Chapter 5
Stability analysis and synthesis of switched
discrete-time systems
5.1 On delay-independent analysis of time-delay
switched discrete-time systems
In this section present a method which is based on Lyapunov stability theorem to
study the problems of stability analysis and switching law design for time-delay
switched discrete-time systems with the state-driven switching. On delay independence
stability analysis, one may extend to the case of arbitrary subsystems of time-delay
switched discrete-time system and develop the simple switching rule to stabilize the
time-delay switched discrete-time system and construct linear matrix inequalities based
design procedures for stability analysis. The particular method can be applied to cases
whose all individual subsystems are unstable. Numerical examples are exploited to
illustrate the proposed schemes.
5.1.1 Stability analysis and switching law design
Consider a time-delay switched discrete-time system given by
x(k  1)  A ( x(k )) x(k )  Ad  ( x(k )) x(k  h)
159
(5.1a)
x( s )   ( s) , s  h,
, 1, 0
(5.1b)
where x(k )  R n is state, A ( x(k )) , Ad ( x(k ))  R nn are known constant matrices
with appropriate dimensions,  :{h,
, 1, 0}  R n denotes the initial function and
h  0 is a positive integer representing the constant time delay,  ( x ( k )) ：
R n  {1, 2,
, N} is a piecewise constant scalar function of state, called a switch
signal, i.e., the matrix A ( x( k )) switches between matrices A1 , A2 ,, AN belonging
to the set { A1, A2 ,
, AN } and Al , l  {1, 2,
between matrices Ad1, Ad 2 ,
Adl , l  {1, 2,
, N }, the matrix Ad  ( x(k )) switches
, Ad N belonging to the set { Ad1, Ad 2 ,
, AdN } and
, N }, i.e., the time-delay switched discrete-time system (5.1) can be
described as follows:
x(k  1)  Al x(k )  Adl x(k  h) , l  1, 2,
,N
(5.2)
Definition 5.1: The time-delay switched discrete-time system (5.1) is said to
asymptotically stable via switching if there exists a switching law  (x) , a positive
definite Lyapunov function V (k )  xT (k ) Px(k ) and a positive constant  such that
V (k )  V (k  1)  V (k )    xT (k ) x(k )
(5.3)
holds for all trajectories of time-delay switched discrete-time system (5.1).
This section, the main approaches used the Lyapunov stability theorem and linear
matrix inequalities (LMIs) to study the problem of delay independence stability and
switching law design for time-delay switched discrete-time systems. In the study of
160
time-delay switched discrete-time systems, the main problems are how to constructively
design a switching rule and how to derive sufficient stability conditions which can
guarantee the stability of the switched system under the switching rule.
Theorem 5.1: Given a positive constant delay h, there exists a switching law for
time-delay switched discrete-time system (5.1) such that the system is asymptotically
stable if there exist matrices P  0 and Q  0 , positive constants  l  0, 1 ,
(1  l  r ) satisfying
N
 l  1
such that the following matrix inequalities
l 1
N
N

T
   l Al PAl  P  Q
  l AlT PAdl 
l 1

l 1

0
N
N
T
T

  l Adl PAl
  l Adl PAdl  Q 

l 1
l 1

(5.4)
Proof: If there exist matrices P  0 and Q  0 , positive constants  l  0, 1 ,
(1  l  r ) satisfying
N
 l  1
such that the inequality (5.4) holds, then, for
l 1
x(k )  R n , x(k )  0 , one may get :
 xT  k 

N
N

T
T
  l Al PAdl 
   l Al PAl  P  Q
l 1
l 1
  x  k  
xT  k  h   

N
N
 x  k  h 
T
T
  l Adl PAl
  l Adl PAdl  Q  

l 1
l 1


 N
  xT  k  xT  k  h     l

 l 1

 AT PA  P  Q
l
 l
T

Adl PAl


T
Adl PAdl  Q 
AlT PAdl
  x  k  

0
x
k

h



 
(5.5)
161
Therefore, it follows that for any k , at least there exists an l  {1, 2,
 AT PA  P  Q
l
 xT  k  xT  k  h    l


T
Adl PAl

  x k  


T
Adl
PAdl  Q   x  k  h  
AlT PAdl
  AT 
T
T


 x  k  x  k  h    l  P  Al

  AT 
  dl 

, N } such that
  P  Q 0    x  k  
Adl   


Q    x  k  h  
 0


T
 xT  k  AlT  xT  k  h  Adl
P  Al x(k )  Adl x(k  h)
 xT  k P x  k T x

k  Q x  k T x  k
 h Q x 0k
(5.6)
h
For time-delay switched discrete-time system (5.1), one may choose a Lyapunov
functional candidate of the form:
V xk   x T k Pxk  
k 1
 xT q Qx q 
(5.7)
qk d
Then, one can find the difference equation of Lyapunov function (5.8):
V ( x(k ))  V ( x(k  1))  V ( x(k ))


T
 xT  k  AlT  xT  k  h  Adl
P  Al x(k )  Adl x(k  h)
 xT  k P x  k T x

k  Q x  k T x  k
 h Q x
k
h
(5.8)
In the light of (5.8), (5.6) also is the difference equation of Lyapunov function.
Therefore, if inequality (5.4) is satisfied then V ( x(k ))  0 . The time-delay switched
162
□
discrete-time system is delay-independent asymptotically stable.
Switching Law: Time-delay switched discrete-time system (5.1) is switched to or stay
at mode i at sampling step k if (5.9) is satisfied at k.


T
V xk   xT k AlT  xT k  d Adl
PAl x(k )  Adl x(k  d )
 xT k P  Q xk   xT k  d Qx k  d   0 ， l  1,2,, N
Now, the above result can simplify parameter, and reduce 1,  2 ,
(5.9)
,  N to  . Thus, it
is easy to analyze the time-delay switched discrete-time systems.
Corollary 5.1: Given a positive constant delay h, there exists a switching law for
time-delay
switched
discrete-time
system
(5.1)
such
that
the
system
is
delay-independent asymptotically stable if there exist matrices P  0 and Q  0 ,
positive constants   0 , such that the following matrix inequalities
 N 1
 l l 1 T
(  (1   )  Al PAl )
 l 1
 (1   )1 N  N 1 AT PA  P  Q
N
N


N 1

 l l 1 T
 (  (1   )  Adl PAl )
l 1

  (1   )1 N  N 1 AT PA
dN
N

N 1


l 1

1 N N 1 T
 (1   )

AN PAdN 

0
N 1

T
(  (1   )  l  l 1 Adl
PAdl )

l 1

1 N N 1 T
 (1   )

AdN PAdN  Q 
(  (1   )  l  l 1 AlT PAdl )
(5.10)
Proof: In the light of Lemma 4.3 and 4.4, then
163
N
N 1
 l  ( 
l 1
l 1 (1  
1
1 l 1
)
(1   )
)
1
(1  
1 N 1
(5.11)
)
Substitute (5.11) into (5.4), then proof of Corollary 1 is completed.
□
Remark 5.1: Theorem 5.1 and Corollary 5.1 show that the delay-independence stability
of a time-delay switched discrete-time system can be shown by find common symmetric
positive definite matrices P, Q. And Theorem 5.1 and Corollary 5.1 give a sufficient
condition for the global stability of time-delay switched discrete-time systems. The
stability condition is reduced to that of normal linear discrete time-delay systems when
N=1.
Remark 5.2: In the light of Corollary 5.1, one can simplify parameter, and reduce
1,  2 , ,  N to  . Thus, it is easy to analyze the time-delay switched discrete-time
system. And one may constructively design a switching rule which can guarantee the
stability of the switched systems.
5.1.2 Numeric examples
Example 5.1: Consider the time-delay switched discrete-time system composed of two
subsystems given as:
0.12 0.12 
Subsystem 1: x(k  1)  A1x(k )  Ad1x(k  h) , A1  
,
 1.11
 0
0 
0.05
Ad1  

 0.01 0.02
0
 1.2
Subsystem 2: x(k  1)  A2 x(k )  Ad 2 x(k  h) , A1  
,
 0.16 0.1
164
(5.12a)
0.03 0.1 
Ad 2  
0.02
 0
(14b)
Via normal tests of stability for individual subsystem, subsystem 1 and subsystem 2
are stable. In view of the stability conditions of Corollary 5.2, inequalities (5.10) can be
written as follows:
(1   ) 1 A1T PA1  (1   ) 1A2T PA2  P  Q


 (1   ) 1 AT PA  (1   ) 1AT PA
d1 2
d2
2

(1   ) 1 A1T PAd1  (1   ) 1A2T PAd 2


0
1 T
1 T
(1   ) Ad1PAd1  (1   ) Ad 2 PAd 2  Q 

(5.13)
By using LMI Tool Box of MATLAB [74], then
1.7740 0.1073
0.2899 0.0673
and Q  
P

 , with   1 .
0.1073 1.7740 
0.0673 0.5197
Therefore, the time-delay switched discrete-time system can be stabilized by the
following switching law.
Switching Law: Switched discrete time-dealy system is switched to or stay at mode i
at sampling step k if (5.14) is satisfied at k.


T
V xk   xT k AlT  xT k  d Adl
PAl x(k )  Adl x(k  d )
 xT k P  Q xk   xT k  d Qx k  d   0 ， l 1, 2
(5.14)
With h  5 , switching during k  0,1,,25 and initial value x(0)   20 10 ,
T
the trajectories of time-delay switched discrete-time system are shown in Fig.5.1a. Via
165
switching law, every subsystem must orchestrate the switching between them in
accordance with the states. The switching signal is shown in Fig. 5.1b. Therefore, if
select positive constant   1 and satisfy inequalities (5.13) then the system is
delay-independent asymptotically stable.
state response :x1,x2
10
x2
5
state
0
-5
-10
-15
-20
x1
0
5
10
15
20
25
k
Fig.5.1a: State responses of x1 and x2 with h=5,   1 in Example 5.1
switching signals
[1]: subsystem
1
[2]:subsystem
2
2
1
0
5
10
15
k
166
20
25
Fig.5.1b: The switching signals with h=5,   1 in Example 5.1
Remark 5.3: This example is exploited to illustrate the proposed schemes, stability
conditions
that
guarantee
the
time-delay
switched
discrete-time
system
is
delay-independent asymptotically stable for construction of stabilizing switching law.
The particular method can be applied to cases whose all individual subsystems are
unstable.
Example 5.2: Consider the time-delay switched discrete-time system composed of two
subsystems given as [40, 41]:
Subsystem 1: x(k  1)  A1x(k )  Ad1x(k  h) ,
 0.545  0.43
 0.24 0.07
, Ad1  
A1  


 0.185  0.61
 0.12 0.09
(5.15a)
Subsystem 2: x(k  1)  A2 x(k )  Ad 2 x(k  h) ,
 0.455  0.37
 0.36 0.13
A2  
, Ad 2  


0.59 
 0.215
 0.08 0.11
(5.15b)
In view of the stability conditions of Corollary 5.1, by using LMI Tool Box of
MATLAB, then
18.2879 4.9332 
8.7802 4.4101
P
and Q  

 , with   2 .
 4.9332 22.6862
 4.4101 7.5462
Therefore, the time-delay switched discrete-time system (5.15) can be stabilized by the
switching law (5.14).
This example is exploited to guarantee the time-delay switched discrete-time system is
167
delay-independent asymptotically stable for construction of stabilizing switching law.
Remark 5.4: The example in [40] is stable of time-delay h=2 for any arbitrary
switching function. Then [41] proved the result of [41] that system stability is
irrespective of the value h of the time-delay for any arbitrary switching function. And
then, the present result demonstrated that guarantee the time-delay switched
discrete-time system is delay-independent asymptotically stable for construction of
stabilizing switching law.
5.1.3 Summary
This section, using Lyapunov stability theorem to study the stability analysis of a
class of time-delay switched discrete-time system. Stability conditions that guarantee
the time-delay switched discrete-time system is delay-independent asymptotically stable
for construction of stabilizing switching law be derived. The main advantages of the
present results are that simplify parameter, and reduce 1,  2 ,
,  N to  then it is
easy to analyze the time-delay switched discrete-time system, can be applied to
individual subsystems whose includes unstable subsystems, can extend to the case of
arbitrary subsystems of switched discrete delay system and develop the simple
switching rule to stabilize the time-delay switched discrete-time system and construct
LMI-based design procedures for stability analysis. The subject is interesting and
important, and it will be attracting increasing attention in future.
5.2 Analysis and Synthesis of switched discrete-time
systems
168
This section attempts to investigate the stabilization and switching law design for the
switched discrete-time systems. A theoretical study of the stabilization and switching
law design has been performed using the Lyapunov stability theorem and genetic
algorithm. The present results demonstrated that can be applied to cases when all
individual subsystems are unstable. Finally, some examples are exploited to illustrate
the proposed schemes.
Notation: The following notations will be used throughout the section: (A) stands for
the eigenvalues of matrix A, A  Max[ ( A)] ( A  Min[ ( A)] ) means the maximum
(minimum) eigenvalue in matrix A,
A  Max[ ( AT A)]1/ 2 ,
A
denotes the norm of matrix A, i.e.,
AT is the transpose of matrix A, diag{…} denotes a
block-diagonal matrix.
5.2.1 Stability analysis and switching law design
Consider a switched discrete-time system given by
x(k  1)  A ( x(k )) x(k ) ,
x(0)  x0
(5.16)
where x(k )  R n is state, A ( x(k ))  Rnn , x0 is the initial state,  ( x ( k )) ：
Rn  {1, 2,
, N} is a piecewise constant scalar function of state, called a switch
signal, N is the number of the individual system, i.e., the matrix A ( x( k )) switches
between matrices A1 , A2 ,, AN belonging to the set A  { A1 , A2 ,, AN } and Ai ,
i  {1, 2,
, N }, essentially, the switched discrete-time system (5.16) can be described
as follows:
169
x(k  1)  Ai x(k ) , i  1, 2,  , N
(5.17)
Lemma 5.1: There exists a switching law for the switched discrete-time system (5.17)
such that the system (5.17) is asymptotically stable if there exist positive constants  i
(1  i  N ) satisfying iN1i  1 and the following system is an asymptotically stable
system.
N
x(k  1)   i Ai x(k )
(5.18)
i 1
Proof: There exists a Lyapunov function V (k )  xT (k ) Px(k ) such that
N
N
i 1
i 1
V ( k )  xT (k )[(  i Ai )T P(  i Ai )  P]x(k )
(5.19)
Using Lemma 4.3 and 4.4, then
N
N
i 1
i 1
V (k )  xT (k )[(  i AiT PAi )  P]x(k )  xT (k )[  i ( AiT PAi  P)]x(k )
N
Since there exist positive numbers  i (1  i  N ) such that x(k  1)   i Ai x(k ) is
i 1
asymptotically stable, there exists a Lyapunov function V (k )  xT (k ) Px(k ) such that
N
xT (k )[  i ( AiT PAi  P)]x(k )  0
i 1
It follows that for any k , at least there exists an i {1, 2,
V (k )  xT (k )[ AiT PAi  P]x(k )  0
, N } such that
(5.20)
170
From (5.20), it implies that a convex combination of the corresponding Lyapunov
function is negative along the trajectory and from (5.20) at least one must be negative.
Thus, the switched discrete-time system (5.17) is asymptotically stable.
In this section, one will present two kinds of methods to derive sufficient conditions
such that the switched discrete-time system is asymptotically stable. One is 
stabilization condition; the other is  stabilization condition.
(I)  stabilization condition
For the switched discrete-time system with two individual systems, assuming that
A1_11 and A2 _ 22 are Schur matrices. In the light of Lemma 5.1, the switched
discrete-time system is equivalent to (5.21)
x(k  1)  ( A1  (1   ) A2 ) x(k )
(5.21)
One may choose the Lyapunov function candidate as
V (k )  xT (k ) Px(k )
where
x   x1T

x2T 

T
(5.22)
and
P  diag{P1, P2}
are
unique
real
symmetric
positive-definite matrices satisfying the Lyapunov equations,
AiT_ ii Pi Ai _ ii  Pi   I , i  1, 2.
(5.23)
The difference of Lyapunov function can be written as blow inequality
171


V (k )  xT (k ) (1   ) 2 A1T PA1  (1   1)(1   )2 A2T PA2  P x(k )
(5.24)
It will be convenient throughout this section to use the following notations:
2
c  1  P1  A1_ 21 P 2  A1_12 P1 A1_11  A1_12 P 2 A1_ 22 ,
2
2
d  A2 _11 P1  A2 _ 21 P 2  A2 _11 P1 A2 _12  A2 _ 22 P 2 A2 _ 21 ,
e  A1_12
2
P1  A1_ 22
f  1  P 2  A2 _12
2
2
P 2  A1_11 P1 A1_12  A1_ 21 P 2 A1_ 22 ,
P1  A2 _11 P1 A2 _12  A2 _ 21 P 2 A2 _ 22 .
Theorem 5.2: The switched discrete-time system (5.17) with two individual systems
(N=2), where assumes that
A1_11 and A2 _ 22 are Schur matrices. There exists a
switching law such that the switched discrete-time system (5.17) is asymptotically
stable, if there matrices P1, P2  0 , and positive constant   (1  2 ) . Where
1 : { | (c  P 1)   (d  P 1)  0}
(5.25a)
2 : { | (e  P 2 )   ( f  P 2 )  0}
(5.25b)
Proof: By the Lemma 4.3, the difference of the Lyapunov function (5.22) is following
as
V (k )  xT (k )[(1   ) 2 A1T PA1  (1   1)(1   ) 2 A2T PA2  P]x(k )
172
In the light of Lemma 4.4, let   (1   )1, then
T
V ( k )  x
 AT
1  1_11
(k )((1   )
 AT
 1_12
T

A1_
21  P1 0   A1_11



T
A1_ 22   0 P2   A1_ 21
 AT
1 1  2 _11
(1   )
 AT
 2 _12
A2T _ 21   P1 0   A2 _11



T
A2 _ 22   0 P2   A2 _ 21
A1_12 

A1_ 22 
A2 _12   P1 0 
) x( k )

A2 _ 22   0 P2 
Using the properties of matrix norm,
V ( k )
 x1 [(1   )1 (1  P1  A1_ 21
2
2
P 2  A1_12 P1 A1_11  A1_12 P 2 A1_ 22 )
(1  (1   )1)( A2 _11 P1  A2 _ 21 P 2  A2 _11 P1 A2 _12  A2 _ 22 P 2 A2 _ 21 )  P 1]
2
 x2 [(1   )1( A1_12
2
2
2
P1  A1_ 22
(1  (1   )1)(1  P 2  A2 _12
2
2
P 2  A1_11 P1 A1_12  A1_ 21 P 2 A1_ 22 )
P1  A2 _11 P1 A2 _12  A2 _ 21 P 2 A2 _ 22 )  P 2 ]
Therefore,
V (k )  x1 [(1   )1(c  d )  d  P 1]  x2 [(1   )1(e  f )  f  P 2 ] (5.26)
2
2
173
If one may choose   (1  2 ) , then V (k )  0 . Hence, the system (5.17) is

asymptotically stable. Proof of the Theorem 5.2 is completed.
In the light of front statements and theorems, there exists some conditions in order to
satisfy V (k )  V (k  1)  V (k ) is negative-definite, then the switched discrete-time
system is asymptotically stable. Therefore, for any sampling steps k  1, 2, , N , there
exists an i 1, 2 such that the difference of Lyapunov function
V (k )  xT (k )( AiT PAi  P) x(k )  0 ,
(5.27)
Switching Law: Switched discrete-time system (5.17) with N=2 is switched to or stay
at individual system i for sampling step k if (5.28) is satisfied at k.
xT (k )( AiT PAi  P ) x(k )  0 ,
i  1, 2 .
(5.28)
Next, one may extend arbitrary N individual systems for the switched discrete-time
system (5.17). Assuming that A1_11 , A2 _ 22 , … , AN _ NN are Schur matrices. By
Lemma 5.1, the switched discrete-time system is equivalent to x(k  1)  iN1i Ai x(k ) ,
iN1i  1 . To investigate the stability of system, one may choose the Lyapunov
function
candidate
as
equation
(5.22),
where
x   x1T

x2T
... xTN 

T
and
P  diag{P1, P2 ,..., PN } are unique real symmetric positive-definite matrices satisfying
the Lyapunov equations (5.23) which i  1,2,
, N.
Theorem 5.3: The switched discrete-time system (5.17) with arbitrary N individual
174
systems, where assumes that A1_11, A2 _ 22 ,…, AN _ NN are Schur matrices. There
exists a switching law such that the switched discrete-time system (5.17) is
asymptotically stable, if there matrices P1, P2 ,
, PN  0 , and positive constant 
satisfying conditions (5.29).
N
N
l 1
j 1
i [1  Pi  (  Ai _ li Pl  Ai _ lj )  Ai _ ii
N
N
N
j 1
j i
l 1
k 1
2
Pi ]
   j [(  A j _ li Pl (  A j _ lk )]  P i  0 , for i  1, 2,
where  i 
 i 1
(1   )i
, i  1, 2,
, N  1 and  N 
 N 1
(1   ) N 1
,N
(5.29)
.
Proof: With the same steps presented in the proof of Theorem 5.2, the Theorem 5.3 can

be proved.
Switching Law: The switching law is the same as inequality (5.28) which i=1, 2, ..., N.
(II)  stabilization condition
Another method for the stabilization of the switched discrete-time system (5.17) can
be derived as following section. For the switched discrete-time system (5.17) with two
subsystems, assuming that A1_11 and A2 _ 22 are Schur matrices. In the light of
Lemma 5.1, the stability of the switched discrete-time system is equivalent to
x(k  1)  ( A1  (1   ) A2 ) x(k ) . One may choose the Lyapunov function candidate as
equation (5.22) and P  diag{P1, P2} are unique real symmetric positive-definite
175
matrices satisfying the Lyapunov equations (5.23).
The difference of Lyapunov function can be written as blow inequality
V (k )  xT (k )
  A  (1  ) A 
T
1
2

P  A1  (1   ) A2   P x(k )
(5.30)
It will be convenient throughout this section to use the following notations:
c1  1  P1  A1_ 21
2
P 2  A1_11 A1_12 P1  A1_ 22 A1_ 21 P 2 ,
d1  2 A1_11 A2 _11 P1  2 A1_ 21 A2 _ 21 P 2  A1_12 A2 _11 P1  A2 _12 A1_11 P1
 A1_ 22 A2 _ 21 P 2  A2 _ 22 A1_ 21 P 2 ,
e1  A2 _11
2
P1  A2 _ 21
2
P 2  A2 _12 A2 _11 P1  A2 _ 21 A2 _ 22 P 2 ,
c2  A1_12
2
P1  A1_ 22
2
P 2  A1_11 A1_12 P1  A1_ 22 A1_ 21 P 2 ,
d2  2 A1_12 A2 _12 P1  2 A1_ 22 A2 _ 22 P 2  A1_12 A2 _11 P1  A2 _12 A1_11 P1
 A1_ 22 A2 _ 21 P 2  A2 _ 22 A1_ 21 P 2 and
e2  1  P 2  A2 _12
2
P1  A2 _12 A2 _11 P1  A2 _ 21 A2 _ 22 P 2 .
176
Theorem 5.4: The switched discrete-time system (5.17) with two individual systems
(N=2), where assumes that A1_11 and A2 _ 22 are Schur matrices. There exists a
switching law such that the switched discrete-time system (5.17) is asymptotically
stable, if there matrices P1, P2  0 , constant   (1  2 ) and 0    1 . Where
1 : { | (c1  d1  e1) 2  (d1  2e1)  (e1  P 1)  0}
(5.31a)
2 : { | (c2  d2  e2 ) 2  (d2  2e2 )  (e2  P 2 )  0}
(5.31b)
Proof: With the same as proof of Theorem 5.2, the Theorem 5.4 can be proving.

Switching Law: The switching law is the same as inequality (5.28).
Consider the switched discrete-time system (5.17) with arbitrary N individual systems,
where assumes that A1_11 , A2 _ 22 , … , AN _ NN are Schur matrices. By Lemma 5.1,
the switched discrete-time system is equivalent to
x(k  1)  iN1i Ai x(k ) ,
iN1i  1 . To investigate the stability of system, one may choose the Lyapunov
function
candidate
as
(5.22)
x   x1T

where
x2T
... xTN 

T
and
P  diag{P1, P2 ,..., PN } are unique real symmetric positive-definite matrices satisfying
the Lyapunov equations (5.23) which i  1,2,
, N.
Theorem 5.5: For the switched discrete-time system (5.17) with arbitrary N individual
systems, where assumes that A1_11, A2 _ 22 ,…, AN _ NN are Schur matrices. There
exists a switching law such that the switched discrete-time system (5.17) is
177
asymptotically stable, if there matrices P1, P2 ,
, PN  0 , and positive constant  i
satisfying conditions (5.32).
i). iN1i  1 and 0   i  1
(5.32a)
N
N


2
ii). i2  1  Pi  (  Ai _ li Pl  Ai _ lj )  Ai _ ii Pi 
l 1
j 1


N
N
N

   2j   A j _ li  A j _ lk Pl 
j 1
k 1
l 1

j i
N N
N
N

    l j   Al _ ki P k  A j _ kh   P i  0
l 1 j 1
h 1
 k 1

for i  1, 2,
(5.32b)
, N.
j l
Proof: With the same as proof of Theorem 5.2, the Theorem 5.5 can be proving.

Switching Law: The switching law is the same as inequality (5.28) which i=1, 2, ..., N.
Remark 5.5: Therefore, to deal with the problem of Theorem 5.5, genetic algorithm
(GA) [75-77] be used to find the solution of inequality (5.32). The procedure of genetic
algorithm is as follows:
1). Generate a random population consisting of n chromosomes. Different
chromosomes represent different possible solutions.
2). Calculate the fitness value of each chromosome. The fitness function is the
performance index of a GA to resolve the viability of each chromosome.
178
3). Choose the parents from the current population using the roulette wheel
proportionate selection for reproduction. The threshold must allow some
chromosomes to continue in order to have parents to produce offspring.
4). GA’s use the operations of reproduction, crossover, and mutation to generate the
next generation (offspring). (a) The crossover procedure is to randomly select a
pair of strings from a mating pool, then randomly determine the crossover position.
(b) The mutation operator is used to avoid the possibility of mistaking a local
optimum for a global one. It is an occasional random change at some string
position based on the mutation probability.
Repeat Steps 1)-4) until one or more of the following conditions have been reached: (a)
the fitness of the best chromosome has reached a desired value; (b) It has no further
improvements; (c) the time limit exceeds.
5.2.1 Numeric example
Example 5.3: Consider the switched discrete-time system composed of two individual
systems given as [30] :
0.2703 0.1266
Mode 1: x(k  1)  A1x(k ) , A1  
1.2214 
 0
(5.33a)
 1.2067 0.0874
Mode 2: x(k  1)  A2 x(k ) , A2  

 0.2622 0.1594
(5.33b)
Via normal tests of stability for the system, two individual subsystems (5.33) all are
unstable. Due to A1_11  0.2703 and A2 _ 22  0.1594 are Schur matrices, Theorem 1
179
can be applied to stabilize the switched discrete-time system (5.33). According to
Lyapunov equations (5.23), P1  1.0788 and P2  1.0261 can be obtained. In view of
the stability conditions of Theorem 5.2, one can calculate that c  0.1157 , d  1.7891,
e  1.5849 and f  0.1910 . And then the inequalities (5.25) can be written as follows:
1 : { | 0.9631  0.7103  0} and 2 : { | 0.5588  0.8351  0}
thus, if 0.6692    1.3559 . The switched discrete-time system (5.33) can be stabilized
by the following switching law.
Switching Law: Switched discrete-time system (5.33) is switched to or stay at
individual system i for sampling step k if (5.34) is satisfied at k.
xT (k )( AiT PAi  P ) x(k )  0 ,
i  1, 2 .
(5.34)
The trajectories of the switched discrete-time system (5.33) are shown in Fig.5.2a-b
with initial value x(0)  100  100T for switching during k  0, 1,
switching signal is shown in Fig. 5.2c.
180
, 20 . The
100
90
80
70
state:x1
60
50
40
30
20
10
0
0
2
4
6
8
10
k
12
14
16
18
20
Fig 5.2a: The trajectory of x1 in Example 5.3
0
-20
state:x2
-40
-60
-80
-100
-120
-140
0
2
4
6
8
10
k
12
14
16
18
20
Fig 5.2b: The trajectory of x2 in Example 5.3
181
switching signals
[1]: subsystem
1
[2]:subsystem
2
2
1
0
2
4
6
8
10
k
12
14
16
18
20
Fig 5.2c: The switching signals in Example 5.3
Example 5.4: Consider the switched discrete-time system composed of three individual
systems as follows:
0
0.0123 0.0048
 0.4872
 0
0.4903
0
0.0274 
Mode 1: x(k  1)  A1x(k ) , A1  
0.0117 0.0035 0.8251 0.0123 


0
1.2125 
 0.0082 0.0028
(5.35a)
0
0.0163 
1.2246 0.0026
0.0072 0.9181 0.0053 0.0035 

Mode 2: x(k  1)  A2 x(k ) , A2  
 0
0.0012 0.4594
0 


0.0054 0.0048 0.0032 0.8867 
(5.35b)
0
0.0024 0.0021 
 1.4414
 0.0031 0.7321
0
0.0098 
Mode 3: x(k  1)  A3 x(k ) , A3  
 0.0054
0
0.8534
0 


 0.0051 0.0013 0.0084 0.5621
(5.35c)
Via normal tests of stability for the system, three individual subsystems 5.35 all are
unstable.
182
0 
0.4872
Due to A1_11  
 , A2 _ 22  0.4594 and A3_ 33  0.5621 are Schur
0
0.4903


matrices, Theorem 5.5 can be applied to stabilize the switched discrete-time system (22).
0 
1.3112
According to Lyapunov equation (5.23), P1  
, P2  1.2675 and
1.3165
 0
P3  1.4619 can be obtained. In view of the stability conditions of Theorem 5.5, one
can calculate
3i 1i  1 and 0   i  1 , i=1, 2, 3.
(5.36a)
0.37112  2.0201 22  2.76932  1.69251 2  4.727 2 3  1.98071 3  1.3112  0
(5.36b)
0.897112  0.2811 22  0.940632  1.0061 2  1.0261 2 3  1.85741 3 1.2675  0
(5.36c)
2.197212  1.1903 22  0.492332  3.23761 2  1.5319 23  2.097713  1.4619  0
(5.36d)
In this section, the values of (5.36) indicate the performance of the system. Hence, α is
defined as 1   2   3  1 ( 3  1 - 1 -  2 ) , one can calculate
0.37112  2.0201 22  2.769(1  1   2 )2
1.69251 2  4.727 2 (1  1   2 )  1.98071 (1  1   2 )  1.3112  0 (5.37a)
0.897112  0.2811 22  0.9406(1  1   2 ) 2
183
1.0061 2  1.0261 2 (1  1   2 )  1.85741 (1  1   2 )  1.2675  0 (5.37b)
2.197212  1.1903 22  0.4923(1  1   2 ) 2
3.23761 2  1.5319 2 (1  1  2 )  2.09771 (1  1  2 )  1.4619  0 (5.37c)
A real-coded GA with arithmetic crossover and mutation will be employed in the GA
operators process for this application example. The parameters 1 ,  2 form the
chromosomes of the GA process. Their initial values are randomly generated with the
minimum and maximum bounds chosen as 0 and 1/3 respectively. The control
parameters of the real coded GA with arithmetic crossover and mutation are as follows.
The probability of crossover is 0.9; the probability of mutation is 0.05; the population
size is 500; the number of training iteration is 40000 generation. After applying the GA
operators as Remark 5.5, one can get 1  0.40201,  2  0.39164, 3  0.20635.
Switching Law: Switched discrete-time system (5.35) is switched to or stay at
individual system i for sampling step k if (5.38) is satisfied at k.
xT (k )( AiT PAi  P ) x(k )  0 ,
i  1, 2,3 .
(5.38)
The trajectories of the switched discrete-time system (5.38) are shown in Fig. 5.3a-d
with initial value x(0)   2 1 2 4
T
for switching during k  0, 1,
switching signal is shown in Fig. 5.3e.
184
, 50. The
4
3
2
state:x1
1
0
-1
-2
-3
-4
0
5
10
15
20
25
k
30
35
40
45
50
Fig 5.3a: The trajectory of x1 in Example 5.4
0.4
0.2
state:x2
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
10
15
20
25
k
30
35
40
45
50
Fig 5.3b: The trajectory of x2 in Example 5.4
185
0.5
0
state:x3
-0.5
-1
-1.5
-2
0
5
10
15
20
25
k
30
35
40
45
50
Fig 5.3c: The trajectory of x3 in Example 5.4
4
3
state:x4
2
1
0
-1
-2
-3
0
5
10
15
20
25
k
30
35
40
45
50
Fig 5.3d: The trajectory of x4 in Example 5.4
186
[3]:subsystem
3
switching signals
[2]:subsystem
2
3
[1]: subsystem
1
2
1
0
5
10
15
20
25
k
30
35
40
45
50
Fig. 5.3e: The switching signals in Example 5.4
5.2.3 Summary
Using Lyapunov stability theorem and gene algorithm to deal with the stability of
switched discrete-time systems, the results are straightforward. By a simple switching
law, the sufficient stability conditions for two kinds of methods have been derived for
switched discrete-time systems with the Lyapunov function. In addition, these methods
can be applied to cases when all individual subsystems are unstable.
5.3 Analysis and Synthesis of large-scale switched
discrete-time systems
In this section, a criterion of stabilization for large-scale switched discrete-time
systems is deduced by employing a state-driven switching method and the Lyapunov
stability theorem. The main analysis is based on the fact that the existence of a linear
convex combination implies quadratic stabilizability of the large-scale switched
187
discrete-time system model studied. A switching rule is proposed to guarantee the
stability. In particular, such a switching rule can be applied to cases when all individual
subsystems are unstable. Finally, an example is demonstrated to illustrate the proposed
schemes.
5.3.1 Stability analysis and switching law design
Consider the large-scale switched discrete-time systems with r individual systems and
each individual system composed of N interconnected subsystems:
xi (k  1)  A ( x( k )) _ ii xi (k )   Nj  i, A ( x ( k )) _ ij x j (k )
(5.39)
j 1
N
where xi (k )  R ni is the state vectors of the ith subsystem, i=1, 2,…,N, n   ni ,
i 1
A ( x(k )) _ ii and A ( x(k )) _ ij are some constant matrices of compatible dimensions,
x(t )   x1T (k ) x2T (k )

constant scalar
T
xTN (k )  ,

 ( x( k )) ： R n  {1, 2,
, r} is a piecewise
function of state, called a switch signal, i.e.,  ( x(k ))  l implies that
the subsystem matrices Al _ ii and Al _ ij . Essentially, the switched discrete-time
system (5.39) can be described as follows:
Individual system l:
xi (k  1)  Al _ ii xi (k )   Nj  i Al _ ij x j (k )
(5.40)
j 1
First, one may discuss the stability of the nominal switched discrete-time system. For
the nominal switched discrete-time system as follows,
188
x(k  1)  Al x(k ) , l  1, 2,
,r .
(5.41)
Lemma 5.2: There exists a switching law for the nominal switched discrete-time system
(5.41) such that the system (5.41) is asymptotically stable if there exist a symmetric
matrix P  0 , positive constants  l (1  l  r ) satisfying lr1l  1 such that
r
T
 l ( Al PAl )  P  0
(5.42)
l 1
Proof: If there exist a symmetric matrix P  0 , positive constants  l (1  l  r )
satisfying lr1l  1 such that the inequality (5.42) holds, and the inequality (5.43) is
equivalent to the following inequality
r
T
 l ( Al PAl )  P  0 ,
l 1
l  1, 2,
,r
(5.43)
Then, for x(k )  R n , x(k )  0
r
xT (k )[  l ( AlT PAl  P)]x(k )  0
(5.44)
l 1
Therefore, it follows that for any k, there at least exists an l  {1, 2,
xT (k )[ AlT PAl  P]x(k )  0
, r} such that
(5.45)
From (5.45), it implies that a convex combination of the corresponding Lyapunov
function (5.44) is negative along the trajectory and from (5.45) at least one must be
negative. Thus, the nominal switched discrete-time system (5.41) is asymptotically

stable.
189
Lemma 5.3 [78]: Tchebyshev inequality holds for any matrix Wi  R nn
(iN1Wi )T (iN1Wi )  N (iN1WiTWi )
(5.46)
In the study of large-scale switched discrete-time systems, the main problems are
how to constructively design a switching rule and how to derive sufficient stability
conditions which can guarantee the stability of the switched system under the switching
rule.
For the large-scale switched discrete-time system (5.40) with r individual systems
and each individual system composed of N interconnected subsystems, assumed that the
large-scale switched discrete-time system (5.40) at least has one interconnected
subsystem which Ak _ ii , i  1, 2,
i
To
investigate
xT (k )   x1T

the
, N and ki {1, 2,
stability
of
system
, r}, are Schur matrices.
(5.39),
one
may
denote
T
x2T ... xTN  and P  diag{P1, P2 ,..., PN } which

AkT _ ii Pi Aki _ ii  Pi   I
(5.47)
i
Theorem 5.6:
Consider the large-scale switched discrete-time system (5.39). There
exists a switching law such that the large-scale switched discrete-time system (5.39) is
asymptotically stable, if there exist matrices P1, P2 ,
(5.47) and the following inequality (5.48) holds.
[ ki (1  P i )  lr1 l AlT_ ii Pi Al _ ii )] 
l  ki
190
Pi
2
, PN  0 satisfying equation
( N  1)(lr1  Nj  i l AlT_ ji Pj Al _ ji )  0
(5.48)
j 1
where l 
1
(1   1)l 1(1   )
, r  1 ),  r 
, ( l  1, 2,
1
(1   1 )r 1
and   0
Proof: By inequality (5.44), one can obtain
T
iN1{[lr1l ( Al _ ii xi (k )   Nj i Al _ ij x j (k ))] [lr1l ( Al _ ii xi (k )   Nj  i Al _ ij x j (k ))]
j 1
j 1
 xiT (k ) Pi xi (k )}
 iN1{lr1l ( Al _ ii xi (k ))T Pi ( Al _ ii xi (k ))  lr1l ( Al _ ii xi (k ))T Pi ( Nj  i Al _ ij x j (k ))
j 1
 lr1l ( Nj  i Al _ ij x j (k ))T Pi ( Al _ ii xi (k )) lr1l ( Nj i Al _ ij x j (k ))T Pi ( Nj i Al _ ij x j (k ))
j 1
j 1
j 1
 xiT (k ) Pi xi (k )}
N
r
r
N
N
l 1
j i
j 1
j i
j 1
  {  l [2( Al _ ii xi (k ))T Pi ( Al _ ii xi (k ))   2l (  Al _ ij x j (k ))T Pi (  Al _ ij x j (k ))]
i 1 l 1
 xiT (k ) Pi xi (k )}
N
r
r
N
l 1
j i
j 1
  {  2l ( Al _ ii xi (k ))T Pi ( Al _ ii xi (k ))   2l ( N  1)(  ( Al _ ij x j (k ))T Pi ( Al _ ij x j (k )))
i 1 l 1
 xiT (k ) Pi xi (k )}
191
N
r
  xiT (k ){2[ ki (1  Pi )   l AlT_ ii Pi Al _ ii )]  Pi
i 1
l 1
l  ki
r
N
2( N  1)(   l AlT_ ji Pj Al _ ji )}xi (k )
(5.49)
l 1 j  i
j 1
If conditions (5.48) are satisfied, then, the inequality (5.49) holds for i=1, 2, …, N and
l=1, 2 ,…, r. Therefore, the large-scale switched discrete-time system is asymptotically

stable.
Remark 5.6: The large-scale switched discrete-time system (5.39) with arbitrary r
individual systems and N state vectors, Theorem 5.6 can be applied to cases when all
individual subsystems are unstable. The particular method also can be applied to cases
when r  N , r  N , r  N.
Switching Law: Large-scale switched discrete-time system (5.39) is switched to or stay
at mode l at sampling step k if (5.50) is satisfied at k.
N
N
N
i 1
j i
j 1
j i
j 1
T
 {[ Al _ ii xi (k )   Al _ ij x j (k )] Pi [ Al _ ii xi (k )   Al _ ij x j (k )]  Pi }  0 ,
l  1, 2,
,r
(5.50)
5.3.2 Numeric example
Example 5.5: Consider the large-scale switched discrete-time system composed of
three individual systems given as:
Individual system 1 (l=1):
192
0 
0.1872
0.0042 0.0013
0.0058 0.0036 
x1(k  1)  
x1(k )  
x2 (k )  
x3 (k )


0.1903
0.0055
 0
 0.0023 0.0021 
 0
0 
0 
 0.0026
0.1251
0.0014 0.0035
x2 (k  1)  
x1(k )  
x2 (k )  


 x3 (k )
0.1242
0.0037 0.0015
 0
0.0025 0.0056
0 
0.0018 0.0062
0.0025 0.0015 
0.1125
x3 (k  1)  
x1(k )  
x2 (k )  
x3 (k )


0.0035
1.0324
 0
0.0035 0.0013
 0
(5.51a)
Individual system 2 (l=2):
0 
0.0025 
1.0246
 0
0.0027 0.0034 
x1(k  1)  
x1(k )  
x2 (k )  


 x3 (k )
0.1181
 0
0.0036 0.0058
0.0014 0.0018
0 
0.0032 0.0025
0.1594
0.0035 0.0037 
x2 (k  1)  
x1(k )  
x2 (k )  
x3 (k )


0 
0.1645
0.0052 
0.0023
 0
 0
0 
0.0036
 0.0068 0.0019
x3 (k  1)  
x1(k )  
x2 (k )

0.0032
0.0024 0.0026
 0
0 
0.1867

x3 (k )
0.1554 
 0
(5.51b)
Individual system 3 (l=3):
0 
0 
0.1414
 0.0035
 0.0014 0.0026 
x1(k  1)  
x1(k )  
x2 (k )  
x3 (k )


0.1321
0.0023
 0
0.0007 0.0025
 0
0 
0.0025 0.0036
1.0534
 0.0045 0.0025
x2 (k  1)  
x1(k )  
x2 (k )  


 x3 (k )
0.1652
0.0012 0.0008 
 0
 0.0005 0.0015
0.0032
0 
0.0009 0.0023
 0
 0.1621
x3 (k  1)  
x1(k )  
x2 (k )  
x3 (k )


0.0065
0.1895
 0
0.0023 0.0025
 0
(5.51c)
193
According to the normal test of stability for the large-scale system, the individual
system 1, the individual systems 2 and 3 are unstable (shown in Fig.5.4a-i). By the
Lyapunov equation AkT _ ii Pi Ak _ ii  Pi   I , furthermore, one may choose k1  3 ,
i
i
0 
0 
1.0204
1.0159
, P2  
and
k2  1 and k3  2 , then one can calculate P1  

1.0178
1.0157 
 0
 0
0 
1.0361
.
P3  
1.0247 
 0
In the light of Theorem 5.6, the stable conditions:
i). 1 
1
1
1
, 2 
and 3 
.

1
(1   )
(1   )(1   )
(1   1)2
(5.52a)
ii). (0.02043  0.03691  1.0712 2 ) (1.5531104 1
8.4660 105 2  1.3619 104 3 ) 0.5089  0
(5.52b)
iii). (0.01591  0.0275 2  1.12733 ) (8.4306 1051
2.0729 104  2  6.52111053 ) 0.5078  0
(5.52c)
iv). (0.0361 2  1.09221  0.03683 ) (2.3009 104 1
1.4396 104  2  8.2404 1053 ) 0.5124  0
(26d)
Therefore, one can select   1.5 ( 1  0.4 ,  2  0.24 and 3  0.36 ), then inequality
(5.48) holds. Hence, the large-scale switched discrete-time system (5.51) is
194
asymptotically stable. The switched discrete-time system (5.51) can be stabilized by the
following switching law.
Switching Law: Large-scale switched discrete-time system (5.51) is switched to or stay
at mode l at sampling step k if (5.53) is satisfied at k.
3
3
3
i 1
j i
j 1
j i
j 1
T
 {[ Al _ ii xi (k )   Al _ ij x j (k )] Pi [ Al _ ii xi (k )   Al _ ij x j (k )]  Pi }  0 ,
l  1, 2,3 .
(5.53)
The trajectory of the large-scale switched discrete-time system (5.51) and the
switching
k  [1,30]
during
x1(0)   20 20
T
is
shown
, x2 (0)   10 10
in
Fig.5.5a-c,
with
T
signal is shown in Fig. 5.5d.
solid-line: x11 dotted-line:x12
3000
Subsystem 1 --- state:x1
2000
1000
0
-1000
-2000
-3000
0
50
100
value
and x3 (0)   20 40 . The switching
T
-4000
initial
150
k
200
250
300
Fig. 5.4a. Unstable Subsystem1: x1 in Example 5.5
195
solid-line: x21 dotted-line:x22
3500
3000
Subsystem 1 --- state:x2
2500
2000
1500
1000
500
0
-500
0
50
100
150
k
200
250
300
Fig. 5.4b. Unstable Subsystem1: x2 in Example 5.5
5
6
solid-line: x31 dotted-line:x32
x 10
Subsystem 1 --- state:x3
5
4
3
2
1
0
-1
0
50
100
150
k
200
250
300
Fig. 5.4c. Unstable Subsystem1: x3 in Example 5.5
196
4
3
solid-line: x11 dotted-line:x12
x 10
Subsystem 2 --- state:x1
2.5
2
1.5
1
0.5
0
0
50
100
150
k
200
250
300
Fig. 5.4d. Unstable Subsystem2: x1 in Example 5.5
solid-line: x21 dotted-line:x22
120
Subsystem 2 --- state:x2
100
80
60
40
20
0
-20
0
50
100
150
k
200
250
300
Fig. 5.4e. Unstable Subsystem2: x2 in Example 5.5
197
solid-line: x31 dotted-line:x32
140
Subsystem 2 --- state:x3
120
100
80
60
40
20
0
0
50
100
150
k
200
250
300
Fig. 5.4f. Unstable Subsystem2: x3 in Example 5.5
5
0.5
solid-line: x11 dotted-line:x12
x 10
Subsystem 3 --- state:x1
0
-0.5
-1
-1.5
-2
-2.5
0
50
100
150
k
200
250
300
Fig. 5.4g. Unstable Subsystem3: x1 in Example 5.5
198
7
0
solid-line: x21 dotted-line:x22
x 10
Subsystem 3 --- state:x2
-1
-2
-3
-4
-5
-6
0
50
100
150
k
200
250
300
Fig. 5.4h. Unstable Subsystem3: x2 in Example 5.5
4
2
solid-line: x31 dotted-line:x32
x 10
Subsystem 3 --- state:x3
0
-2
-4
-6
-8
-10
-12
0
50
100
150
k
200
250
300
Fig. 5.4i. Unstable Subsystem3: x3 in Example 5.5
199
solid-line: x11 dotted-line:x12
20
15
state:x1
10
5
0
-5
0
5
10
15
k
20
25
30
Fig. 5.5a. State response for x1 in Example 5.5
solid-line: x21 dotted-line:x22
2
0
state:x2
-2
-4
-6
-8
-10
-12
0
5
10
15
k
20
25
30
Fig. 5.5b. State response for x2 in Example 5.5
200
solid-line: x31 dotted-line:x32
40
35
30
25
state:x3
20
15
10
5
0
-5
-10
0
5
10
15
k
20
25
30
Fig. 5.5c. State response for x3 in Example 5.5
3
2
[1]: subsystem
1
[2]:subsystem
2
[3]:subsystem
3
switching signals
1
0
5
10
15
k
20
25
30
Fig. 5.5d. The switching times of the system during [1, 30] in Example 5.5
5.3.3 Summary
By the Lyapunov stability theorem to deal with the stabilization analysis of
large-scale switched discrete-time systems, the results are straightforward. By a simple
201
switching law, the sufficient stability conditions have been derived for the large-scale
switched discrete-time systems via the state-driven switching. The results can be applied
to cases when r  N ,
r  N,
r  N. for the large-scale switched discrete-time
system with arbitrary r individual systems and N state vectors. In addition, the particular
method can be applied to cases when all individual subsystems are unstable.
202