Stability of Arbitrarily Switched Discrete-time Linear Singular Systems of Index-1

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77

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 [1- 4], 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 [1, 3-5], 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:

________

 Tel.: 84-336112702.

Email: linhpt1803@gmail.com

https//doi.org/ 10.25073/2588-1124/vnumap.4312

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(_k 1) ( 1) ( )_k ( )

E_ _ x k A_ x k (1)

where : N {0} I_N : {1, 2,.., },N N N,is a switching signal taking values in the finite set I_N; , R^{n n}

i i

E A ^ are given matrices, and x k( )Rⁿare unknown vector for al kN. Suppose that the matrices E_iare singular for all i1,2,..,N.

We remark that in some works on SDLS [4, 6], 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 1 System (1) is called an arbitrarily switched singular system of index-1 (shortly, index- 1 SDLS) if it satisfies the following conditions

(i) rankE_i  r n;

(ii) S_ij kerE_i {0}i j, , where S_ij A_i^¹(ImE_j) { :  A_iIm }E_j . From condition (ii) in Definition 1 we show that

ker Rⁿ , {1,2,.., }.

ij i

S  E  i j N

Indeed, put W_ij ImA_iImE_j. Then consider linear operators T_ij:S_ij W_ij, defined by T_ijxA_ijx, we can easily show that kerT_ij kerA_i. According to [7] we have

dimS_ij dimW_ij dim kerT_ij dimW_ij dim kerA_i. 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 dimS_ij r.Moreover, from condition (ii) in Definition 1 we have dimS_ij r. Hence dimS_ij r, i.e., S_ijkerE_i Rⁿ.

Define the matrix V_ij { ,..., ,s¹_ij s h_ij^r _i^r^¹,...,h_iⁿ}, whose columns form bases of S_ijand kerE_i, respectively, and Qdiag(O I_r, _{n r}_ ), PI_n-Q. Here O_ris the r r zero matrix and I_mstands for the

m m identity matrix.

Then the matrix Q_ij V QV_ij _ij^¹ defines a projection onto kerE_ialong S_ijand P_ij I_nQ_ijis the projection onto S_ijalong kerE_i.

Using similar arguments as in [8-11] we can prove the following results.

Theorem 1 For index-1 SDLS (1), the following assertions hold.

(i) G_ijk E_j AV QV_i _ij _jk^¹is non-singular for all i j k, , {1, 2,.., }N ;

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(ii) E P_j _jk E_j; (iii) P_jk G E_ijk^¹ _j; (iv) V G AV Q_jk^¹ _ijk^¹ _i _ij Q. Proof.

(i) Assume that xkerG_ijk, we have 0G x_ijk (E_j AV QV_{i ij} _jk^¹)xE x_j AV QV x_{i ij} _jk^¹ .Then

1

j i ij jk

E x AV QV x^ , thus V QV x_ij _jk^¹ S_{i j}_, . Furthermore, V QV x_ij _jk^¹ V QV V V x_ij _ij^¹ _ij _jk^¹  Q V V x_ij _ij _jk^¹ kerE_i. Since S_ijkerE_i{0}we get V QV x_ij _jk^¹ 0, thus E x_j  AV QV x_{i ij} _jk^¹ 0, hence xkerE_jImQ_jk, i.e., xQjkx. On the other hand, from the relation Q x V V V QV x_jk  _jk _ij^¹ _ij _jk^¹ 0, we have xQ x_jk 0. It means that kerG_ijk{0}, i.e., the matrix G_ijkis non-singular.

(ii) Since Q_jkis the projection onto kerE_jthen we have E Q_j _jk 0, i.e.,

( )

j j jk jk j jk

E E P Q E P .

(iii) From relation G P_ijk _jk  (E_j AV QV_i _ij _jk^¹)V PV_jk _jk^¹  E P_j _jk AV QPV_i _ij _jk^¹ E_j, we get

1 .

jk ijk j

P G E^

(iv) From formula of G_ijk E_jAV QV_{i ij} _jk^¹we have G_ijkV_jk E_jV_jk AV Q_i _ij , thus

i ij ijkVjk j jk

AV QG E V .

The last assertion follows from relations:

1 1 1 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 1 is proved.

Using items (iii), and (iv) of Theorem 1, we get

1

1 1

: ^ijk ;

ijk jk ijk i ij

n r

A O

A V G AV

O I

 



 

   

 

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1 1

: ^r

ijk jk ijk j jk

n r

I O

E V G E V

O O

 



 

   

 .

Theorem 2 The index-1 SDLS (1) has a unique solution with x(0) x₀ Rif and only if x₀S__{(0) (1)}_ , i.e., the initial condition x₀is consistent. In this case, the following solution formula holds.

1 ( ) ( 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 V_^₍¹_k__{1) (}__k_₂₎G_^¹_{( ) (}_k_ _k__{1) (}__k_₂₎, and using the transformation

1 ( ) ( 1)

( ) _k _k ( )

x k V_^ _ _ x k , we get

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E__{( ) (}_k_ _k__{1) (}__k_₂₎x k(  1) A__{( ) (}_k_ _k__{1) (}_ _k_₂₎x k( ). (3) Putting x k( ) : ( ( ) , ( ) ) v k ^T w k ^{T T}, where v k( )R^r, w k( )R^{n r}^ , we can reduce system (3) to the following systems

1

( ) ( 1) ( 2)

( 1) ( ),

( ) 0.

k k k

v k A v k

w k

    

  

 

 (4)

System (4) has the solution

1 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) (0) (1) (2)

1 ( ) ( 1) ( 1) ( ) ( 1) (0) (1) (2) (0) (1)

( ) ( )

... (0)

0

... (0).

k k

k k k k k

x k V x k

V v k

w k

V A A v

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 x₀ is consistent.

Definition 2 System (1) is called exponentially stable if there exist a positive constant  and a constant 0  1such that such that for all switching signals and all solutions x of (1) the following inequality holds

( ) ^k 0 0.

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

x k V x k

V v k

w k

V A x k

V A V x k

 

    

      



  



   



 

  

 

 

Then put _( ,k k1) : _₍_k__{1) ( ) (}_ _k_ _k_₁₎V__{( ) (}_k__k_₁₎A_₍_k__{1) ( ) (}_ _k_ _k_₁₎V_^₍¹_k__{1) ( )}_ _k we get the following one-step solution operator

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( ) ( , 1) ( 1).

x k  _ k k x k

Hence we can define the state transition matrix as

( 1) ( ) ( 1) ( ) ( 1) ( 2)

( , ) :i j _i _i _i ... _j _j _j , i j 0.

          

      

Definition 3 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 _Bfor system (1) we will prove that the set

{ R : _w: ( , ) _w. ^{i j}, , 0},

S w M ‖ _ i j‖ M w^  i j is non-empty and bounded from below.

Indeed, from the formula  _ijk V A V_jk _ijk _ij^¹, , ,i j k{1,2,.., },N we see that the set of matrices _ijkis finite, then there exists a positive constant  0such 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 wSwe have w0. It follows that the set S is non-empty and bounded from below.

Lemma 1 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 1 in 3 steps.

Step 1.We show the existence of the limit in (5) Put ai max _( , 0)i

  ‖  ‖ . Then we have a_i__j a a_i _jfor all ,i j0.According to Polya-Szego [12]

we obtain that

1

lim _iⁱ

i a

 exists. It means that the limit in (5) exists.

Step 2.Put

1 1 lim max ( ,0) ⁱ .

i _ i

 

  ‖  ‖ We prove  ₁ _B.

Since _B infSthen for all 0there exists w Ssuch that w _B , i.e., there exists M_w such that

( ,0)i M_w( _B )^{i j}, ,i 0.

  ^ 

    

‖ ‖

It follows

1

lim max ( ,0) ⁱ _B .

i _ i

 

 ‖  ‖  

Then we have

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Step 3. We prove _B ₁.

From the definition of ₁, for all 0there exists T >0 such that

1

|a_iⁱ 1|  , i T, i.e.,

( ,0)i ( 1 ) ,ⁱ i T, .

  

     

‖ ‖ (6)

We will show that there exists M0such that

( , )i j M( 1 )^{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 1

( , ) ( ) .

i j

i j i j

 i j   





   

     

‖ ‖

Choosing

1

max{1, }

T

M 



 

    , we get the inequality (7). It means that

1

B .

Thus we obtain _B ₁. Lemma 1 is proved.

Theorem 2 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 M0and 0  1such that

( , )i j M ^{i j} ,i j 0.

 ^ 

    

Thus, _B1.

Sufficiency. Assume that _B1. Then there exist 0and M0such that

B 1

    and _( , )i j  M^{i j}^ ,i j 0.It shows that system (1) is exponentially stable.

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,...,x_r) ,^T x_i0}be a positive octant in R^r, Int( )be the interior of . Consider an order unit norm . _u, defined in [13], [14], and the corresponding order unit space

(R , , . ).^r ‖ ‖_u

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Theorem 3 Assume that the matrices A_ijk¹ , determined by (2), are positive definite, and there exists a vector ˆvInt( )such that vˆA v_ijk¹ ˆInt P( )for all , ,i j k. Then system (4) is exponentially stable, hence system (1) is also exponentially stable.

Proof.

Since vˆA v_ijk¹ ˆInt P( )then there exists a _ijk(0,‖ ‖uˆ _u)such that the closed ball ˆ 1 ˆ

[ _ijk , _ijk]

B vA v  . Since _ˆ ¹ _ˆ _ˆ _[_ˆ ¹ _ˆ_, _] ˆ

ijk

ijk ijk ijk

u

v A v v B v A v v

 

   

‖ ‖ we get ˆ ¹ ˆ ˆ 0

ˆ

ijk ijk

u

v A v v

v

   

‖ ‖ . Let

(0,1) ˆ

ijk ijk

v u

  

‖ ‖ , then A v_ijk¹ ˆ (1 _ij)vˆ.

Put inf{_ijk, , ,i j k{1, 2,...,N}}, we obtain A v_ijk¹ ˆ (1 )vˆfor all , ,i j k. Using the positive definiteness of matrices A_ijk¹ and the monotonicity of ˆ‖ ‖v _uwe get

ˆ

1 1

( 1) ( ) ( 1) (0) (1) (2) (R , . )

1 1

ˆ ( 1) ( ) ( 1) (0) (1) (2)

1 1

( 1) ( ) ( 1) (1) (2) (3) ˆ

1 1

ˆ ( 1) ( ) ( 1) (1) (2) (3)

ˆ

...

... ˆ

... (1 )ˆ

(1 ) ˆ

...

(1 ) ˆ

r

k k k v

k v

A A

A A v

v

     

 

 

 

 

‖ ‖ ‖ ‖

=‖ ‖

‖ ‖

‖ ‖  (1 ) .^k

According to [15], system (4) is exponentially stable. It follows that there exist finite positive constants 0  1and  0such that

( ) ^k (0) .

v k   v

‖ ‖ ‖ ‖

Furthermore since the corresponding solution of system (1) is x k( )V__{( ) (}_k_ _k_₁₎( ( ) ,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

 

    

   





  











‖ ‖

‖ ‖ ‖ ‖

‖ ‖ ‖ ‖ ‖ ‖

Putting ¹

,max1,2,.., _ij _ij

i j N V V

  ^

  ‖ ‖ ‖ ‖ , we have

The last relation shows that the solution of system (1) is exponentially stable.

Example 1 Put

1 2

2 3 0 3 2 0

0 2 0 , 1 6 0 ,

0 0 0 0 0 0

E E

    

   

   

   

   

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1 2

1 1 0 2 1 0

0 1 0 ,

1 1 2 1 ,

0 0 1 0 0

A A



   

   

   

   

   

11 12

1 0 0

0 0

0 0 1 1 3 V V

 

 

   

 

 

,

21 22

2 0 0 0 2 0 0 0 1 V V

 

 

   

 

 

.

We calculate the matrices A_ijk, , ,i j k{1, 2}as

111 112

0

0 0

0 0 1

1 2 1 12 1 2

A A

 

 

   

 

 

,

121 122

3 16 1 12 0 0

0 0

1 6 1 1 4

A A



 

 

   

 

 

,

211 212

7 8 1 3 1 4

0 0

0 0 1

1 6

A A

 

 

   

 

 

,

221 222

5 8 1 8 1 16

0 0

0 0 1

5 16

A A

 

 

   

 

 

.

Clearly all the matrices A_ijkare positive definite. We choose ˆv(9,3)^TInt( ) and find that ˆ _ijk1 ˆ

vA vare also inside Int( ). It means that this system satisfies all the condition of Theorem 3, thus it is exponentially stable.

Acknowledgments

The author is grateful to Professors Pham Ky Anh, Do Duc Thuan and Stefan Trenn for useful discussions leading to the formulation of the problem as well as the results obtained in this paper.

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