Tadeusz Kaczorek
Reduction and Decomposition of Singular Fractional Discrete-Time Linear Systems
REDUCTION AND DECOMPOSITION
OF SINGULAR FRACTIONAL DISCRETE-TIME LINEAR SYSTEMS
Tadeusz KACZOREK*
*
Faculty of Electrical Engineering, Białystok University of Technology, ul. Wiejska 45D, 15-351 Białystok
[email protected]
Abstract: Reduction of singular fractional systems to standard fractional systems and decomposition of singular fractional
discrete-time linear systems into dynamic and static parts are addressed. It is shown that if the pencil of singular fractional linear discrete-time system is regular then the singular system can be reduced to standard one and it can be decomposed into
dynamic and static parts The proposed procedures are based on modified version of the shuffle algorithm and illustrated
by numerical examples.
1. INTRODUCTION
Singular (descriptor) linear systems have been addressed in many papers and books (Dodig and Stosic, 2009;
Dai, 1989; Fahmy and O’Reill, 1989; Gantmacher, 1960;
Kaczorek, 1992, 2007a; Kucera and Zagalak, 1988).
The eigenvalues and invariants assignment by state
and output feedbacks have been investigated in (Dodig
and Stosic, 2009; Dai, 1989; Fahmy and O’Reill, 1989;
Kucera and Zagalak, 1988; Kaczorek, 2004) and the realization problem for singular positive continuous-time systems with delays in Kaczorek (2007b). The computation
of Kronecker’s canonical form of a singular pencil has been
analyzed in Van Dooren (1979). The fractional differential
equations have been considered in the monograph
(Podlubny, 1999). Fractional positive linear systems have
been addressed in (Kaczorek, 2008, 2010) and in the monograph (Kaczorek, 2011). Luenberger in (Luenberger, 1978)
has proposed the shuffle algorithm to analysis of the singular linear systems.
In this paper a modified version of the shuffle algorithm
will be proposed for the reduction of the singular fractional
system to equivalent standard fractional system
and for decomposition of the singular fractional system into
dynamic and static parts.
The paper is organized as follows. In section 2
it is shown that if the pencil of the singular system is regular then the singular system can be reduced to equivalent
standard fractional system. The decomposition of singular
fractional system into dynamic and static parts is addressed
in section 4. Concluding remarks are given in section 5.
To the best of the author’s knowledge the reduction
and the decomposition of singular fractional linear discretetime systems have not been considered yet.
The following notation be used in the paper.
The set of n × m real matrices will denoted by ℜ௡×௠
and ℜ௡ ≔ ℜ௡×ଵ . The set of × real matrices
with nonnegative entries will be denoted by ℜ௠௫௡
ା
and ℜ௡ା ≔ ℜ௡௫ଵ
ା . The set of nonnegative integers will be
denoted by ା and the × identity matrix by ௡ .
62
2. REDUCTION OF SINGULAR FRACTIONAL
SYSTEMS TO EQUIVALENT STANDARD
FRACTIONAL SYSTEMS
Consider the singular fractional discrete-time linear system described by the state equation:
E∆α xi +1 = Axi + Bui , i ∈ Z + = {0,1,...}
(1)
where, ௜ ∈ ℜ௡ , ௜ ∈ ℜ௠ are the state and input vectors,
∈ ℜ௡×௡ , ∈ ℜ௡×௡ , ∈ ℜ௡×௠ and the fractional difference of the order α is defined by:
∆α xi =
i
α 
∑ (−1) k  k  xi − k , 0 < α < 1
(2)
for k = 0
 1
α  
α
(
α
−
1
)...(
α
−
k
+
1
)
=
  
for k = 1, 2,...
k 

k!
(3)
k =0
It is assumed that:
det = 0
(4a)
det[
− ] ≠ 0
(4b)
and
for some ∈ (the field of complex numbers).
Substituting (2) into (1) we obtain:
i +1
∑ Eck xi − k +1 = Axi + Bui , i ∈ Z +
(5)
k =0
where:
α 
ck = (−1) k  
k 
(6)
Applying the row elementary operations to (5) we obtain:
i +1
E 
A 
B 
∑  01 ck xi − k +1 =  A12  xi + B12 ui , i ∈ Z +
k =0
(7)
acta mechanica et automatica, vol.5 no.4 (2011)
where ଵ ∈ ℜ௡భ×௡ is full row rank and ଵ ∈ ℜ௡భ×௡ ,
ଶ ∈ ℜ(௡ି௡భ)×௡ , ଵ ∈ ℜ௡భ×௠ , ଶ ∈ ℜ(௡ି௡భ)×௠ . The equation (7) can be rewritten as:
i +1
∑ E1ck xi − k +1 = A1xi + B1ui
(8a)
k =0
(17)
If the matrix:
and
0 = A2 xi + B2ui
(8b)
Substituting in (8b) i by i + 1 we obtain:
A2 xi +1 = − B2ui +1
(9)
The equations (8a) and (9) can be written in the form:
 E1 
 A1 − c1 E1 
c E 
xi −  2 1  xi −1
 A  xi +1 = 

0


 0 
 2
 0 
c E 
B 
− ... −  i +1 1  x0 +  1 ui + 
ui +1
0
0


 
− B2 
(10)
 E1 
A 
 2
 E2 


 A20 
(18)
is nonsingular then premultiplying the equation (17)
by its inverse we obtain the standard system:
xi +1 = Aˆ 0 xi + Aˆ1 xi −1 + ... + Aˆ i x0
+ Bˆ 0 ui + Bˆ1ui +1 + Bˆ 2ui + 2
−1
 A20  ˆ  E 2 
− A , A1 =  A 
 21 
 20 
is nonsingular then premultiplying the equation (10) by the
inverse matrix ଵ ିଵ we obtain the standard system:
ଶ
(12)
where:
−1
(13)
If the matrix (11) is singular then applying the row elementary operations to (10) we obtain:
 A20 
 A21 
 E2 
 0  xi +1 =  A  xi +  A  xi −1
 
 20 
 21 
A
 2, i 
 B20 
 B21 
+ ... + 
 x0 + 
ui + 
 u i +1
A
B
 20 
 B21 
 2, i 
(14)
0 = A20 xi + A21 xi −1 + ... + A2, i x0 + B20ui + B21ui +1
(15)
where ଶ ∈ ℜ௡మ×௡ is full row rank with ଶ ≥ ଵ and ଶ,௝ ∈
ℜ௡మ×௡ , ̅ଶ,௝ ∈ ℜ(௡ି௡మ)×௡ , = 0,1, … , , ଶ,௞ ∈ ℜ௡మ×௠ ,
̅ଶ,௞ ∈ ℜ(௡ି௡మ)×௠ , = 0,1.
From (14) we have:
Substituting in (15) i by i + 1 (in state vector x
and in input u) we obtain:
A20 xi +1 = − A21 xi − ... − A2, i x1 − B20 ui +1 − B21ui + 2
From (2.14) and (2.16) we have:
 A21 
− A 
 22 
 E  A 
,..., Aˆi =  2   2,i ,
 A20   0 
−1
−1
 E  c E 
 A1 − c1E1 
, A1 = −  1   2 1 


A
0


 2   0 
−1
 E  c
E 
,..., Ai = −  1   i + 1 1 ,
 A2   0 
−1
−1
 E  B 
E   0 
B0 =  1   1 , B1 =  1  
.
A
A
−B 
 2   0 
 2   2 
−1
−1
(11)
xi +1 = A0 xi + A1 xi −1 + ... + Ai x0 + B0ui + B1ui +1
(19)
where:
E 
Aˆ 0 =  2 
 A20 
If the matrix:
E 
A0 =  1 
 A2 
 E2 
 A20 
 A21 
 A2, i 
 A  xi +1 = 
 xi + 
 xi −1 + ... + 
 x0
−
A
−
A
 0 
 20 
 21 
 22 
 B 
 0 
B 
+  20 ui +  21 ui +1 + 
 ui + 2
−
B
0


 20 
− B21 
(16)
 E  B 
E 
Bˆ 0 =  2   20 , Bˆ1 =  2 
A
0

 A20 
 20  
−1
(20)
 B21 
− B ,
 20 
−1
E   0 
Bˆ 2 =  2  

 A20  − B21 
If the matrix (18) is singular we repeat the procedure.
Continuing this procedure after at most n steps we finally
obtain a nonsingular matrix and the desired fractional system. The procedure can be justified as follows. The elementary row operations do not change the rank of the matrix
[
− ]. The substitution in the equations (8b) and (15)
i by i + 1 also does not change the rank of the matrix
[
− ] since it is equivalent to multiplication of tits lower
rows by z and by assumption (4b) holds. Therefore,
the following theorem has been proved.
Theorem 1. The singular fractional linear system (5) satisfying the assumption (4) can be reduced to the standard fractional linear system
~
~
~
~
xi +1 = A0 xi + A1 xi −1 + ... + Ai x0 + B0 ui
(21)
~
~
+ B1ui +1 + ... + B p ui + p
where ௝ ∈ ℜ௡×௡ , = 0,1, … , , ௞ ∈ ℜ௡×௠ , = 0,1,
… , < whose dynamics depends on the future inputs
௜ାଵ , … , ௜ା௣ .
Example 1. Consider the singular fractional linear system
(1) for = 0,5 with:
5 0 2
 0.2 2 − 2
1 2
E = 2 0 1, A =  2
1 0 , B = − 1 2  . (22)
1 0 0
− 1.8 0 − 1
 2 − 1
In this case the conditions (4) are satisfied since:
det E = 0 and
63
Tadeusz Kaczorek
Reduction and Decomposition of Singular Fractional Discrete-Time Linear Systems
5 z − 0.2 − 2 2 z + 2
det[ Ez − A] = 2 z − 2 − 1
z = z − 0.2
z + 1.8
0
0
0 
1
 E2  

=
2
0
1
A  

 20  − 3 − 1 − 0.5


1
(27)
Applying to the matrices (22) the following elementary
row operations L[1 + 2 × (−2)] , L[3 + 1× (−1)] we obtain:
is nonsingular and
with the matrices:
5 0 2 0.2 2 − 2 1 2 
[ E A B] = 2 0 1
2
1 0 − 1 2 
1 0 0 − 1.8 0 − 1 2 − 1
1 0 0 − 3.8 0 − 2 3 − 2
→ 2 0 1
2
1 0 − 1 2 
0 0 0
2
0 1 − 1 1 
 E A1 B1 
= 1

 0 A2 B2 
0 
1 0
−1
ˆA =  E2   A20  =  2 0
1 
0 
 
 
 A20  − A21  − 3 − 1 − 0.5


0
−2 
− 3.3

=  4.85 − 0.5 3.625
 9.6
1
4.5 
M
(23)
and the equations (8) have the form:
i +1
1 0 0
− 3.8 0 − 2
 3 − 2
xi − k +1 = 
xi + 


u i
0 1
1 0 
 2
− 1 2 
(24a)
∑ ck 2
k =0
and
0 = [2 0 1]xi + [− 1 1]ui
α 
 α  α (α − 1)
1
c1 = −  = −α = −0.5 , c2 = (−1) 2   =
=− ,
1
2
2
!
8
 
 
α (α − 1)...(α − i )
…, ci +1 = (−1)i −1
(i + 1)!
α = 0.5
and the equation (10) has the form:
0
1 xi −1
0
0
0 ui +1
− 1
(25)
1 0 0
The matrix 2 0 1 is singular and we perform
2 0 1
the elementary row operation L[3 + 2 × ( −1)] on (25) obtaining the following:
−2 
1 0 0
 − 3 .3 0
1
1
 2 0 1 x =  3

1
0.5  xi +  2

 i +1 
8
0 0 0
 − 3 − 1 − 0.5
− 2
1 0 0
 3 − 2
0
− ... − ci +1  2 0 1  x0 + − 1 2 ui + 0
− 2 0 − 1
 1 − 2
1
0 0
0 1  xi −1
0 − 1
0
0 ui +1
− 1
(26)
The matrix:
64
obtain
the
0 
1 0
−1

ˆA =  E2  − A2,i  =  2 0
1 
i 
 
 
A
0
 − 3 − 1 − 0.5
 20  


0 
1 0
= − 3 0 − 0.5,
 0 0
1 
−1
−1
equation
(19)
− 3.3 0 − 2 
 3
1 0.5 

 0.25 0 0.125
1 0 0


 2 0 1
0 0 0
(24b)
Using (6) we obtain:
1 0 0
− 3.3 0 − 2
1 0
 2 0 1 x =  3
 x + 1 2 0
1
0
.
5

 i +1 
 i 8
2 0 1
 0
0 0
0 0 
1 0 0
 3 − 2
0
− ... − ci +1 2 0 1 x0 + − 1 2 ui + 0
0 0 0
 0 0 
1
we
(28)
0
0 
1
−1
 E 2   B20  
ˆ
B0 = 
0
1 
 
=2
A
0
 − 3 − 1 − 0.5
 20  


− 2
 3


= − 5.5 3 
 − 7
6 
−1
0
0 
1
−1
 E 2   B21  

Bˆ1 = 
=
2
0
1 
 − B  
A
 20   20  − 3 − 1 − 0.5


0 0 
= 1 − 2
0 0 
0
0 
1
−1
E   0  
=
Bˆ 2 =  2  
2
0
1 
 
−
A
B
20
21




− 3 − 1 − 0.5
0 0 
= 1 − 1
0 0 
 3 − 2
− 1 2 


 0
0 
−1
−1
 0 0


 0 0
− 1 2
 0 0
 0 0


− 1 1
3. DECOMPOSITION OF SINGULAR
FRACTIONAL SYSTEM INTO DYNAMIC
AND STATIC PARTS
Consider the singular fractional system (5) satisfying
the assumptions (4). Applying the procedure presented
in section 2 after p steps we obtain:
acta mechanica et automatica, vol.5 no.4 (2011)
 A p ,0 
 A p,1 
 A pi 
E p 
 0  xi +1 =  A  xi +  A  xi −1 + ... +  A  x0
 
 p ,0 
 p,2 
 pi 
 B p,0 
 B p ,1 
 B p , p −1 
+
ui + 
ui +1 + ... + 


ui + p −1
 B p,0 
 B p ,1 
 B p , p −1 
(29)
 Ep 
n× n

∈ℜ
A
p
,
0


(30)
Using the elementary column operations we may reduce
the matrix (30) to the form:

( n − n p )× n p
, A21 ∈ ℜ
I n − n p 

0
(31)
and performing the same elementary operations on the
matrix ௡ we can find the matrix ∈ ℜ௡×௡ such that:
I n p
 Ep 

Q =  A
 A p,0 
 21
0 
I n − n p 

(32)
Taking into account (32) and defining the new sate vector:
~
x (1)  (1)
n
n−n
~
xi = Q−1xi =  i(2) , ~
xi ∈ ℜ p , ~
xi(2) ∈ ℜ p , i ∈ Z + (33)
~
 xi 
from (29) we obtain:
~
xi(+1)1 = E p xi +1 = E p QQ−1xi +1 = Ap,0QQ−1 xi + Ap,1QQ−1xi −1
+ ... + ApiQQ−1 x0 + B p,0ui + B p,1ui +1 + ... + B p, p −1ui + p −1
~
~
x (1) 
x (1) 
A(p2,0) ] i(2)  + [ A(p1,)1 A(p2,1) ] i(−21) 
xi 
~
xi −1 
~
~
x (1) 
+ ... + [ A(pi1) A(pi2) ] 0(2)  + B p,0ui + B p,1ui +1
x0 
~
+ ... + B p, p −1ui + p −1
+ B p,0ui + B p,1ui +1 + ... + B p, p −1ui + p −1 , i ∈ Z +
(34)
(1) ~ (1)
(2) ~ (2)
~
xi(2) = − A21~
xi(1) − Ap(1,1) ~
xi(−1)1 − Ap(2,1) ~
xi(−21) − ... − Api
x0 − Api
x0
− B p,0ui − ... − B p, p −1ui + p −1, i ∈ Z +
(35)
where:
Apj Q
( 2)
A pj
],
1 0 0
Q =  0 0 − 1
− 2 1 0 
and
0
0  1 0 0   1
0
0
1
 E2 





1   0 0 − 1 =  0
1
0,
 A Q =  2 0
20


− 3 − 1 − 0.5 − 2 1 0  − 2 − 0.5 1
A21 = [− 2 − 0.5], n2 = 2.
(39)
In this case the equations (34) and (35) have the forms:
and
(1)
Apj = [ A pj
The standard system described by the equation (37)
is called the dynamic part of the system (5) and the system
described by the equation (35) is called the static part
of the system (5).
Therefore, the following theorem has been proved.
Theorem 2. The singular fractional linear system (5) satisfying the assumption (4) can be decomposed into the dynamical part (37) and static part (35) whose dynamics depend
on the future inputs ௜ାଵ , … , ௜ା௣ିଵ .
Example 2. Consider the singular fractional system (1)
for = 0,5 with the matrices (22). The matrix (27)
is nonsingular. To reduce this matrix to the form (31)
we perform the elementary operations R[1 + 3 × (−2)] ,
R[2 × (−1)] , R[2,3] . The matrix Q has the form:
1 0 0  x1, i 
x (1) 
 ~


−1
~
xi = Q xi = 2 0 1  x2, i  =  i( 2) ,
~

  xi 
0 − 1 0  x3, i 
x1, i

 ~ ( 2)
~
xi(1) = 
, xi = − x2, i .
2
x
x
+
3, i 
 1, i
= A(p1,)0 ~
xi(1) + A(p2,0) ~
xi(2) + ... + A(pi1) ~
x0(1) + A(pi2) ~
x0(2)
A(pj2) ],
~
~
A p,0 = A(p1,)0 − A(p2,0) A21 ,..., A pi = A(pi1) − A(p2,0) A(pi1)
~
~
B p,0 = B p ,0 − A(p2,0) B p ,0 ,..., B p , p −1 = B p , p −1 − A(p2,0) B p , p −1
(38)
The new state vector (33) is:
= [ A(p1,)0
= [ A(pj1)
(37)
where
where ௣ ∈ ℜ௡೛ ×௡ is full row rank, ௣௝ ∈ ℜ௡೛×௡ , ̅௣௝ ∈
ℜ(௡ି௡೛)×௡ , = 0,1, … , and ௣௞ ∈ ℜ௡೛×௠ , ௣௞ ∈
ℜ(௡ି௡೛)×௠ , = 0,1, … , − 1 with nonsingular matrix:
I n p
A
 21
~
~ (1) ~
~
xi(+1)1 = A p ,0 ~
xi(1) + ... + A pi ~
x0 + B p ,0 ui
~
+ ... + B p, p −1ui + p −1 , i ∈ Z +
j = 0,1,...,i (36)
0.7 − 2 ~ (1)  0  ~ ( 2) 1 ~ (1)
~
xi(+1)1 = 
 xi +   xi + xi −1
8
 2 0 .5 
− 1
 3 − 2
− ... − ci +1~
x0(1) + 
 ui
− 1 2 
(40)
and
~
xi( 2) = [2 0.5]~
xi(1) + [0.25 0]~
xi(−1)1
+ ... + ci +1[− 2 0]~
x0(1) − [1 − 2]u i − [1 − 1]ui +1
(41)
Substituting (41) into (40) we obtain:
Substitution of (34) into (35) yields:
65
Tadeusz Kaczorek
Reduction and Decomposition of Singular Fractional Discrete-Time Linear Systems
0.7 − 2 ~ (1) 1 1 0  ~ (1)
~
xi(+1)1 = 
x + 
x
0 i
8 0 − 1 i −1
0
(42)
 1 0 ~ (1) 3 − 2
0 0 
− ... − ci +1 
 x0 + 0 0 ui + 1 − 1ui +1
− 2 1




The dynamic part of the system is described by (42)
and the static part by (41).
4. CONCLUDING REMARKS
The singular fractional linear discrete-time systems with
regular pencil have been addressed. It has been shown that
if the assumption (4) are satisfied then: 1) the singular fractional linear system can be reduced to equivalent standard
fractional system (Theorem 1), 2) the singular fractional
linear system can be decomposed into dynamic and static
parts (Theorem 2). The proposed procedures have been
illustrated by numerical examples. The considerations can
be easily extended to singular fractional linear continuoustime systems.
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Acknowledgments: This work was supported by National Centre
of Sciences in Poland under work N N514 6389 40.
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