It should be noted that although there are a few works devoted to stability analysis of SDLS, see [5, 13, 15], to our best of knowledge, the problem of investigating the stability for [r]
(1)84
Stability of Arbitrarily Switched Discrete-time Linear Singular Systems of Index-1
Pham Thi Linh*
VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
Received 22 December 2018
Revised 27 December 2018; Accepted 28 December 2018
Abstract: In this paper, the index-1 notion for arbitrarily switched discrete-time linear singular systems (SDLS) has been introduced Based on the Bohl exponents of SDLS as well as properties of associated positive switched systems, some necessary and sufficient conditions have been established for exponential stability
Keywords: Switched system, linear discrete-time singular system, positive system, index-1 system
1 Introduction
Recently there has been a great interest in arbitrarily switched discrete-time linear singular systems due to their importance in both theoretical and practical aspects, see [5, 10, 13, 15], and the references therein Consider a switched system consisting of a set of subsystems and a rule that describes switching among them It is well known that, even if all linear descriptor subsystems are stable but inappropriate switching may make the whole system unstable On the other hand, since abrupt changes in system dynamics may be caused by unpredictable environmental factors or component failures, it is important to require the stability for some real-life switched systems under arbitrary switching It should be noted that although there are a few works devoted to stability analysis of SDLS, see [5, 13, 15], to our best of knowledge, the problem of investigating the stability for such switched systems via their Bohl exponents or properties of associated positive switched systems has not yet been studied before Thus, this work was intended as an attempt to fill this gap
2 Switched discrete-time linear singular systems of index-1
Consider the following autonomous SDLS of the form:
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(2)(k 1) ( 1) ( )k ( )
E x k A x k (1)
where : N {0} IN: {1, 2, , },N N N, is a switching signal taking values in the finite set IN
; E Ai, iRn n are given matrices, and x k( )Rn are unknown vector for al kN Suppose that the matrices E are singular for all i i1,2, ,N
We remark that in some works on SDLS [14, 15], instead of (1), a simpler system of the form ( )k ( 1) ( )k ( )
E x k A x k ,
can be considered Moreover, all the techniques developed in this paper can easily be applied to the above mentioned SDLS
Definition System (1) is called an arbitrarily switched singular system of (shortly,
index-1 SDLS) if it satisfies the following conditions (i) rankEi r n;
(ii) Sij kerEi {0}i j, , where 1(Im ) { : Im }
ij i j i j
S A E A E
From condition (ii) in Definition we show that ker Rn , {1, 2, , }
ij i
S E i j N
Indeed, put Wij ImAiImEj Then consider linear operators Tij:Sij Wij, defined by TijxAijx
, we can easily show that kerTij kerAi According to [9] we have dimSij dimWij dim kerTij dimWij dim kerAi
On the other hand
dim dim(Im Im )
dim Im dim Im dim(Im Im )
ij i j
i j i j
W A E
A E A E
From last the relation we get
dim dim Im dim Im dim(Im Im ) dim(ker ) dim(Im Im )
ij i j i j i
i j
S A E A E A
n r A E
This relation shows that dimSij r Moreover, from condition (ii) in Definition we have dimSij r Hence dimSij r, i.e., ker R
n
ij i
S E
Define the matrix 1 { , , r, r , , n}
ij ij ij i i
V s s h h , whose columns form bases of S and kerij E , i
respectively, and Qdiag(O Ir, n r ), PIn-Q Here O is the r rr zero matrix and I stands for the m m m identity matrix
Then the matrix Qij V QVij ij1 defines a projection onto kerE along i S and ij Pij InQij is the projection onto S along kerij E i
Using similar arguments as in [1-3, 7] we can prove the following results
Theorem For index-1 SDLS (1), the following assertions hold
(i)
ijk j i ij jk
(3)(ii) E Pj jk Ej; (iii) Pjk G Eijk1 j; (iv) 1
jk ijk i ij
V G AV Q Q
Proof
(i) Assume that xkerGijk, we have 0G xijk
(EjAV QVi ij jk)xE xj AV QV xi ij jk1 Then
j i ij jk
E x AV QV x , thus V QV xij jk1 Si j, Furthermore,V QV xij jk1 V QV V V xij ij1 ij jk1 Q V V xij ij jk1 kerEi Since SijkerEi{0} we get V QV xij jk1 0, thus E xj AV QV xi ij jk1 0, hence xkerEj ImQjk, i.e., xQjkx On the other hand, from the relation 1 0
jk jk ij ij jk
Q x V V V QV x , we have xQ xjk 0 It
means that kerGijk{0}, i.e., the matrix G is non-singular ijk
(ii) Since Q is the projection onto kerjk E then we have j E Qj jk 0, i.e.,
( )
j j jk jk j jk
E E P Q E P
(iii) From relation G Pijk jk ( 1) j i ij jk jk jk
E AV QV V PV
j jk i ij jk j
E P AV QPV E , we get
1 jk ijk j
P G E
(iv) From formula of ijk j i ij jk
G E AV QV we have GijkVjk EjVjkAV Qi ij , thus
i ij ijkVjk j jk
AV QG E V
The last assertion follows from relations:
1 1
1 1
1
( )
jk ijk i ij jk ijk ijk jk j jk
jk jk jk ijk j jk
n jk jk jk
V G AV Q V G G V E V
V V V G E V
I V P V
Q
Theorem is proved
Using items (iii), and (iv) of Theorem 1, we get
1
1
: ijk ;
ijk jk ijk i ij
n r
A O
A V G AV
O I
(2)
1
: r
ijk jk ijk j jk
n r
I O
E V G E V
O O
Theorem The index-1 SDLS (1) has a unique solution with x(0) x0 R if and only if (0) (1)
x S , i.e., the initial condition x is consistent In this case, the following solution formula holds 0
( ) ( 1) ( 1) ( ) ( 1) (0) (1) (2) (0) (1)
( ) k k k k k (0)
x k V A A V x
Proof
Multiplying both sides of system (1) by 1
(k 1) (k 2) ( ) (k k 1) (k 2)
V G , and using the transformation
( ) ( 1) ( ) k k ( )
(4)E( ) (k k1) ( k2)x k( 1) A( ) (kk1) ( k2)x k( ) (3) Putting x k( ) : ( ( ) , ( ) ) v k T w k T T, where v k( )Rr, w k( )Rn r , we can reduce system (3) to the following systems
1
( ) ( 1) ( 2)
( 1) ( ),
( )
k k k
v k A v k
w k
(4)
System (4) has the solution
1
( 1) ( ) ( 1) (0) (1) (2)
( ) (0),
( ) 0,
k k k
v k A A v
w k
hence the solution of system (1) can be written as ( ) ( 1)
( ) ( 1)
( ) ( 1) ( 1) ( ) ( 1) (0) (1) (2) ( ) ( 1) ( 1) ( ) ( 1) (0) (1) (2) (0) (1)
( ) ( )
( ) ( )
(0)
0
(0)
k k
k k
k k k k k
k k k k k
x k V x k
v k V
w k
v
V A A
V A A V x
3 Stability of linear switched singular systems of index-1
Suppose that system (1) is of index-1 and the initial condition x is consistent 0
Definition System (1) is called exponentially stable if there exist a positive
constant and a constant 0 1 such that such that for all switching signals and all solutions x of (1) the following inequality holds
0
( ) k
x k x k
3.1 Bohl exponents and exponential stability
To define Bohl exponent for system (1), we first construct the so-called one-step solution operator ( ,k k 1)
from x k 1 to x(k) ( ) ( 1)
( ) ( 1)
( ) ( 1) ( 1) ( ) ( 1) ( ) ( 1) ( 1) ( ) ( 1) ( 1) ( )
( ) ( )
( ) ( )
( 1) ( 1) k k
k k
k k k k k
k k k k k k k
x k V x k
v k V
w k
V A x k
V A V x k
Then put ( ,k k 1) : (k1) ( ) ( k k1)V( ) (k k 1)A(k 1) ( ) ( k k 1)V(1k 1) ( ) k
we get the following one-step solution operator
( ) ( , 1) ( 1)
(5)Hence we can define the state transition matrix as ( 1) ( ) ( 1) ( ) ( 1) ( 2)
( , ) :i j i i i j j j , i j
Definition Assume that system (1) is of index-1 and ( , )i j is the state transition matrix Then Bohl exponent for system (1) is defined as follows:
inf{ R : : ( , ) i j, , 0}
B w Mw i j M ww i j
‖ ‖
To show the existence of Bohl exponent B for system (1) we will prove that the
set
{ R : : ( , ) i j, , 0},
w w
S w M ‖ i j‖M w i j
is non-empty and bounded from below
Indeed, from the formula 1$,$ , , {1, 2, , },
ijk V A Vjk ijk ij i j k N
we see that the set of matrices ijk is finite, then there exists a positive constant 0 such that
, ,max{1,2, , } ijk i j k N
‖ ‖
Thus we obtain that
( , )i j i j, ,i j 0,
‖ ‖
hence S Besides, for all wS we have w0 It follows that the set S is non-empty and bounded from below
Lemma Assume that system (1) is of index-1 and ( , )i j is the state transition matrix Then
1 lim max ( ,0) i B
i i
‖ ‖ (5)
Proof
We carry the proof of Lemma in steps Step We show the existence of the limit in (5)
Put aimax ‖ ( , 0)i ‖ Then we have aija ai j for all ,i j0 According to Polya-Szego [11] we obtain that
1 lim i i
ia exists It means that the limit in (5) exists Step Put
1 lim maxi ( ,0)i i
‖ ‖ We prove 1 B
Since B infS then for all 0 there exists w S such that w B , i.e., there exists Mw such that
( ,0)i Mw( B )i j, ,i
‖ ‖
It follows
1
lim max ( ,0) i B i ‖ i ‖ Then we have
(6)From the definition of 1, for all 0 there exists T > such that
1
| i | , ,
i
a i T
i.e.,
‖ ( ,0)i ‖(1 ) ,i i T, (6) We will show that there exists M0 such that
1
( , )i j M( )i j, i T,
‖ ‖ (7)
Indeed, when i j T, for every we always have switching signal * such that
*
( , )i j (i j,0)
Hence we have
* 1
( , )i j (i j,0) ( )i j, i j T,
‖ ‖ ‖ ‖
When i j T, we have the following estimate
1
( , ) ( )
i j
i j i j
i j
‖ ‖
Choosing
1
max{1, }
T
M
, we get the inequality (7) It means that
B
Thus we obtain B 1 Lemma is proved
Theorem An index-1 SDLS (1) is exponentially stable if and only if B 1
Proof
Necessity Assume that system (1) is exponentially stable It follows that there exist a positive constant M0 and 0 such that
( , )i j M i j ,i j
Thus, B1
Sufficiency Assume that B1 Then there exist 0 and M0 such that
B
and ( , )i j Mi j ,i j It shows that system (1) is exponentially stable Theorem is proved
3.2 Stability of positive linear switched singular systems of index-1
In this Subsection, we investigate the stability of index-1 SDLS satisfying some positivity condition Let : { x( ,x x1 2, ,xr) ,T xi 0} be a positive octant in Rr, Int( ) be the interior of Consider an order unit norm
(7)Theorem Assume that the matrices ijk
A , determined by (2), are positive definite, and there exists
a vector ˆvInt( ) such that ˆ ˆ
( ) ijk
vA vInt P for all , ,i j k Then system (4) is exponentially stable,
hence system (1) is also exponentially stable
Proof
Since ˆ ˆ
( ) ijk
vA vInt P then there exists a ijk(0,‖ ‖uˆ u) such that the closed ball
1
ˆ ˆ
[ ijk , ijk]
B vA v Since ˆ ˆ ˆ ˆ ˆ
[ , ]
ˆ ijk
ijk ijk ijk
u
v A v v B v A v
v
‖ ‖ we get
1
ˆ ˆ ˆ
ˆ ijk ijk
u
v A v v
v
‖ ‖ Let
(0,1) ˆ ijk ijk u v
‖ ‖ , then
1 ˆ ˆ
(1 )
ijk ij
A v v
Put inf{ijk, , ,i j k{1, 2, , }}N , we obtain A vijk1 ˆ (1 )vˆ for all , ,i j k Using the positive
definiteness of matrices ijk
A and the monotonicity of ˆ‖ ‖ we get v u
ˆ
1
( 1) ( ) ( 1) (0) (1) (2) (R , )
1
ˆ
( 1) ( ) ( 1) (0) (1) (2)
1
ˆ
( 1) ( ) ( 1) (1) (2) (3)
1
ˆ
( 1) ( ) ( 1) (1) (2) (3)
ˆ
ˆ
ˆ
(1 )
ˆ
(1 )
ˆ
(1 )
r v
k k k
k k k v
k k k v
k k k v
k v
A A
A A v
A A v
A A v
v ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
‖ ‖ (1 ) k
According to [6], system (4) is exponentially stable It follows that there exist finite positive constants 0 and 0 such that
( ) k (0)
v k v
‖ ‖ ‖ ‖
Furthermore since the corresponding solution of system (1) is x k( )V( ) (k k1)( ( ) ,0)v k T T, we have ( ) ( 1)
( 1) ( ) ( 1) ( ) ( 1)
1
( ) ( 1) (0) (1)
1
( ) ( 1) (0) (1)
( ) ( ( ) ,0)
( (0) ,0)
(0) (0)
T T
k k
k T T
k k k k k
k k k k
k k
x k V v k
V D V v
V V x
V V x
‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖
Putting
,max1,2, , ij ij
i j N V V
‖ ‖ ‖ ‖ , we have
The last relation shows that the solution of system (1) is exponentially stable Theorem is proved
Example Put
1
2
0
0 0
E ,
3
1
0 0
(8)1
1
0
0
A
,
2
2
1
0
A
,
11 12
1 0
0
0
1
V V
, 21 22
2 0 0
V V
We calculate the matrices Aijk, , ,i j k{1, 2} as
111 112
0
0
0
1 12
A A
, 121 122
3 16 12 0
0
1 1
A A
,
211 212
7
0
0
1
A A
, 221 222
5 8 16
0
0
5 16
A A
Clearly all the matrices A are positive definite We choose ˆijk v(9,3)TInt( ) and find that
ˆ ijkˆ
vA v are also inside the Int( ) It means that this system satisfies all the condition of Theorem 3, thus it is exponentially stable
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