the case of (simply)
continuous initial conditions.
To prove the claim, by contradiction, let us suppose that
the inequality (17) does not hold for certain continuous initial
conditions (¯ξ0, ¯ x0) (satisfying ¯ x0(0) = g(¯ξ0, ¯ x0)). So, there exist a
measurable locally essentially bounded input, ¯ u, and a time t1 ≥
0, such that the following inequality holds for the corresponding
solution (¯ξ(t), ¯ x(t)):
?????
?????
>
functionals f and g with respect to the second arguments.
Letηbeanarbitrarilychosenpositivereal.Since C1([−∆,0];Rd)
×C1([−∆,0];Rn) is dense in C([−∆,0];Rd)×C([−∆,0];Rn), there
exists a function
z
?y −¯ξ0?∞+ ?z − ¯ x0?∞ <
be a positive real such that a > h > 0, hsups∈[−∆,0]|˙ z(s)| <η
|g(¯ξ0, ¯ x0) − z(−h)| <η


Since
2 is obvious. Let us prove the
?¯ξ(t1)
¯ x(t1)
Let ? > 0 such that the following inequality holds
?¯ξ(t1)
¯ x(t1)
Let [−a,0], a
??????> β(?¯ξ0?∞+ ?¯ x0?∞,t1) + γ(?¯ u[0,t1)?∞).
??????> β(?¯ξ0?∞+ ?¯ x0?∞+ ?,t1) + γ(?¯ u[0,t1)?∞) + ?.
(20)
(21)
0, be the interval of independence of the
?y
?
in C1([−∆,0];Rd) × C1([−∆,0];Rn) such that
η
6, and |g(y,z) − g(¯ξ0, ¯ x0)| <
η
6. Let h
6and
6. Let p ∈ W1,∞([−∆,0];Rn) be defined as
s ∈ [−∆,−h)
h+ g(y,z)s + h
h
p(s) =

z(s)
−z(−h)s
s ∈ [−h,0].
(22)
sup
s∈[−h,0]|z(s) − p(s)| ≤ h sup
s∈[−h,0]|˙ z(s)| + |g(y,z) − z(−h)|,
(23)
it follows that ?y −¯ξ0?∞+ ?p − ¯ x0?∞< η. Therefore, there exists
a sequence of functions
¯ xj
0];Rn) (the superscript j = 0,1,... is the index of the term in the
sequence)satisfyingthematchingcondition ¯ xj
that
?¯ξj
0
0
?
∈ W1,∞([−∆,0];Rd) × W1,∞([−∆,
0(0) = g(¯ξj
0, ¯ xj
0),such
lim
j→+∞?¯ξj
Let (¯ξj(t), ¯ xj(t)) be the solution corresponding to the initial
conditions (¯ξj
integer such that (l + 1)a > t1. By Theorem 2.2, p. 43, in Hale and
Lunel (1993) (see also Section 2.6, p. 58 in Hale and Lunel (1993)),
there exist l + 2 positive reals δ0< δ1< ··· < δl< δl+1= ?, such
that, if ?¯ξj
sup
0−¯ξ0?∞+ ?¯ xj
0− ¯ x0?∞= 0.
(24)
0, ¯ xj
0) and to the input ¯ u. Let l be the first nonnegative
ia−¯ξia?∞+ ?¯ xj
ia− ¯ xia?∞< δi, then
τ∈[ia,min{(i+1)a,t1}]?¯ξj(τ) −¯ξ(τ)?∞+ ?¯ xj(τ) − ¯ x(τ)?∞< δi+1,
i = 0,1,...,l. Let¯j such that ?¯ξ
the following inequality holds
?????
?????
?????
????
Theorem 7. Let the system described by the Eq. (16) be ISS. Let
there exist a locally Lipschitz functional V
C([−∆,0];Rn) → R+, functions α1, α2of class K∞, and functions
α3, ρ of class K, such that, with a suitable Nafunctional, the following
hypotheses are satisfied:



u ∈ Rm: ψ(0) = g(φ,ψ),
(25)
¯j
0−¯ξ0?∞+ ?¯ x
¯j
0− ¯ x0?∞< δ0. Then,
?¯ξ¯j(t)
¯ x¯j(t)
From (21) and (26) and from (17) it follows that
?¯ξ(t1)
¯ x(t1)
+γ(?¯ u[0,t1)?∞) + ? ≤ β(?¯ξ0?∞+ ?¯ x0?∞+ ?,t1)
?¯ξ(t1)
¯ x(t1)
Therefore, if (20) were true, the contradiction would follow
?¯ξ(t1)
?
−
?¯ξ(t)
¯ x(t)
??????< ?,
?¯ξ¯j(t1)
t ∈ [−∆,t1].
(26)
??????≤
?????
¯ x¯j(t1)
??????+ ? ≤ β
?
?¯ξ
¯j
0?∞+ ?¯ x
¯j
0?∞,t1
?
+γ(?¯ u[0,t1)?∞) + ? <
??????.
(27)
¯ x(t1)
?????<
????
?¯ξ(t1)
¯ x(t1)
?????.
?
:
C([−∆,0];Rd) ×
(H1)

α1(|φ(0)|) ≤ V
∀
ψ
with ψ(0) = g(φ,ψ);
D+V
ψ
?φ
ψ
??φ
ψ
??
≤ α2
?
Na
??φ
ψ
???
,
?φ
?
∈ W1,∞([−∆,0];Rd) × W1,∞([−∆,0];Rn)
(H2)


??φ
?
?
,u
?
≤ −α3
?
Na
??φ
ψ
???
,
∀∈ W1,∞([−∆,0];Rd) × W1,∞([−∆,0];Rn),
Na
??φ
ψ
??
≥ ρ(|u|).
Then, system (2) is ISS.
Proof. Let us consider system (2). By Lemma 6, we sup-
pose that the initial conditions belong to W1,∞([−∆,0];Rd) ×
W1,∞([−∆,0];Rn).Let
Let w(t) = V
From Lemma 5 it follows that the function t → w(t) is locally ab-
solutely continuous. Therefore, we can adopt here the well-known
reasoning used in the main Theorem in Sontag (1989). Let the in-
put u(t) be such that esssupt≥0|u(t)| = v, for a suitable v ∈ R+.
?ξ(t)
x(t)
?
bethesolutionon[0,b),0 < b ≤ +∞.
. From Lemma 4, D+w(t) = D+V
??ξt
xt
????ξt
xt
?
,u(t)
?
,a.e.

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P. Pepe et al. / Automatica 44 (2008) 2266–2273
From H1, H2it follows that there exist a function β1of class KL and
a function γ1of class K such that the inequality
?
holds ∀t
Remark 1, it follows that the solution
following inequality holds
|ξ(t)| ≤ β1
Na
??ξ0
[0,b). From inequalities (1), taking into account
?ξ(t)
x0
??
,t
?
+ γ1(v)
(28)
∈
x(t)
?
exists ∀t ≥ 0 and the
|ξ(t)| ≤ β1(¯ γa(?ξ0?∞+ ?x0?∞),t) + γ1(v),
From the hypothesis that system (16) is ISS, it follows that there
exist a function β2of class KL and a function γ2of class K such that
the following inequality holds
t ≥ 0.
(29)
|x(t)| ≤ β2(?x0?∞,t) + γ2( sup
τ∈[−∆,t)|ξ(τ)|),
t ≥ 0.
(30)
From (29) and (30), it follows that the inequalities hold
?ξt?∞≤ ?ξ0?∞+ β1(¯ γa(?ξ0?∞+ ?x0?∞),0) + γ1(v),
?xt?∞ ≤ ?x0?∞+ β2(?x0?∞,0)
+γ2(2?ξ0?∞+ 2β1(¯ γa(?ξ0?∞+ ?x0?∞),0))
+γ2(2γ1(v)),
|x(t)| ≤ β2(?x¯t?∞,t −¯t) + γ2(β1(¯ γa(?ξ0?∞+ ?x0?∞),¯t − ∆)
+γ1(v)),
Setting in (32)¯t =
of the second inequality in (31), it follows that (29), (31), (32) yield
the ISS inequality (17).
∀t ≥ 0;
(31)
∀t,¯t : t ≥¯t ≥ ∆;
2t, t ≥ 2∆, and substituting?x¯t?∞with the r.h.s.
(32)
1
Remark 8. Whether the ISS of the system described by the Eq.
(16) is necessary for the ISS of the overall system (2) is to be
investigated. Here we point out that, for the following example
(case of the neutral system 1.6 in Hale and Lunel (2002), rewritten
in the coupled form)
˙ξ(t) = −ξ(t) + x(t − 1) + u(t),
x(t) = ξ(t) − x(t − 1),
it happens that the system described by the corresponding
equation (16) is not ISS (it is not asymptotically stable), but
the overall system is globally asymptotically stable and, for any
given constant input u(t) = v, v ∈ R, the following equalities
hold: limt→+∞x(t) =
the performed simulations, bounded inputs yield bounded state
variables.
(33)
v, limt→+∞ξ(t) = 2v. Moreover, in all
Remark 9. If in Theorem 7 the hypothesis that the system
described by the Eq. (16) is ISS is not introduced, and, instead of
the hypothesis H1, the following hypothesis
?
ψψ
?φ
ψ
is introduced, then, the following results hold: the solution of (2)
exists ∀t ≥ 0; the input-state inequality
??ξt
holds∀t ≥ 0,forsuitablefunctionsβofclass KLandγ ofclass K.For
instance, if the N2norm (by which the critical derivation of x(t) can
beavoided)isusedasNafunctional,thenonemayachieveaninput-
stateinequalitywhichinvolvesthe L2normof xt(seetheresultsfor
the L2-Stability given in Pepe (2005) and Pepe and Verriest (2003)).
α1
Na
??φ
?
???
≤ V
??φ
??
≤ α2
?
Na
??φ
ψ
???
,
∀
∈ C([−∆,0];Rd) × C([−∆,0];Rn)
Na
xt
??
≤ β
?
Na
??ξ0
x0
??
,t
?
+ γ(?u[0,t)?∞)
(34)
Remark 10. The overall system (2) cannot be regarded as a cas-
cade of two subsystems. Actually it consists of the interconnection
of a differential system and a difference one (the variable ξ of the
differential part forces the difference part and the variable x of the
difference part forces the differential part). If the hypotheses H1,
H2are satisfied, then the differential part of the system is ISS with
respect to the input u (when the initial condition x0 = 0, see the
inequality (29)), though forced by the variable x(t) ?= 0,t ≥ 0,
too (this is a key point, note that the hypotheses H1,H2involve the
overall system equations). Only when this result is achieved, then
the overall system could be regarded, at the aim of studying the
ISS, as a cascade of ISS systems, provided that the hypothesis that
the difference part of the system is ISS with respect to the input ξ
(first variable) is introduced.
Remark 11. The Nafunctional is introduced in order to yield as
much generality as possible for Theorem 7. For instance, the Na
functional may be a seminorm. This allows also a lot of freedom in
the choice of the Liapunov–Krasovskii functional. As an illustrating
example, let us consider the following system described by scalar
coupled delay differential and difference equations
˙ξ(t) = −ξ(t) + (1 + x2(t − ∆))(−ξ3(t) + u(t))
x(t) =1
2x(t − ∆) + ξ(t)ξ2(t − ∆).
(35)
InthiscasetheISScanbeprovedbythefunctionalV
??φ
ψ
??
= φ2(0).
The Nafunctional defined as Na
that the second variable is not involved at all. The function ρ of
class K defined as ρ(|u|) = |u|
??φ
ψ
??
= |φ(0)| can be used. Note
1
3 can be used.
5. The linear case
As far as the linear case is concerned, let us consider a system
described by the following linear equations
˙ξ(t) = A0ξ(t) +
p
?
p
?
i=1
Aiξ(t − ∆i) +
p
?
p
?
i=1
Bix(t − ∆i) + Gu(t),
x(t) = D0ξ(t) +
i=1
Diξ(t − ∆i) +
i=1
Cix(t − ∆i),
(36)
where A0, D0, Ai, Bi, Ci, Di, i = 1,2,...,p, G are real matrices
of suitable dimension, 0 < ∆1 < ∆2 < ··· < ∆p are the
(arbitrary, noncommensurate) delays. The methodology proposed
in this paper yields LMI conditions for the input-to-state stability
ofthegeneralmultiplenoncommensuratedelayscase,asshownin
the following:
Corollary 12. Let the LMIs (1) (2) in (Corollary 5, Pepe, 2005) be
feasible. Then, for any given delays 0 < ∆1 < ∆2 < ··· < ∆p,
the resulting system (36) is ISS.
Proof. The proof is achieved by applying Theorem 7, with the
functional (18) (19) in Pepe (2005) and using the N2norm as Na
functional.
?
In the paper (Pepe & Jiang, 2006) it is proved for systems
described by retarded functional differential equations that the
asymptotic stability of the trivial solution in the unforced case
is equivalent to the input-to-state stability. In the following an
analogous result for linear systems described by neutral functional
differential equations in Hale’s form (special case of system
(36)) is given. It is to be noted that, while for linear retarded
functional differential equations studied in Pepe and Jiang (2006)
the asymptotic stability of the trivial solution in the unforced case
implies the exponential stability, this is not true in general for