Pointwise Completeness and Pointwise Degeneracy of Descriptor Linear Discrete-Time Systems with Different Fractional Orders

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, ℜ^n×m - the set of n × m real matrices and ℜⁿ = ℜ^n×1, Z₊ - the set of nonnegative integers, I_n - the n × n identity matrix, imgP – the image of the matrix P.

2.

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 $\begin{array}{l} E [\begin{array}{l} Δ^{α} x_{1} (i + 1) \\ Δ^{β} x_{2} (i + 1) \end{array}] = A [\begin{array}{l} x_{1} (i) \\ x_{2} (i) \end{array}] and \\ E = [\begin{matrix} E_{1} & 0 \\ 0 & E_{2} \end{matrix}], A = [\begin{array}{l} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}], \end{array}$ where $0 < α, β < 2, i \in Z_{+} = {0, 1, 2, \dots}, x_{1} (i) \in ℜ^{n_{1}}$ and $x_{2} (i) \in ℜ^{n_{2}}$ are the state vectors and $E_{k}, A_{k j} \in ℜ^{n_{k} \times n_{j}}; k, j = 1, 2.$

The fractional difference of α(β) order is defined by [11, 13] 2 $\begin{array}{l} Δ^{α} x (i) = \sum_{j = 0}^{i} c_{α} (j) x (i - j), \\ c_{α} (j) = {(- 1)}^{j} (\begin{matrix} α \\ j \end{matrix}) = {(- 1)}^{j} \frac{α (α - 1) \dots (α - j + 1)}{j!}, c_{α} (0) = 1, j = 1, 2, \dots \end{array}$

In descriptor systems it is assumed that det E = 0 and the pencil is regular, i.e. 3 $d e t [[\begin{matrix} E_{1} z_{1} & 0 \\ 0 & E_{2} z_{2} \end{matrix}] - [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}]] \neq 0 for some z_{1}, z_{2} \in C$ where C is the field of complex numbers.

Premultiplying (1) by the matrix ${[E diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - A]}^{- 1}$ we obtain 4 $\bar{E} [\begin{array}{l} Δ^{α} x_{1} (i + 1) \\ Δ^{β} x_{2} (i + 1) \end{array}] = \bar{A} [\begin{array}{l} x_{1} (i) \\ x_{2} (i) \end{array}], i \in Z_{+}$ where 5 $\begin{array}{l} \bar{E} = {[E diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - A]}^{- 1} E = [\begin{array}{l} {\bar{E}}_{11} & {\bar{E}}_{12} \\ {\bar{E}}_{21} & {\bar{E}}_{22} \end{array}], \\ \bar{A} = {[E diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - A]}^{- 1} A = [\begin{array}{l} {\bar{A}}_{11} & {\bar{A}}_{12} \\ {\bar{A}}_{21} & {\bar{A}}_{22} \end{array}] . \end{array}$

The equation (1) and (4) have the same solution $x (i) = [\begin{array}{l} x_{1} (i) \\ x_{2} (i) \end{array}] .$

Lemma 1. If there exist c₁, c₂ ∈ C such that 6 $\bar{E} [diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2})] \bar{E} = {\bar{E}}^{2} [diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2})]$ then 7 $\bar{E} \bar{A} = \bar{A} \bar{E} .$

Proof. From (5) we have 8 $\bar{E} diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - \bar{A} = [E diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - A]^{- 1} \times {[E diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - A]}^{- 1} A = I_{n}$ and 9 $\bar{A} = \bar{E} diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - I_{n} .$

Using (9) we obtain 10 $\bar{E} \bar{A} = \bar{E} {\bar{E} diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - I_{n}} = {\bar{E}}^{2} diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - \bar{E}$ and 11 $\bar{A} \bar{E} = {\bar{E} diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) - I_{n}} \bar{E} = \bar{E} diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2}) \bar{E} - \bar{E} .$

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 $\bar{E} [diag (I_{n_{1}} c_{1}, I_{n_{2}} c_{2})] = \bar{E} c = c \bar{E} .$

Lemma 2. If the condition (7) is satisfied then 13 $\bar{E} {\bar{A}}^{D} = {\bar{A}}^{D} \bar{E},$ 14 ${\bar{E}}^{D} \bar{A} = \bar{A} {\bar{E}}^{D},$ 15 ${\bar{E}}^{D} {\bar{A}}^{D} = {\bar{A}}^{D} {\bar{E}}^{D} .$

Proof is given in [13].

Remark 2. If A ≠ 0 and we assume c₁ = c₂ = 0 then 16 $\bar{E} = [^{- A] - 1} E, \bar{A} = - I_{n}$ in this case the condition (7) is satisfied.

Substituting (2) into (4) we obtain 17 $\bar{E} [\begin{array}{l} Δ^{α} x_{1} (i + 1) \\ Δ^{β} x_{2} (i + 1) \end{array}] = [\begin{array}{l} {\bar{A}}_{1 α} & {\bar{A}}_{12} \\ {\bar{A}}_{21} & {\bar{A}}_{2 β} \end{array}] [\begin{array}{l} x_{1} (i) \\ x_{2} (i) \end{array}] + \sum_{j = 2}^{i + 1} [\begin{matrix} I_{n_{1}} c_{α} (j + 1) & 0 \\ 0 & I_{n_{2}} c_{β} (j + 1) \end{matrix}] [\begin{array}{l} x_{1} (i - j + 1) \\ x_{2} (i - j + 1) \end{array}],$ where ${\bar{A}}_{1 α} = {\bar{A}}_{11} + α I_{n_{1}}$ , ${\bar{A}}_{2 β} = {\bar{A}}_{22} + β I_{n_{2}}$ .

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

Theorem 1.

The fractional discrete-time linear system (4) with Ē = I_n and initial conditions $x (0) = [\begin{array}{l} x_{1} (0) \\ x_{2} (0) \end{array}]$ has the solution 18 $x (i) = Φ_{i} [\begin{array}{l} x_{1} (0) \\ x_{2} (0) \end{array}],$ where 19a $Φ_{i} = {\begin{matrix} I_{n} for i = 0 \\ \bar{A} Φ_{i - 1} - D_{1} Φ_{i - 2} - \dots - D_{i - 1} Φ_{0} for i = 1, 2, \dots \end{matrix}$ 19b $\begin{array}{l} \bar{A} = [\begin{array}{l} {\bar{A}}_{11} & {\bar{A}}_{12} \\ {\bar{A}}_{21} & {\bar{A}}_{22} \end{array}], D_{k} = [\begin{matrix} I_{n_{1}} c_{α} (k + 1) & 0 \\ 0 & I_{n_{2}} c_{β} (k + 1) \end{matrix}], \\ k = 1, 2, \dots \end{array}$

Proof is given in [13].

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

Definition 1. A matrix Ē^D is called the Drazin inverse of Ē ∈ ℜ^n×n if it satisfies the conditions 20a $\bar{E} {\bar{E}}^{D} = {\bar{E}}^{D} \bar{E},$ 20b ${\bar{E}}^{D} \bar{E} {\bar{E}}^{D} = {\bar{E}}^{D},$ 20c ${\bar{E}}^{D} {\bar{E}}^{q + 1} = {\bar{E}}^{q},$ where q is the smallest nonnegative integer (called index of Ē), satisfying the condition rank Ē^q = rank Ē^q+1.

The Drazin inverse Ē^D of a square matrix Ē always exists and is unique. If det Ē ≠ 0 then Ē^D = Ē⁻¹. The Drazin inverse matrix Ē^D 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 $x (0) = [\begin{array}{l} x_{1} (i) \\ x_{2} (i) \end{array}] \in I m ({\bar{E}}^{D} \bar{E}) = {\bar{E}}^{D} \bar{E} v$ , x ∈ ℜⁿ has the solution 21 $x (i) = [\begin{array}{l} x_{1} (i) \\ x_{2} (i) \end{array}] = {\hat{Φ}}_{i} {\bar{E}}^{D} \bar{E} v,$ where 22a $\begin{array}{l} {\hat{Φ}}_{i} = {\begin{matrix} I_{n} for i = 0 \\ \hat{A} {\hat{Φ}}_{i - 1} - {\hat{D}}_{1} {\hat{Φ}}_{i - 2} - \dots - {\hat{D}}_{i - 1} {\hat{Φ}}_{0} for i = 1, 2, \dots \end{matrix} \\ \hat{A} = {\bar{E}}^{D} \bar{A} = [\begin{array}{l} {\hat{A}}_{11} & {\hat{A}}_{12} \\ {\hat{A}}_{21} & {\hat{A}}_{22} \end{array}], \end{array}$ 22b ${\hat{D}}_{k} = {\bar{E}}^{D} [\begin{matrix} I_{n_{1}} c_{α} (k + 1) & 0 \\ 0 & I_{n_{2}} c_{β} (k + 1) \end{matrix}], k = 1, 2, \dots$

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 $\bar{E} {\hat{Φ}}_{i + 1} {\bar{E}}^{D} \bar{E} v = \bar{E} (\hat{A} {\hat{Φ}}_{i} - {\hat{D}}_{1} {\hat{Φ}}_{i - 1} - \dots - {\hat{D}}_{1} {\hat{Φ}}_{0}) {\bar{E}}^{D} \bar{E} v = \bar{E} {\bar{E}}^{D} \hat{A} (\hat{A} {\hat{Φ}}_{i - 1} - {\hat{D}}_{1} {\hat{Φ}}_{i - 2} - \dots - {\hat{D}}_{i - 1} {\hat{Φ}}_{0}) {\bar{E}}^{D} \bar{E} v = \hat{A} (\hat{A} {\hat{Φ}}_{i - 1} - {\hat{D}}_{1} {\hat{Φ}}_{i - 2} - \dots - {\hat{D}}_{i - 1} {\hat{Φ}}_{0}) {({\bar{E}}^{D} \bar{E})}^{2} v = \hat{A} {\hat{Φ}}_{i} {\bar{E}}^{D} \bar{E} v$ since (14) and (Ē^D Ē)² = Ē^D Ē. Therefore, the solution of the equation (1) has the form (21).

3.

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 x_f ∈ ℜⁿ, there exists an boundary condition x(0) ∈ $I \bar{m} E {\bar{E}}^{D}$ such that 24 $x (q) = x_{f} \in I \bar{m} E {\bar{E}}^{D} .$

Theorem 3.

The descriptor fractional discrete-time linear system (1) is pointwise complete for i = q and every $x_{f} \in ℜ^{n} \in I \bar{m} E {\bar{E}}^{D}$ if and only if 25a $d e t {\hat{Φ}}_{q} \neq 0$ where 25b ${\hat{Φ}}_{q} = \hat{A} {\hat{Φ}}_{q - 1} - {\hat{D}}_{1} {\hat{Φ}}_{q - 2} - \dots - {\hat{D}}_{q - 1} {\hat{Φ}}_{0}$ and Â, ${\hat{D}}_{k}$ are defined by (22b).

Proof. From (21) for i = q we obtain 26 $x_{f} = x (q) = {\hat{Φ}}_{q} {\bar{E}}^{D} \bar{E} x (0) .$

For given $x_{f} \in ℜ^{n} \in I \bar{m} E {\bar{E}}^{D}$ we may find $x (0) \in I \bar{m} E {\bar{E}}^{D}$ 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 $\begin{array}{l} E = [\begin{matrix} E_{1} & 0 \\ 0 & E_{2} \end{matrix}] = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}], A = [\begin{array}{l} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}] = [\begin{array}{l} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array}], \\ n_{1} =1, n_{2} =2 . \end{array}$

We choose c₁ = c₂ = 1 and using (5), (27) we obtain 28 $\begin{array}{l} \bar{E} = {[E diag (c_{1}, c_{2}) - A]}^{- 1} E = [\begin{array}{l} {\bar{E}}_{11} & {\bar{E}}_{12} \\ {\bar{E}}_{21} & {\bar{E}}_{22} \end{array}] = [\begin{array}{l} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}], \\ \bar{A} = {[E diag (c_{1}, c_{2}) - A]}^{- 1} A = [\begin{matrix} {\bar{A}}_{11} & {\bar{A}}_{12} \\ {\bar{A}}_{21} & {\bar{A}}_{22} \end{matrix}] = [\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{matrix}] \end{array}$

The Drazin inverse matrix of Ē has the form 29 ${\bar{E}}^{D} = [\begin{matrix} 1 & 0 & - 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] and \bar{E} {\bar{E}}^{D} = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}] .$

In this case 30 ${\hat{Φ}}_{1} = \hat{A} = {\bar{E}}^{D} \bar{A} = [\begin{matrix} 1 & 0 & - 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{matrix}] = [\begin{array}{l} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] .$

And $x (0) \in I \bar{m} E {\bar{E}}^{D} = [\begin{matrix} x_{11} (0) \\ 0 \\ x_{21} (0) \end{matrix}] and x_{11} (0)$ , x₂₁(0) are arbitrary.

Note that the matrix ${\hat{Φ}}_{1}$ is singular and by Theorem 2 the descriptor fractional system with (27) is not pointwise complete for q = 1 and every x_f ∈ ℜ³ of the form $x_{f} = [\begin{matrix} x_{11} (t_{f}) \\ 0 \\ x_{21} (t_{f}) \end{matrix}]$ and x₁₁(t_f), x₂₁(t_f) are arbitrary.

Using (25b) for q = 2 we obtain 31 ${\hat{Φ}}_{2} = {\hat{A}}^{2} - {\hat{D}}_{1} = {[\begin{matrix} 0 & 0 & 1 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{matrix}]}^{2} - [\begin{matrix} I_{2} c_{α} (2) & 0 \\ 0 & c_{β} (2) \end{matrix}] = [\begin{matrix} 0.12 & 0 & 0 \\ 0 & 0.88 & 0 \\ 0 & 0 & 0.08 \end{matrix}]$ and 32 $\hat{d e t Φ_{2}} = 8.448 * 10^{- 3} \neq 0.$

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

4.

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 = q_f if there exists a vector v ∈ ℜⁿ such that for all initial conditions $x (0) \in I \bar{m} E {\bar{E}}^{D}$ the solution of (1) for q = q_f satisfy the condition 33 $v^{T} x_{f} = 0.$

Theorem 3.

The descriptor fractional continuous-time linear system (1) is pointwise degenerated in the direction v ∈ ℜⁿ for q = q_f if and only if 34 $d e t {\hat{Φ}}_{q} = 0,$ where ${\hat{Φ}}_{q}$ is defined by (25b).

Proof. From (4.1) and (26) for q = q_f we have 35 $v^{T} {\hat{Φ}}_{q} x (0) = 0.$

There exists a nonzero vector v ∈ ℜⁿ such that (35) holds for all $x (0) \in I \bar{m} E {\bar{E}}^{D}$ 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 = q_f 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 $v^{T} {\hat{Φ}}_{q} = 0 .$

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 ${\hat{Φ}}_{q}$ 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 ${\hat{Φ}}_{2}$ is nonsingular. Therefore, by Theorem 3 the system (1) with (27) is not pointwise degenerated for i = q = 2.

5.

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.

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Pointwise Completeness and Pointwise Degeneracy of Descriptor Linear Discrete-Time Systems with Different Fractional Orders

Tadeusz KACZOREK

Publicado en línea: 26 jun 2025

Páginas: 288 - 291

Recibido: 13 jul 2024

Aceptado: 06 jun 2025

DOI: https://doi.org/10.2478/ama-2025-0035

Palabras clavedrazin inverse, descriptor, discrete-time, linear, system, pointwise completeness, pointwise degeneracy, fractional, different orders

© 2025 Tadeusz KACZOREK, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Palabras clave
drazin inverse, descriptor, discrete-time, linear, system, pointwise completeness, pointwise degeneracy, fractional, different orders