INTRODUCTION

Descriptor (singular) linear systems have been considered in [3,5,7,15,19]. The fundamentals of fractional calculus have been given in [22, 23, 13]. The linear systems with fractional orders have been analyzed in [4, 6, 9, 10] and with different fractional orders in [1, 12, 15, 23, 24]. The analysis of differential algebraic equations and its numerical solutions have been analyzed in [20] and the numerical and symbolic computations of generalized inverses in [29]. The T-Jordan canonical form and the T-Drazin inverse based on the T-product have been addressed in [23]. In [21] The multilinear time-invariant descriptor systems have been analyzed in [21]. The descriptor and standard positive linear systems by the use of Drazin inverse has been addressed in [2, 8, 15]. The pointwise degeneracy of autonomous control systems have been considered in [20] and of linear delay-differential systems with nonnilpotent passive matrices in [16]. The pointwise completeness and degeneracy of fractional descriptor discrete-time linear systems by the use of the Drazin inverse matrices have been addressed in [9, 11, 12] and of fractional different orders in [14, 15, 26]. Analysis of the differential-algebraic equations has been analyzed in [19] and the numerical and symbolic computations of the generalized inverses in [27]. The T-Jordan canonical form and T-Drazin inverse based on the T-Jordan canonical form and T-Drazin inverse based on the T-product has been investigated in [21, 22]. The numerical and symbolic computation of the generalized inverses have been analyzed in [27].

In this paper the pointwise completeness and the pointwise degeneracy of descriptor linear discrete-time systems with different orders will be analyzed.

The paper is organized as follows. In Section 2 the Drazin inverse of matrices is applied to find the solution to descriptor linear discrete-time systems with different fractional orders. Necessary and sufficient conditions for the pointwise completeness of the systems with fractional orders are established in Section 3 and the pointwise degeneracy of the systems in Section 4. Concluding remarks are given in Section 5.

The following notation will be used: ℜ - the set of real numbers, ℜn×m - the set of n × m real matrices and ℜn = ℜn×1, Z+ - the set of nonnegative integers, In - the n × n identity matrix, imgP – the image of the matrix P.

 SOLUTION OF THE STATE EQUATIONS OF FRACTIONAL DESCRIPTOR DISCRETE-TIME LINEAR SYSTEMS

Consider the descriptor fractional discrete-time linear system with two different fractional orders

1
E[Δαx1(i+1)Δβx2(i+1)]=A[x1(i)x2(i)] andE=[E100E2],A=[A11A12A21A22],
where
0<α,β<2, i∈Z+={0,1,2,…}, x1(i)∈ℜn1
and
x2(i)∈ℜn2
are the state vectors and
Ek,Akj∈ℜnk×nj;k,j=1,2.

The fractional difference of α(β) order is defined by [11, 13]

2
Δαx(i)=∑j=0icα(j)x(i−j),cα(j)=(−1)j(αj)=(−1)jα(α−1)…(α−j+1)j!,cα(0)=1,j=1,2,…

In descriptor systems it is assumed that det E = 0 and the pencil is regular, i.e.

3
det[[E1z100E2z2]−[A11A12A21A22]]≠0 for some z1,z2∈C
where C is the field of complex numbers.

Premultiplying (1) by the matrix

[Ediag(In1c1,In2c2)−A]−1
we obtain
4
E¯[Δαx1(i+1)Δβx2(i+1)]=A¯[x1(i)x2(i)],i∈Z+
where
5
E¯=[Ediag(In1c1,In2c2)−A]−1E=[E¯11E¯12E¯21E¯22],A¯=[Ediag(In1c1,In2c2)−A]−1A=[A¯11A¯12A¯21A¯22].

The equation (1) and (4) have the same solution

x(i)=[x1(i)x2(i)].

Lemma 1. If there exist c1, c2 ∈ C such that

6
E¯[diag(In1c1,In2c2)]E¯=E¯2[diag(In1c1,In2c2)]
then
7
E¯A¯=A¯E¯.

Proof. From (5) we have

8
E¯diag(In1c1,In2c2)−A¯=[Ediag(In1c1,In2c2)−A]−1×[Ediag(In1c1,In2c2)−A]−1A=In
and
9
A¯=E¯diag(In1c1,In2c2)−In.

Using (9) we obtain

10
E¯A¯=E¯{E¯diag(In1c1,In2c2)−In}=E¯2diag(In1c1,In2c2)−E¯
and
11
A¯E¯={E¯diag(In1c1,In2c2)−In}E¯=E¯diag(In1c1,In2c2)E¯−E¯.

Therefore, if the condition (6) is satisfied then the equation (7) holds.

Remark 1. If c1 = c2 ∈ C then the equality (6) is always satisfied

12
E¯[diag(In1c1,In2c2)]=E¯c=cE¯.

Lemma 2. If the condition (7) is satisfied then

13
E¯A¯D=A¯DE¯,
14
E¯DA¯=A¯E¯D,
15
E¯DA¯D=A¯DE¯D.

Proof is given in [13].

Remark 2. If A ≠ 0 and we assume c1 = c2 = 0 then

16
E¯=[−A]−1E,A¯=−In
in this case the condition (7) is satisfied.

Substituting (2) into (4) we obtain

17
E¯[Δαx1(i+1)Δβx2(i+1)]=[A¯1αA¯12A¯21A¯2β][x1(i)x2(i)]+∑j=2i+1[In1cα(j+1)00In2cβ(j+1)][x1(i−j+1)x2(i−j+1)],
where A¯1α=A¯11+αIn1, A¯2β=A¯22+βIn2.

In particular case when Ē = In we have the following theorem.

Theorem 1.

The fractional discrete-time linear system (4) with Ē = In and initial conditions

x(0)=[x1(0)x2(0)]
has the solution
18
x(i)=Φi[x1(0)x2(0)],
where
19a
Φi={In for i=0A¯Φi−1−D1Φi−2−…−Di−1Φ0 for i=1,2,…
19b
A¯=[A¯11A¯12A¯21A¯22],Dk=[In1cα(k+1)00In2cβ(k+1)],k=1,2,…

Proof is given in [13].

If Ē ≠ In then the Drazin inverse of matrix Ē will be applied to find the solution to the equation (4).

Definition 1. A matrix ĒD is called the Drazin inverse of Ē ∈ ℜn×n if it satisfies the conditions

20a
E¯E¯D=E¯DE¯,
20b
E¯DE¯E¯D=E¯D,
20c
E¯DE¯q+1=E¯q,
where q is the smallest nonnegative integer (called index of Ē), satisfying the condition rank Ēq = rank Ēq+1.

The Drazin inverse ĒD of a square matrix Ē always exists and is unique. If det Ē ≠ 0 then ĒD = Ē−1. The Drazin inverse matrix ĒD can be computer by the one of known methods [2, 3, 13].

Theorem 2.

The descriptor fractional discrete-time linear system (4) with initial conditions x(0)=[x1(i)x2(i)]∈Im(E¯DE¯)=E¯DE¯v, x ∈ ℜn has the solution

21
x(i)=[x1(i)x2(i)]=Φ^iE¯DE¯v,
where
22a
Φ^i={In for i=0A^Φ^i−1−D^1Φ^i−2−…−D^i−1Φ^0 for i=1,2,…A^=E¯DA¯=[A^11A^12A^21A^22],
22b
D^k=E¯D[In1cα(k+1)00In2cβ(k+1)],k=1,2,…

Proof. Taking into account that the equations (1) and (4) have the same solution the proof will be accomplisched by showing that the solution (21) satisfies the equation (4).

Using (21) and (22) we obtain

23
E¯Φ^i+1E¯DE¯v=E¯(A^Φ^i−D^1Φ^i−1−…−D^1Φ^0)E¯DE¯v=E¯E¯DA^(A^Φ^i−1−D^1Φ^i−2−…−D^i−1Φ^0)E¯DE¯v=A^(A^Φ^i−1−D^1Φ^i−2−…−D^i−1Φ^0)(E¯DE¯)2v=A^Φ^iE¯DE¯v
since (14) and (ĒD Ē)2 = ĒD Ē. Therefore, the solution of the equation (1) has the form (21).

 THE POINTWISE COMPLETENESS OF DESCRIPTOR FRACTIONAL DISCRETE-TIME LINEAR SYSTEMS WITH DIFFERENT FRACTIONAL ORDERS

In this section necessary and sufficient conditions for the pointwise completeness of the descriptor discrete-time linear systems with different fractional orders will be established.

Definition 2. The descriptor fractional discrete-time linear system (1) is called pointwise complete at the point i = q if for every final state xf ∈ ℜn, there exists an boundary condition x(0) ∈ Im¯EE¯D such that

24
x(q)=xf∈Im¯EE¯D.

Theorem 3.

The descriptor fractional discrete-time linear system (1) is pointwise complete for i = q and every xf∈ℜn∈Im¯EE¯D if and only if

25a
detΦ^q≠0
where
25b
Φ^q=A^Φ^q−1−D^1Φ^q−2−…−D^q−1Φ^0
and Â, D^k are defined by (22b).

Proof. From (21) for i = q we obtain

26
xf=x(q)=Φ^qE¯DE¯x(0).

For given xf∈ℜn∈Im¯EE¯D we may find x(0)∈Im¯EE¯D if and only if the condition (25) is satisfed. Therefore, the descriptor fractional system (1) is pointwise complete at the point i = q if and only if the condition (25) is satisfied.

Example 1

Consider the descriptor fractional system (1) for α = 0.6, β = 0.8 with the matrices

27
E=[E100E2]=[100001000],A=[A11A12A21A22]=[011000010],n1=1, n2=2.

We choose c1 = c2 = 1 and using (5), (27) we obtain

28
E¯=[Ediag(c1,c2)−A]−1E=[E¯11E¯12E¯21E¯22]=[101000001],A¯=[Ediag(c1,c2)−A]−1A=[A¯11A¯12A¯21A¯22]=[0010−10000]

The Drazin inverse matrix of Ē has the form

29
E¯D=[10−1000001] and E¯E¯D=[100000001].

In this case

30
Φ^1=A^=E¯DA¯=[10−1000001][0010−10000]=[001000000].

And x(0)∈Im¯EE¯D=[x11(0)0x21(0)]  and  x11(0), x21(0) are arbitrary.

Note that the matrix Φ^1 is singular and by Theorem 2 the descriptor fractional system with (27) is not pointwise complete for q = 1 and every xf ∈ ℜ3 of the form xf=[x11(tf)0x21(tf)] and x11(tf), x21(tf) are arbitrary.

Using (25b) for q = 2 we obtain

31
Φ^2=A^2−D^1=[0010−10000]2−[I2cα(2)00cβ(2)]=[0.120000.880000.08]
and
32
detΦ2^=8.448*10−3≠0.

Therefore, by Theorem 2 the descriptor fractional system with (27) is pointwise complete for q = 2.

 THE POINTWISE DEGENERACY OF FRACTIONAL DESCRIPTOR LINEAR DISCRETE-TIME SYSTEMS

In this section necessary and sufficient conditions for the pointwise degeneracy of the descriptor discrete-time linear systems with different fractional orders will be established.

Definition 4.1. The descriptor fractional discrete-time linear system (1) is called pointwise degenerated in the direction v for q = qf if there exists a vector v ∈ ℜn such that for all initial conditions x(0)∈Im¯EE¯D the solution of (1) for q = qf satisfy the condition

33
vTxf=0.

Theorem 3.

The descriptor fractional continuous-time linear system (1) is pointwise degenerated in the direction v ∈ ℜn for q = qf if and only if

34
detΦ^q=0,
where Φ^q is defined by (25b).

Proof. From (4.1) and (26) for q = qf we have

35
vTΦ^qx(0)=0.

There exists a nonzero vector v ∈ ℜn such that (35) holds for all x(0)∈Im¯EE¯D if and only if the condition (34) is satisfied. Therefore, the descriptor fractional system (1) is pointwise degenerated in the direction v ∈ ℜn for q = qf if the condition (34) is satisfied.

Remark 2. The vector v ∈ ℜn in which the descriptor fractional discrete-time linear system (1) is pointwise degenerated can be computed from the equation

36
vTΦ^q=0.

Example 2.

(Continuation of Example 1) Consider the system (1) for α = 0.6, β = 0.8 with the matrices (27). In Example 1 it was shown that the matrix Φ^q for q = 1 is nonsingular. Therefore, the descriptor fractional system (1) with (27) is pointwise degenerated for q = 1 and any direction v.

From (31) and Theorem 3 it follows that the matrix Φ^2 is nonsingular. Therefore, by Theorem 3 the system (1) with (27) is not pointwise degenerated for i = q = 2.

 CONCLUDING REMARKS

The Drazin inverse of matrices has been applied to investigation of the pointwise completeness and the pointwise degeneracy of the descriptor linear discrete-time systems with different fractional orders. Necessary and sufficient conditions for the pointwise completeness (Theorem 2) and for the pointwise degeneracy (Theorem 3) of the fractional linear discrete–time systems have been established. The considerations have been illustrated by numerical examples. The presented methods can be extended to the descriptor linear systems with many different fractional orders.