VECTOR DISSIPATIVITY THEORY FOR DISCRETE-TIME LARGE-SCALE NONLINEAR DYNAMICAL SYSTEMS WASSIM M. ppt

VECTOR DISSIPATIVITY THEORY FOR DISCRETE-TIME
LARGE-SCALE NONLINEAR DYNAMICAL SYSTEMS
WASSIM M. HADDAD, QING HUI, VIJAYSEKHAR CHELLABOINA,
AND SERGEY NERSESOV
Received 15 October 2003
In analyzing large-scale systems, it is often desirable to treat the overall system as a col-
lection of interconnected subsystems. Solution properties of the large-scale system are
then deduced from the solution properties of the individual subsystems and the na-
ture of the system interconnections. In this paper, we develop an analysis framework for
discrete-time large-scale dynamical systems based on vector dissipativity notions. Specif-
ically, using vector storage functions and vector supply rates, dissipativity properties of
the discrete-time composite large-scale system are shown to be determined from the dissi-
pativity properties of the subsystems and their interconnections. In particular, extended
Kalman-Yakubovich-Popov conditions, in terms of the subsystem dynamics and inter-
connection constraints, characterizing vector dissipativeness via vector system storage
functions are derived. Finally, these results are used to develop feedback interconnection
stability results for discrete-time large-scale nonlinear dynamical systems using vector
Lyapunov functions.
1. Introduction
Modern complex dynamical systems are highly interconnected and mutually interdepen-
dent, both physically and through a multitude of information and communication net-
work constraints. The sheer size (i.e., dimensionality) a nd complexity of these large-scale
dynamical systems often necessitate a hierarchical decentralized architecture for analyz-
ing and controlling these systems. Specifically, in the analysis and control-system design
of complex large-scale dynamical systems, it is often desir able to treat the overall system
as a collection of interconnected subsystems. The behavior of the aggregate or compos-
ite (i.e., large-scale) system can then be predicted from the behaviors of the individual
subsystems and their interconnections. T he need for decentralized analysis and control
design of large-scale systems is a direct consequence of the physical size and complexity
of the dynamical model. In particular, computational complexity may be too large for
model analysis while severe constraints on communication links between system sensors,

actuators, and processors may render centralized control architectures impractical.
Copyright © 2004 Hindawi Publishing Corporation
Advances in Difference Equations 2004:1 (2004) 37–66
2000 Mathematics Subject Classification: 93A15, 93D30, 93C10, 70K20, 93C55
URL: />38 Vector dissipativity and discrete-time large-scale systems
An approach to analyzing large-scale dynamical systems was introduced by the pio-
neering work of
ˇ
Siljak [19] and involves the notion of connective stability.Inparticular,
the large-scale dynamical system is decomposed into a collection of subsystems with local
dynamics and uncertain interactions. Then, each subsystem is considered independently
so that the stabilit y of each subsystem is combined with the interconnection constraints
to obtain a vector Lyapunov function for the composite large-scale dynamical system guar-
anteeing connective stability for the overall system. Vector Lyapunov functions were first
introduced by Bellman [2] and Matrosov [17] and further developed by Lakshmikan-
tham et al. [11], with [7, 14, 15, 16, 18, 19, 20] exploiting their utility for analyzing large-
scale systems. The use of vector Lyapunov functions in large-scale system analysis offers
a very flexible framework since each component of the vector Lyapunov function can
satisfy less-rigid requirements as compared to a single scalar Lyapunov function. More-
over, in large-scale systems, several Lyapunov functions arise naturally from the stability
properties of each subsystem. An alternative approach to vector Lyapunov functions for
analyzing large-scale dynamical systems is an input-output approach wherein stability
criteria are der ived by assuming that each subsystem is either finite gain, passive, or conic
[1, 12, 13, 21].
Since most physical processes evolve naturally in continuous time, it is not surprising
that the bulk of large-scale dynamical system theory has been developed for continuous-
time systems. Nevertheless, it is the overwhelming trend to implement controllers digi-
tally. Hence, in this paper we extend the notions of dissipativity theory [22, 23]tode-
velop vector dissipativity notions for large-scale nonlinear discrete-time dynamical sys-
tems; a notion not previously considered in the literature. In particular, we introduce

a generalized definition of dissipativity for large-scale nonlinear discrete-time dynami-
cal systems in terms of a vector inequality involving a vector supply rate,avector storage
function, and a nonnegative, semistable dissipation matrix. Generalized notions of vector
available storage and vector required supply are also defined and shown to be element-
by-element ordered, nonnegative, and finite. On the subsystem level, the proposed ap-
proach prov ides a discrete energy flow balance in terms of the stored subsystem energy,
the supplied subsystem energy, the subsystem energy gained from all other subsystems
independent of the subsystem coupling strengths, and the subsystem energy dissipated.
Furthermore, for large-scale discrete-time dynamical systems decomposed into intercon-
nected subsystems, dissipativity of the composite system is shown to be determined from
the dissipativity properties of the individual subsystems and the nature of the intercon-
nections. In particular, we develop extended Kalman-Yakubovich-Popov conditions, in
terms of the local subsystem dynamics and the interconnection constraints, for charac-
terizing vector dissipativeness via vector storage functions for large-scale discrete-time
dynamical systems. Finally, using the concepts of vector dissipativity and vector storage
functions as candidate vector Lyapunov functions, we develop feedback interconnection
stability results of large-scale discrete-time nonlinear dynamical systems. General stability
criteria are given for Lyapunov and asymptotic stability of feedback interconnections of
large-scale discrete-time dynamical systems. In the case of vector quadratic supply rates
involving net subsystem powers and input-output subsystem energies, these results pro-
vide a positivity and small gain theorem for large-scale discrete-time s ystems predicated
on vector Lyapunov functions.
Wassim M. Haddad et al. 39
2. Mathematical preliminaries
In this section, we introduce notation, several definitions, and some key results needed
for analyzing discrete-time large-scale nonlinear dynamical systems. Let R denote the set
of real numbers, let Z
+
denote the set of nonnegative integers, let R
n

denote the set of
n × 1columnvectors,letS
n
denote the set of n × n symmetric matrices, let N
n
(resp.,
P
n
) denote the set of n × n nonnegative (resp., positive) definite matrices, let (·)
T
denote
transpose, and let I
n
or I denote the n × n identity matrix. For v ∈ R
q
,wewritev≥≥0
(resp., v0) to indicate that ever y component of v is nonnegative (resp., positive). In
this case we say that v is nonnegative or positive, respectively. Let R
q
+
and R
q
+
denote the
nonnegative and positive orthants of R
q
; that is, if v ∈ R
q
,thenv ∈ R
q

+
and v ∈ R
q
+
are
equivalent, respectively, to v≥≥0andv0. Finally, we write ·for the Euclidean vector
norm, spec(M) for the spectrum of the square mat rix M, ρ(M) for the spectral radius of
the square matrix M, ∆V(x( k)) for V (x( k +1))− V(x(k)), Ꮾ
ε
(α), α ∈ R
n
, ε>0, for the
open ball centered at α with radius ε,andM ≥ 0(resp.,M>0) to denote the fact that the
Hermitian matrix M is nonnegative (resp., positive) definite. The following definition
introduces the notion of nonnegative matrices.
Definit ion 2.1 (see [3, 4, 9]). Let W
∈ R
q×q
. The matrix W is nonnegative (resp., positive)
if W
(i, j)
≥ 0(resp.,W
(i, j)
> 0), i, j = 1, , q. (In this paper it is important to distinguish
between a square nonnegative (resp., positive) matrix and a nonnegative-definite (resp.,
positive-definite) matrix.)
The following definition introduces the notion of class ᐃ functions involving nonde-
creasing functions.
Definit ion 2.2. A function w
= [w

1
, ,w
q
]
T
: R
q
→ R
q
is of class ᐃ if w
i
(r

) ≤ w
i
(r

),
i = 1, ,q,forallr

,r

∈ R
q
such that r

j
≤ r

j

, j = 1, , q,wherer
j
denotes the jth com-
ponent of r.
Note that if w(r) = Wr,whereW ∈ R
q×q
, then the function w(·)isofclassᐃ if and
only if W is nonnegative. The following definition introduces the notion of nonnegative
functions [9].
Definit ion 2.3. Let w
= [w
1
, ,w
q
]
T
: ᐂ → R
q
,whereᐂ is an open subset of R
q
that con-
tains R
q
+
.Thenw is nonnegative if w(r)≥≥0forallr ∈ R
q
+
.
Note that if w : R
q

→ R
q
is such that w(·) ∈ ᐃ and w(0)≥≥0, then w is nonnegative.
Note that, if w(r) = Wr,thenw(·) is nonnegative if and only if W ∈ R
q×q
is nonnegative.
Proposition 2.4 (see [9]). Suppose R
q
+
⊂ ᐂ. Then R
q
+
is an invariant set with respect to
r(k +1)= w

r(k)

, r(0) = r
0
, k ∈ Z
+
, (2.1)
if and only if w : ᐂ → R
q
is nonnegative.
The following lemma is needed for developing several of the results in later sections.
For the statement of this lemma, the following definition is required.
40 Vector dissipativity and discrete-time large-scale systems
Definit ion 2.5. The equilibrium solution r(k) ≡ r
e

of (2.1)isLyapunov stable if, for ev-
ery ε>0, there exists δ = δ(ε) > 0suchthatifr
0
∈ Ꮾ
δ
(r
e
) ∩ R
q
+
,thenr(k) ∈ Ꮾ
ε
(r
e
) ∩ R
q
+
,
k ∈ Z
+
. The equilibrium solution r(k) ≡ r
e
of (2.1)issemistable if it is Lyapunov stable
and there exists δ>0suchthatifr
0
∈ Ꮾ
δ
(r
e
) ∩ R

q
+
, then lim
k→∞
r(k) exists and con-
verges to a Lyapunov stable equilibrium point. The equilibrium solution r(k) ≡ r
e
of
(2.1)isasymptotically stable if it is Lyapunov stable and there exists δ>0suchthatif
r
0
∈ Ꮾ
δ
(r
e
) ∩ R
q
+
, then lim
k→∞
r(k) = r
e
. Finally, the equilibrium solution r(k) ≡ r
e
of
(2.1)isglobally asymptotically stable if the previous statement holds for all r
0
∈ R
q
+

.
Recall that a matrix W ∈ R
q×q
is semistable if and only if lim
k→∞
W
k
exists [9] while
W is asymptotically stable if and only if lim
k→∞
W
k
= 0.
Lemma 2.6. Suppose W ∈ R
q×q
is nonsingular and nonnegative. If W is semistable (resp.,
asymptotically stable), then there exist a scalar α ≥ 1 (resp., α>1) and a nonne gative vector
p ∈ R
q
+
, p = 0, (resp., positive vector p ∈ R
q
+
) such that
W
−T
p = αp. (2.2)
Proof. Since W is semistable, i t follows from [9, Theorem 3.3] that |λ| < 1orλ = 1and
λ = 1 is semisimple, where λ ∈ spec(W). Since W
T

≥≥0, it follows from the Perron-
Frobenius theorem that ρ(W) ∈ spec(W) and hence there exists p≥≥0, p = 0, such that
W
T
p = ρ(W)p. In addition, since W is nonsingular, ρ(W) > 0. Hence, W
T
p = α
−1
p,
where α  1/ρ(W), which proves that there exist p≥≥0, p = 0, and α ≥ 1suchthat(2.2)
holds. In the case where W is asymptotically stable, the result is a direct consequence of
the Perron-Frobenius theorem. 
Next, we present a stability result for discrete-time large-scale nonlinear dynamical
systems using vector Lyapunov functions. In particular, we consider discrete-time non-
linear dynamical systems of the form
x(k +1)= F

x(k)

, x

k
0

=
x
0
, k ≥ k
0
, (2.3)

where F : Ᏸ → R
n
is continuous on Ᏸ, Ᏸ ⊆ R
n
is an open set with 0 ∈ Ᏸ,andF(0) = 0.
Here, we assume that (2.3) characterizes a discrete-time large-scale nonlinear dynami-
cal system composed of q interconnected subsystems such that, for all i = 1, , q,each
element of F(x)isgivenbyF
i
(x) = f
i
(x
i
)+Ᏽ
i
(x), where f
i
: R
n
i
→ R
n
i
defines the vec-
tor field of each isolated subsystem of (2.3), Ᏽ
i
: Ᏸ → R
n
i
defines the structure of inter-

connection dynamics of the ith subsystem with all other subsystems, x
i
∈ R
n
i
, f
i
(0) = 0,
Ᏽ
i
(0) = 0, and

q
i=1
n
i
= n. For the discrete-time large-scale nonlinear dynamical system
(2.3), we note that the subsystem states x
i
(k), k ≥ k
0
,foralli = 1, ,q,belongtoR
n
i
as
long as x(k)  [x
T
1
(k), ,x
T

q
(k)]
T
∈ Ᏸ, k ≥ k
0
. The next theorem presents a stability result
for (2.3) via vector Lyapunov functions by relating the stabilit y properties of a compari-
son system to the stability properties of the discrete-time large-scale nonlinear dynamical
system.
Theorem 2.7 (see [11]). Consider the discrete-time large-scale nonlinear dynamical system
given by (2.3). Suppose there exist a continuous vector function V : Ᏸ
→ R
q
+
and a positive
Wassim M. Haddad et al. 41
vector p ∈ R
q
+
such that V(0) = 0, the scalar function v : Ᏸ → R
+
defined by v(x) = p
T
V(x),
x ∈ Ᏸ, is such that v(0) = 0, v(x) > 0, x = 0,and
V

F(x)

≤≤w


V(x)

, x ∈ Ᏸ, (2.4)
where w : R
q
+
→ R
q
is a class ᐃ function such that w(0) = 0. Then the stability properties of
the zero solution r(k) ≡ 0 to
r(k +1)= w

r(k)

, r

k
0

=
r
0
, k ≥ k
0
, (2.5)
imply the corresponding stability properties of the zero solution x(k) ≡ 0 to (2.3). That is, if
the zero solution r(k) ≡ 0 to (2.5) is Lyapunov (resp., asymptotically) stable, then the zero
solution x(k) ≡ 0 to (2.3) is Lyapunov (resp., asymptotically) stable. If, in addition, Ᏸ = R
n

and V(x) →∞as x→∞, then global asymptotic stabilit y of the zero solution r(k) ≡ 0 to
(2.5) implies global asymptotic stability of the zero solution x(k) ≡ 0 to (2.3).
If V : Ᏸ → R
q
+
satisfies the conditions of Theorem 2.7,wesaythatV(x), x ∈ Ᏸ,isavec-
tor Lyapunov function for the discrete-time large-scale nonlinear dynamical system (2.3).
Finally, we recall the notions of dissipativity [6] and geometric dissipativity [8, 9]for
discrete-time nonlinear dynamical systems Ᏻ of the form
x(k +1)= f

x(k)

+ G

x(k)

u(k), x

k
0

= x
0
, k ≥ k
0
, (2.6)
y(k) = h

x(k)


+ J

x(k)

u(k), (2.7)
where x ∈ Ᏸ ⊆ R
n
, u ∈ ᐁ ⊆ R
m
, y ∈ ᐅ ⊆ R
l
, f : Ᏸ → R
n
satisfies f (0) = 0, G : Ᏸ →
R
n×m
, h : Ᏸ → R
l
satisfies h(0) = 0, and J : Ᏸ → R
l×m
. For the discrete-time nonlinear dy-
namical system Ᏻ, we assume that the required properties for the existence and unique-
ness of solutions are satisfied; that is, u(·) satisfies sufficient regularity conditions such
that (2.6) has a unique solution forward in time. Note that since all input-output pairs
u ∈ ᐁ, y ∈ ᐅ of the discrete-time nonlinear dynamical system Ᏻ are defined on Z
+
,the
supply rate [22] satisfy ing s(0,0) = 0 is local ly summable for all input-output pairs satis-
fying (2.6), (2.7); that is, for all input-output pairs u ∈ ᐁ, y ∈ ᐅ satisfying (2.6), (2.7),

s(·,·) satisfies

k
2
k=k
1
|s(u(k), y(k))| < ∞, k
1
,k
2
∈ Z
+
.
Definit ion 2.8 (see [6, 8]). The discrete-time nonlinear dynamical system Ᏻ given by (2.6),
(2.7)isgeometrically dissipative (resp., dissipative) with respect to the supply rate s(u, y)
if there exist a continuous nonnegative-definite function v
s
: R
n
→ R
+
,calledastorage
function,andascalarρ>1(resp.,ρ = 1) such that v
s
(0) = 0 and the dissipation inequality
ρ
k
2
v
s


x

k
2

≤ ρ
k
1
v
s

x

k
1

+
k
2
−1

i=k
1
ρ
i+1
s

u(i), y(i)


, k
2
≥ k
1
, (2.8)
is satisfied for all k
2
≥ k
1
≥ k
0
,wherex(k), k ≥ k
0
, is the solution to (2.6)withu ∈ ᐁ.The
discrete-time nonlinear dynamical system Ᏻ given by (2.6), (2.7)islossless with respect to
the supply rate s(u, y) if the dissipation inequality is satisfied as an equality with ρ = 1for
all k
2
≥ k
1
≥ k
0
.
42 Vector dissipativity and discrete-time large-scale systems
An equivalent statement for dissipativity of the dynamical system (2.6), (2.7)is
∆v
s

x(k)


≤ s

u(k), y(k)

, k ≥ k
0
, u ∈ ᐁ, y ∈ ᐅ. (2.9)
Alternatively, an equivalent statement for geometric dissipativity of the dynamical system
(2.6), (2.7)is
ρv
s

x(k +1)

− v
s

x(k)

≤ ρs

u(k), y(k)

, k ≥ k
0
, u ∈ ᐁ, y ∈ ᐅ. (2.10)
3. Vector dissipativity theory for discrete-time large-scale nonlinear dynamical
systems
In this section, we extend the notion of dissipative dynamical systems to develop the gen-
eralized notion of vector dissipativity for discrete-time large-scale nonlinear dynamical

systems. We begin by considering discrete-time nonlinear dynamical systems Ᏻ of the
form
x(k +1)= F

x(k),u(k)

, x

k
0

= x
0
, k ≥ k
0
, (3.1)
y(k)
= H

x(k),u(k)

, (3.2)
where x ∈ Ᏸ ⊆ R
n
, u ∈ ᐁ ⊆ R
m
, y ∈ ᐅ ⊆ R
l
, F : Ᏸ × ᐁ → R
n

, H : Ᏸ × ᐁ → ᐅ, Ᏸ is
an open set with 0 ∈ Ᏸ,andF(0,0) = 0. Here, we assume that Ᏻ represents a discrete-
time large-scale dynamical system composed of q interconnected controlled subsystems
Ᏻ
i
such that, for all i = 1, ,q,
F
i

x, u
i

=
f
i

x
i

+ Ᏽ
i
(x)+G
i

x
i

u
i
,

H
i

x
i
,u
i

=
h
i

x
i

+ J
i

x
i

u
i
,
(3.3)
where x
i
∈ R
n
i

, u
i
∈ ᐁ
i
⊆ R
m
i
, y
i
 H
i
(x
i
,u
i
) ∈ ᐅ
i
⊆ R
l
i
,(u
i
, y
i
) is the input-output pair
for the ith subsystem, f
i
: R
n
i

→ R
n
i
and Ᏽ
i
: Ᏸ → R
n
i
are continuous and satisfy f
i
(0) = 0
and Ᏽ
i
(0) = 0, G
i
: R
n
i
→ R
n
i
×m
i
is continuous, h
i
: R
n
i
→ R
l

i
satisfies h
i
(0) = 0, J
i
: R
n
i
→
R
l
i
×m
i
,

q
i=1
n
i
= n,

q
i=1
m
i
= m,and

q
i=1

l
i
= l. Furthermore, for the system Ᏻ we as-
sume that the required properties for the existence and uniqueness of solutions are sat-
isfied. We define the composite input and composite output for the discrete-time large-
scale system Ᏻ as u
 [u
T
1
, ,u
T
q
]
T
and y  [y
T
1
, , y
T
q
]
T
, respectively. Note that, in this
case, the set ᐁ = ᐁ
1
×···×ᐁ
q
contains the set of input values and ᐅ = ᐅ
1
×···×ᐅ

q
contains the set of output values.
Definit ion 3.1. For the discrete-time large-scale nonlinear dynamical system Ᏻ given by
(3.1), (3.2), a vector function S
= [s
1
, ,s
q
]
T
: ᐁ × ᐅ → R
q
such that S(u, y)  [s
1
(u
1
, y
1
),
,s
q
(u
q
, y
q
)]
T
and S(0,0) = 0iscalledavector supply rate.
Note that, since all input-output pairs (u
i

, y
i
) ∈ ᐁ
i
× ᐅ
i
, i = 1, ,q, satisfying (3.1),
(3.2)aredefinedonZ
+
, s
i
(·,·) satisfies

k
2
k=k
1
|s
i
(u
i
(k), y
i
(k))| < ∞, k
1
,k
2
∈ Z
+
.

Definit ion 3.2. The discrete-time large-scale nonlinear dynamical system Ᏻ given by (3.1),
(3.2)isvector dissipative (resp., geometrically vector dissipative) with respect to the vector
Wassim M. Haddad et al. 43
supply rate S(u, y) if there exist a continuous, nonnegative definite vector function V
s
=
[v
s1
, ,v
sq
]
T
: Ᏸ → R
q
+
,calledavector storage function, and a nonsingular nonnegative
dissipation matrix W ∈ R
q×q
such that V
s
(0) = 0, W is semistable (resp., asymptotically
stable), and the vector dissipation inequality
V
s

x(k)

≤≤W
k−k
0

V
s

x

k
0

+
k−1

i=k
0
W
k−1−i
S

u(i), y(i)

, k ≥ k
0
, (3.4)
is satisfied, where x(k), k ≥ k
0
, is the solution to (3.1)withu ∈ ᐁ. The discrete-time large-
scale n onlinear dynamical system Ᏻ given by (3.1), (3.2)isvector lossless with respect to
the vector supply rate S(u, y) if the vector dissipation inequality is satisfied as an equality
with W semistable.
Note that if the subsystems Ᏻ
i

of Ᏻ are disconnected, that is, Ᏽ
i
(x) ≡ 0foralli =
1, ,q,andW ∈ R
q×q
is diagonal, positive definite, and semistable, then it follows from
Definition 3.2 that each of the isolated subsystems Ᏻ
i
is dissipative or geometrically dis-
sipative in the sense of Definition 2.8. A similar remark holds in the case where q = 1.
Next, define the vector available storage of the discrete-time large-scale nonlinear dynam-
ical system Ᏻ by
V
a

x
0

 sup
K≥k
0
, u(·)

−
K−1

k=k
0
W
−(k+1−k

0
)
S

u(k), y(k)


, (3.5)
where x(k), k ≥ k
0
, is the solution to (3.1)withx(k
0
) = x
0
and admissible inputs u ∈
ᐁ. The supremum in (3.5) is taken componentwise, which implies that, for different
elements of V
a
(·), the supremum is calculated separately. Note that V
a
(x
0
)≥≥0, x
0
∈ Ᏸ,
since V
a
(x
0
) is the supremum over a set of vectors containing the zero vector (K = k

0
). To
state the main results of this section, the following definition is required.
Definit ion 3.3 (see [9]). The discrete-time large-scale nonlinear dynamical system Ᏻ given
by (3.1), (3.2)iscompletely reachable if, for all x
0
∈ Ᏸ ⊆ R
n
, there exist a k
i
<k
0
and
a square summable input u(·)definedon[k
i
,k
0
] such that the state x(k), k ≥ k
i
,can
be driven from x(k
i
) = 0tox(k
0
) = x
0
. A discrete-time large-scale nonlinear dynamical
system Ᏻ is zero-state observable if u(k) ≡ 0andy(k) ≡ 0implyx(k) ≡ 0.
Theorem 3.4. Consider the discrete-time large-scale nonlinear dy namical system Ᏻ given
by (3.1), (3.2) and assume that Ᏻ is completely reachable. Let W ∈ R

q×q
be nonsingular,
nonnegative, and semistable (resp., asymptotically stable). Then
K−1

k=k
0
W
−(k+1−k
0
)
S

u(k), y(k)

≥≥0, K ≥ k
0
, u ∈ ᐁ, (3.6)
for x(k
0
) = 0 if and only if V
a
(0) = 0 and V
a
(x) is finite for all x ∈ Ᏸ.Moreover,if(3.6)
holds, then V
a
(x) , x ∈ Ᏸ, is a vector storage function for Ᏻ and hence Ᏻ is vector dissipative
(resp., geometrically vector dissipative) w ith respect to the vector supply rate S(u, y).
44 Vector dissipativity and discrete-time large-scale systems

Proof. Su ppose V
a
(0) = 0andV
a
(x), x ∈ Ᏸ,isfinite.Then
0 = V
a
(0) = sup
K≥k
0
, u(·)

−
K−1

k=k
0
W
−(k+1−k
0
)
S

u(k), y(k)


, (3.7)
which implies (3.6).
Next, suppose (3.6) holds. Then, for x(k
0

) = 0,
sup
K≥k
0
, u(·)

−
K−1

k=k
0
W
−(k+1−k
0
)
S

u(k), y(k)


≤≤0, (3.8)
which implies that V
a
(0)≤≤0. However, since V
a
(x
0
)≥≥0, x
0
∈ Ᏸ, it follows that V

a
(0) =
0. Moreover, since Ᏻ is completely reachable, it follows that, for every x
0
∈ Ᏸ, there exists
ˆ
k>k
0
and an admissible input u(·)definedon[k
0
,
ˆ
k]suchthatx(
ˆ
k) = x
0
. Now, since
(3.6)holdsforx(k
0
) = 0, it follows that, for all admissible u(·) ∈ ᐁ,
K−1

k=k
0
W
−(k+1−k
0
)
S


u(k), y(k)

≥≥
0, K ≥
ˆ
k, (3.9)
or, equivalently, multiplying (3.9) by the nonnegative matrix W
ˆ
k−k
0
,
ˆ
k>k
0
,yields
−
K−1

k=
ˆ
k
W
−(k+1−
ˆ
k)
S

u(k), y(k)

≤≤

ˆ
k−1

k=k
0
W
−(k+1−
ˆ
k)
S

u(k), y(k)

≤≤Q

x
0

∞, K ≥
ˆ
k, u ∈ ᐁ,
(3.10)
where Q : Ᏸ → R
q
.Hence,
V
a

x
0


=
sup
K≥
ˆ
k, u(·)

−
K−1

k=
ˆ
k
W
−(k+1−
ˆ
k)
S

u(k), y(k)


≤≤Q

x
0

∞, x
0
∈ Ᏸ, (3.11)

which implies that V
a
(x
0
), x
0
∈ Ᏸ,isfinite.
Finally, since (3.6) implies that V
a
(0) = 0andV
a
(x), x ∈ Ᏸ, is finite, it follows from
the definition of the vector available storage that
−V
a

x
0

≤≤
K−1

k=k
0
W
−(k+1−k
0
)
S


u(k), y(k)

=
k
f
−1

k=k
0
W
−(k+1−k
0
)
S

u(k), y(k)

+
K−1

k=k
f
W
−(k+1−k
0
)
S

u(k), y(k)


, K ≥ k
0
.
(3.12)
Wassim M. Haddad et al. 45
Now, multiplying (3.12) by the nonnegative matrix W
k
f
−k
0
, k
f
>k
0
, it follows that
W
k
f
−k
0
V
a

x
0

+
k
f
−1


k=k
0
W
−(k+1−k
f
)
S

u(k), y(k)

≥≥ sup
K≥k
f
, u(·)

−
K−1

k=k
f
W
−(k+1−k
f
)
S

u(k), y(k)



= V
a

x

k
f

,
(3.13)
which implies that V
a
(x), x ∈ Ᏸ, is a vector storage function and hence Ᏻ is vector dis-
sipative (resp., geometrically vector dissipative) with respect to the vector supply rate
S(u, y). 
It follows from Lemma 2.6 that if W ∈ R
q×q
is nonsingular, nonnegative, and semi-
stable (resp., asymptotically stable), then there exist a scalar α ≥ 1(resp.,α>1) and a
nonnegative vector p ∈ R
q
+
, p = 0, (resp., p ∈ R
q
+
)suchthat(2.2) holds. In this case,
p
T
W
−k

= αp
T
W
−(k−1)
=··· = α
k
p
T
, k ∈ Z
+
. (3.14)
Using (3.14), we define the (scalar) available storage for the discrete-time large-scale non-
linear dynamical system Ᏻ by
v
a

x
0

 sup
K≥k
0
, u(·)

−
K−1

k=k
0
p

T
W
−(k+1−k
0
)
S

u(k), y(k)


=
sup
K≥k
0
, u(·)

−
K−1

k=k
0
α
k+1−k
0
s

u(k), y(k)


,

(3.15)
where s : ᐁ × ᐅ → R defined as s(u, y)  p
T
S(u, y) is the (scalar) supply rate for the
discrete-time large-scale nonlinear dynamical system Ᏻ.Clearly,v
a
(x) ≥ 0forallx ∈ Ᏸ.
As in standard dissipativity theory, the available storage v
a
(x), x ∈ Ᏸ, denotes the maxi-
mum amount of (scaled) energy that can be extracted from the discrete-time large-scale
nonlinear dynamical system Ᏻ at any instant K.
The following theorem relates vector storage functions and vector supply rates to scalar
storage functions and scalar supply rates of discrete-time large-scale dynamical systems.
Theorem 3.5. Consider the discrete-time large-scale nonlinear dynamical system Ᏻ given by
(3.1), (3.2). Suppose Ᏻ is vector dissipative (resp., geometrically vector dissipative) with re-
spect to the vector supply rate S : ᐁ
× ᐅ → R
q
and with vector storage funct ion V
s
: Ᏸ → R
q
+
.
Then there exists p ∈ R
q
+
, p = 0,(resp.,p ∈ R
q

+
) such that Ᏻ is dissipative (resp., geometri-
cally dissipative) with respect to the scalar supply rate s(u, y) = p
T
S(u, y) and with storage
function v
s
(x)  p
T
V
s
(x) , x ∈ Ᏸ. Moreover, in this case, v
a
(x) , x ∈ Ᏸ, is a storage function
for Ᏻ and
0 ≤ v
a
(x) ≤ v
s
(x), x ∈ Ᏸ. (3.16)
46 Vector dissipativity and discrete-time large-scale systems
Proof. Su ppose Ᏻ is vector dissipative (resp., geometrically vector dissipative) with re-
specttothevectorsupplyrateS(u, y). Then there exist a nonsingular, nonnegative, and
semistable (resp., asymptotically stable) dissipation matrix W and a vector storage func-
tion V
s
: Ᏸ → R
q
+
such that the dissipation inequality (3.4) holds. Furthermore, it follows

from Lemma 2.6 that there exist α ≥ 1(resp.,α>1) and a nonzero vector p ∈ R
q
+
(resp.,
p ∈ R
q
+
) satisfying (2.2). Hence, premultiplying (3.4)byp
T
and using (3.14), it follows
that
v
s

x(k)

≤ α
−(k−k
0
)
v
s

x

k
0

+
k−1


i=k
0
α
−(k−1−i)
s

u(i), y(i)

, k ≥ k
0
, u ∈ ᐁ, (3.17)
where v
s
(x) = p
T
V
s
(x), x ∈ Ᏸ, which implies dissipativity (resp., geometric dissipativ-
ity) of Ᏻ with respect to the supply rate s(u, y) and with storage function v
s
(x), x ∈ Ᏸ.
Moreover, since v
s
(0) = 0, it follows from (3.17)thatforx(k
0
) = 0,
k−1

i=k

0
α
i+1−k
0
s

u(i), y(i)

≥ 0, k ≥ k
0
, u ∈ ᐁ, (3.18)
which, using (3.15), implies that v
a
(0) = 0. Now, it can easily be shown that v
a
(x), x ∈ Ᏸ,
satisfies (3.17), and hence the available storage defined by (3.15) is a storage function for
Ᏻ. Finally, it follows from (3.17)that
v
s

x

k
0

≥ α
k−k
0
v

s

x(k)

−
k−1

i=k
0
α
i+1−k
0
s

u(i), y(i)

≥−
k−1

i=k
0
α
i+1−k
0
s

u(i), y(i)

, k ≥ k
0

, u ∈ ᐁ,
(3.19)
which implies that
v
s

x

k
0

≥ sup
k≥k
0
, u(·)

−
k−1

i=k
0
α
i+1−k
0
s

u(i), y(i)


=

v
a

x

k
0

, (3.20)
and hence (3.16)holds. 
Remark 3.6. It follows from Theorem 3.4 that if (3.6)holdsforx(k
0
) = 0, then the vector
available storage V
a
(x), x ∈ Ᏸ, is a vector storage function for Ᏻ. In this case, it follows
from Theorem 3.5 that there exists p ∈ R
q
+
, p = 0, such that v
s
(x)  p
T
V
a
(x)isastorage
function for Ᏻ that satisfies (3.17), and hence, by (3.16), v
a
(x) ≤ p
T

V
a
(x), x ∈ Ᏸ.
Remark 3.7. It is impor tant to note that it follows from Theorem 3.5 that if Ᏻ is vector
dissipative, then Ᏻ can either be (scalar) dissipative or (scalar) geometrically dissipative.
The follow ing theorem provides sufficient conditions guaranteeing that all scalar stor-
age functions defined in terms of vector storage functions, that is, v
s
(x) = p
T
V
s
(x), of a
given vector dissipative discrete-time large-scale nonlinear dynamical system are positive
definite.
Wassim M. Haddad et al. 47
Theorem 3.8. Consider the discrete-time large-scale nonlinear dy namical system Ᏻ given
by (3.1), (3.2) and assume that Ᏻ is zero-state observable. Furthermore, assume that Ᏻ is
vector dissipative (resp., geometrically vector dissipative) with respect to the vector supply
rate S(u, y) and there exist α ≥ 1 and p ∈ R
q
+
such that (2.2) holds. In addition, assume that
there exist functions κ
i
: ᐅ
i
→ ᐁ
i
such that κ

i
(0) = 0 and s
i
(κ
i
(y
i
), y
i
) < 0, y
i
= 0,forall
i = 1, ,q. Then, for all vector storage functions V
s
: Ᏸ → R
q
+
, the storage function v
s
(x) 
p
T
V
s
(x) , x ∈ Ᏸ, is positive definite; that is, v
s
(0) = 0 and v
s
(x) > 0, x ∈ Ᏸ, x = 0.
Proof. It follows from Theorem 3.5 that v

a
(x), x ∈ Ᏸ, is a storage function for Ᏻ that
satisfies (3.17). Next, suppose, ad absurdum, there exists x
∈ Ᏸ such that v
a
(x) = 0, x = 0.
Then it follows from the definition of v
a
(x), x ∈ Ᏸ,thatforx(k
0
) = x,
K−1

k=k
0
α
k+1−k
0
s

u(k), y(k)

≥ 0, K ≥ k
0
, u ∈ ᐁ. (3.21)
However, for u
i
= k
i
(y

i
), we have s
i
(κ
i
(y
i
), y
i
) < 0, y
i
= 0, for all i = 1, , q, and since
p0, it follows that y
i
(k) = 0, k ≥ k
0
, i = 1, ,q, which further implies that u
i
(k) = 0,
k ≥ k
0
, i = 1, , q.SinceᏳ is zero-state observable, it follows that x = 0 and hence v
a
(x) =
0ifandonlyifx = 0. The result now follows from (3.16). Finally, for the geometrically
vector dissipative case, it fol lows from Lemma 2.6 that p0 with the rest of the proof
being identical as above. 
Next, we introduce the concept of vector required supply of a discrete-time large-scale
nonlinear dynamical system. Specifically, define the vector required supply of the discrete-
time large-scale dynamical system Ᏻ by

V
r

x
0

 inf
K≥−k
0
+1, u(·)
k
0
−1

k=−K
W
−(k+1−k
0
)
S

u(k), y(k)

, (3.22)
where x(k), k ≥−K, is the solution to (3.1)withx(−K) = 0andx(k
0
) = x
0
.Notethat
since, with x(k

0
) = 0, the infimum in (3.22)isthezerovector,itfollowsthatV
r
(0) = 0.
Moreover, since Ᏻ is completely reachable, it follows that V
r
(x)∞, x ∈ Ᏸ. Using the
notion of the vector required supply, we present necessary and sufficient conditions for
dissipativity of a large-scale dynamical system with respect to a vector supply rate.
Theorem 3.9. Consider the discrete-time large-scale nonlinear dy namical system Ᏻ given
by (3.1), (3.2) and assume that Ᏻ is completely reachable. Then Ᏻ is vector dissipative (resp.,
geometrically vector dissipative) with respect to the vector supply rate S(u, y) if and only if
0
≤≤V
r
(x)∞, x ∈ Ᏸ. (3.23)
Moreover, if (3.23) holds, then V
r
(x) , x ∈ Ᏸ, is a vector storage funct ion for Ᏻ.Finally,ifthe
vector available storage V
a
(x) , x ∈ Ᏸ, is a vector storage function for Ᏻ, then
0≤≤V
a
(x)≤≤V
r
(x)∞, x ∈ Ᏸ. (3.24)
48 Vector dissipativity and discrete-time large-scale systems
Proof. Su ppose (3.23)holdsandletx(k), k ∈ Z
+

, satisfy (3.1) with admissible inputs
u(k) ∈ ᐁ, k ∈ Z
+
,andx(k
0
) = x
0
. Then it follows from the definition of V
r
(·)thatfor
−K ≤ k
f
≤ k
0
− 1andu(·) ∈ ᐁ,
V
r

x
0

≤≤
k
0
−1

k=−K
W
−(k+1−k
0

)
S

u(k), y(k)

=
k
f
−1

k=−K
W
−(k+1−k
0
)
S

u(k), y(k)

+
k
0
−1

k=k
f
W
−(k+1−k
0
)

S

u(k), y(k)

,
(3.25)
and hence,
V
r

x
0

≤≤W
k
0
−k
f
inf
K≥−k
f
+1, u(·)

k
f
−1

k=−K
W
−(k+1−k

f
)
S

u(k), y(k)


+
k
0
−1

k=k
f
W
−(k+1−k
0
)
S

u(k), y(k)

= W
k
0
−k
f
V
r


x

k
f

+
k
0
−1

k=k
f
W
k
0
−1−k
S

u(k), y(k)

,
(3.26)
which shows that V
r
(x), x ∈ Ᏸ, is a vector storage function for Ᏻ and hence Ᏻ is vector
dissipative with respect to the vector supply rate S(u, y).
Conversely, suppose that Ᏻ is vector dissipative with respect to the vector supply rate
S(u, y). Then there exists a nonnegative vector storage function V
s
(x), x ∈ Ᏸ,suchthat

V
s
(0) = 0. Since Ᏻ is completely reachable, it follows that for x(k
0
) = x
0
, there exist
K>−k
0
and u(k), k ∈ [−K, k
0
], such that x(−K) = 0. Hence, it follows from the vec-
tor dissipation inequality (3.4)that
0≤≤V
s

x

k
0

≤≤W
k
0
+K
V
s

x(−K)


+
k
0
−1

k=−K
W
k
0
−1−k
S

u(k), y(k)

, (3.27)
which implies that for all K ≥−k
0
+1andu ∈ ᐁ,
0≤≤
k
0
−1

k=−K
W
−(k+1−k
0
)
S


u(k), y(k)

(3.28)
or, equivalently,
0≤≤ inf
K≥−k
0
+1, u(·)
k
0
−1

k=−K
W
−(k+1−k
0
)
S

u(k), y(k)

=
V
r

x
0

. (3.29)
Since, by complete reachability, V

r
(x)∞, x ∈ Ᏸ, it follows that (3.23)holds.
Wassim M. Haddad et al. 49
Finally, suppose that V
a
(x), x ∈ Ᏸ, is a vector storage function. Then, for x(−K) = 0,
x(k
0
) = x
0
,andu ∈ ᐁ, it follows that
V
a

x

k
0

≤≤W
k
0
+K
V
a

x(−K)

+
k

0
−1

k=−K
W
k
0
−1−k
S

u(k), y(k)

, (3.30)
which implies that
0≤≤V
a

x

k
0

≤≤ inf
K≥−k
0
+1, u(·)
k
0
−1


k=−K
W
−(k+1−k
0
)
S

u(k), y(k)

=
V
r

x

k
0

, x ∈ Ᏸ.
(3.31)
Since x(k
0
) = x
0
∈ Ᏸ is arbitr ary and, by complete reachability, V
r
(x)∞, x ∈ Ᏸ,(3.31)
implies (3.24). 
The next result is a direct consequence of Theorems 3.4 and 3.9.
Proposition 3.10. Consider the discrete-time large-scale nonlinear dynamical system Ᏻ

given by (3.1), (3.2). Let M = diag[µ
1
, ,µ
q
] be such that 0 ≤ µ
i
≤ 1, i = 1, , q.IfV
a
(x),
x ∈ Ᏸ,andV
r
(x) , x ∈ Ᏸ, are vector storage functions for Ᏻ, then
V
s
(x) = MV
a
(x)+

I
q
− M

V
r
(x), x ∈ Ᏸ, (3.32)
is a vector storage function for Ᏻ.
Proof. First note that M≥≥0andI
q
− M≥≥0ifandonlyifM = diag[µ
1

, ,µ
q
]and
µ
i
∈ [0, 1], i = 1, ,q. Now, the result is a direct consequence of the vector dissipation in-
equality (3.4) by noting that if V
a
(x)andV
r
(x) satisfy (3.4), then V
s
(x) satisfies (3.4). 
Next, recall that if Ᏻ is vector dissipative (resp., geometrically vector dissipative), then
there exist p ∈ R
q
+
, p = 0, and α ≥ 1(resp.,p ∈ R
q
+
and α>1) such that (2.2)and(3.14)
hold. Now, define the (scalar) required supply for the large-scale nonlinear dynamical
system Ᏻ by
v
r

x
0

 inf

K≥−k
0
+1, u(·)
k
0
−1

k=−K
p
T
W
−(k+1−k
0
)
S

u(k), y(k)

=
inf
K≥−k
0
+1, u(·)
k
0
−1

k=−K
α
k+1−k

0
s

u(k), y(k)

, x
0
∈ Ᏸ,
(3.33)
where s(u, y) = p
T
S(u, y)andx(k), k ≥−K, is the solution to (3.1)withx(−K) = 0and
x(k
0
) = x
0
.Itfollowsfrom(3.33) that the required supply of a discrete-time large-scale
nonlinear dynamical system is the minimum amount of generalized energy which can be
delivered to the discrete-time large-scale system in order to transfer it from an initial state
x(
−K) = 0toagivenstatex(k
0
) = x
0
. Using the same arguments as in case of the vector
required supply, it follows that v
r
(0) = 0andv
r
(x) < ∞, x ∈ Ᏸ.

50 Vector dissipativity and discrete-time large-scale systems
Next, u sing the notion of required supply, we show that all storage functions of the
form v
s
(x) = p
T
V
s
(x), where p ∈ R
q
+
, p = 0, are bounded from above by the required sup-
ply and bounded from below by the available storage. Hence, a dissipative discrete-time
large-scale nonlinear dynamical system can only deliver to its surroundings a fraction of
all of its stored subsystem energies and can only store a fraction of the work done to all of
its subsystems.
Corollary 3.11. Consider the discrete-time large-scale nonlinear dynamical system Ᏻ given
by (3.1), (3.2). Assume that Ᏻ is vector dissipative with respect to a vector supply rate S(u, y)
and with vector storage function V
s
: Ᏸ → R
q
+
. Then v
r
(x) , x ∈ Ᏸ, is a storage funct ion for Ᏻ.
Moreover, if v
s
(x)  p
T

V
s
(x) , x ∈ Ᏸ,wherep ∈ R
q
+
, p = 0, then
0 ≤ v
a
(x) ≤ v
s
(x) ≤ v
r
(x) < ∞, x ∈ Ᏸ. (3.34)
Proof. It follows from Theorem 3.5 that if Ᏻ is vector dissipative with respect to the vector
supply rate S(u, y) and with a vector storage function V
s
: Ᏸ → R
q
+
, then there exists p ∈
R
q
+
, p = 0, such that Ᏻ is dissipative with respect to the supply rate s(u, y) = p
T
S(u, y)
and with storage function v
s
(x) = p
T

V
s
(x), x ∈ Ᏸ.Hence,itfollowsfrom(3.17), with
x(−K) = 0andx(k
0
) = x
0
,that
k
0
−1

k=−K
α
k+1−k
0
s

u(k), y(k)

≥ 0, K ≥−k
0
, u ∈ ᐁ, (3.35)
which implies that v
r
(x
0
) ≥ 0, x
0
∈ Ᏸ. Furthermore, it is easy to see from the definition

of a required supply that v
r
(x), x ∈ Ᏸ, satisfies the dissipation inequality (3.17). Hence,
v
r
(x), x ∈ Ᏸ, is a storage function for Ᏻ. Moreover, it follows from the dissipation in-
equality (3.17), w ith x(−K) = 0, x(k
0
) = x
0
,andu ∈ ᐁ,that
α
k
0
v
s

x

k
0

≤ α
−K
v
s

x(−K)

+

k
0
−1

k=−K
α
k+1
s

u(k), y(k)

=
k
0
−1

k=−K
α
k+1
s

u(k), y(k)

,
(3.36)
which implies that
v
s

x


k
0

≤ inf
K≥−k
0
+1, u(·)
k
0
−1

k=−K
α
k+1−k
0
s

u(k), y(k)

=
v
r

x

k
0

. (3.37)

Finally, it follows from Theorem 3.5 that v
a
(x), x ∈ Ᏸ, is a storage function for Ᏻ,and
hence, using (3.16)and(3.37), (3.34)holds. 
Remark 3.12. It follows from Theorem 3.9 that if Ᏻ is vector dissipative with respect to
the vector supply rate S(u, y), then V
r
(x), x ∈ Ᏸ, is a vector storage function for Ᏻ and,
Wassim M. Haddad et al. 51
by Theorem 3.5, there exists p ∈ R
q
+
, p = 0, such that v
s
(x)  p
T
V
r
(x), x ∈ Ᏸ,isastorage
function for Ᏻ satisfying (3.17). Hence, it follows from Corollary 3.11 that p
T
V
r
(x) ≤
v
r
(x), x ∈ Ᏸ.
The next result relates vector (resp., scalar) available storage and vector (resp., scalar)
required supply for vector lossless discrete-time large-scale dynamical systems.
Theorem 3.13. Consider the discrete-time large-scale nonlinear dynamical system Ᏻ given

by (3.1), (3.2). Assume that Ᏻ is completely reachable to and from the origin. If Ᏻ is vector
lossless with respect to the vector supply rate S(u, y) and V
a
(x) , x ∈ Ᏸ, is a vector storage
function, then V
a
(x) = V
r
(x) , x ∈ Ᏸ.Moreover,ifV
s
(x) , x ∈ Ᏸ, is a vector storage function,
then all (scalar) storage functions of the form v
s
(x) = p
T
V
s
(x) , x ∈ Ᏸ,wherep ∈ R
q
+
, p = 0,
are given by
v
s

x
0

=
v

a

x
0

=
v
r

x
0

=−
K−1

k=k
0
α
k+1−k
0
s

u(k), y(k)

=
k
0
−1

k=−K

α
k+1−k
0
s

u(k), y(k)

,
(3.38)
where x(k), k ≥ k
0
, is the solution to (3.1)withu ∈ ᐁ, x(−K) = 0, x(K) = 0, x(k
0
) = x
0
∈
Ᏸ,ands(u, y) = p
T
S(u, y).
Proof. Su ppose Ᏻ is vector lossless with respect to the vector supply rate S(u, y). Since
Ᏻ is completely reachable to and from the origin, it follows that, for every x
0
= x(k
0
) ∈
Ᏸ, there exist K
+
>k
0
, −K

−
<k
0
,andu(k) ∈ ᐁ, k ∈ [−K
−
,K
+
], such that x(−K
−
) = 0,
x(K
+
) = 0, and x(k
0
) = x
0
. Now, it follows from the dissipation inequality (3.4)whichis
satisfied as an equality that
0 =
K
+
−1

k=−K
−
W
K
+
−1−k
S


u(k), y(k)

, (3.39)
or, equivalently,
0 =
K
+
−1

k=−K
−
W
−(k+1−k
0
)
S

u(k), y(k)

=
k
0
−1

k=−K
−
W
−(k+1−k
0

)
S

u(k), y(k)

+
K
+
−1

k=k
0
W
−(k+1−k
0
)
S

u(k), y(k)

≥≥ inf
K≥−k
0
+1, u(·)
k
0
−1

k=−K
W

−(k+1−k
0
)
S

u(k), y(k)

+inf
K≥k
0
, u(·)
K−1

k=k
0
W
−(k+1−k
0
)
S

u(k), y(k)

=
V
r

x
0


− V
a

x
0

,
(3.40)
which implies that V
r
(x
0
)≤≤V
a
(x
0
), x
0
∈ Ᏸ.However,itfollowsfromTheorem 3.9 that if
Ᏻ is vector dissipative and V
a
(x), x ∈ Ᏸ, is a v ector storage function, then V
a
(x)≤≤V
r
(x),
x ∈ Ᏸ, which along with (3.40) implies that V
a
(x) = V
r

(x), x ∈ Ᏸ. Furthermore, since Ᏻ
52 Vector dissipativity and discrete-time large-scale systems
is vector lossless, there exist a nonzero vector p ∈ R
q
+
and a scalar α ≥ 0 satisfying (2.2).
Now, it follows from (3.39)that
0 =
K
+
−1

k=−K
−
p
T
W
−(k+1−k
0
)
S

u(k), y(k)

=
K
+
−1

k=−K

−
α
k+1−k
0
s

u(k), y(k)

=
k
0
−1

k=−K
−
α
k+1−k
0
s

u(k), y(k)

+
K
+
−1

k=k
0
α

k+1−k
0
s

u(k), y(k)

≥ inf
K≥−k
0
+1, u(·)
k
0
−1

k=−K
α
k+1−k
0
s

u(k), y(k)

+inf
K≥k
0
, u(·)
K−1

k=k
0

α
k+1−k
0
s

u(k), y(k)

= v
r

x
0

− v
a

x
0

, x
0
∈ Ᏸ,
(3.41)
which along with (3.34) implies that for any (scalar) storage function of the form v
s
(x) =
p
T
V
s

(x), x ∈ Ᏸ, the equality v
a
(x) = v
s
(x) = v
r
(x), x ∈ Ᏸ,holds.Moreover,sinceᏳ is
vector lossless, the inequalities (3.17)and(3.36) are satisfied as equalities and
v
s

x
0

=−
K−1

k=k
0
α
k+1−k
0
s

u(k), y(k)

=
k
0
−1


k=−K
α
k+1−k
0
s

u(k), y(k)

, (3.42)
where x(k), k
≥ k
0
, is the solution to (3.1)withu ∈ ᐁ, x(−K) = 0, x(K) = 0, and x(k
0
) =
x
0
∈ Ᏸ. 
The next proposition presents a characterization for vector dissipativity of discrete-
time large-scale nonlinear dynamical systems.
Proposition 3.14. Consider the discrete-time large-scale nonlinear dynamical system Ᏻ
given by (3.1), (3.2) and assume V
s
= [v
s1
, ,v
sq
]
T

: Ᏸ → R
q
+
is a continuous vector storage
function for Ᏻ. Then Ᏻ is vector dissipative with respect to the vector supply rate S(u, y) if
and only if
V
s

x(k +1)

≤≤WV
s

x(k)

+ S

u(k), y(k)

, k ≥ k
0
, u ∈ ᐁ. (3.43)
Proof. T he proof is immediate from (3.4) and hence is omitted. 
As a special case of vector dissipativity theory, we can analyze the stability of discrete-
time large-scale nonlinear dynamical systems. Specifically, assume that the discrete-time
large-scale dynamical system Ᏻ is vector dissipative (resp., geometrically vector dissipa-
tive) w ith respect to the vector supply r ate S(u, y) and with a continuous vector storage
function V
s

: Ᏸ → R
q
+
. Moreover, assume that the conditions of Theorem 3.8 are satisfied.
Then it follows from Proposition 3.14,withu(k) ≡ 0andy(k) ≡ 0, that
V
s

x(k +1)

≤≤WV
s

x(k)

, k ≥ k
0
, (3.44)
where x(k), k ≥ k
0
,isasolutionto(3.1)withx(k
0
) = x
0
and u(k) ≡ 0. Now, it follows
from Theorem 2.7,withw(r) = Wr, that the zero solution x(k) ≡ 0to(3.1), with u(k) ≡
0, is Lyapunov (resp., asymptotically) stable.
Wassim M. Haddad et al. 53
More generally, the problem of control system design for discrete-time large-scale non-
linear dynamical systems can be addressed within the framework of vector dissipativity

theory. In particular, suppose that there exists a continuous vector function V
s
: Ᏸ → R
q
+
such that V
s
(0) = 0and
V
s

x(k +1)

≤≤Ᏺ

V
s

x(k)

,u(k)

, k ≥ k
0
, u ∈ ᐁ, (3.45)
where Ᏺ : R
q
+
× R
m

→ R
q
and Ᏺ(0,0) = 0. Then the control system design problem for
a discrete-time large-scale dynamical system reduces to constructing an energy feedback
control law φ : R
q
+
→ ᐁ of the form
u = φ

V
s
(x)



φ
T
1

V
s
(x)

, ,φ
T
q

V
s

(x)

T
, x ∈ Ᏸ, (3.46)
where φ
i
: R
q
+
→ ᐁ
i
, φ
i
(0) = 0, i = 1, ,q, such that the zero solution r(k) ≡ 0tothe
comparison system
r(k +1)= w

r(k)

, r

k
0

=
V
s

x


k
0

, k ≥ k
0
, (3.47)
is rendered asymptotically stable, where w(r)  Ᏺ(r, φ(r)) is of class ᐃ. In this case, if
there exists p ∈ R
q
+
such that v
s
(x)  p
T
V
s
(x), x ∈ Ᏸ, is positive definite, then it fol-
lows from Theorem 2.7 that the zero solution x(k) ≡ 0to(3.1), with u given by (3.46), is
asymptotically stable.
As can be seen from the above discussion, using an energy feedback control architec-
ture and exploiting the comparison system within the control design for discrete-time
large-scale nonlinear dynamical systems can significantly reduce the dimensionality of a
control synthesis problem in terms of a number of states that need to be stabilized. It
should be noted however that, for stability analysis of discrete-time large-scale dynamical
systems, the comparison system need not be linear as implied by (3.44). A discrete-time
nonlinear compar ison system would still guarantee stability of a discrete-time large-scale
dynamical system provided that the conditions of Theorem 2.7 are satisfied.
4. Extended Kalman-Yakubovich-Popov conditions for discrete-time large-scale non-
linear dynamical systems
In this section, we show that vector dissipativeness (resp., geometric vector dissipative-

ness) of a discrete-time large-scale nonlinear dynamical system Ᏻ of the form (3.1), (3.2)
can be characterized in terms of the local subsystem functions f
i
(·), G
i
(·), h
i
(·), and
J
i
(·), along with the interconnection structures Ᏽ
i
(·)fori = 1, , q. For the results in this
section, we consider the special case of dissipative systems with quadratic vector supply
rates and set Ᏸ = R
n
, ᐁ
i
= R
m
i
,andᐅ
i
= R
l
i
. Specifically, let R
i
∈ S
m

i
, S
i
∈ R
l
i
×m
i
,and
Q
i
∈ S
l
i
be given and assume S(u, y)issuchthats
i
(u
i
, y
i
) = y
T
i
Q
i
y
i
+2y
T
i

S
i
u
i
+ u
T
i
R
i
u
i
, i =
1, ,q. For the statement of the next result, recall that x = [x
T
1
, ,x
T
q
]
T
, u = [u
T
1
, ,u
T
q
]
T
,
y = [y

T
1
, , y
T
q
]
T
, x
i
∈ R
n
i
, u
i
∈ R
m
i
, y
i
∈ R
l
i
, i = 1, ,q,

q
i=1
n
i
= n,


q
i=1
m
i
= m,and
54 Vector dissipativity and discrete-time large-scale systems

q
i=1
l
i
= l.Furthermore,for(3.1), (3.2), define Ᏺ : R
n
→ R
n
, G : R
n
→ R
n×m
, h : R
n
→
R
l
,andJ : R
n
→ R
l×m
by Ᏺ(x)  [Ᏺ
T

1
(x), ,Ᏺ
T
q
(x)]
T
,whereᏲ
i
(x)  f
i
(x
i
)+Ᏽ
i
(x), i =
1, ,q, G(x)  diag[G
1
(x
1
), ,G
q
(x
q
)], h(x)  [h
T
1
(x
1
), ,h
T

q
(x
q
)]
T
,andJ(x) 
diag[J
1
(x
1
), ,J
q
(x
q
)]. In addition, for all i = 1, , q,define
ˆ
R
i
∈ S
m
,
ˆ
S
i
∈ R
l×m
,and
ˆ
Q
i

∈ S
l
such that each of these mat rices consists of zero blocks except, respectively, for
the matrix blocks R
i
∈ S
m
i
, S
i
∈ R
l
i
×m
i
,andQ
i
∈ S
l
i
on (i,i) position. Finally, we intro-
duce a more general definition of vector dissipativity involving an underlying nonlinear
comparison system.
Definit ion 4.1. The discrete-time large-scale nonlinear dynamical system Ᏻ given by (3.1),
(3.2)isvector dissipative (resp., geometrically vector dissipative) with respect to the vector
supply rate S(u, y) if there exist a continuous, nonnegative definite vector function V
s
=
[v
s1

, ,v
sq
]
T
: Ᏸ → R
q
+
,calledavector storage function,andaclassᐃ function w : R
q
+
→
R
q
such that V
s
(0) = 0, w(0) = 0, the zero solution r(k) ≡ 0 to the comparison system
r(k +1)= w

r(k)

, r

k
0

= r
0
, k ≥ k
0
, (4.1)

is Lyapunov (resp., asymptotically) stable, and the vector dissipation inequality
V
s

x(k +1)

≤≤w

V
s

x(k)

+ S

u(k), y(k)

, k ≥ k
0
, (4.2)
is satisfied, where x(k), k ≥ k
0
, is the solution to (3.1)withu ∈ ᐁ. The discrete-time large-
scale n onlinear dynamical system Ᏻ given by (3.1), (3.2)isvector lossless with respect to
the vector supply rate S(u, y) if the vector dissipation inequality is satisfied as an equality
with the zero solution r(k)
≡ 0to(4.1) being Lyapunov stable.
Remark 4.2. If in Definition 4.1 the function w : R
q
+

→ R
q
is such that w(r) = Wr,where
W ∈ R
q×q
,thenW is nonnegative and Definition 4.1 collapses to Definition 3.2.
Theorem 4.3. Consider the discrete-time large-scale nonlinear dy namical system Ᏻ given
by (3.1), (3.2). Let R
i
∈ S
m
i
, S
i
∈ R
l
i
×m
i
,andQ
i
∈ S
l
i
, i = 1, ,q. If there exist functions
V
s
= [v
s1
, ,v

sq
]
T
: R
n
→ R
q
+
, P
1i
: R
n
→ R
1×m
, P
2i
: R
n
→ N
m
, w = [w
1
, ,w
q
]
T
: R
q
+
→

R
q
, 
i
: R
n
→ R
s
i
,andᐆ
i
: R
n
→ R
s
i
×m
, such that v
si
(·) is continuous, v
si
(0) = 0, i = 1, ,q,
w ∈ ᐃ, w(0) = 0,
v
si

Ᏺ(x)+G(x)u

=
v

si

Ᏺ(x)

+ P
1i
(x) u + u
T
P
2i
(x) u, x ∈ R
n
, u ∈ R
m
, (4.3)
the zero solution r(k) ≡ 0 to (4.1) is Lyapunov (resp., asymptotically) stable, and, for all
x ∈ R
n
and i = 1, ,q,
0
= v
si

Ᏺ(x)

− h
T
(x)
ˆ
Q

i
h(x) − w
i

V
s
(x)

+ 
T
i
(x) 
i
(x),
0 =
1
2
P
1i
(x) − h
T
(x)

ˆ
S
i
+
ˆ
Q
i

J(x)

+ 
T
i
(x) ᐆ
i
(x),
0 =
ˆ
R
i
+ J
T
(x)
ˆ
S
i
+
ˆ
S
T
i
J(x)+J
T
(x)
ˆ
Q
i
J(x) − P

2i
(x) − ᐆ
T
i
(x)ᐆ
i
(x),
(4.4)
Wassim M. Haddad et al. 55
then Ᏻ is vector dissipat ive (resp., geometrically vector dissipative) with respect to the vector
quadratic supply rate S(u, y),wheres
i
(u
i
, y
i
) = u
T
i
R
i
u
i
+2y
T
i
S
i
u
i

+ y
T
i
Q
i
y
i
, i = 1, ,q.
Proof. Suppose that there exist functions v
si
: R
n
→ R
+
, 
i
: R
n
→ R
s
i
, ᐆ
i
: R
n
→ R
s
i
×m
,

w : R
q
+
→ R
q
, P
1i
: R
n
→ R
1×m
,andP
2i
: R
n
→ N
m
,suchthatv
si
(·) is continuous and
nonnegative-definite, v
si
(0) = 0, i = 1, ,q, w(0) = 0, w ∈ ᐃ, the zero solution r(k) ≡ 0
to (4.1 ) is Lyapunov (resp., asymptotically) stable, and (4.3)and(4.4) are satisfied. Then
for any u ∈ ᐁ and x ∈ R
n
, i = 1, ,q,itfollowsfrom(4.3)and(4.4)that
s
i


u
i
, y
i

= u
T
ˆ
R
i
u +2y
T
ˆ
S
i
u + y
T
ˆ
Q
i
y
= h
T
(x)
ˆ
Q
i
h(x)+2h
T
(x)


ˆ
S
i
+
ˆ
Q
i
J(x)

u
+ u
T

J
T
(x)
ˆ
Q
i
J(x)+J
T
(x)
ˆ
S
i
+
ˆ
S
T

i
J(x)+
ˆ
R
i

u
= v
si

Ᏺ(x)

− w
i

V
s
(x)

+ P
1i
(x) u + 
T
i
(x) 
i
(x)+2
T
i
(x)ᐆ

i
(x)u
+ u
T
P
2i
(x) u + u
T
ᐆ
T
i
(x) ᐆ
i
(x) u
= v
si

Ᏺ(x)+G(x)u

+


i
(x)+ᐆ
i
(x) u

T



i
(x)+ᐆ
i
(x)u

− w
i

V
s
(x)

≥ v
si

Ᏺ(x)+G(x)u

− w
i

V
s
(x)

,
(4.5)
where x(k), k
≥ k
0
, satisfies (3.1). Now, the result follows from (4.5)withvectorstorage

function V
s
(x) = [v
s1
(x), ,v
sq
(x)]
T
, x ∈ R
n
. 
Using (4.4), it follows that for k ≥ k
0
and i = 1, ,q,
s
i

u
i
(k), y
i
(k)

+

w
i

V
s


x(k)

− v
si

x(k)

=
∆v
si

x(k)

+


i

x(k)

+ ᐆ
i

x(k)

u(k)

T



i

x(k)

+ ᐆ
i

x(k)

u(k)

,
(4.6)
where V
s
(x) = [v
s1
(x), ,v
sq
(x)]
T
, x ∈ R
n
, which can be interpreted as a generalized en-
ergy balance equation for the ith subsystem of Ᏻ where ∆v
si
(x( k)) is the change in energy
between consecutive discrete times, the two discrete terms on the left are, respectively,
the external supplied energy to the ith subsystem and the energy gained by the ith sub-

system from the net energy flow between all subsystems due to subsystem coupling, and
the second discrete term on the right corresponds to the dissipated energy from the ith
subsystem.
Remark 4.4. Note that if Ᏻ with u(k)
≡ 0 is vector dissipative (resp., geometrically vector
dissipative) with respect to the vector quadratic supply rate where Q
i
≤ 0, i = 1, ,q,then
it follows from the vector dissipation inequality that
V
s

x(k +1)

≤≤w

V
s

x(k)

+ S

0, y(k)

≤≤w

V
s


x(k)

, k ≥ k
0
, (4.7)
56 Vector dissipativity and discrete-time large-scale systems
where S(0, y)=[s
1
(0, y
1
), ,s
q
(0, y
q
)]
T
, s
i
(0, y
i
(k))= y
T
i
(k)Q
i
y
i
(k) ≤ 0, k ≥ k
0
, i = 1, ,q,

and x(k), k ≥ k
0
, is the solution to (3.1)withu(k) ≡ 0. If, in addition, there exists p ∈ R
q
+
such that p
T
V
s
(x), x ∈ R
n
, is positive definite, then it follows from Theorem 2.7 that the
undisturbed (u(k) ≡ 0) large-scale nonlinear dynamical system (3.1)isLyapunov(resp.,
asymptotically) stable.
Next, we extend the notions of passivity and nonexpansivity to vector passivity and
vector nonexpansivity.
Definit ion 4.5. The discrete-time large-scale nonlinear dynamical system Ᏻ given by (3.1),
(3.2)withm
i
= l
i
, i = 1, ,q,isvector passive (resp., geometrically vector passive)ifitis
vector dissipative (resp., geometrically vector dissipative) with respect to the vector supply
rate S(u, y), where s
i
(u
i
, y
i
) = 2y

T
i
u
i
, i = 1, ,q.
Definit ion 4.6. The discrete-time large-scale nonlinear dynamical system Ᏻ given by (3.1),
(3.2)isvector nonexpansive (resp., geometr ically vector nonexpansive) if it is vector dissipa-
tive (resp., geometrically vector dissipative) with respect to the vector supply rate S(u, y),
where s
i
(u
i
, y
i
) = γ
2
i
u
T
i
u
i
− y
T
i
y
i
, i = 1, ,q,andγ
i
> 0, i = 1, , q,aregiven.

Remark 4.7. Note that a mixed vector passive nonexpansive formulation of Ᏻ can also
be considered. Specifically, one can consider discrete-time large-scale nonlinear dynam-
ical systems Ᏻ which are vector dissipative with respect to vector supply rate S(u, y),
where s
i
(u
i
, y
i
) = 2y
T
i
u
i
, i ∈ Z
p
, s
j
(u
j
, y
j
) = γ
2
j
u
T
j
u
j

− y
T
j
y
j
, γ
j
> 0, j ∈ Z
ne
,andZ
p
∪
Z
ne
={1, , q}. Furthermore, vector supply rates for vector input strict passivity, vector
output strict passivity, and vector input-output strict passivity, generalizing the passivity
notions given in [10], can also be considered. However, for simplicity of exposition, we
do not do so here.
The next result presents constructive sufficient conditions guaranteeing vector dissipa-
tivity of Ᏻ withrespecttoavectorquadraticsupplyrateforthecasewherethevectorstor-
age function V
s
(x), x ∈ R
n
, is component decoupled; that is, V
s
(x) =[v
s1
(x
1

), ,v
sq
(x
q
)]
T
,
x ∈ R
n
.
Theorem 4.8. Consider the discrete-time large-scale nonlinear dy namical system Ᏻ given
by (3.1), (3.2). Assume that there exist functions V
s
= [v
s1
, ,v
sq
]
T
: R
n
→ R
q
+
, P
1i
: R
n
→
R

1×m
i
, P
2i
: R
n
→ N
m
i
, w = [w
1
, ,w
q
]
T
: R
q
+
→ R
q
, 
i
: R
n
→ R
s
i
,andᐆ
i
: R

n
→ R
s
i
×m
i
such that v
si
(·) is continuous, v
si
(0) = 0, i = 1, ,q, w ∈ ᐃ, w(0) = 0,thezerosolution
r(k) ≡ 0 to (4.1) is Lyapunov (resp., asymptotically) stable, and, for all x ∈ R
n
and i =
1, ,q,
0 ≤ v
si

Ᏺ
i
(x)

− v
si

Ᏺ
i
(x)+G
i


x
i

u
i

+ P
1i
(x) u
i
+ u
T
i
P
2i
(x)u
i
,
0 ≥ v
si

Ᏺ
i
(x)

− h
T
i

x

i

Q
i
h
i

x
i

− w
i

V
s
(x)

+ 
T
i

x
i


i

x
i


,
0 =
1
2
P
1i
(x) − h
T
i

x
i

S
i
+ Q
i
J
i

x
i

+ 
T
i

x
i


ᐆ
i

x
i

,
0 ≤ R
i
+ J
T
i

x
i

S
i
+ S
T
i
J
i

x
i

+ J
T
i


x
i

Q
i
J
i

x
i

− P
2i
(x) − ᐆ
T
i

x
i

ᐆ
i

x
i

.
(4.8)
Wassim M. Haddad et al. 57

Then Ᏻ is vector dissipative (resp., geometr ically vector dissipative) with respect to the vector
supply rate S(u, y),wheres
i
(u
i
, y
i
) = u
T
i
R
i
u
i
+2y
T
i
S
i
u
i
+ y
T
i
Q
i
y
i
, i = 1, ,q.
Proof. For any admissible input u = [u

T
1
, ,u
T
q
]
T
such that u
i
∈ R
m
i
, k ∈ Z
+
,andi =
1, ,q,itfollowsfrom(4.8)that
s
i

u
i
(k), y
i
(k)

= u
T
i
(k)R
i

u
i
(k)+2y
T
i
(k)S
i
u
i
(k)+y
T
i
(k)Q
i
y
i
(k)
= h
T
i

x
i
(k)

Q
i
h
i


x
i
(k)

+2h
T
i

x
i
(k)

S
i
+ Q
i
J
i

x
i
(k)

u
i
(k)
+ u
T
i
(k)


J
T
i

x
i
(k)

Q
i
J
i

x
i
(k)

+ J
T
i

x
i
(k)

S
i
+ S
T

i
J
i

x
i
(k)

+ R
i

u
i
(k)
≥ v
si

Ᏺ
i

x(k)

+ P
1i

x(k)

u
i
(k)+

T
i

x
i
(k)


i

x
i
(k)

+2
T
i

x
i
(k)

ᐆ
i

x
i
(k)

u

i
(k)+u
T
i
(k)P
2i

x(k)

u
i
(k)
+ u
T
i
(k)ᐆ
T
i

x
i
(k)

ᐆ
i

x
i
(k)


u
i
(k) − w
i

V
s

x(k)

≥ v
si

x
i
(k +1)

+


i

x
i
(k)

+ ᐆ
i

x

i
(k)

u
i
(k)

T
×


i

x
i
(k)

+ ᐆ
i

x
i
(k)

u
i
(k)

− w
i


V
s

x(k)

≥ v
si

x
i
(k +1)

− w
i

V
s

x(k)

,
(4.9)
where x(k), k ≥ k
0
, satisfies (3.1). Now, the result follows from (4.9)withvectorstorage
function V
s
(x) = [v
s1

(x
1
), ,v
sq
(x
q
)]
T
, x ∈ R
n
. 
Finally, we provide necessary and sufficient conditions for the case where the discrete-
time large-scale nonlinear dynamical system Ᏻ is vector lossless with respect to a vector
quadratic supply rate.
Theorem 4.9. Consider the discrete-time large-scale nonlinear dy namical system Ᏻ given
by (3.1), (3.2). Let R
i
∈ S
m
i
, S
i
∈ R
l
i
×m
i
,andQ
i
∈ S

l
i
, i = 1, ,q. Then Ᏻ is vector lossless
w ith respect to the vector quadratic supply rate S(u, y),wheres
i
(u
i
, y
i
) = u
T
i
R
i
u
i
+2y
T
i
S
i
u
i
+
y
T
i
Q
i
y

i
, i = 1, , q, if and only if there exist functions V
s
= [v
s1
, ,v
sq
]
T
: R
n
→ R
q
+
, P
1i
:
R
n
→ R
1×m
, P
2i
: R
n
→ N
m
,andw = [w
1
, ,w

q
]
T
: R
q
+
→ R
q
such that v
si
(·) is continuous,
v
si
(0) = 0, i = 1, ,q, w ∈ ᐃ, w(0) = 0,thezerosolutionr(k) ≡ 0 to (4.1) is Lyapunov
stable, and, for all x ∈ R
n
, i = 1, ,q,(4.3)holdsand
0 = v
si

Ᏺ(x)

− h
T
(x)
ˆ
Q
i
h(x) − w
i


V
s
(x)

,
0 =
1
2
P
1i
(x) − h
T
(x)

ˆ
S
i
+
ˆ
Q
i
J(x)

,
0 =
ˆ
R
i
+ J

T
(x)
ˆ
S
i
+
ˆ
S
T
i
J(x)+J
T
(x)
ˆ
Q
i
J(x) − P
2i
(x).
(4.10)
Proof. Sufficiency follows as in the proof of Theorem 4.3. To show necessity, suppose that
Ᏻ is lossless w ith respect to the vector quadratic supply rate S(u, y). Then, there exist
continuous functions V
s
= [v
s1
, ,v
sq
]
T

: R
n
→ R
q
+
and w = [w
1
, ,w
q
]
T
: R
q
+
→ R
q
such
58 Vector dissipativity and discrete-time large-scale systems
that V
s
(0) = 0, the zero solution r(k) ≡ 0to(4.1) is Lyapunov stable, and
v
si

Ᏺ(x)+G(x)u

= w
i

V

s
(x)

+ s
i

u
i
, y
i

=
w
i

V
s
(x)

+ u
T
ˆ
R
i
u +2y
T
ˆ
S
i
u + y

T
ˆ
Q
i
y
= w
i

V
s
(x)

+ h
T
(x)
ˆ
Q
i
h(x)+2h
T
(x)

ˆ
Q
i
J(x)+
ˆ
S
i


u
+ u
T

ˆ
R
i
+
ˆ
S
T
i
J(x)+J
T
(x)
ˆ
S
i
+ J
T
(x)
ˆ
Q
i
J(x)

u, x ∈ R
n
, u ∈ R
m

.
(4.11)
Since the rig ht-hand side of (4.11)isquadraticinu, it follows that v
si
(Ᏺ(x)+G(x)u)is
quadratic in u and hence there exist P
1i
: R
n
→ R
1×m
and P
2i
: R
n
→ N
m
such that
v
si

Ᏺ(x)+G(x)u

= v
si

Ᏺ(x)

+ P
1i

(x) u + u
T
P
2i
(x) u, x ∈ R
n
, u ∈ R
m
. (4.12)
Now, using (4.12) and equating coefficients of equal powers in (4.11)yield(4.10). 
5. Specialization to discrete-time large-scale linear dynamical systems
In this section, we specialize the results of Section 4 tothecaseofdiscrete-timelarge-scale
linear dynamical systems. Specifically, we assume that w ∈ ᐃ is linear so that w(r) =
Wr,whereW ∈ R
q×q
is nonnegative, and consider the discrete-time large-scale linear
dynamical system Ᏻ given by
x(k +1)= Ax(k)+Bu(k), x

k
0

=
x
0
, k ≥ k
0
,
y(k) = Cx(k)+Du(k),
(5.1)

where A ∈ R
n×n
and A is partitioned as A  [A
ij
], i, j = 1, , q, A
ij
∈ R
n
i
×n
j
,

q
i=1
n
i
=
n, B = block− diag[B
1
, ,B
q
], C = block −diag[C
1
, ,C
q
], D = block −diag[D
1
, ,D
q

],
B
i
∈ R
n
i
×m
i
, C
i
∈ R
l
i
×n
i
,andD
i
∈ R
l
i
×m
i
, i = 1, ,q.
Theorem 5.1. Consider the discrete-time large-scale linear dynamical system Ᏻ given by
(5.1). Let R
i
∈ S
m
i
, S

i
∈ R
l
i
×m
i
,andQ
i
∈ S
l
i
, i = 1, , q. Then Ᏻ is vector dissipative (resp.,
geometrically vector dissipative) with respect to the vector supply rate S(u, y),wheres
i
(u
i
, y
i
)
= u
T
i
R
i
u
i
+2y
T
i
S

i
u
i
+ y
T
i
Q
i
y
i
, i = 1, , q, and with a three-times continuously differentiable
vector storage function if and only if there exist W ∈ R
q×q
, P
i
∈ N
n
, L
i
∈ R
s
i
×n
,andZ
i
∈
R
s
i
×m

, i = 1, , q, such that W is nonnegative and semistable (resp., asymptotically stable),
and, for all i = 1, ,q,
0 = A
T
P
i
A − C
T
ˆ
Q
i
C −
q

j=1
W
(i, j)
P
j
+ L
T
i
L
i
,
0 = A
T
P
i
B − C

T

ˆ
S
i
+
ˆ
Q
i
D

+ L
T
i
Z
i
,
0 =
ˆ
R
i
+ D
T
ˆ
S
i
+
ˆ
S
T

i
D + D
T
ˆ
Q
i
D − B
T
P
i
B − Z
T
i
Z
i
.
(5.2)
Wassim M. Haddad et al. 59
Proof. Sufficiency fol lows from Theorem 4.3 with Ᏺ(x)= Ax, G(x) = B, h(x) = Cx, J(x) =
D, P
1i
(x) = 2x
T
A
T
P
i
B, P
2i
(x) = B

T
P
i
B, w(r) = Wr, 
i
(x) = L
i
x, ᐆ
i
(x) = Z
i
,andv
si
(x) =
x
T
P
i
x, i = 1, ,q. To show necessity, suppose Ᏻ is vector dissipative with respect to the
vector supply rate S(u, y), where s
i
(u
i
, y
i
) = u
T
i
R
i

u
i
+2y
T
i
S
i
u
i
+ y
T
i
Q
i
y
i
, i = 1, , q.Then,
with w(r) = Wr, there exists V
s
: R
n
→ R
q
+
such that W is nonnegative and semistable
(resp., asymptotically stable), V
s
(x)  [v
s1
(x), ,v

sq
(x)]
T
, x ∈ R
n
, V
s
(0) = 0, and for all
x ∈ R
n
, u ∈ R
n
,
V
s
(Ax + Bu) − WV
s
(x)≤≤S(u, y). (5.3)
Next, it follows from (5.3) that there exists a three-times continuously differentiable vec-
tor function d = [d
1
, ,d
q
]
T
: R
n
× R
m
→ R

q
such that d(x,u)≥≥0, d(0,0) = 0, and
0
= V
s
(Ax + Bu) − WV
s
(x) − S(u,Cx + Du)+d(x, u). (5.4)
Now, expanding v
si
(·)andd
i
(·,·) via Taylor series expansion about x = 0, u = 0, and us-
ing the fact that v
si
(·)andd
i
(·,·) are nonnegative and v
si
(0) = 0, d
i
(0,0) = 0, i = 1, ,q,
it follows that there exist P
i
∈ N
n
, L
i
∈ R
s

i
×n
,andZ
i
∈ R
s
i
×m
, i = 1, ,q,suchthat
v
si
(x) = x
T
P
i
x + v
sri
(x),
d
i
(x, u) =

L
i
x + Z
i
u

T


L
i
x + Z
i
u

+ d
ri
(x, u), x ∈ R
n
, u ∈ R
m
, i = 1, ,q,
(5.5)
where v
sri
: R
n
→ R and d
ri
: R
n
× R
m
→ R contain the higher-order terms of v
si
(·), d
i
(·,·),
respectively. Using the above expressions, (5.4) can be written componentwise as

0 = (Ax + Bu)
T
P
i
(Ax + Bu) −
q

j=1
W
(i, j)
x
T
P
j
x
−

x
T
C
T
ˆ
Q
i
Cx +2x
T
C
T
ˆ
Q

i
Du + u
T
D
T
ˆ
Q
i
Du +2x
T
C
T
ˆ
S
i
u +2u
T
D
T
ˆ
S
i
u + u
T
ˆ
R
i
u

+


L
i
x + Z
i
u

T

L
i
x + Z
i
u

+ δ(x,u),
(5.6)
where δ(x, u)issuchthat
lim
x
2
+u
2
→0


δ(x,u)


x

2
+ u
2
= 0. (5.7)
Now, viewing (5.6) as the componentwise Taylor series expansion of (5.4)aboutx = 0
and u = 0, it follows that for all x ∈ R
n
and u ∈ R
m
,
0
= x
T

A
T
P
i
A −
q

j=1
W
(i, j)
P
j
− C
T
ˆ
Q

i
C + L
T
i
L
i

x
+2x
T

A
T
P
i
B − C
T
ˆ
S
i
− C
T
ˆ
Q
i
D + L
T
i
Z
i


u
+ u
T

Z
T
i
Z
i
− D
T
ˆ
Q
i
D − D
T
ˆ
S
i
−
ˆ
S
T
i
D −
ˆ
R
i
+ B

T
P
i
B

u, i = 1, ,q.
(5.8)
Now, equating coefficients of equal powers in (5.8)yields(5.2).

60 Vector dissipativity and discrete-time large-scale systems
Remark 5.2. Note that the equations in (5.2)areequivalentto

Ꮽ
i
Ꮾ
i
Ꮾ
T
i
Ꮿ
i

=−

L
T
i
Z
T
i



L
i
Z
i

≤ 0, i = 1, , q, (5.9)
where, for all i = 1, ,q,
Ꮽ
i
= A
T
P
i
A − C
T
ˆ
Q
i
C −
q

j=1
W
(i, j)
P
j
,
Ꮾ

i
= A
T
P
i
B − C
T

ˆ
S
i
+
ˆ
Q
i
D

,
Ꮿ
i
=−

ˆ
R
i
+ D
T
ˆ
S
i

+
ˆ
S
T
i
D + D
T
ˆ
Q
i
D − B
T
P
i
B

.
(5.10)
Hence, vector dissipativity of discrete-time large-scale linear dynamical systems with re-
spect to vector quadratic supply rates can be characterized via (cascade) linear matrix
inequalities (LMIs) [5]. A similar remark holds for Theorem 5.3 below.
The next result presents sufficient conditions guaranteeing vector dissipativity of Ᏻ
with respect to a vector quadratic supply rate in the case where the vector storage function
is component decoupled.
Theorem 5.3. Consider the discrete-time large-scale linear dynamical system Ᏻ given by
(5.1). Let R
i
∈ S
m
i

, S
i
∈ R
l
i
×m
i
,andQ
i
∈ S
l
i
, i = 1, , q,begiven.Assumethereexistma-
trices W ∈ R
q×q
, P
i
∈ N
n
i
, L
ii
∈ R
s
ii
×n
i
, Z
ii
∈ R

s
ii
×m
i
, i = 1, ,q, L
ij
∈ R
s
ij
×n
i
,andZ
ij
∈
R
s
ij
×n
j
, i, j = 1, , q, i = j, such that W is nonnegative and semistable (resp., asymptotically
stable), and, for all i = 1, ,q,
0 ≥ A
T
ii
P
i
A
ii
− C
T

i
Q
i
C
i
− W
(i,i)
P
i
+ L
T
ii
L
ii
+
q

j=1, j=i
L
T
ij
L
ij
,
0
= A
T
ii
P
i

B
i
− C
T
i
S
i
− C
T
i
Q
i
D
i
+ L
T
ii
Z
ii
,
0 ≤ R
i
+ D
T
i
S
i
+ S
T
i

D
i
+ D
T
i
Q
i
D
i
− B
T
i
P
i
B
i
− Z
T
ii
Z
ii
,
(5.11)
and for j
= 1, ,q, l = 1, ,q, j = i, l = i, l = j,
0 = A
T
ij
P
i

B
i
,
0 = A
T
ij
P
i
A
il
,
0 = A
T
ii
P
i
A
ij
+ L
T
ij
Z
ij
,
0 ≤ W
(i, j)
P
j
− Z
T

ij
Z
ij
− A
T
ij
P
i
A
ij
.
(5.12)
Then Ᏻ is vector dissipative (resp., geome trically vector dissipative) with respect to the vec-
tor supply rate S(u, y)  [s
1
(u
1
, y
1
), ,s
q
(u
q
, y
q
)]
T
,wheres
i
(u

i
, y
i
) = u
T
i
R
i
u
i
+2y
T
i
S
i
u
i
+
y
T
i
Q
i
y
i
, i = 1, ,q.
Proof. Since P
i
∈ N
n

i
, the function v
si
(x
i
)  x
T
i
P
i
x
i
, x
i
∈ R
n
i
, is nonnegative definite and
v
si
(0) = 0. Moreover, since v
si
(·) is continuous, it follows from (5.11)and(5.12)thatfor
Wassim M. Haddad et al. 61
all u
i
∈ R
m
i
, i = 1, ,q,andk ≥ k

0
,
v
si

x
i
(k +1)

=

q

j=1
A
ij
x
j
(k)+B
i
u
i
(k)

T
P
i

q


j=1
A
ij
x
j
(k)+B
i
u
i
(k)

≤ x
T
i
(k)

W
(i,i)
P
i
+ C
T
i
Q
i
C
i
− L
T
ii

L
ii
−
q

j=1, j=i
L
T
ij
L
ij

x
i
(k)
−
q

j=1, j=i
2x
T
i
(k)L
T
ij
Z
ij
x
j
(k)+2x

T
i
(k)C
T
i
S
i
u
i
(k)+2x
T
i
(k)C
T
i
Q
i
D
i
u
i
(k)
− 2x
T
i
(k)L
T
ii
Z
ii

u
i
(k)+
q

j=1, j=i
x
T
j
(k)

W
(i, j)
P
j
− Z
T
ij
Z
ij

x
j
(k)
+ u
T
i
(k)R
i
u

i
(k)+2u
T
i
(k)D
T
i
S
i
u
i
(k)
+ u
T
i
(k)D
T
i
Q
i
D
i
u
i
(k) − u
T
i
(k)Z
T
ii

Z
ii
u
i
(k)
=
q

j=1
W
(i, j)
v
s j

x
j
(k)

+ u
T
i
(k)R
i
u
i
(k)+2y
T
i
(k)S
i

u
i
(k)+y
T
i
(k)Q
i
y
i
(k)
−

L
ii
x
i
(k)+Z
ii
u
i
(k)

T

L
ii
x
i
(k)+Z
ii

u
i
(k)

−
q

j=1, j=i

L
ij
x
i
(k)+Z
ij
x
j
(k)

T

L
ij
x
i
(k)+Z
ij
x
j
(k)


≤ s
i

u
i
(k), y
i
(k)

+
q

j=1
W
(i, j)
v
s j

x
j
(k)

,
(5.13)
or, equivalently, in vector form,
V
s

x(k +1)


≤≤WV
s

x(k)

+ S(u, y), u ∈ ᐁ, k ≥ k
0
, (5.14)
where V
s
(x)  [v
s1
(x
1
), ,v
sq
(x
q
)]
T
, x ∈ R
n
.Now,itfollowsfromProposition 3.14 that
Ᏻ is vector dissipative (resp., geometrically vector dissipative) with respect to the vector
supply rate S(u, y) and with vector storage function V
s
(x), x ∈ R
n
. 

6. Stability of feedback interconnections of discrete-time large-scale nonlinear
dynamical systems
In this section, we consider stability of feedback interconnections of discrete-time large-
scale nonlinear dynamical systems. Specifically, for the discrete-time large-scale dynam-
ical system Ᏻ given by (3.1), (3.2), we consider either a dynamic or static discrete-time
large-scale feedback system Ᏻ
c
. Then, by appropriately combining vector storage func-
tions for each system, we show stability of the feedback interconnection. We begin by
considering the discrete-time large-scale nonlinear dynamical system (3.1), (3.2)with