A Note on Controllability of Time Invariant Linear Fractional h-Difference Equations

Abstract

In this paper, we establish and prove two main results: (i) a Kalman-like controllability criterion, and (ii) a rank condition on the controllability matrix, defined via the discrete Mittag–Leffler function, for time-invariant linear fractional-order h-discrete systems. Using some properties of the Mittag–Leffler-type function within the framework of fractional h-discrete calculus, we state and prove the variation of constants formula for an initial value problem. Then we use this formula to prove the equivalence between two notions of controllability: complete controllability and controllability to the origin.

1. Introduction

A practical criterion for assessing the controllability of linear systems of ordinary differential equations was first introduced by Kalman [1,2]. This criterion relies on the rank conditions of the coefficient matrices in the linear system. Subsequently, analogous criteria have been developed for discrete-time systems, fractional-order systems, fractional-order discrete systems [3,4,5,6,7,8].

Fractional-order systems, also known as non-integer order systems, are increasingly used to model a wide variety of real-world phenomena with greater precision and flexibility. Historically, such systems have been predominantly formulated in the continuous-time domain [9,10,11,12,13,14]. However, recent advancements in discrete fractional calculus have facilitated the development of discrete-time fractional-order models, prompting a growing body of research focused on their theoretical properties and applications [15,16,17,18,19].

Mozyrska et al. [17] studied the properties of the non-integer order h-difference linear control systems and stated the rank conditions for controllability and observability of the discrete systems with Caputo-type fractional h-difference operator by introducing the controllability matrix, which consists of a matrix discrete Mittag–Leffler function. In [6], the authors study the local controllability and observability of nonlinear discrete-time systems considering the Caputo, the Riemann–Liouville, and the Grünwald–Letnikov-type fractional h-difference operators. In 2019, Atıcı and Nguyen [4] studied the controllability and observability of the discrete $\Delta$-fractional time-invariant linear systems. In all these papers, one can observe that the criterion for the controllability of the non-integer order system relies on the rank of a controllability matrix or Gramian matrix.

On the other hand, several papers [5,20,21,22] present the controllability criterion based on the rank of a matrix constructed directly from the studied linear system. This criterion is widely known as the Kalman condition for the controllability of a given linear system. In this paper, we employ both approaches to derive a controllability criterion: one based on the introduction of a Gramian matrix, and the other on the construction of a matrix using the coefficient matrices of the considered non-integer order h-difference equation system.

Motivated by foundational results in the controllability of linear integer-order systems, we seek to extend these concepts to linear fractional-order h-discrete systems by leveraging recent theoretical advances, ultimately establishing a Kalman-like criterion for controllability. By choosing the time domain $h\mathbb{N}$, our results will coincide with the controllability criterion for the linear discrete fractional systems when $h=1$ and for the linear continuous fractional systems when $h\to 0$.

We structure the paper is as follows: In the preliminaries section, we give basic definitions, some theorems, and notations in fractional h-discrete calculus. Additionally, we formulate and prove the variation of constants formula for the associated initial value problem. Section 2 introduces the time-invariant fractional h-discrete system and establishes the main results of the paper, with a particular focus on deriving a Kalman-like controllability criterion. We introduce a controllability matrix which includes the matrix discrete Mittag–Leffler function and obtain the controllability criterion for the system with the rank condition on the controllability matrix. The variation of constants formula is a main tool to prove the equivalency of notions of controllability: complete controllability and controllability to the origin. We close the paper with a concluding remark in the final section.

For a comprehensive source in the area of discrete fractional calculus, we direct the reader to a book by Goodrich and Peterson [23].

2. Preliminaries

Represent by $h{\mathbb{N}}_a=\{a,a+h,a+2h,\dots \}$, $h{\mathbb{N}}_a^b=\{a,a+h,a+2h,\dots ,b\}$ for a, $b\in \mathbb{R}$, $h\in {\mathbb{R}}^{+}$ with ${\displaystyle \frac{b-a}{h}}\in {\mathbb{N}}_1$.

Definition 1

([24]). For $w:h{\mathbb{N}}_a\to \mathbb{R}$, the backward h-difference operator is

$${\nabla }_{h}w\left(t\right)={\displaystyle \frac{w\left(t\right)-w(t-h)}{h}},t\in h{\mathbb{N}}_{a+h}.$$

Remark 1.

Observe that

1. 

If we take $h=1$ in Definition 1, we obtain the backward difference (nabla) operator

$$\nabla w\left(t\right)=w\left(t\right)-w(t-1),t\in {\mathbb{N}}_{a+1};$$

2. 

If ${\displaystyle \underset{h\to 0}{lim}\frac{w\left(t\right)-w(t-h)}{h}}$ exists, then we have ${\displaystyle \underset{h\to 0}{lim}{\nabla }_{h}w\left(t\right)=w^{\prime }\left(t\right)}$.

Definition 2

([24]). The h-rising factorial function is

$$t_h^{\overline{r}}=h^r{\displaystyle \frac{\Gamma (\frac{t}{h}+r)}{\Gamma \left(\frac{t}{h}\right)}},t,r\in \mathbb{R},h\in {\mathbb{R}}^{+},$$

provided the RHS is well-defined. Here $\Gamma (·)$ is the gamma function.

Definition 3

([24]). For $w:h{\mathbb{N}}_a\to \mathbb{R}$, the nabla h-fractional sum of order α is

$${\nabla }_{h,a}^{-\alpha }w\left(t\right):={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{s=a/h}^{t/h}{(t-\rho \left(sh\right))}_h^{\overline{\alpha -1}}w\left(sh\right)h,\alpha \in {\mathbb{R}}^{+},a\in \mathbb{R},t\in h{\mathbb{N}}_a,$$

where $h\in {\mathbb{R}}^{+}$ and $\rho \left(t\right)=t-h$.

Definition 4

([24]). For $w:h{\mathbb{N}}_a\to \mathbb{R}$, the αth order Riemann–Liouville nabla h-fractional difference is

$${\nabla }_{h,a}^{\alpha }w\left(t\right):={\nabla }_h^n{\nabla }_{h,a}^{-(n-\alpha )}w\left(t\right),h\in {\mathbb{R}}^{+},n-1<\alpha \le n,n\in \mathbb{N},t\in h{\mathbb{N}}_{a+nh}.$$

Theorem 1

([25]). For $w:h{\mathbb{N}}_a\to \mathbb{R}$, $\alpha \notin {\mathbb{N}}_1$, $n-1<\alpha <n$, and $n\in {\mathbb{N}}_1$,

$${\nabla }_{h,a}^{\alpha }w\left(t\right):={\displaystyle \frac{1}{\Gamma (-\alpha )}}\sum _{s=a/h}^{t/h}{(t-\rho \left(sh\right))}_h^{\overline{-\alpha -1}}w\left(sh\right)h,t\in h{\mathbb{N}}_a.$$

Definition 5

([26]). The two-parameter Mittag–Leffler function in discrete-time is defined by

$$E_{\lambda ,\nu ,\mu }^h(t,\rho \left(a\right))={\displaystyle \frac{1-\lambda h^{\nu }}{h^{\mu }}}\sum _{k=0}^{\infty }{\lambda }^k{\displaystyle \frac{{(t-\rho \left(a\right))}_h^{\overline{\nu k+\mu }}}{\Gamma (\nu k+\mu +1)}},t\in h{\mathbb{N}}_a.$$

Here λ, μ, $a\in \mathbb{R}$ and h, $\nu \in {\mathbb{R}}^{+}$ with $|\lambda h^{\nu }|<1$. Clearly, $E_{\lambda ,\nu ,\mu }^h(a,\rho \left(a\right))=1$.

Definition 6

([26]). Let $M_n$ represents all $n\times n$ matrices over $\mathbb{R}$, and ${\mathbb{R}}^n$ represents all ordered n-tuples of real numbers. For every $y\in {\mathbb{R}}^n$ and $D\in M_n$, an operator norm on $M_n$ is

$$\parallel D\parallel =\underset{\parallel y\parallel =1}{max}\parallel Dy\parallel ,$$

which corresponds to each vector norm on ${\mathbb{R}}^n$. $\parallel I_n\parallel =1$, where $I_n$ is the $n\times n$ identity matrix.

Remark 2

([26]). For $D\in M_n$, $h>0$, $0<\alpha <1$ with $\varrho \left(D\right)<h^{-\alpha }$, the matrix power series

$$\sum _{k=0}^{\infty }{D}^k{\displaystyle \frac{{(t-\rho \left(a\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}$$

converges.

Remark 3.

If $\varrho \left(D\right)<h^{-\alpha }$, we note that the inverse of $\left(I_n-h^{\alpha }D\right)$ exists. To see this, let λ be an eigenvalue of D. Then, $h^{\alpha }\lambda$ is an eigenvalue of $h^{\alpha }D$, and therefore $1-h^{\alpha }\lambda$ is an eigenvalue of $I_n-h^{\alpha }D$. We know that the matrix $I_n-h^{\alpha }D$ is invertible if and only if none of its eigenvalues are zero, i.e., $1-h^{\alpha }\lambda \ne 0$ for all $\lambda \in \mathrm{spectrum}\left(D\right)$. Since $\varrho \left(D\right)<h^{-\alpha }$, we obtain $\left|\lambda \right|<h^{-\alpha }$ for all eigenvalues λ of D. That is, $|h^{\alpha }\lambda |<1$ for all eigenvalues λ of D. Consequently, $1-h^{\alpha }\lambda \ne 0$ for all eigenvalues λ of D. Therefore, $I_n-h^{\alpha }D$ is invertible.

Definition 7.

Consequently, by Remarks 2 and 3, define by

$$E_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right)):=\left(I_n-h^{\alpha }D\right){\displaystyle \frac{1}{h^{\alpha -1}}}\sum _{k=0}^{\infty }{D}^k{\displaystyle \frac{{(t-\rho \left(a\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}},t\in h{\mathbb{N}}_a,$$

(1)

where $D\in M_n$, $h>0$, $0<\alpha <1$ with $\varrho \left(D\right)<h^{-\alpha }$.

Theorem 2

([26]). For $D\in M_n$, $h>0$, $0<\alpha <1$ with $\varrho \left(D\right)<h^{-\alpha }$, the initial value problem (IVP)

$$\left\{\begin{array}{c}{\nabla }_{h,a}^{\alpha }x\left(t\right)=Dx\left(t\right),t\in h{\mathbb{N}}_{a+h},\hfill \\ {\nabla }_{h,a}^{-(1-\alpha )}x\left(t\right){|}_{t=a}=x\left(a\right)=x_0,\hfill \end{array}\right.$$

has the solution

$$x\left(t\right)=E_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))x_0,t\in h{\mathbb{N}}_a,$$

where $x_0$, $x\in {\mathbb{R}}^n$.

Proposition 1

([26]). The following are valid.

1. 

$E_{D,\alpha ,\alpha -1}^h(a,\rho \left(a\right))=I_n$.

2. 

${\nabla }_{h,a}^{\alpha }{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))=D{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))$, $t\in h{\mathbb{N}}_{a+h}$.

We close this section by stating the variation of constants formula for the system of fractional difference equations.

Theorem 3.

For $D\in M_n$, $h>0$, $0<\alpha <1$ with $\varrho \left(D\right)<h^{-\alpha }$ and $g:h{\mathbb{N}}_a\to {\mathbb{R}}^n$, the unique solution of the IVP

$$\left\{\begin{array}{c}{\nabla }_{h,a}^{\alpha }x\left(t\right)=Dx\left(t\right)+g\left(t\right),t\in h{\mathbb{N}}_{a+h},\hfill \\ {\nabla }_{h,a}^{-(1-\alpha )}x\left(t\right){|}_{t=a}=x\left(a\right)=x_0,\hfill \end{array}\right.$$

(2)

is given by

$$\begin{array}{cc}x\left(t\right)=E_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))x_0\hfill & \\ & \hfill +{\left(I_n-h^{\alpha }D\right)}^{-1}\sum _{s=a/h+1}^{t/h}{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(sh\right))g\left(sh\right)h^{\alpha },t\in h{\mathbb{N}}_a.\end{array}$$

Proof. 

In view of Theorem 2, it suffices to show that

$${\left(I_n-h^{\alpha }D\right)}^{-1}\sum _{s=a/h+1}^{t/h}{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(sh\right))g\left(sh\right)h^{\alpha },$$

is a particular solution of (2). Denote by

$$f\left(t\right)={\left(I_n-h^{\alpha }D\right)}^{-1}\sum _{s=a/h+1}^{t/h}{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(sh\right))g\left(sh\right)h^{\alpha },t\in h{\mathbb{N}}_a.$$

It is enough to show that

$${\nabla }_{h,a}^{\alpha }f\left(t\right)=Df\left(t\right)+g\left(t\right),t\in h{\mathbb{N}}_{a+h}.$$

To see this, for $t\in h{\mathbb{N}}_a$, consider

$$\begin{array}{cc}\hfill f\left(t\right)& ={\left(I_n-h^{\alpha }D\right)}^{-1}\sum _{s=a/h+1}^{t/h}{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(sh\right))g\left(sh\right)h^{\alpha }\hfill \\ & ={\left(I_n-h^{\alpha }D\right)}^{-1}\sum _{s=a/h+1}^{t/h}\left[\left(I_n-h^{\alpha }D\right){\displaystyle \frac{1}{h^{\alpha -1}}}\sum _{k=0}^{\infty }{D}^k{\displaystyle \frac{{(t-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}\right]g\left(sh\right)h^{\alpha }\hfill \\ & =\sum _{k=0}^{\infty }{D}^k\left[\sum _{s=a/h+1}^{t/h}{\displaystyle \frac{{(t-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}g\left(sh\right)h\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{k=0}^{\infty }{D}^k\left[\sum _{s=a/h}^{t/h}{\displaystyle \frac{{(t-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}g\left(sh\right)h-{\displaystyle \frac{{(t-\rho \left(a\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}g\left(a\right)h\right]\hfill \\ & =\sum _{k=0}^{\infty }{D}^k\left[{\nabla }_{h,a}^{-(\alpha k+\alpha )}g\left(t\right)-{\displaystyle \frac{{(t-\rho \left(a\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}g\left(a\right)h\right]\hfill \\ & =\sum _{k=0}^{\infty }{D}^k\left[{\nabla }_{h,a}^{-(\alpha k+\alpha )}g\left(t\right)\right]-{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right).\hfill \end{array}$$

Now, consider

$$\begin{array}{cc}\hfill {\nabla }_{h,a}^{\alpha }f\left(t\right)& ={\nabla }_{h,a}^{\alpha }\left[\sum _{k=0}^{\infty }{D}^k\left[{\nabla }_{h,a}^{-(\alpha k+\alpha )}g\left(t\right)\right]-{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\right]\hfill \\ & ={\nabla }_h^n{\nabla }_{h,a}^{-(n-\alpha )}\left[\sum _{k=0}^{\infty }{D}^k\left[{\nabla }_{h,a}^{-(\alpha k+\alpha )}g\left(t\right)\right]\right]\hfill \\ & -{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }{\nabla }_{h,a}^{\alpha }{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\hfill \\ & =\sum _{k=0}^{\infty }{D}^k{\nabla }_h^n\left[{\nabla }_{h,a}^{-(n-\alpha )}{\nabla }_{h,a}^{-(\alpha k+\alpha )}g\left(t\right)\right]\hfill \\ & -{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }D{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\hfill \\ & =\sum _{k=0}^{\infty }{D}^k\left[{\nabla }_h^n{\nabla }_{h,a}^{-(\alpha k+n)}g\left(t\right)\right]-{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }D{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\hfill \\ & =\sum _{k=0}^{\infty }{D}^k{\nabla }_{h,a}^{-\alpha k}g\left(t\right)-{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }D{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\hfill \\ & =g\left(t\right)+\sum _{k=1}^{\infty }{D}^k{\nabla }_{h,a}^{-\alpha k}g\left(t\right)-h^{\alpha }D{\left(I_n-h^{\alpha }D\right)}^{-1}{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\hfill \\ & =g\left(t\right)+D\left[\sum _{k=0}^{\infty }{D}^k{\nabla }_{h,a}^{-(\alpha k+\alpha )}g\left(t\right)-{\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }{E}_{D,\alpha ,\alpha -1}^h(t,\rho \left(a\right))g\left(a\right)\right]\hfill \\ & =Df\left(t\right)+g\left(t\right),\hfill \end{array}$$

where we used the fact that

$$\begin{array}{cc}\hfill {\left(I_n-h^{\alpha }D\right)}^{-1}{h}^{\alpha }D& ={\left(I_n-h^{\alpha }D\right)}^{-1}\left[I_n-\left(I_n-h^{\alpha }D\right)\right]\hfill \\ & ={\left(I_n-h^{\alpha }D\right)}^{-1}{I}_n-{\left(I_n-h^{\alpha }D\right)}^{-1}\left(I_n-h^{\alpha }D\right)\hfill \\ & ={\left(I_n-h^{\alpha }D\right)}^{-1}-I_n\hfill \\ & =I_n{\left(I_n-h^{\alpha }D\right)}^{-1}-\left(I_n-h^{\alpha }D\right){\left(I_n-h^{\alpha }D\right)}^{-1}\hfill \\ & =\left[I_n-\left(I_n-h^{\alpha }D\right)\right]{\left(I_n-h^{\alpha }D\right)}^{-1}\hfill \\ & =h^{\alpha }D{\left(I_n-h^{\alpha }D\right)}^{-1}.\hfill \end{array}$$

The proof is complete. □

3. Controllability Results

This section establishes novel conditions on the controllability of the following system

$${\nabla }_{h,t_0}^{\alpha }x\left(t\right)=Ax\left(t\right)+Bv\left(t\right),t\in h{\mathbb{N}}_{t_0+h}^{t_1}.$$

(3)

We consider

$$\begin{array}{cc}\hfill & x\left(t_0\right)=x_0,\mathrm{the}\mathrm{initial}\mathrm{state},\hfill \\ \hfill & A,n\times n\mathrm{constant}\mathrm{matrix},\hfill \\ \hfill & x,n\times 1\mathrm{state}\mathrm{vector},\hfill \\ \hfill & B,n\times m\mathrm{constant}\mathrm{matrix},\hfill \\ \hfill & v,m\times 1\mathrm{control}\mathrm{vector}\mathrm{with}m\le n,\hfill \end{array}$$

and $\alpha \in (0,1)$. Since output has no influence on controllability, it is omitted in the analysis. By Theorem 3, the solution of (3) is

$$\begin{array}{c}x\left(t\right)=E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))x_0\hfill \\ \hfill +{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t/h}{E}_{A,\alpha ,\alpha -1}^h(t,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha },t\in h{\mathbb{N}}_{t_0}^{t_1}.\end{array}$$

(4)

The following definitions assume $t_0$ and $t_1$ are positive real numbers with

$$\begin{array}{cc}\hfill & x\left(t_0\right)=x_0,\mathrm{the}\mathrm{initial}\mathrm{state},\hfill \\ \hfill & x\left(t_1\right)=x_1,\mathrm{the}\mathrm{final}\mathrm{state},\hfill \\ \hfill & \overline{x}\left(t_1\right)=0_{n\times 1},\mathrm{the}\mathrm{zero}\mathrm{state},\hfill \\ \hfill & I=h{\mathbb{N}}_{t_0}^{t_1-h},\mathrm{a}\mathrm{finite}\mathrm{and}\mathrm{discrete}\mathrm{set}\mathrm{of}\mathrm{time}\mathrm{instances}.\hfill \end{array}$$

Definition 8

([7]). System (3) (or the pair $\{A,B\}$) is completely controllable if, in the classical sense, there exists a control input v that can drive $x\left(t_0\right)$ to $x\left(t_1\right)$ within I. If this is not possible, (3) (or the pair $\{A,B\}$) is not controllable.

Definition 9

([7]). System (3) (or the pair $\{A,B\}$) is controllable to the origin in the classical sense if there exists a control input v that can drive $x\left(t_0\right)$ to $\overline{x}\left(t_1\right)$ within I.

In order to establish conditions for the controllability of (3), we introduce the concepts of the controllability matrix and the Gramian controllability matrix. The controllability matrix [7] ${\widehat{W}}_{n\times \left(nm\right)}$ associated with (3) in the classical sense is

$$\widehat{W}:=[BAB\dots A^{n-1}B].$$

The Gramian controllability matrix ${\mathcal{P}}_{n\times n}$ associated with (3) is

$$\mathcal{P}(t,t_0):={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t/h}{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B{B}^T{\left[E_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))\right]}^T{h}^{\alpha }.$$

Theorem 4.

The following two statements are equivalent:

1. 

System (3) is completely controllable on I.

2. 

Rank of $\mathcal{P}(t_1,t_0)$ is n.

Proof. 

First, we show that if (3) is completely controllable then $\mathcal{P}(t_1,t_0)$ of (3) has rank n. Let us prove this by contradiction. Suppose that rank of $\mathcal{P}(t_1,t_0)$ is less than n. Then, there is a nonzero n-tuple real vector $\eta$ with ${\eta }^T\mathcal{P}(t_1,t_0)$ is equal to $0_{1\times n}$. Consequently,

$$\begin{array}{cc}\hfill 0& ={\eta }^T\mathcal{P}(t_1,t_0)\eta \hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{\eta }^T{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B{B}^T{\left[E_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))\right]}^T\eta h^{\alpha }\hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{\parallel {\eta }^T{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B\parallel }_2^2{h}^{\alpha },\hfill \end{array}$$

where ${\parallel ·\parallel }_2^2$ defines the Euclidean norm. Hence,

$${\eta }^T{E}_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))B=0_{1\times m},t\in h{\mathbb{N}}_{t_0+h}^{t_1}.$$

(5)

Assuming controllability, $x\left(t_0\right)=x_0$ is transferred to $x\left(t_1\right)=x_f=E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0+\eta$ via a control input v. The system’s solution can be written as follows by substituting the starting and final states into (4).

$$\begin{array}{c}{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0+\eta \hfill \\ \hfill =E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0+{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha },\end{array}$$

that is,

$$\eta ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha }.$$

Using (5), we obtain

$${\eta }^T\eta ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{\eta }^T{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha }=0,$$

a contradiction. Therefore, the rank of $\mathcal{P}(t_1,t_0)$ is n.

Conversely, if rank of $\mathcal{P}(t_1,t_0)$ is n, then $\mathcal{P}(t_1,t_0)$ is invertible. Therefore, for the given any initial state $x\left(t_0\right)=x_0$ and final state $x\left(t_1\right)=x_f$, we can choose the control signal v as

$$v\left(t\right)=B^T{\left[E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t\right))\right]}^T{\left[\mathcal{P}(t_1,t_0)\right]}^{-1}\left[x_f-E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0\right].$$

The solution at $t=t_1$ becomes

$$\begin{array}{cc}\hfill x\left(t_1\right)& =E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0+{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha }\hfill \\ & =E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0+{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))\hfill \\ & \times B\left[B^T{\left[E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))\right]}^T\right]{\left[\mathcal{P}(t_1,t_0)\right]}^{-1}\left[x_f-E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0\right]h^{\alpha }.\hfill \end{array}$$

By performing the above last summation, we obtain

$$\begin{array}{cc}\hfill & {\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))B{B}^T{\left[E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(sh\right))\right]}^T{h}^{\alpha }\hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B{B}^T{\left[E_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))\right]}^T{h}^{\alpha }\hfill \\ & =\mathcal{P}(t_1,t_0).\hfill \end{array}$$

Hence, we have

$$x\left(t_1\right)=E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0+\mathcal{P}(t_1,t_0){\left[\mathcal{P}(t_1,t_0)\right]}^{-1}\left[x_f-E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))x_0\right]=x_f,$$

a contradiction. Therefore, (3) is completely controllable on I. □

Theorem 5.

The following two statements are equivalent:

1. 

System (3) is completely controllable on I.

2. 

Rank of $\widehat{W}$ is n.

Proof. 

Clearly, the rank of

$$\widehat{W}\left(N\right):=[BAB{A}^{2}B\dots A^{N-1}B],$$

is same as the rank of $\widehat{W}$ for all $N\ge n$. Then,

$$A^n=\sum _{s=0}^{n-1}{p}_s{A}^s.$$

The values $-p_s$ represent the coefficients of the characteristic polynomial for matrix A. Then, we obtain

$$A^{n}B=\sum _{s=0}^{n-1}{p}_s{A}^{s}B.$$

Therefore, the columns of the matrix $A^{n}B$ are linearly dependent on the columns of $\widehat{W}$. Hence, $\mathrm{rank}\left(\widehat{W}(n+1)\right)=\mathrm{rank}\left(\widehat{W}\right)$. By multiplying the last equation with A, we get

$$A^{n+1}B=\sum _{s=0}^{n-1}{p}_s{A}^{s+1}B.$$

As a result, ranks of $\widehat{W}$, $\widehat{W}(n+1)$ and $\widehat{W}(n+2)$ are same. Continuing this process, we deduce that the ranks of $\widehat{W}\left(N\right)$ and $\widehat{W}$ are the same for each $N\ge n$.

To show that the rank of $\widehat{W}$ is n if (3) is completely controllable on I, let us consider a set of time instances of $nh$. Because (3) is completely controllable on I, a control signal v exists that can move $x\left(t_0\right)=x_0\in {\mathbb{R}}^n$ to any final state $x(nh+t_0)=x_f\in {\mathbb{R}}^n$. Putting $t_1=nh+t_0$ into (4) yields

$$\begin{array}{c}{x}_f=E_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(t_0\right))x_0\hfill \\ \hfill +{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{(nh+t_0)/h}{E}_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha },t\in h{\mathbb{N}}_{t_0}^{nh+t_0}.\end{array}$$

By performing the sum, we obtain

$$\begin{array}{cc}\hfill & x(nh+t_0)-E_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(t_0\right))x_0\hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{(nh+t_0)/h}{E}_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(sh\right))Bv\left(sh\right)h^{\alpha }\hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=1}^n{E}_{A,\alpha ,\alpha -1}^h(nh,\rho \left(sh\right))Bv(t_0+sh)h^{\alpha }\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{s=1}^n\left[\sum _{k=0}^{\infty }{A}^k{\displaystyle \frac{{(nh-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}\right]Bv(t_0+sh)h\hfill \\ & =\sum _{k=0}^{\infty }\left[\sum _{s=1}^n{A}^k{\displaystyle \frac{{(nh-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}Bv(t_0+sh)h\right]\hfill \\ & =\underset{N\to \infty }{lim}\sum _{k=0}^N\left[\sum _{s=1}^n{A}^k{\displaystyle \frac{{(nh-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}Bv(t_0+sh)h\right]\hfill \\ & =\underset{N\to \infty }{lim}\sum _{k=0}^N{A}^{k}B\left[\sum _{s=1}^n{\displaystyle \frac{{(nh-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}v(t_0+sh)h\right].\hfill \end{array}$$

Each term in the previous series is a linear combination of the columns of B, $AB$, $A^{2}B,\dots ,A^{N}B$. From the above discussion, it follows that any of these matrices is a linear combination of B, $AB$, $A^{2}B,\dots ,A^{n-1}B$. Hence, the vector

$$\sum _{k=0}^N{A}^{k}B\left[\sum _{s=1}^n{\displaystyle \frac{{(nh-\rho \left(sh\right))}_h^{\overline{\alpha k+\alpha -1}}}{\Gamma (\alpha k+\alpha )}}v(t_0+sh)h\right]$$

is a linear combination of the columns of B, $AB$, $A^{2}B,\dots ,A^{n-1}B$, i.e., it belongs to the range space of the Kalman matrix $\widehat{W}$. Therefore,

$$x(nh+t_0)-E_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(t_0\right))x_0\in range\left(\widehat{W}\right).$$

Assume $\mathrm{rank}\left(\widehat{W}\right)<n$. Then, there is a nonzero n-tuple real vector $\eta$ with

$${\eta }^T\widehat{W}=0_{1\times mn}.$$

This implies

$${\eta }^T\left(x_f-E_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(t_0\right))x_0\right)=0_{1\times n},$$

independent of the choice of control input v. However, since the system is completely controllable, we can select $x_f=E_{A,\alpha ,\alpha -1}^h(nh+t_0,\rho \left(t_0\right))x_0+\eta$ implying that ${\eta }^T\eta =0$, a contradiction. Hence, $\widehat{W}$ must have rank n.

Conversely, assume that $\mathrm{rank}\left(\widehat{W}\right)=n$. To derive a contradiction, suppose (3) is uncontrollable. In this case, $\mathcal{P}(t_0+nh,t_0)$ must have rank strictly less than n. Consequently, there is a nonzero n-tuple real vector $\eta$ with ${\eta }^T\mathcal{P}(t_0+nh,t_0)=0_{1\times n}$. Then,

$$\begin{array}{cc}\hfill 0& ={\eta }^T\mathcal{P}(t_0+nh,t_0)\eta \hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_0/h+n}{\eta }^T{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B{B}^T{\left[E_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))\right]}^T\eta h^{\alpha }\hfill \\ & ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_0/h+n}{\parallel {\eta }^T{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B\parallel }_2^2{h}^{\alpha },\hfill \end{array}$$

which implies that

$${\eta }^T{E}_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))B=0_{1\times m},t\in {\mathbb{N}}_{t_0}^{t_0+nh}.$$

Set $t=t_0$. By Proposition 1 (1),

$${\eta }^{T}B=0_{1\times m}.$$

Using Proposition 1, we have

$${\eta }^{T}A{E}_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))B=0_{1\times m},t\in {\mathbb{N}}_{t_0}^{t_0+nh}.$$

Set $t=t_0$. By Proposition 1 (1),

$${\eta }^{T}AB=0_{1\times m}.$$

Repeating the same step up to $n-1$ times, we have

$${\eta }^T{A}^{k}B=0_{1\times m},k=0,1,\dots ,n-1.$$

Then, we have

$${\eta }^T[BAB\dots A^{n-1}B]={\eta }^T\widehat{W}=0_{1\times mn},$$

a contradiction. Therefore, (3) is completely controllable. □

Remark 4.

For every n-tuple real vector η,

$${\eta }^T\mathcal{P}(t_1,t_0)\eta ={\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{\parallel {\eta }^T{E}_{A,\alpha ,\alpha -1}^h(sh,\rho \left(t_0\right))B\parallel }_2^2{h}^{\alpha }.$$

Thus, $\mathcal{P}(t_1,t_0)$ is symmetric and non-negative.

To show that complete controllability to origin and controllability are equivalent for (3), we need to add another assumption about x.

Theorem 6.

Assume $E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))$ in (4) is nonsingular over the discrete set of time instances $h{\mathbb{N}}_{t_0+h}^{t_1}$. Then, (3) is completely controllable if and only if it is controllable to the origin.

Proof. 

Assume (3) is completely controllable. By selecting $x\left(t_1\right)=0_{n\times 1}$, it follows from Definition 9 that it is controllable to the origin. Conversely, suppose $E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))$ in (3) is nonsingular over the discrete set of time instances $h{\mathbb{N}}_{t_0+h}^{t_1}$ and that (3) is controllable to origin. Given $x\left(t_0\right)$ and $x(t_1$), define

$$y\left(t_0\right):=x\left(t_0\right)-{\left[E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))\right]}^{-1}x\left(t_1\right),y\left(t_1\right):=0_{n\times 1}.$$

Thus, we get a system with initial state $y\left(t_0\right)$ and final state $y\left(t_1\right)$. By assumption, a control input v exists over I with $y\left(t_0\right)$ that can be driven to $y\left(t_1\right)$. By Theorem 3,

$$\begin{array}{cc}\hfill & y\left(t_1\right)=y\left(t_0\right)E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))+{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h\left(t_{,}\rho \left(sh\right)\right)Bv\left(sh\right)h^{\alpha },\hfill \\ \hfill & 0_{n\times 1}=\left[x\left(t_0\right)-{\left[E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))\right]}^{-1}x\left(t_1\right)\right]E_{A,\alpha ,\alpha -1}^h(t_1,\rho \left(t_0\right))\hfill \\ & +{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h\left(t_{,}\rho \left(sh\right)\right)Bv\left(sh\right)h^{\alpha },\hfill \\ \hfill & 0_{n\times 1}=x\left(t_0\right)E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))-x\left(t_1\right)+{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h\left(t_{,}\rho \left(sh\right)\right)Bv\left(sh\right)h^{\alpha },\hfill \\ \hfill & x\left(t_1\right)=x\left(t_0\right)E_{A,\alpha ,\alpha -1}^h(t,\rho \left(t_0\right))+{\left(I_n-h^{\alpha }A\right)}^{-1}\sum _{s=t_0/h+1}^{t_1/h}{E}_{A,\alpha ,\alpha -1}^h\left(t_{,}\rho \left(sh\right)\right)Bv\left(sh\right)h^{\alpha },\hfill \end{array}$$

there exists control vector v for any $x\left(t_0\right)$ and $x\left(t_1\right)$. Therefore, the given system is completely controllable. □

4. A Concluding Remark

Replacing the ordinary derivative (difference) operator with a fractional derivative (difference) operator when $0<\alpha <1$ in a system can lead to equations that no longer retain the system’s physical validity [27]. To preserve the physical meaning of the model during fractionalization, it is crucial to maintain dimensional consistency across both sides of the equation. This is achieved by raising the model’s parameters to the power of $\alpha$ corresponding to the order of the fractional derivative. The reader may refer to recent scholarly works addressing this issue [9,13].

With this note, the following result stated in [20] can be revisited:

Theorem 7.

The standard linear discrete-time system

$$x_{t+1}=A{x}_t+B{u}_t,$$

for $t\in {\mathbb{N}}_0$, is controllable in the discrete interval $[0,q]$ if and only if the fractional discrete-time linear system

$${\Delta }^{\alpha }{x}_{t+1}=A{x}_t+B{u}_t$$

for $t\in {\mathbb{N}}_0$, is controllable in the discrete interval $[0,q]$.

• Similar statements appear in other works, such as [21]. Assuming the dimensional consistency of both sides of the equation holds, we can restate it as follows:

• A linear system in discrete-time is controllable ⇎ A fractionalized linear system in discrete-time is controllable.

Next we demonstrate the validity of this statement with some examples.

Example 1.

$${\nabla }_{h,t_0}^{\alpha }y\left(t\right)=Ay\left(t\right)+Bu\left(t\right),t\in h{\mathbb{N}}_{t_0+h}^{t_1},$$

$$A=\left(\begin{array}{cc}-2^{\alpha }& 2^{\alpha }\\ 2^{\alpha }& 1\end{array}\right),B=\left(\begin{array}{c}1\\ 2^{\alpha }\end{array}\right).$$

The system is controllable for all $\alpha \in [0,1)$ except $\alpha =1$.

Example 2.

$${\nabla }_{h,t_0}y\left(t\right)=Ay\left(t\right)+Bu\left(t\right),t\in h{\mathbb{N}}_{t_0+h}^{t_1},$$

$$A=\left(\begin{array}{cc}1& 1\\ 9& {\displaystyle \frac{9}{4}}\end{array}\right),B=\left(\begin{array}{c}1\\ 4\end{array}\right).$$

The above discrete system is controllable. On the other hand, the fractionalized system

$${\nabla }_{h,t_0}^{\alpha }y\left(t\right)=\left(\begin{array}{cc}{1}^{\alpha }& 1^{\alpha }\\ 9^{\alpha }& {\left({\displaystyle \frac{9}{4}}\right)}^{\alpha }\end{array}\right)y\left(t\right)+\left(\begin{array}{c}{1}^{\alpha }\\ 4^{\alpha }\end{array}\right)u\left(t\right),t\in h{\mathbb{N}}_{t_0+h}^{t_1},$$

is not controllable for $\alpha =0.5$.

In this note, we study controllability of a fractional h-discrete system. We state and prove Kalman type controllability criterion. We also emphasize how the fractionalizing process can change the controllability of the system. We support our restatement with some empirical examples.