Stability Analysis of Switched Time-Delay Systems
Peng Yan and Hitay
¨
Ozbay
Abstract— This paper addresses the asymptotic stability of
switched time delay systems with heterogenous t ime invariant
time delays. Piecewise Lyapunov-Razumikhin functions are
introduced for the switching candidate systems to investigate
the stability in the presence of infinite number of switchings.
We provide sufficient conditions in terms of the minimum dwell
time to gu arantee asymptotic stability under the assumptions
that each switching candidate is d el ay-in dependently or delay-
dependentl y stable. Conservatism analysis is also provided by
comparing with the dwell time conditions for switched delay
free systems.
I. INTRODUCTION
Switching control offers a new look into the design of
complex control systems (e.g. nonlinear systems, parameter
varying systems and uncertain systems) [1], [8], [9], [19],
[17], [21], [28]. Unlike th e conventional adaptive control
techniques th a t rely on continuous tuning, the switching
control method updates the controller parame te rs in a discrete
fashion based on the switching logic. The resulting closed-
loop systems have hybrid behaviors (e.g. co ntinuous dynam-
ics, discrete time dynamics and jump pheno mena, etc.). One
of the most challenging issues in the area of hybrid systems
is the stability analysis in the presence of contr ol switching.
We refer to [9] for a general review on switching control
methods.
In particular, we are in terested in the stability analy sis of
switched time delay sy stems. In fact, time delay systems
are ubiquitous in chemical processes, aerody namics, and
communication networks [3], [14]. To further comp lica te
the situation, the time delays are usually time varying and
uncertain [24], [25]. It has been shown that robust H
∞
con-
trollers can be designed for such infinite dimensional plants,
where robustness can be guara nteed within some uncertainty
bounds [4]. In order to inco rporate larger operating range
or better robustness, controller switching can be introduced,
which results in switched closed-loop systems with time
delays. For delay free systems, stability analysis and design
methodology have been investigated recently in the frame-
work of hybrid dynamical systems [1], [2], [8], [11], [19],
[21], [26]. In particular, [21] provided sufficient conditions
on the stability of the switching control systems based on
Filippov solutions to discontinuous differential equations and
Lyapunov functionals; [19] proposed a dwell-tim e based
switching control, where a sufficiently large dwell-time can
This work is supported in part by by T
¨
UB
˙
ITAK under grant no. EEEAG-
105E156.
P. Yan is with Enterprise Design Center, Seagate Technology LL C, 1280
Disc Drive, Shakopee, MN 55379, US A Peng.Yan@seagate.com
H.
¨
Ozbay is with Dept. of Electrical & Electronics Engineering, Bilkent
University, Ankara 06800, Turkey hitay@bilkent.edu.tr
guarante e the system stability. A more flexible result was
obtained in [10], where the average dwell- time was intro-
duced for switching control. In [26] the results o f [10] w e re
extended to LPV sy stems. LaSalle’s invariance principle was
extended to a class of switched linear systems for stability
analysis [8]. Despite the variety and significance of the many
results on hybrid system stability, stability of switched time
delay systems hasn ’t been adequately addressed due to the
general difficulty of infinite dimensional systems [7].
Two important approaches in the stability analysis of
time de lay systems are (1) Lyapunov-Krasovskii method,
and (2) Lyapunov- Razumikhin method [6], [20]. Various
sufficient conditions with re spect to th e stability of time
delay systems have been given using Riccati-typ e inequalities
or LMIs [3], [12], [14], [24]. In the mean w hile, stability
analysis in the presence of switching ha s been discussed
in some rec e nt works [16], [18], [22]. In [18] stability and
stabilizability were discussed for discrete time switched time
delay systems; [16] considered similar stability problem in
continuous time doma in. Note that [18] and [16] are trajec-
tory dependent re sults without taking admissible switching
signals into considerations.
The main c ontribution of this paper is a collection of
results on the trajectory independ ent stability of continuous
time switched time delay systems using piecewise Lyapunov-
Razumikhin functions. The dwell time of the switching
signals is constructively given, which guarantees asymptotic
stability for th e delay independent case and the delay de-
pendent case, respectively. Note that the asympto tic stability
of finite dim e nsional linear systems indicates exponential
stability, whe reas this is not the case for infinite dimensional
systems, [7], [ 15]. This poses the key challenge in the
analysis of switched time delay systems.
The paper is organized as follows. The problem is defined
in Section II. In Section III, the main results on the stability
of switch e d time delay systems are presented in terms of the
dwell time of the switching signals. Conservatism analysis
is provided by comparing w ith the dwell time conditions
for switching delay free systems in Section IV, followed by
conclud ing rema rks in Section V.
II. PROBLEM DEFINITION
For convenience, we would like to employ the following
notation. The general Retarded Functional D ifferential E qua-
tions (RFDE) with time delay r can be described as
˙x(t) = f (t, x
t
) (1)
with initial condition φ(·) ∈ C([−r, 0], R
n
), where x
t
denotes the state defined by x
t
(θ) = x(t + θ), −r ≤ θ ≤ 0.
Proceedings of the
47th IEEE Conference on Decision and Control
Cancun, Mexico, Dec. 9-11, 2008
WeB07.5
978-1-4244-3124-3/08/$25.00 ©2008 IEEE 2740
We use k·k to denote the Euclidean norm of a vector in R
n
,
and |f|
[t−r,t]
for the ∞- norm of f, i.e.
|f|
[t−r,t]
:= sup
t−r≤θ≤t
kf(θ)k,
where f is an element of the Banach spac e C([t −r, t], R
n
).
Consider the following switched time delay systems:
Σ
t
:
˙x(t) = A
q(t)
x(t) +
¯
A
q(t)
x(t − τ
q(t)
), t ≥ 0
x
0
(θ) = φ(θ), ∀θ ∈ [−τ
max
, 0]
(2)
where x(t) ∈ R
n
and q(t) is a piecewise switching signal
taking values on the set F := {1, 2, ..., l} , i.e. q(t) = k
j
,
k
j
∈ F, for ∀t ∈ [t
j
, t
j+1
), whe re t
j
, j ∈ Z
+
∪ {0}, is the
j
th
switching time instant. It is clear that the trajectory of
Σ
t
in any arbitrary switching interval t ∈ [t
j
, t
j+1
) obeys:
Σ
k
j
:
˙x(t) = A
k
j
x(t) +
¯
A
k
j
x(t − τ
k
j
), t ∈ [t
j
, t
j+1
)
x
t
j
(θ) = φ
j
(θ), ∀θ ∈ [−τ
k
j
, 0],
(3)
where φ
j
(θ) is defined a s:
φ
j
(θ) =
x(t
j
+ θ) −τ
k
j
≤ θ < 0
lim
h→0
− x(t
j
+ h), θ = 0
(4)
We introduce the triplet Σ
i
:= (A
i
,
¯
A
i
, τ
i
) ∈ R
n×n
×
R
n×n
× R
+
to describe the i
th
candidate system of (2).
Thus for ∀t ≥ 0, we have Σ
t
∈ A := {Σ
i
: i ∈ F},
where A is th e family of can didate systems of (2). In (2),
φ(·) : [−τ
max
, 0] → R
n
is a continuous and bounded vector-
valued function, where τ
max
= max
i∈F
{τ
i
} is the maximal
time delay of the candidate systems in A.
Similar to [8], we say that the switched time-delay system
Σ
t
described by (2) is stable if there exists a function ¯α of
class K
1
such that
kx(t)k ≤ ¯α(|x|
[t
0
−τ
max
,t
0
]
), ∀t ≥ t
0
≥ 0, (5)
along the trajectory of (2) . Furthermore, Σ
t
is asymptotically
stable when Σ
t
is stable and lim
t→+∞
x(t) = 0.
Lemma 2.1: ([3], [14]) Suppose for a given triplet Σ
i
∈
A, i ∈ F, there exists symmetric and positive-definite P
i
∈
R
n×n
, such that the following LM I with respect to P
i
is
satisfied for some p
i
> 1 and α
i
> 0:
P
i
A
i
+ A
T
i
P
i
+ p
i
α
i
P
i
P
i
¯
A
i
¯
A
T
i
P
i
−α
i
P
i
< 0. (6)
Then Σ
i
is asymptotically stable independent of delay.
If all candidate systems of (2), Σ
i
∈ A, are d elay-
indepen dently asymptotically stable satisfying (6), we denote
A by
˜
A.
Lemma 2.2: ([3], [14]) Suppose for a given triplet Σ
i
∈
A, i ∈ F, there exists symmetric and positive-definite P
i
∈
R
n×n
, and a scalar p
i
> 1, such that
τ
−1
i
Ω
i
P
i
¯
A
i
M
i
M
T
i
¯
A
T
i
P
i
−R
i
< 0 (7)
1
A continuous function ¯α(·) : R
+
→ R
+
is a class K function if it is
strictly increasing and ¯α(0) = 0.
where
Ω
i
= (A
i
+
¯
A
i
)
T
P
i
+ P
i
(A
i
+
¯
A
i
) + τ
i
p
i
(α
i
+ β
i
)P
i
,
M
i
= [A
i
¯
A
i
],
R
i
= diag(α
i
P
i
, β
i
P
i
),
and α
i
> 0, β
i
> 0 are scalars. Then Σ
i
is asymptotically
stable dependent of delay.
Similarly we de note A by
˜
A
d
if all candidate systems of (2)
are delay-d ependen tly asy mptotically stable satisfying (7).
In what follows, we will establish sufficient co nditions
to guar a ntee stability of switched system (2) for the delay
indepen dent case and the delay depende nt case. Therefore,
we will assume that A =
˜
A and A =
˜
A
d
respectively
in the corresponding sections in this paper. An important
method in stability a nalysis of switched systems is based o n
the construction of the common Lyapunov function (CLF),
which allows for arbitrary switching. However, this method
is too conservative from the perspective of controller de-
sign because it is usu a lly difficult to find the CLF for all
the candidate systems, pa rticularly for time delay systems
whose stability criteria are only sufficient in most of the
circumstances. A recent paper [29] explored the CLF method
for switched time delays systems with three very strong as-
sumptions: (i) each candidate system has the same time delay
τ; ( ii) each candidate is assumed to be delay independently
stable; (iii) The A-matr ix is always symmetric and the
¯
A-
matrix is always in the form of δI. In the present paper, w e
consider an alternative method using p iecewise Lyapunov-
Razumikhin functions for a general class of systems (2)
and ob ta in stability conditions in terms of the dwell time
of the switching signal. This method can be used for the
case with delay indepe ndent criterion (6) an d the case with
delay dependent criterion (7).
III. MAIN RESULTS ON DWELL TIME BASED SWITCHING
For a given p ositive con stant τ
D
, the switching signal set
based on the dwell time τ
D
is denoted by S[τ
D
], where
for any switching signal q(t) ∈ S[τ
D
], the distance between
any co nsecutive discontinuities of q(t), t
j+1
−t
j
, j ∈ Z
+
∪
{0}, is larger than τ
D
[10], [1 9]. Sufficient condition on the
minimum dwell time to guarantee the stable switching will be
given using piecewise Lyapunov-Razumikhin functions. No te
that the dwell time based switching is tra je c tory-independent
[8].
Before presenting the main result of this paper, we recall
the following lemma [7] for general Retarded Functional
Differential Equations (1).
Lemma 3.1: [7] Suppose u, v, w, p : R
+
→ R
+
are
continuous,nondecreasing functions, u(0) = v(0) = 0,
u(s), v(s), w(s), p(s) positive for s > 0, p(s) > s, and
v(s) strictly increasing. If there is a con tinuous function
V : R × R
n
→ R such that
u(kx(t)k) ≤ V (t, x) ≤ v(kx(t)k), t ∈ R, x ∈ R
n
, (8)
and
˙
V (t, x(t)) ≤ −w(kx(t)k), (9)
47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeB07.5
2741
if
V (t + θ, x(t + θ)) < p(V (t, x(t))) ∀θ ∈ [−r, 0], (10)
then th e solution x = 0 of the RFDE is uniformly asymp-
totically stable.
A particular case of (1) is a linear time delay system
Σ
i
, i ∈ F, where we can construct the corresponding
Lyapunov-Razumikhin function in the quadratic form
V
i
(t, x) = x
T
(t)P
i
x(t), P
i
= P
T
i
> 0. (11)
Apparen tly V
i
can be bounded b y
u
i
(kx(t)k) ≤ V
i
(t, x) ≤ v
i
(kx(t)k), ∀x ∈ R
n
, (12)
where
u
i
(s) := κ
i
s
2
, v
i
(s) := ¯κ
i
s
2
, (13)
in which κ
i
:= σ
min
[P
i
] > 0 denote s the smallest singular
value of P
i
and ¯κ
i
:= σ
max
[P
i
] > 0 the largest singular
value of P
i
.
Propo sition 3.2: For each time delay systems Σ
i
with
Lyapunov-Razumikhin function defined by (11) assume (9)
and (10) are satisfied for some w
i
(s). Then we have
|x|
[t
m
−τ
i
,t
m
]
≤
¯κ
i
κ
i
1/2
|x|
[t
n
−τ
i
,t
n
]
, ∀t
m
≥ t
n
≥ 0. (14)
Proof. Define
¯
V
i
(t, x) := sup
−τ
i
≤θ≤0
V
i
(t + θ, x(t + θ)) (15)
for t ≥ 0, we have
κ
i
(|x|
[t−τ
i
,t]
)
2
≤
¯
V
i
(t, x) ≤ ¯κ
i
(|x|
[t−τ
i
,t]
)
2
, t ≥ 0 (16)
The definition o f
¯
V
i
(t, x) implies ∃θ
0
∈ [−τ
i
, 0], such that
¯
V
i
(t, x) = V (t + θ
0
, x(t + θ
0
)). Introduce the upper right-
hand derivative of
¯
V
i
(t, x) as
˙
¯
V
+
i
= lim sup
h→0
+
1
h
[
¯
V
i
(t + h, x(t + h)) −
¯
V
i
(t, x(t))],
we have
(i). If θ
0
= 0, i.e. V
i
(t + θ, x(t + θ)) ≤ V
i
(t, x(t)) <
p(V
i
(t, x(t))), we have
˙
V
i
(t, x) < 0 by (9). Therefore
˙
¯
V
+
i
≤ 0.
(ii). If −τ
i
< θ
0
< 0, we have
¯
V
i
(t+ h, x(t+ h)) =
¯
V
i
(t, x)
for h > 0 sufficiently small, which results in
˙
¯
V
+
i
= 0.
(iii). If θ
0
= −τ
i
, the continuity of V
i
(t, x) imp lies
˙
¯
V
+
i
≤ 0.
The above analysis shows that
¯
V
i
(t
m
) ≤
¯
V
i
(t
n
), ∀t
m
≥ t
n
≥ 0. (17)
Recall (16), we have
κ
i
(|x|
[t
m
−τ
i
,t
m
]
)
2
≤
¯
V
i
(t
m
) ≤
¯
V
i
(t
n
) ≤ ¯κ
i
(|x|
[t
n
−τ
i
,t
n
]
)
2
,
(18)
for any t
m
≥ t
n
≥ 0. Th is im plies (14) and proves th e result.
Suppose all of the conditions of Lemma 3.1 are satisfied
for general RFDE (1), we also have the following result.
Lemma 3.3: [7] Suppose |φ|
[t
0
−r,t
0
]
≤
¯
δ
1
,
¯
δ
1
> 0, and
¯
δ
2
> 0 such that v(
¯
δ
1
) = u(
¯
δ
2
). For all η satisfying 0 <
η ≤
¯
δ
2
, we have
V (t, x) ≤ u(η), ∀ t ≥ t
0
+ T. (19)
Here
T =
Nv(
¯
δ
1
)
γ
(20)
is defined by γ = inf
v
−1
(u(η))≤s≤
¯
δ
2
w(s) and N = ⌈ (v(
¯
δ
1
)−
u(η))/a⌉, where ⌈·⌉ is the ceiling integer function and a > 0
satisfies p(s) − s > a for u(η) ≤ s ≤ v(
¯
δ
1
).
A. The Case with Delay Ind e pendent Criterion
Consider the switched time de lay systems Σ
t
defined
by (2) and assume each candidate system Σ
i
, i ∈ F
delay-independently asymptotically stable satisfying (6) (i.e.
A =
˜
A). A sufficient condition on the minimum d well
time to guarantee the asymptotic stability can be derived
using multiple piecewise Lyapunov-Razumik hin functions.
In order to state the main result we make some preliminary
definitions.
For the switched delay system s (2), first assume τ
D
>
τ
max
. Consider an arbitrary switching interval [t
j
, t
j+1
) of
the piecewise switching signal q(t) ∈ S[τ
D
], where q(t) =
k
j
, k
j
∈ F for ∀t ∈ [t
j
, t
j+1
) and t
j
is th e j
th
switching time
instant for j ∈ Z
+
∪{0} and t
0
= 0. The state variable x
j
(t)
defined on this interval obeys (3). For the convenience of
using “sup”, we defin e x
j
(t
j+1
) = lim
h→0
−
x
j
(t
j+1
+ h) =
x
j+1
(t
j+1
) based on the fact that x(t) is contin uous for
t ≥ 0. Therefore x
j
(t) is now defined on a compact set
[t
j
, t
j+1
]. Recall ( 4), the initial condition φ
j
(t) of Σ
k
j
is
φ
j
(t) = x(t) = x
j−1
(t), t ∈ [t
j
−τ
k
j
, t
j
] fo r j ∈ Z
+
, which
is true because τ
D
> τ
max
.
Construct the Lyapunov-Razumikhin function
V
k
j
(x
j
, t) = x
T
j
(t)P
k
j
x
j
(t), t ∈ [t
j
, t
j+1
] (21)
for (3), then we have
κ
k
j
kx
j
(t)k
2
≤ V
k
j
(t, x
j
) ≤ ¯κ
k
j
kx
j
(t)k
2
, ∀x
j
∈ R
n
. (22)
A straightforward calculation gives the time derivative of
V
k
j
(t, x
j
(t)) along the tra je c tory of (3)
˙
V
k
j
(t, x
j
) = x
T
j
(A
T
k
j
P
k
j
+ P
k
j
A
k
j
)x
j
+2x
T
j
(t)P
k
j
¯
A
k
j
x
j
(t − τ
k
j
), (23)
where
2x
T
j
(t)P
k
j
¯
A
k
j
x
j
(t − τ
k
j
)
≤ α
k
j
x
T
j
(t − τ
k
j
)P
k
j
x
j
(t − τ
k
j
)
+α
−1
k
j
x
T
j
(t)P
k
j
¯
A
k
j
P
−1
k
j
¯
A
T
k
j
P
k
j
x
j
(t), ∀α
k
j
> 0.
Applying Razumikhin condition with p(s) = p
k
j
s, p
k
j
> 1,
we obtain
x
T
j
(t − τ
k
j
)P
k
j
x
j
(t − τ
k
j
) ≤ p
k
j
x
T
j
(t)P
k
j
x
j
(t) (24)
for
V
k
j
(t + θ, x
j
(t + θ)) < p
k
j
V
k
j
(t, x
j
(t)) ∀θ ∈ [−τ
k
j
, 0].
47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeB07.5
2742
Let
S
k
j
:= −(A
T
k
j
P
k
j
+ P
k
j
A
k
j
+ p
k
j
α
k
j
P
k
j
+α
−1
k
j
P
k
j
¯
A
k
j
P
−1
k
j
¯
A
T
k
j
P
k
j
) (25)
we have
˙
V
k
j
(t, x
j
) ≤ −x
T
j
(t)S
k
j
x
j
(t). (26)
Because Σ
t
∈
˜
A, we have S
k
j
> 0 from Lemma 2.1.
Furthermore we can select w(s) = w
k
j
s
2
in Lemma 3.1,
such that (9 ) is satisfied, where w
k
j
:= σ
min
[S
k
j
] > 0.
Define
λ := max
i∈F
¯κ
i
κ
i
, (27)
and
µ := max
i∈F
¯κ
i
w
i
. (28)
Now we are ready to state the main result.
Theorem 3.4: Let the dwell time be defined by τ
D
:=
T
∗
+ τ
max
, where
T
∗
:= λµ⌊
λ − 1
¯p − 1
+ 1⌋, (29)
with ¯p := min
i∈F
{p
i
} > 1, and ⌊·⌋ being the floo r integer
function. Then the system (2) with Σ
t
∈
˜
A is asymptotically
stable for any switching rule q(t) ∈ S[τ
D
].
Proof. First we claim that for all τ > τ
D
, there exist
0 < β < 1 a nd 0 < α < 1, such that τ ≥
¯
T + τ
max
, where
¯
T :=
λµ
α
2
⌈
λ − α
2
α
2
β(¯p − 1)
⌉. (30)
For a given τ , to find such α and β defin e
˜
T + τ
max
:= τ >
τ
D
= T
∗
+ τ
max
, and consider two cases below.
1) If ⌊(λ − 1)/(¯p − 1)⌋ =: k < (λ −1)/(¯p − 1) < k + 1,
then can find ∆
1
> 0 and ∆
2
> 0 small enou gh, such
that
⌈
λ − α
2
1
α
2
1
β(¯p − 1 )
⌉ = ⌈
λ − 1
¯p − 1
⌉ = k + 1 = ⌊
λ − 1
¯p − 1
+ 1⌋
with α
1
= (1 + ∆
1
)
−
1
2
< 1 and β = (1 + ∆
2
)
−
1
2
< 1.
Let
˜
T = T
∗
+ ǫ, ǫ > 0. It is easy to ch eck that
λµ
α
2
2
⌈
λ − α
2
1
α
2
1
β(¯p − 1 )
⌉ =
λµ
α
2
2
(k + 1) ≤ (k + 1)λµ + ǫ =
˜
T ,
(31)
where 0 < α
2
= (1 + ∆
3
)
−
1
2
< 1 with 0 < ∆
3
≤
ǫ
(k+1)λµ
. Now choosing 0 < α = max{α
1
, α
2
} < 1,
we have
¯
T ≤
˜
T , which is straightfo rward from (30)
and (31).
2) If (λ − 1)/(¯p − 1) = k > 0 is an integer. We can
similarly find 0 < α
1
< 1 and 0 < β < 1 such that
⌈
λ − α
2
1
α
2
1
β(¯p − 1 )
⌉ = ⌈
λ − 1
¯p − 1
+ 1⌉ = k + 1 = ⌊
λ − 1
¯p − 1
+ 1⌋
In the same fashion as 1), we can constructively have
0 < α < 1 and 0 < β < 1 such that
¯
T ≤
˜
T .
This proves the first claim.
The second claim we make is that kx
j
(t)k ≤ αδ
j
for any t ≥ t
j
+
¯
T , t ∈ [t
j
, t
j+1
], where we assume
|φ
j
(t)|
[t
j
−τ
k
j
,t
j
]
≤ δ
j
. To show this fact, we ca n choose
¯
δ
1
= δ
j
,
¯
δ
2
=
¯
δ
1
p
¯κ
k
j
/κ
k
j
≥
¯
δ
1
, an d select η = α
¯
δ
1
in
Lemma 3. 3. It is straightforward that 0 < η <
¯
δ
1
≤
¯
δ
2
.
Recall (19) and ( 20), we have
V
k
j
(t, x
j
) ≤ κ
k
j
η
2
, fo r t ≥ t
j
+ T, (32)
where
T =
Nv(
¯
δ
1
)
γ
⌈(v(
¯
δ
1
) − u(η))/a⌉v(
¯
δ
1
)
inf
v
−1
(u(η))≤s≤
¯
δ
2
w(s)
=
¯κ
2
k
j
⌈(v(
¯
δ
1
) − u(η))/a⌉
α
2
w
k
j
κ
k
j
(33)
Combining (22 ) and (32) yields
kx
j
(t)k ≤ αδ
j
, for t ≥ t
j
+ T. (34)
Now cho osing a = β(p
k
j
− 1)κ
k
j
η
2
, we have
T =
¯κ
2
k
j
⌈
¯κ
k
j
κ
k
j
−α
2
α
2
β(p
k
j
−1)
⌉
α
2
w
k
j
κ
k
j
≤
¯
T (35)
Therefore from (34) and (35) we have
|x
j
|
[t
j
+
¯
T ,t
j+1
]
≤ αδ
j
, (36)
as claimed.
Now recall that t
j+1
− t
j
> τ
D
. Therefore t
j+1
− t
j
≥
¯
T +τ
max
≥
¯
T +τ
k
j+1
. Also notice that φ
j+1
(t) = x
j
(t), t ∈
[t
j+1
− τ
k
j+1
, t
j+1
]. We have
|φ
j+1
|
[t
j+1
−τ
k
j+1
,t
j+1
]
= |x
j
|
[t
j+1
−τ
k
j+1
,t
j+1
]
≤ |x
j
|
[t
j
+
¯
T ,t
j+1
]
≤ αδ
j
:= δ
j+1
(37)
and δ
0
is defined as δ
0
:= |φ|
[−τ
max
,0]
≥ |φ|
[−τ
k
0
,0]
. There-
fore we obtain a convergent sequence {δ
i
}, i = 0, 1, 2, . . . ,
where δ
i
= α
i
δ
0
.
Meanwhile, (14) implies
|x
j
|
[t−τ
k
j
,t]
≤
s
¯κ
k
j
κ
k
j
|x
j
|
[t
j
−τ
k
j
,t
j
]
, ∀t ∈ [t
j
, t
j+1
]. (38)
Hence
sup
t∈[t
j
,t
j+1
]
kx
j
(t)k
≤ sup
t∈[t
j
,t
j+1
]
|x
j
|
[t−τ
k
j
,t]
≤
√
λ|x
j
|
[t
j
−τ
k
j
,t
j
]
≤
√
λδ
j
= α
j
√
λδ
0
, (39)
which implies the asymptotic stability of the switched time
delay system Σ
t
with the switching signal q(t) ∈ S
[τ
D
]
.
B. The Case with Delay Dependent Criterion
In a similar fashion, w e can investigate the stability of the
switched time delay sy stem Σ
t
of (2) under the assumption
that Σ
t
∈
¯
A
d
. Hence each candidate system Σ
i
, i ∈ F is
delay-de pendently asymptotically stable satisfying (7 ). We
assume τ
d
D
> 2τ
max
in this scenario. Similar to the pro of
of Theor e m 3.4, we consider an arbitrary switching interval
[t
j
, t
j+1
) of the piecewise switching signal q(t) ∈ S[τ
d
D
] ,
47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeB07.5
2743
where the state variable x
j
(t) defined on this interval obeys
(3). The first order model transformation [7] of (3) results in
˙x
j
(t) = (A
k
j
+
¯
A
k
j
)x
j
(t)
−
¯
A
k
j
Z
0
−τ
k
j
[A
k
j
x
j
(t + θ) +
¯
A
k
j
x(t + θ − τ
k
j
)]dθ(40)
where the initial condition ψ
j
(t) is defined as ψ
j
(t) =
x
j−1
(t), t ∈ [t
j
−2τ
k
j
, t
j
] for j ∈ Z
+
, and ψ
0
(t) defined by
ψ
0
(t) =
φ(t), t ∈ [−τ
max
, 0]
φ(−τ
max
), t ∈ [−2τ
max
, −τ
max
)
By using the Lyap unov-Razumikhin function (21), we obtain
the time derivative of V
k
j
(t, x
j
(t)) along the trajectory of
(40)
˙
V
k
j
(t, x
j
) = x
T
j
(t)[P
k
j
(A
k
j
+
¯
A
k
j
) + (A
k
j
+
¯
A
k
j
)
T
P
k
j
]x
j
(t)
−
Z
0
−τ
k
j
[2x
T
j
(t)P
k
j
¯
A
k
j
(A
k
j
x
j
(t + θ)
+
¯
A
k
j
x
j
(t + θ − τ
k
j
)]dθ.
Assume V
k
j
(t + θ, x
j
(t + θ)) < p(V
k
j
(t, x
j
(t))) for ∀θ ∈
[−2τ
k
j
, 0], where p(s) = p
k
j
s, p
k
j
> 1, we have [3], [14]
˙
V
k
j
(t, x
j
) ≤ −x
T
j
(t)S
d
k
j
x
j
(t), (41)
where
S
d
k
j
:= − {P
k
j
(A
k
j
+
¯
A
k
j
) + (A
k
j
+
¯
A
k
j
)
T
P
k
j
+ τ
k
j
[α
−1
k
j
P
k
j
¯
A
k
j
A
k
j
P
−1
k
j
¯
A
T
k
j
A
T
k
j
P
k
j
+ β
−1
i
P
k
j
(
¯
A
k
j
)
2
P
−1
k
j
(
¯
A
T
k
j
)
2
P
k
j
+ p
k
j
(α
k
j
+ β
k
j
)P
k
j
]}. (42)
Because Σ
t
∈
˜
A
d
, we have S
d
k
j
> 0 from Lemma 2 .2.
Therefore we can select w(s) = w
d
k
j
s
2
in Lemm a 3.1, such
that (9) holds, where w
d
k
j
:= σ
min
[S
d
k
j
] > 0.
Theorem 3.5: Let the dwell time be τ
d
D
:= T
∗
d
+ 2τ
max
,
where
T
∗
d
:= λµ
d
⌊
λ − 1
¯p − 1
+ 1⌋, (43)
with
µ
d
:= max
i∈F
¯κ
i
w
d
i
(44)
and the other parameters are the same as those defined
in Theorem 3.4. Then, the system (2) with Σ
t
∈
˜
A
d
is
asymptotically stable for any switching rule q(t) ∈ S[τ
d
D
].
Proof. We can apply similar arguments used in the proof
of Theor e m 3.4 to o btain the fo llowing in equality:
sup
t∈[t
j
,t
j+1
]
kx
j
(t)k ≤
√
λδ
d
j
, (45)
where |ψ
j
(t)|
[t
j
−2τ
k
j
,t
j
]
≤ δ
d
j
, and δ
d
j+1
= αδ
d
j
. Note that
δ
d
0
can be selected as
δ
d
0
:= |ψ|
[−2τ
max
,0]
= |φ|
[−τ
max
,0]
= δ
0
.
It is clear that |ψ|
[−2τ
k
0
,0]
≤ δ
d
0
, which further implies δ
d
j
=
δ
j
, j ∈ Z
+
∪ { 0}. The upper bound of the state variable
x(t) of the switched time delay system s Σ
t
is bounded by
a decre asing sequence {δ
i
}, i = 0, 1, 2 , . . . converging to
zero, which implies the asymptotic stability an d proves this
theorem.
The dwell time ba sed stability analysis proposed in this
paper is g eneral in the sense that it can be used for other
stability re sults based on Razumikhin theorems as long as
the corr e spondingly Lyapu nov fu nctions are in quadratic
forms. Particularly, Theorem 3.5 can be extended easily
to the case where Σ
t
has time-varying time delays and
parameter uncertainties, which has important applications
such as TCP (Transmission Control Pro tocol) congestion
control of computer networks [13], [25].
IV. CONSERVATISM ANALYSIS
The dw e ll time based stability results had been obtained
for switched linear systems f ree of delays [10], [19]. It
is in te resting to compare the conservatism of the results
presented in this paper with those for de la y free systems.
In fact, one extreme case of the switched system Σ
t
is
τ
i
= 0 and
¯
A
i
= 0 for i ∈ A, which corresponds to the
delay free scenario. For each candidate system ˙x = A
i
x, a
sufficient and nec e ssary co ndition to guarantee asymptotic
stability is ∃P
i
= P
T
i
> 0, such that Q
i
:= −(A
T
i
P
i
+
P
i
A
i
) > 0. Correspondingly a dwell time based stability for
such switched delay free system is q(t) ∈ S
[˜τ
D
]
, where
˜τ
D
= ˜µ ln λ, (46)
where λ is defined by (27) and
˜µ := max
i∈F
¯κ
i
˜w
i
, (47)
where ˜w
i
:= σ
min
[Q
i
] > 0.
On the other hand in our case, for τ
i
= 0 and
¯
A
i
= 0, we
observe that
lim
α
i
→0
+
S
i
= lim
α
i
,β
i
→0
+
S
d
i
= Q
i
, i ∈ F (48)
from (25) and (42), which indicates µ = µ
d
= ˜µ by (28),
(44), and (47). Accordingly we can select p
i
> 1 , i ∈ F
sufficiently large such that ⌊
λ−1
¯p−1
+ 1⌋ = 1 in (29) and (43),
and obtain
τ
D
= T
∗
= λµ = λµ
d
= T
∗
d
= τ
d
D
. (49)
Therefore
τ
D
= τ
d
D
= λ˜µ > ˜µ ln λ = ˜τ
D
. (50)
The dwell times derived for switched time delay sy stems
are proportional to λ, as opposite to the logarithm of λ
for switched delay free systems. This gap is due to the
fact that asymptotic stability for linear delay free systems
implies exponential stability. However, for time delay sys-
tems, the sufficient stability conditions based on Lyapunov-
Razumikhin the orem do not guarantee exponential stability.
As a m atter of fact, the exp onential estimates for time delay
systems require additional assumptions besides asymptotic
stability [15].
47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008 WeB07.5
2744