Chapter 3
3-2.
Consider the switched time-delay large-scale systems
N
N
j i
j 1
j 1
xi (t )  A (t ) _ ii xi (t )   A (t ) _ ij x j (t )   B (t ) _ ij x j (t   j (t ))
(3.1a)
xi (t )   i (t ), t  [ , 0],
(3.1b)
N
where xi (t )  R ni is the state vectors of the ith subsystem, i=1, 2,…,N, n   ni , A (t ) _ ii ,
i 1
A (t ) _ ij ,

and
x (t )  x1T (t )
B (t ) _ ij ,
are
some
constant
matrices
of
compatible
dimensions,
T
x2T (t )  x TN (t ) ,  (t ) ： R n  {1, 2, , r} is a piecewise constant function of
time, called a switch signal, i.e., the matrix
A1 _ ii , A2 _ ii ,, Ar _ ii
Al _ ii , l  {1, 2,  , r}.
belonging
to
the
A (t ) _ ii ,
set
switches between matrices
A  {A1 _ 11, A2 _ ii ,, Ar _ ii }
and
The matrix B (t ) _ ij switches between matrices B1 _ ij , B2 _ ij ,, Br _ ij
belonging to the set B  {B1 _ ij , B2 _ ij ,, Br _ ij } and
Bi , i, j  {1, 2,, N } .
A (t ) _ ii ,
A (t ) _ ij and B (t ) _ ij , are matrices of compatible dimensions.   0 is the time-delay duration.
 i (t ) is a vector-valued initial continuous function defined on the interval [  , 0] , and finally
 i (t ) , defined on    t  0, is the initial condition of the state.
As we have indicated, very few research articles have ever discussed the stability of the
switched time-delay large-scale system (3.1). To facilitate our analysis later, some helpful lemmas
are given below first.
In the light of Lemma 2.3, it is obvious that the stability of the switched time-delay system (3.1)
is equivalent to that of the switched large-scale system
1
N
w i (t )  ( Al _ ii  zBl _ ii ) xi (t )   ( Al _ ij  zBl _ ij ) w j (t )
j i
j 1
N
 Al _ ii wi (t )   Al _ ij w j (t )
z 1
j i
j 1
(3.2)
where Al _ ii  Al _ ii  zBl _ ii and Al _ ij  Al _ ij  zBl _ ij .
To facilitate our analysis later, one helpful lemma is given below first.
Lemma 3.1 [
]： For matrix A  Rnn , B  Rnn , the following relation holds:
exp[( A  zB)t ]  exp[ ( A  zB)t ]
 exp[(  ( A)  B )t ] ,
z 1
(3.3)
Without loss of generality, we assume that the switched time-delay large-scale system (3.1) at
least has one individual system whose  ( Al _ ii )  Bl _ ii
value, i=1, 2, …, N, are less than zero,
and
i  min (  ( Al _ ii )  Bl _ ii ) when  ( Al _ ii )  Bl _ ii  0
(3.4a)
i  max (  ( Al _ ii )  Bl _ ii ) when  ( Al _ ii )  Bl _ ii  0
(3.4b)
1 l  r1
r1  l  r
we assume Ti (t ) (or Ti (t ) ) is the total activation time of the ith subsystem whose
 ( Al _ ii )  Bl _ ii
values are no less than zero (total activation time of the ith subsystem whose
 ( Al _ ii )  Bl _ ii
values are less than zero). The switching law of the ith subsystem can be
definited as following:
inf [Ti (t ) / Ti (t )]  (i  *i ) /( i  *i )
(3.5)
t  t0
where i  (0, i ) ,
*i  (i , i ), 1  i  N .
Furthermore, we assume T  (t ) (or T  (t ) ) is the total activation time of individual systems
whose  ( Al _ ii )  Bl _ ii
values are less than zero for i=1, 2,…, N (the total activation time of
individual systems at least has one subsystem whose  ( Al _ ii )  Bl _ ii
values is not less than
zero). The total activation time ratio between T  (t ) and T  (t ) can be called a switching law of
the switched time-delay large-scale system (3.1). Therefore, this paper is concerned with the
following problem: find the total activation time ratio between T  (t ) and T  (t ) which
guarantee that the switched time-delay large-scale system (3.1) is globally exponentially stable with
2
stability margin  .
Furthermore, we definite an auxiliary matrix as following:
Y (t )  CY (t )
(3.6)
where
Y (t )   y1 (t )
y 2 (t )  y N (t )T ,
and C  [cij ] , cii  i , i=j; cij  aij* , i  j
yi ( t )  R
which aij*  max ( Al _ ij  Bl _ ij )
for i, j=1, 2, …,N
l 1,2,... r
A sufficient condition for stability of the switched time-delay large-scale systems (3.1) is
established in the following theorem by using the time-switched method.
Theorem 3.1：Suppose the switched time-delay large-scale system (3.1) at least has one individual
system whose  ( Al _ ii )  Bl _ ii
value, i=1, 2, …, N, are less than zero. The switched time-delay
large-scale system (3.1) is globally exponentially stable with stability margin  , if the system (3,6)
is stable and the system (3.1) satisfies the following switching law:
inf [T  (t ) / T  (t )]  max (Ti (t ) / Ti (t ))
t  t0
(3.7)
1 i  N
where   min (i ).
1 i  N
Proof：The trajectory response of system (3.2) can be written as follows:
wi (t )  e
Ap
d 1 _ ii
 tt1 e
Ap
 tt1 e
Ap
( t  td ) Ap _ ii ( td  td 1 )
A
(t  t )
e d
 e p1 _ ii 1 0 wi (t0 )
d 1 _ ii
( t  t d ) AP
0
d 1 _ ii
(t
d _ ii d
e
(t  t d ) AP
(t
d _ ii d
e
0
 t d 1 )
 t d 1 )
e
e
Ap
1
_ ii ( t1  s )
N
 Ap _ ij w j ( s )ds
1
j i
j 1
A p 2 _ ii ( t1  s ) N
 Ap2 _ ij w j ( s )ds
j i
j 1

 tt e
Ap
d 1 _ ii
(t  s ) N
d
 Ap _ ij w j ( s )ds
d 1
(3.8)
j i
j 1
Where pd  {1, 2,  , N }.
Let
vi (t )  e
Ap
d 1 _ ii
( t  t d ) A p _ ii ( t d  td i 1 )
d
e
 e
A p _ ii ( t1  t 0 )
1
vi (t0 )
Performing the norm on both sides of Eq. (3.9) and in view of lemma 3.1, we can obtain
vi (t )  e
Ap
d 1 _ ii
(t  t d )
e
Ap
(t
d _ ii d
 t d 1 )
 e
Ap
1
3
_ ii ( t1  t 0 )
vi (t0 )
(3.9)
e
(  ( Ap
d 1 _ ii
) B p
d 1 _ ii
)( t  t d ) (  ( A p
e
)
d _ ii
Bp
d _ ii
)( t d  t d 1 )
 e
(  ( Ap
1 _ ii
) B p
1
_ ii
 
 
 ei T (t0 ,t )  i T (t0 ,t ) vi (t0 )
)( t1  t 0 )
(3.10)
Furthermore, the switching law (3.5) of the ith subsystem can be become as
iTi (t )  iTi (t )  *i (Ti (t )  Ti (t ))
 *i (t  t0 )
(3.11)
Hence, the following inequivalent equation can be obtained
vi (t )  e  i (t  t0 ) vi (t0 )  e i (t t0 ) vi (t0 )
*
(3.12)
and
N
N
j i
j 1
j i
j 1
wi (t )  e  i (t  t0 ) wi (t0 )  tt1 e  i (t  s )  aij* w j ( s ) ds  tt 2 e  i (t  s )  aij* w j ( s ) ds
0
1
N
   tt e  i (t  s )  aij* w j ( s ) ds
d
j i
j 1
N
 e  i (t  t0 ) wi (t0 )  tt e  i (t  s )  aij* w j ( s ) ds
0
j i
j 1
(3.13)
Let
N
yi (t )  e  i (t  t0 ) yi (t0 )  tt e  i (t  s )  aij* y j ( s ) ds
0
j i
j 1
(3.14)
Then, the Eq.(3.14) is the solution of system.(3.15).
N
y i (t )  i yi (t )   aij* y j (t )
(3.15)
j i
j 1
The system (3.15) also can be expressed as the system (3.6). Hence, the exponentially stable of
yi (t ) implies that of wi (t ) by the comparison theorem.
It is clear that if the system (3.6) is stable and the switching law is as (3.7), then the switched
time-delay large-scale system (3.1) is also globally exponentially stable with stability margin  . 
The above method can be extended to the large-scale switched system without time-delay.
Consider the switched large-scale systems
N
xi (t )  A (t ) _ ii xi (t )   A (t ) _ ij x j (t )
(3.16)
j i
j 1
4
vi (t0 )
i.e.,
N
xi (t )  Al _ ii xi (t )   Al _ ij x j (t )
(3.17)
j i
j 1
Without loss of generality, we assume that the switched large-scale system (3.17) at least has one
individual system whose  ( Al _ ii ) value, i=1, 2, …, N, are less than zero, and
i  min (  ( Al _ ii ) ) when  ( Al _ ii )  0
(3.18a)
i  max ( ( Al _ ii )) when  ( Al _ ii )  0
(3.18b)
1 l  r1
r1 1 l  r
we assume Ti (t ) (or Ti (t ) ) is the total activation time of the ith subsystem whose
 ( Al _ ii ) values are no less than zero (total activation time of the ith subsystem whose
 ( Al _ ii )
values are less than zero). The switching law of the ith subsystem can be definited as following:
inf [Ti (t ) / Ti (t )]  (i  i* ) /( i  i* )
(3.19)
t  t0
where i  (0, i ) ,
i*  (i , i ), 1  i  N .
Furthermore, we assume T  (t ) (or T  (t ) ) is the total activation time of individual systems
whose  ( Al _ ii ) values are less than zero for i=1, 2,…, N (the total activation time of individual
systems at least has one subsystem whose  ( Al _ ii ) values is not less than zero). The total
activation time ratio between T  (t ) and T  (t ) can be called a switching law of the switched
large-scale system (3.17). Therefore, this paper is concerned with the following problem: find the
total activation time ratio between T  (t ) and T  (t ) which guarantee that the switched
large-scale system (3.17) is globally exponentially stable with stability margin  .
Furthermore, we definite an auxiliary matrix as following:
Y (t )  C Y (t )
(3.20)
where
Y (t )   y1 (t )
y2 (t )  y N (t )T ,
and C  [cij ] , cii  i , i=j; cij  aij* , i  j
yi ( t )  R
for i, j=1, 2, …,N which aij*  max ( Al _ ij )
l 1,2,... r
A sufficient condition for stability of the switched large-scale systems (3.17) is established in the
following theorem by using the time-switched method.
Corollary 3.1: The nominal large-scale switched systems (3.17) without time-delay, i.e., Bl _ ij  0 .
Suppose the switched time-delay large-scale system (3.17) at least has one individual system whose
5
 ( Al _ ii ) value, i=1, 2, …, N, are less than zero. The switched time-delay large-scale system (3.17)
is globally exponentially stable with stability margin  , if the system (3.20) is stable and the
system (3.17) satisfies the following switching law:
inf [T  (t ) / T  (t )]  max (Ti (t ) / Ti (t ))
t  t0
(3.21)
1 i  N
where   min (i ).
1i  N
The above method can be extended to the time-delay switched system. A sufficient condition for
stability of the switched time-delay systems is established in the following theorem by using the
time-switched method.
Consider the switched time-delay systems
x (t )  A (t ) x(t )  B (t ) x(t   )
(3.22a)
x(t )   (t ), t  [  , 0],
(3.22b)
Without loss of generality, we assume that the switched time-delay system (3.22) at least has
one individual subsystem whose  ( Al )  Bl
value is less than zero, the that of remaining
individual subsystem are no less than zero, i.e.,
( Al )  Bl  0, 1  l  r1 ,
(3.23a)
 ( Al )  Bl  0, r1  1  l  r ,
(3.23b)
  min(  ( Al )  Bl ), 1  l  r1
(3.24a)
  max(  ( Al )  Bl ), r1  1  l  r
(3.24b)
and
Furthermore, we assume T  (t ) (or T  (t ) ) is the total activation time of individual subsystems
whose  ( Al )  Bl
 ( Al )  Bl
values are no less than zero (total activation time of individual systems whose
values are less than zero). The total activation time ratio between T  (t ) and T  (t )
can be called a switching law of the time-delay switched system (3.22). Therefore, this paper is
concerned with the following problem: find the total activation time ratio between T  (t ) and
T  (t ) which guarantee that the switched time-delay system (3.22) is globally exponentially stable
with stability margin  for an arbitrary switching singal  (t ).
Corollary 3.2 [ ]: Suppose the switched time-delay system (3.22) exists at least an individual
subsystem whose  ( Al )  Bl
value is less than zero. The switched time-delay system (3.22) is
globally exponentially stable with stability margin  , if the system (3.22) satisfies the following
switching law:
6
inf [T  (t ) / T  (t )]  (  * ) /(   * )
t t0
(3.25)
where   (0,  ) and *  ( ,  ).
3-3. Numerical example
Individual system 1 (l=1):
 0.1
 10 0 
0.5 1 
 0
x1 (t )  
x1 (t )  
x2 (t )  


 x3 ( t )
0

9
0
.
5
0
.
2

0
.
2
0
.
3






0.5 0.4 
0.8 1 
  1  0.9

x1 (t   )  
x2 (t   )  


 x3 ( t   )
0.1  0.3
0.1 0.5
 0.1  2 
0.2 0.7
 10 2 
 1  0.3
x 2 (t )  
x1 (t )  
x2 (t )  
x3 ( t )


 12
0 
 0.1 0.3
 0
0.4
0.4 
 1.1 0.2
 0.8
  1 0.5

x
(
t


)

x
(
t


)

1
2

 0.5  0.7
0.7 0.9 x3 (t   )
 0.5 0.7




 0.1 0.7
0 0.4
 13 1 
x3 (t )  
x1 (t )  
x2 (t )  


 x3 (t )
0.9  1
1 0 
 0.5  12
 2 0.3
 0.1  1
1.1 0.1

x1 (t   )  
x2 (t   )  


 x3 ( t   )
0.5 0.7
 0.7 2 
 0 0.5
(3.26a)
Individual system 2 (l=2):
0 
 1 0.5
 0.2
  1  0.2
x1 (t )  
x
(
t
)

x
(
t
)

x3 ( t )
1
2

 0
 0.5
 0.3
1 
0.8 1.2


 0.5 1 
0.6 0.7
 0.1 0.2

x1 (t   )  
x2 (t   )  
x3 ( t   )


0.5
 0.7 0.9
 0.1 1 
 1
0.7  0.1
 1 1 
0.2 0.1
x 2 (t )  
x1 (t )  
x2 (t )  


 x3 (t )
 1  0.2
0.5 0.8
0.9 0.7
 0.1 0.2
 2 1
0.1 1 

x1 (t   )  
x2 (t   )  


 x3 ( t   )
 0.2 1.1
 0.1 1
0.8 0.9
 1.1 0.7 
 2 0.5 
 10 0 
x3 (t )  
x1 (t )  
x2 (t )  
x3 ( t )


 2
 0.1  0.4
  1  0.8
 1
1.1 0.2
0.7 0.8 
0.1 0.4 

x1 (t   )  
x2 (t   )  


 x3 (t   )
0
.
7
0
.
9
0
.
1

0
.
5
0
.
3

0
.
5






Individual system 3 (l=3):
 8 0 
 0.5 0.1
0.1 0.4
x1 (t )  
x
(
t
)

x
(
t
)

1
2

 0.9  2
0.5 0.6 x3 (t )
 0  9




7
(3.26b)
 0.1 0.1
 0.2 0.3
 1  0.2

x1 (t   )  
x2 (t   )  


 x3 ( t   )
 0.2 0.3
 0.2 0.5
0.1 0.3 
0.4 
1
0.1 0.4
 1 0.7
x 2 (t )  
x1 (t )  
x2 (t )  


 x3 ( t )
 1  0.1
0.6 1 
0.5  1
 0.1 0.8 
 0.1  0.1
0.1  0.2

x
(
t


)

x
(
t


)

x3 ( t   )
1
2

0.2 0.4 
1
1 
0.7  0.1



 0.1  0.1
0.2 0.3
0.4 0.1
x3 (t )  
x1 (t )  
x2 (t )  


 x3 ( t )
1 
 0.5
 0.1 0.2
  1  2
 0.1 0.5
0.1 0.1 
  1  2

x1 (t   )  
x2 (t   )  


 x3 (t   )
 0.1 0.8
  1  0.5
 1 0.5
(3.26c)
According to the normal test of stability for the large-scale system, the individual system 1 is
stable, the individual systems 2 and 3 are both unstable. We can easily calculate r1  7.6382,
r1  3.1088,
r2  8.3814,
r2  3.3419,
2.8021 2.6298
 5

matrix c  2.3730
6
2.6822


3.5487 3.3522  6.5 
r3  8.1258 and r3  2.9037. Hence, the stable
can be obtained by choosing r1*  5,
r2*  6 and
r3*  6.5. Furthermore, the switching law of the ith subsystem, i=1, 2, 3, can be calculated as
T1 / T1  3.0736, T2 / T2  3.9229, T3 / T3  5.784. Therefore, the total activation time ratio
for the switching law is
T  (t ) / T  (t )  5.784
(3.27)
Finally, the switched large-scale time-delay system (3.26) is globally exponentially stable with
stability margin   (0, 5) under the switching law (3.27). In order to satisfy the switching law
(3.27), we choose the total activation time ratio is 6：1. The activation time of individual system 1 is
0.6 sec and the total activation time of individual system 2 and 3 is 0.1 sec, respectively. The
trajectory of the time- delay switched system is shown in figure 3-1 with initial state 1 2
T
and
time-delay 0.1 sec.
3-4. Conclusion
According to the time-switched method and the comparison theorem, the sufficient conditions
of the switched laws are presented and the total activation time ratio under the switching laws is
8
required to be no less than a specified constant, such that the switched time-delay system is
delay-dependent exponentially stable with stability margin  . The results can be also extended to
the uncertain and interval cases.
9