# Global Uniform Asymptotic Stability of a Class of Switched Linear Systems with an Infinite Number of Subsystems

Authors: L. F. Araghi; A. A. Suratgar; E. Feizi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Stability of switching systems with an infinite number of subsystems is important in some structure of systems, like fuzzy systems, neural networks, and so forth. Because of the relationship between stability of a set of matrices and switching systems, this paper first studies the stability of a set of matrices, then and the results are applied for stability of switching systems. Some new conditions for globally uniformly asymptotically stability (GUAS) of discrete-time switched linear systems with an infinite number of subsystems are proposed.
The paper considers some examples and simulation results.


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## Body

## 1. Introduction

A switched system is a dynamical system that consists of a finite or infinite number of subsystems and a logical rule that orchestrates switching between these subsystems. Mathematically, these subsystems are usually described by a collection of indexed differential or difference equations. One convenient way to classify switched systems is based on the dynamics of their subsystems, for example, continuous-time or discrete-time, linear or nonlinear, and so on [1]. In the recent years, many researchers have investigated on the stability of switching systems. Lin and Antsaklis proposed a necessary and sufficient condition for asymptotic stabilization of switched linear systems [1]. In [2] absolute asymptotic stability of discrete linear inclusions in Banach (both finite and infinite dimensional) space was studied and the relation between absolute asymptotic stability, asymptotic stability, uniform asymptotic stability, and uniform exponential stability was established.The concept of state-norm estimators for switched nonlinear systems under averaged well-time switching signals was studied in [3]. Liberzon and Trenn showed that switching between stable subsystems may lead to instability and that the presence of algebraic constraints leads to a larger variety of possible instability mechanisms [4].Some new sufficient conditions for exponential stability of switched linear systems under arbitrary switching were proposed in [5]. Hien et al. considered the problem of exponential stability and stabilization of switched linear time-delay systems. They assumed system parameter uncertainties are time varying and unknown but norm bounded [6].A new sufficient condition that guarantees global exponential stability of switched linear systems based on Lyapunov-Metzler inequalities was proposed in [7]. A class of uncertain impulsive systems with delayed input was studied in [8]. It was shown that these systems can be transformed into switched systems without time delay.Hante and Sigalotti considered switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators. They provided necessary and sufficient conditions for their global exponential stability, uniform with respect to the switching signal, in terms of the existence of a Lyapunov function common to all modes [9].Du et al. proposed finite-time stability and stabilization problems for switched linear systems. Firstly, they extended the concept of finite-time stability to switched linear systems. Then, a necessary and sufficient condition for finite-time stability of switched linear systems was presented based on the state transition matrix of the system [10]. Stability and L
2 gain properties for a class of switched systems which are composed of normal discrete-time subsystems were studied in [11]. It was shown that when all subsystems are Schur stable, V
(
x
)
=
x
T
x is a common quadratic Lyapunov function for the subsystems and the switched normal system is exponentially stable under arbitrary switching.The exponential stability and stabilization problems of a class of nonlinear impulsive switched systems with time-varying disturbances were studied in [12], and some sufficient conditions were obtained by using the switched Lyapunov function method as algebraic inequality constraints and linear matrix inequalities.Santarelli studied switched state feedback control law for the stabilization of LTI systems of arbitrary dimension [13]. A unified approach to stability analysis for switched linear descriptor systems under arbitrary switching in both continuous-time and discrete-time domains was studied in [14]. The proposed approach was based on common quadratic Lyapunov functions incorporated with linear matrix inequalities (LMIs). The boundedness and attractiveness of nonlinear switched delay systems whose subsystems have different equilibria were studied in [15]. It was shown that the nonlinear switched delay system is attractive and obtained the attractive region. The authors of the paper proposed three methods for existence of a common quadratic Lyapunov function for robust stability analysis of fuzzy Elman neural network [16].Since switched systems are used in many applications such as robotics, integrated circuit design, manufacturing, power electronics, switched-capacitor networks, chaos generators, and automated highway systems, some applications for linear or nonlinear switched systems have been further investigated, and many valuable results and simulations have been obtained; see [17–20] and some references therein.In [17], an optimal control problem with dynamics that switch between several subsystems of nonlinear differential equations was considered. It was shown that an approximate solution for this optimal control problem can be computed by solving a sequence of conventional dynamic optimization problems.An optimal control problem was considered in which the control takes values from a discrete set, the state and control are subject to continuous inequality constraints, and a new computational method was proposed for solving optimal discrete-valued control problems [18].In [19] an approach with an existing hybrid power system model was explained. The problem of choosing an operating schedule to minimize generator, battery, and switching costs was first posed as a mixed discrete dynamic optimization problem. Some of the latest computational techniques for generating optimal control laws for switched systems with nonlinear dynamics and continuous inequality constraints were reviewed in [20].In [21, 22], some theorems for stability analysis of TSK and linguistic fuzzy models were proposed by the authors.In the recent years, there are many articles about switched systems and stability of them with a finite number of switching but there is no research about the stability of switched systems when the number of subsystems is infinite. Since the study of this class of switching systems is important in some structure of systems, like fuzzy systems, and so forth, so in this paper, stability analysis of discrete-time switched linear systems with an infinite number of switching is investigated and some new conditions are proposed for globally uniformly asymptotically stability (GUAS) of these systems.The paper is organized as follows.Section2 introduces basic definitions and concepts. In Section 3, we derive some conditions for convergence to zero of infinite product of matrices. In Section 4, we extract some new methods for stability analysis of linear switching systems with infinite number of subsystems and provide some illustrative examples. Section 5 gives conclusions.


## 2. Problem Statement

Consider a setΣ of matrices A
i
k and pick an initial point x
0, at t
=
0. A switched linear system is a dynamical system of the type:
(1)
x
k
+
1
=
A
i
k
x
k
,
A
i
k
∈
Σ
,
i
k
∈
I
,
k
∈
Z
+
,
where I is aninfinite index set, i
k is switching law, and the state x
∈
R
n and A
i
k
∈
R
n
×
n for all i
k
∈
I [23, 24].This note means that, at every instant, the matrixA
i
k defined to evolve the system can be replaced by another one from the set Σ.The stability of a switching system when there is no restriction on the switching signals is usually called stability analysis under arbitrary switching. For this analysis, it is necessary that all the subsystems are asymptotically stable. However, even when all the subsystems of a switching system are exponentially stable, it is still possible to construct a divergent trajectory from any initial condition. So, in general, the assumption of subsystems’ stability is not sufficient to ensure stability of switching systems under arbitrary switching, except for some special cases, such as pairwise commutative systems and symmetric or normal systems (all subsystems). Consequently, if there exists a common Lyapunov function for all the subsystems, then the stability of the switching system is guaranteed under arbitrary switching [23].Definition 1.
The linear switching system (1) is globally uniformly asymptotically stable (GUAS) if for any initial condition x
0
∈
R
n and any switching law i
k [23, 25]:
(2)
lim
⁡
k
→
∞
⁡
x
k
=
0
,
∀
(
i
k
)
.
As this is supposed to hold for any initial vectorx
0, it is equivalent to saying that all matrix products taken from Σ converge to the zero matrix; that is,
(3)
lim
⁡
k
→
∞
⁡
A
i
k
A
i
k
-
1
⋯
A
i
1
=
0
,
∀
(
i
k
)
.The GUAS problem is closely related to determining the joint spectral radius (JSR) of the set of matricesΣ
=
{
A
1
,
…
,
A
n
}, denoted by ρ
(
Σ
) [26].The joint spectral radius characterizes the maximal asymptotic growth rate of a point submitted to a switching linear system in discrete time. On the other hand, it characterizes the maximal asymptotic growth rate of the norms of long products of matrices taken in as a setΣ. The stability of the system is ensured when the growth rate is less than one [25, 27].Let∥
·
∥
:
R
n
→
R
+ denote the Euclidean vector norm and denote [27]
(4)
ρ
k
(
Σ
)
=
max
⁡
⁡
{
∥
A
i
1
A
i
2
⋯
A
i
k
∥
1
/
k
,
i
j
∈
{
0
,
1
,
…
,
n
}
}
,
and then the joint spectral radius is defined as
(5)
ρ
(
Σ
)
=
lim
⁡
k
→
∞
⁡
ρ
k
(
Σ
)
.The switching system (1) is GUAS if and only if ρ
(
Σ
)
<
1 [26].Some results show that computing or even approximating the JSR is extremely hard [25]. Here it is proposed some matrix structures for the subsystems in which the stability of a switching system is guaranteed.Consider the discrete-time switching linear system as follows:(6)
x
(
k
+
1
)
=
Σ
k
x
(
k
)
,
k
=
1
,
2
,
…
,
where Σ
k
∈
Σ, Σ
=
{
Σ
1
,
Σ
2
,
…
}.A switching dynamical system of the form (6) is stable if for any initial condition x
0
∈
R
n and any sequence of matrices Σ
k
≜
{
Σ
1
Σ
2
⋯
Σ
k
}, lim
⁡
k
→
∞
⁡
x
(
k
)
=
0 [25].Lemma 2.
Let{
A
k
}
k
∈
N be a set of square matrices. If there exists α
<
1 such that for all k
∈
N, ∥
A
k
∥
≤
α, then
(7)
lim
⁡
k
→
∞
⁡
A
k
A
k
-
1
⋯
A
1
=
0
.Proof.
From the submultiplicity property of norms,(8)
∥
A
B
∥
≤
∥
A
∥
∥
B
∥
the following equation is correct:
(9)
∥
A
k
A
k
-
1
⋯
A
1
∥
≤
∥
A
k
∥
∥
A
k
-
1
∥
⋯
∥
A
1
∥
.
Since, for allk
∈
N, ∥
A
k
∥
≤
α, therefore,
(10)
lim
⁡
k
→
∞
⁡
∥
A
k
∥
∥
A
k
-
1
∥
⋯
∥
A
1
∥
≤
lim
⁡
k
→
∞
α
k
⟶
0
.
So,(11)
lim
⁡
k
→
∞
⁡
∥
A
k
A
k
-
1
⋯
A
1
∥
⟶
0
,
by continuity of the norms, it is extracted that
(12)
lim
⁡
k
→
∞
⁡
A
k
A
k
-
1
⋯
A
1
⟶
0
.The next proof is based on the geometric series. It proves the sufficient condition for invertibility of matrices. The space ofn
×
n-matrices with real or complex numbers as elements is considered.Lemma 3.
LetA be a real or complex n
×
n-matrix and let I be the identity matrix. If ∥
I
-
A
∥
<
1, then A is invertible.Proof.
LetB
=
I
-
A, A
∈
M
n
×
n
(
C
), and ∥
B
∥
=
γ
<
1.
Since∥
B
n
∥
≤
∥
B
∥
n
=
γ
n, ∑
n
=
0
∞
∥
B
n
∥
≤
∑
n
=
0
∞
γ
n
<
∞, hence, C
=
∑
n
=
0
∞
B
n converges in M
n
×
n
(
C
).
IfC
n
=
I
+
B
+
B
2
+
⋯
+
B
n, then
(13)
C
n
(
I
-
B
)
=
(
I
+
B
+
B
2
+
⋯
+
B
n
)
-
(
B
+
B
2
+
⋯
+
B
n
+
1
)
=
I
-
B
n
+
1
,
whereas
(14)
∥
B
n
+
1
∥
≤
γ
n
+
1
,
so
B
n
+
1
⟶
0
as
n
⟶
∞
.
As a result,(15)
C
(
I
-
B
)
=
lim
⁡
⁡
C
n
(
I
-
B
)
=
I
.
Similarly,(
I
-
B
)
C
=
I.
So(
I
-
B
) is invertible and (
I
-
B
)
-
1
=
∑
n
=
0
∞
B
n.
Finally,I
-
B
=
I
-
(
I
-
A
)
=
A (for more details see [28]).


## 3. Infinite Product of Matrices

In this section some conditions are studied for convergence to zero of infinite product of matrices.Theorem 4.
LetA
1
,
A
2
,
… be a sequence of n
×
n matrices and ∥
·
∥ a matrix norm. If the sequence satisfies(1) and (
2):(1)
∑
k
=
1
∞
(
∥
A
k
∥
+
-
1
)
c
o
n
v
e
r
g
e
s,
(2)
∑
k
=
1
∞
(
1
-
∥
A
k
∥
-
)
d
i
v
e
r
g
e
s,where ∥
A
∥
+
=
max
⁡
⁡
{
∥
A
∥
,
1
}, ∥
A
∥
-
=
min
⁡
⁡
{
∥
A
∥
,
1
}, and A
i
1
,
A
i
2
,
… is any rearrangement of the sequence, then ∏
k
=
1
∞
A
i
k
=
0.
Thus, this sequence converges to 0, and soA
i
1
,
A
i
2
A
i
1
,
… converges to 0 [27].Theorem 5.
LetΣ be a compact matrix set as Σ
=
{
Σ
1
,
Σ
2
,
…
}. Then every infinite product taken from Σ converges to 0 iff ρ
^
(
Σ
)
<
1, where ρ
^
(
Σ
) is the joint spectral radius of a set of matrices [27].Corollary 6.
LetΣ be a compact matrix set. Then every infinite product, taken from Σ, converges to 0 iff there is a norm ∥
·
∥ such that ∥
A
∥
≤
γ
,
γ
<
1
,
f
o
r
a
l
l
A
∈
Σ [27].The next theorem refers to a specific structure of matrices.Consider the discrete-time switching linear system (6) such that
(16)
Σ
k
=
[
A
k
B
k
0
C
k
]
,
where Σ
k
∈
Σ, Σ
=
{
Σ
1
,
Σ
2
,
…
} and k is an infinite index set, k
=
1,2
,
….   [
A
k
]
∈
R
n
1
×
n
2, [
B
k
]
∈
R
n
1
×
n
2 and [
C
k
]
∈
R
n
2
×
n
2, and the state x
∈
R
n
,
Σ
k
∈
R
n
×
n.Theorem 7.
Let(
Σ
k
)
k
∈
N be a sequence of Matrices of the form (16) and let there exist two numbers α and γ such that γ
<
1, α
<
1, and ∥
A
i
∥
≤
α, ∥
C
i
∥
≤
γ for some matrix norm ∥
·
∥. The sequence P
k
=
Σ
1
Σ
2
⋯
Σ
k (infinite product of matrices) converges to zero if and only if A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
c
o
n
v
e
r
g
e
s
t
o
z
e
r
o.Proof.
To prove the sufficient condition, by constructionP
k from Σ
k as follows:
(17)
P
k
=
Σ
1
Σ
2
⋯
Σ
k
,
Σ
k
∈
Σ
,
k
∈
N
,
P
k
=
[
A
1
A
2
⋯
A
k
X
k
0
C
1
C
2
⋯
C
k
]
,
where
(18)
X
k
=
A
1
A
2
⋯
A
k
-
1
B
k
+
A
1
A
2
⋯
A
k
-
2
B
k
-
1
C
k
+
A
1
A
2
⋯
A
k
-
3
B
k
-
2
C
k
-
1
C
k
+
⋯
.
By hypothesis of the theorem and Lemma2, since ∥
A
i
∥
≤
α
<
1
,
∥
C
i
∥
≤
γ
<
1, where i
∈
N,
(19)
lim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
=
lim
⁡
k
→
∞
⁡
C
1
C
2
⋯
C
k
=
0
.
Thereforelim
⁡
k
→
∞
⁡
P
k
=
0 implies that lim
⁡
k
→
∞
⁡
X
k
→
0 and so
(20)
lim
⁡
k
→
∞
⁡
X
k
-
X
k
-
1
=
0
.
By some calculations, betweenX
k and X
k
-
1, the following relation is achieved:
(21)
X
k
=
X
k
-
1
C
k
+
A
1
A
2
⋯
A
k
-
1
B
k
,
(
k
∈
N
)
.
By subtractionX
k
-
1, the above equation is
(22)
X
k
-
X
k
-
1
=
X
k
-
1
C
k
+
A
1
A
2
⋯
A
k
-
1
B
k
-
X
k
-
1
=
X
k
-
1
(
C
k
-
I
)
+
A
1
A
2
⋯
A
k
-
1
B
k
.
According to Lemma3 and because of ∥
C
i
∥
≤
γ
<
1, clearly, (
C
k
-
I
) is invertible. Therefore,
(23)
(
X
k
-
X
k
-
1
)
(
C
k
-
I
)
-
1
=
X
k
-
1
+
A
1
A
2
⋯
A
k
-
1
B
k
(
C
k
-
I
)
-
1
,
(
X
k
-
X
k
-
1
)
(
I
-
C
k
)
-
1
=
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
-
X
k
-
1
.
From (20) and (23),
(24)
lim
⁡
k
→
∞
⁡
X
k
=
lim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
,
and consequently
(25)
lim
⁡
k
→
∞
⁡
P
k
=
[
0
lim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
0
0
]
.
Namely, by considering the assumed conditions of the Theorem7, P
k converges to zero if A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
c
o
n
v
e
r
g
e
s
t
o
z
e
r
o.
To prove the necessary condition of the theorem, it must be verified that iflim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1 converges to zero, then lim
⁡
k
→
∞
⁡
P
k
=
0.
To prove this, first it must be proved that(26)
lim
⁡
k
→
∞
∥
X
k
∥
=
lim
⁡
k
→
∞
∥
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
∥
.
By definingD
k as the difference between X
k in (18) and A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1,
(27)
D
k
=
X
k
-
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
;
k
∈
N
and Y
k as
(28)
Y
k
=
A
1
A
2
⋯
A
k
B
k
+
1
(
I
-
C
k
+
1
)
-
1
-
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
.
From (27) and (28) it is obtained that
(29)
D
k
+
1
=
X
k
+
1
-
A
1
A
2
⋯
A
k
B
k
+
1
(
I
-
C
k
+
1
)
-
1
=
(
D
k
-
Y
k
)
C
k
+
1
.
So,∥
D
k
+
1
∥
≤
(
∥
D
k
∥
+
∥
Y
k
∥
)
∥
C
k
+
1
∥, such that ∥
C
i
∥
≤
γ, and therefore,
(30)
∥
D
k
+
1
∥
≤
(
∥
D
k
∥
+
∥
Y
k
∥
)
γ
,
∥
D
k
∥
≤
(
∥
D
k
-
1
∥
+
∥
Y
k
-
1
∥
)
γ
,
∥
D
k
-
i
∥
≤
(
∥
D
k
-
i
-
1
∥
+
∥
Y
k
-
i
-
1
∥
)
γ
.
By repeating the above inequalities it is obtained that(31)
∥
D
k
∥
≤
∥
D
k
-
i
∥
γ
i
+
∥
Y
k
-
i
∥
γ
i
+
∥
Y
k
-
i
+
1
∥
γ
i
-
1
+
⋯
+
∥
Y
k
-
1
∥
γ
1
,
∥
D
k
∥
≤
∥
D
1
∥
γ
k
-
1
+
∑
i
=
1
k
-
1
∥
Y
k
-
i
∥
γ
i
,
i
=
1
,
2
,
…
,
k
-
1
.
Therefore,(32)
lim
⁡
k
→
∞
⁡
∥
D
k
∥
≤
lim
⁡
k
→
∞
∑
i
=
1
k
-
1
∥
Y
k
-
i
∥
γ
i
,
0
≤
γ
<
1
,
because of lim
⁡
k
→
∞
⁡
∥
D
1
∥
γ
k
-
1
=
0.
By consideringS
=
lim
⁡
⁡
sup
⁡
k
→
∞
⁡
∥
D
k
∥
<
∞ and since lim
⁡
k
→
∞
⁡
∥
Y
k
∥
=
0, so sequence {
∥
Y
k
∥
}
k
=
1
∞ is bounded; that is, there exist M
>
0 such that for all k, ∥
Y
k
∥
≤
M. Moreover ∑
i
=
1
k
-
1
∥
Y
k
-
i
∥
γ
i
≤
M
∑
i
=
1
k
-
1
γ
i
≤
M
γ
/
(
1
-
γ
) for all k.
Thus the sequence{
∑
i
=
1
k
-
1
∥
Y
k
-
i
∥
γ
i
}
k
=
1
∞ is bounded. So there exist S
∈
R such that lim
⁡
⁡
sup
⁡
k
→
∞
⁡
∑
i
=
1
k
-
1
∥
Y
k
-
i
∥
γ
i
=
S.
Now, let(33)
S
k
+
1
=
∑
i
=
1
k
∥
Y
k
+
1
-
i
∥
γ
i
=
∥
Y
k
∥
γ
+
∑
i
=
2
k
∥
Y
k
+
1
-
i
∥
γ
i
=
∥
Y
k
∥
γ
+
γ
∑
i
=
1
k
-
1
∥
Y
k
-
i
∥
γ
i
=
∥
Y
k
∥
γ
+
γ
S
k
.
In fact, we haveS
k
+
1
=
∥
Y
k
∥
γ
+
γ
S
k. By taking the limit superior of both sides of the above equation, it is obtained that lim
⁡
⁡
sup
⁡
k
→
∞
S
k
+
1
=
γ
lim
⁡
⁡
sup
⁡
k
→
∞
S
k, hence S
=
γ
S, and since γ
≠
1, then it implies that S
=
0.
The above equation means thatlim
⁡
k
→
∞
⁡
D
k
=
0 and from (27),
(34)
lim
⁡
k
→
∞
X
k
=
lim
⁡
k
→
∞
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
,
k
∈
N
.
So, iflim
⁡
k
→
∞
∥
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
∥
=
0, then lim
⁡
k
→
∞
⁡
∥
X
k
∥
=
0, and also by hypothesis of Theorem 7 and Lemma 2,
(35)
lim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
=
lim
⁡
k
→
∞
⁡
C
1
C
2
⋯
C
k
=
0
.
Therefore,(36)
lim
⁡
k
→
∞
⁡
P
k
=
0
.
So it completes the proof.


## 4. Stability of Switching Linear Systems with Infinite Number of Subsystems

The following conditions are extracted for stability of a switching linear system when it comprises an infinite number of subsystems. They are reached from the conditions for convergence to zero of infinite product of matrices.Corollary 8.
The switching linear system (6) is GUAS if the following conditions are satisfied: (1)
∑
k
=
1
∞
(
∥
Σ
k
∥
+
-
1
)
c
o
n
v
e
r
g
e
s,
(2)
∑
k
=
1
∞
(
1
-
∥
Σ
k
∥
-
)
d
i
v
e
r
g
e
s.Proof.
Thereby establishing these two conditions, from Theorem4, it is obtained that ∏
k
=
1
∞
Σ
i
k
=
0. From Corollary 6, Lemma 2, and by Definition 1, ∥
Σ
k
∥
≤
γ, γ
<
1
,
for
all
Σ
k
∈
Σ. Therefore, according to Theorem 5, ρ
^
(
Σ
)
<
1, and so the system (6) is GUAS.Corollary 9.
Let(
Σ
k
)
k
∈
N be a sequence of switching systems of the form (16) with ∥
A
i
∥
≤
α
<
1, ∥
C
i
∥
≤
γ
<
1. The discrete time switching linear system (6) with an infinite number of switching system is GUAS under arbitrary switching if
(37)
lim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
=
0
.
Arbitrary switching refers to switching systems that there are no restrictions on the discrete event dynamics.Proof.
By Definition1 and Theorem 7, any infinite product of this kind of switching linear system under these conditions converges to 0. By definition of the joint spectral radius mentioned in (5),
(38)
ρ
(
Σ
)
=
lim
⁡
k
→
∞
⁡
ρ
k
(
Σ
)
=
lim
⁡
⁡
max
⁡
⁡
{
∥
A
i
1
A
i
2
⋯
A
i
k
∥
1
/
k
:
i
j
∈
{
0
,
1
,
…
,
n
}
}
,
and because of ∥
A
i
∥
≤
α
<
1,
(39)
∥
∏
j
=
1
k
A
i
j
∥
1
/
k
≤
∏
j
=
1
k
∥
A
i
j
∥
1
/
k
≤
α
k
<
1
,
∀
k
.
Then,(40)
ρ
k
(
Σ
)
≤
α
<
1
⟹
lim
⁡
k
→
∞
⁡
ρ
k
(
Σ
)
≤
α
<
1
.
So,ρ
(
Σ
)
<
1 and the switching linear system is GUAS.Lemma 10.
Consider the discrete-time switching linear system (6) with matrix structure of the form (16) and B
k
=
0. The switching system is GUAS if and only if {
A
1
,
A
2
,
…
,
A
k
,
…
}, {
C
1
,
C
2
,
…
,
C
k
,
…
} are GUAS.Proof.
Let{
Σ
k
} of the form [
A
k
0
0
C
k
] where [
A
k
]
∈
R
n
1
×
n
1 and [
C
k
]
∈
R
n
2
×
n
2.
Then,(41)
P
k
=
Σ
1
Σ
2
⋯
Σ
k
=
[
A
1
A
2
⋯
A
k
0
0
C
1
C
2
⋯
C
k
]
,
such that the state x
∈
R
n
,
Σ
k
∈
R
n
×
n. Solim
⁡
k
→
∞
⁡
P
k
→
0 if and only if,
lim
⁡
k
→
∞
A
1
A
2
⋯
A
k
→
0,  lim
⁡
k
→
∞
C
1
C
2
⋯
C
k
→
0.
In fact, the discrete-time switching linear system (6) with B
k
=
0 is GUAS if and only if {
A
1
,
A
2
,
…
,
A
k
,
…
}, {
C
1
,
C
2
,
…
,
C
k
,
…
} are GUAS.Result 1.
The discrete time switching linear system (6) with an infinite number of switching systems of the form (16) is GUAS under arbitrary switching if there exist two numbers α and γ such that, ∥
A
i
∥
≤
α
<
1, ∥
C
i
∥
≤
γ
<
1, and B
i
≤
β
<
∞ is bounded.Result 2.
Because ofρ
(
·
)
≤
∥
·
∥, the discrete time switching linear system (6) with an infinite number of switching systems with the form (16) is GUAS under arbitrary switching if ρ
(
A
i
)
<
1, ρ
(
C
i
)
<
1.Example 11.
Consider switching linear system (6) with two subsystems of the form (16) and let Σ be a set of triangular block matrices with blocks on the diagonal:
(42)
Σ
=
{
Σ
1
,
Σ
2
}
=
{
[
-
.
1
.
3
5
-
.
5
.
1
2
0
0
.
8
]
,
[
-
.
4
-
.
2
4
-
.
5
.
3
5
0
0
.
2
]
}
.
So(43)
Σ
k
=
[
A
k
B
k
0
C
k
]
,
k
=
1
,
2
.
Because of(44)
∥
A
1
∥
=
∥
-
.
1
.
3
-
.
5
.
1
∥
=
0.5414
<
1
,
∥
A
2
∥
=
∥
-
.
4
-
.
2
-
.
5
.
3
∥
=
0.6531
<
1
,
∥
C
1
∥
=
0.8
<
1
,
∥
C
2
∥
=
0.2
<
1
,
∥
B
1
∥
=
∥
5
2
∥
<
∞
,
∥
B
2
∥
=
∥
4
5
∥
<
∞
,
the switching linear system under arbitrary switching is GUAS.Figure1 shows the simulation result of Example 11.Figure 1
Phase portrait of switching linear system with two subsystems.Example 12.
Consider a discrete time switching linear system of the form (16) that Σ
=
{
Σ
1
,
Σ
2
,
…
}, and
(45)
Σ
k
=
[
1
3
k
cos
⁡
⁡
(
k
)
4
.
5
1
4
k
3
sin
⁡
(
k
)
0
0
1
2
k
]
,
k
=
1
,
2
,
…
.
According to Result2 of this paper, this switching system with infinite number of subsystems is GUAS under arbitrary switching because of
(46)
ρ
(
A
k
)
=
ρ
(
[
1
3
k
cos
⁡
⁡
(
k
)
.
5
1
4
k
]
)
<
1
,
ρ
(
1
2
k
)
<
1
,
for
k
=
1,2
,
…
.
Solim
⁡
k
→
∞
⁡
A
1
A
2
⋯
A
k
-
1
B
k
(
I
-
C
k
)
-
1
=
0 for all k
∈
N. Figure 2 shows some simulation results for k
=
10, 100, 1000, and 2000 subsystems that switch in a switching system.Phase portrait and switching surfaces with 10, 100, 1000, and 2000 subsystems.
(a)
k
=
10
(b)
k
=
100
(c)
k
=
1000
(d)
k
=
20
00


## 5. Conclusion

This paper proposed a new necessary and sufficient condition for convergence to zero of infinite product of matrices. Using the conditions to convergence to zero of matrices, stability of switching linear system with an infinite number of subsystems was investigated. Some sufficient conditions were extracted for globally uniformly asymptotically stability (GUAS) of the discrete time switching linear systems.



---

*Source: 302542-2014-05-19.xml*