**2.1. Definitions of non-integer derivatives**

There are different definitions of the non-integer derivatives [1–3, 45]. Let us consider a function *f*(*t*) of the real variable *t*, continuous and integrable on [*a*, +∞[, where *a* is the origin of *t*. Indeed, we usually deal with dynamic systems and *f*(*t*) is let to be a causal function of *t*, with *t* ≥ *a*. Thus *f*(*t*) = 0 if *t* < *a*.

Let us introduce the positive integer number *m* such that (*m* − 1) < *α* < *m*. We obtain the definition due to A.V. Letnikov

$$\, \_a^L D\_t^a f(t) = \frac{1}{\Gamma(m - a + 1)} \int\_a^t (t - \tau)^{m - a} f^{(m + 1)}(\tau) d\tau + \sum\_{k = 0}^m \frac{f^{(k)}(a)(t - a)^{k - a}}{\Gamma(k - a + 1)}, \text{ for } t < a. \tag{1}$$

The definition given by (1) assumes that function *f* is sufficiently differentiable and that *f* (*<sup>k</sup>*)(*a*) < ∞, *k* = 0, 1, ..., *m*. Γ is the Euler gamma function

$$
\Gamma(\beta) = \int\_0^\infty z^{\beta - 1} e^{-z} dz \tag{2}
$$

in which *z* is a complex variable. A new version is the Letnikov-Riemann-Liouville definition (LRL) given by

$$\, \_a^L D\_t^a f(t) = \frac{d^m}{dt^m} \{ \frac{1}{\Gamma(m-a)} \int\_a^t \frac{f(\tau)}{(t-\tau)^{a-m+1}} d\tau \} \tag{3}$$

Naturally, as physical systems are modeled by differential equations containing eventually fractional derivatives, it is necessary to give to these equations initial conditions that must be physically interpretable. Unfortunately, the LRL definition leads to initial conditions containing the value of the fractional derivative at the initial conditions. To overcome this difficulty, Caputo proposed another definition given by

$$\, \_a^C D\_t^a f(t) = \frac{1}{\Gamma(m-a)} \int\_a^t \frac{f^{(m)}(\tau)}{(t-\tau)^{a-m+1}} d\tau \tag{4}$$

Note the remarkable fact that the LRL fractional derivative of a constant function *f*(*t*) = *C* is not zero

$${}^{L}\_{a}D\_{t}^{\alpha}\mathbb{C} = \frac{\mathbb{C}t^{-\alpha}}{\Gamma(1-\alpha)}\tag{5}$$

whereas Caputo's fractional derivative of a constant is identically zero.

#### **2.2. FOS differential equation**

2 Advances in Discrete Time Systems

systems.

the general trend to use digital computer-aided simulations and implementations in control engineering and signal processing. In the particular case of FOS, the discretization methods lead to new models and enlarge the possibilities of representation and simulation of such

More specifically, the familiar state-space representation and its well-established results in the integer-order case receive here an extension to the non-integer case. This extension induces noticeable changes: there appears a neccessity of interpretation of the initial conditions [31] and a neccesity to take into account the history (or memory) of the system from the initial instant, since we have to deal, instead of the classical state, with a so-called "'pseudo state"', that is an expanded state [32, 33]. This infinite-dimensional system, as we expose it in the following sections, has been studied, from the point of view of its structural properties, i.e.,

In the beginning, most of the works consecrated to the study of FOS focused on continuous-time representations. The continuous-time state-space FOS representation was introduced in [24, 37–40]. It has been employed in analysis of system performances. The solution of the state-space equation was derived by using the Mittag-Lefller function [41]. Next, the stability of FOS was investigated [42] and a condition based on the argument principle was established to guarantee the asymptotic stability of the fractional-order system. Further, the controllability and the observability properties have been defined and some algebraic criteria of these two properties have been derived in [43]. Another contribution to the analysis of the controllability and the observability with commensurate FOS modeled by

There are different definitions of the non-integer derivatives [1–3, 45]. Let us consider a function *f*(*t*) of the real variable *t*, continuous and integrable on [*a*, +∞[, where *a* is the origin of *t*. Indeed, we usually deal with dynamic systems and *f*(*t*) is let to be a causal

Let us introduce the positive integer number *m* such that (*m* − 1) < *α* < *m*. We obtain the

The definition given by (1) assumes that function *f* is sufficiently differentiable and that

 <sup>∞</sup> 0

in which *z* is a complex variable. A new version is the Letnikov-Riemann-Liouville definition

*zβ*−1*e*

(*τ*)*dτ* +

*m* ∑ *k*=0 *f*(*<sup>k</sup>*)(*a*)(*t* − *a*)*k*−*<sup>α</sup>*

<sup>Γ</sup>(*<sup>k</sup>* <sup>−</sup> *<sup>α</sup>* <sup>+</sup> <sup>1</sup>) , for *<sup>t</sup>* <sup>&</sup>lt; *<sup>a</sup>*. (1)

<sup>−</sup>*zdz* (2)

(*<sup>t</sup>* − *<sup>τ</sup>*)*m*−*<sup>α</sup> <sup>f</sup>* (*m*+1)

Γ(*β*) =

controllability, observability [34, 35] and, to some extent, of its stability [36].

**2. LTI continuous-time fractional-order modeling**

fractional state-space equations is brought in [44].

**2.1. Definitions of non-integer derivatives**

function of *t*, with *t* ≥ *a*. Thus *f*(*t*) = 0 if *t* < *a*.

 *t a*

*f* (*<sup>k</sup>*)(*a*) < ∞, *k* = 0, 1, ..., *m*. Γ is the Euler gamma function

definition due to A.V. Letnikov

Γ(*m* − *α* + 1)

*<sup>t</sup> <sup>f</sup>*(*t*) = <sup>1</sup>

*L <sup>a</sup> D<sup>α</sup>*

(LRL) given by

Let us now consider a SISO LTI FOS. By means of its dynamic input-output relation and using (4), we can derive its continuous-time models. In all what follows, we use Caputo's definition of a fractional derivative with initial time *t* = 0, *i.e.*, *a* = 0. The derived differential equation is then expressed by [45]

$$\sum\_{i=1}^{n\_a} a\_i D^{a\_i} y(t) = \sum\_{j=1}^{n\_b} b\_j D^{\beta\_j} u(t) \tag{6}$$

where *ai*, *bj* ∈ **R**, *αi*, *β<sup>j</sup>* ∈ **R**+, *u*(*t*) ∈ **R** is the input of the system and *y*(*t*) ∈ **R** its output. *na* and *nb* ∈ **N** are the number of terms of each side of the differential equation. The differential equation is said with commensurate order if all the differentiation orders *α<sup>i</sup>* , *β<sup>j</sup>* are multiple integers of a same base order *α*. In this case, it can be expressed by

$$\sum\_{i=1}^{n\_d} a\_i D^{i\alpha} y(t) = \sum\_{j=1}^{n\_b} b\_j D^{j\alpha} u(t) \tag{7}$$

#### **2.3. State-space model derived from a non-integer-order transfer function**

By using Laplace transform, assuming null initial conditions, from (6) we derive the following transfer function

$$G(s) = \frac{Y(s)}{U(s)} = \frac{\sum\_{j=0}^{n\_b} b\_j s^{\beta\_j}}{\sum\_{i=0}^{n\_d} a\_i s^{a\_i}} \tag{8}$$

To facilitate the development of the so-called construction procedure, (8) is rewritten with coefficients *ai* and *bj* denoted differently by *a*′ <sup>2</sup>*k*+<sup>1</sup> and *<sup>a</sup>*′ <sup>2</sup>*<sup>k</sup>* respectively [35]. This gives

$$G(s) = \frac{Y(s)}{U(s)} = \frac{a\_0' + a\_2' s^{a\_2} + a\_4' s^{a\_4} + \dots + a\_{2n}' s^{a\_{2n}}}{1 + a\_1' s^{a\_1} + a\_3' s^{a\_3} + \dots + a\_{2n+1}' s^{a\_{2n+1}}},\tag{9}$$

The transposition in the time domain yields a state-space representation of System (9), with *xi*(*t*), *u*(*t*) and *y*(*t*) the respective inverse Laplace transforms of X*i*(*s*), *U*(*s*) and *Y*(*s*)

[*u*(*t*) − *x*1(*t*) − *a*′

where *x*(*t*)=[*x*1(*t*) *x*2(*t*)... *x*2*n*+1(*t*)]*<sup>T</sup>* ∈ **R**2*n*+<sup>1</sup> is the column state vector and *x*(0) its initial value. Note that here *x*(0) can be assumed non null, instead of the assumption of null

> *Dγ*<sup>1</sup> *x*1(*t*) . . . *Dγ*2*n*+<sup>1</sup> *x*2*n*+1(*t*)

initial conditions necessary to define the transfer function model. Besides, we have

 

*x*(*t*) =

in which *<sup>γ</sup><sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**⋆+, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>1</sup> for *<sup>i</sup>* <sup>=</sup> 1 and *<sup>γ</sup><sup>i</sup>* = (*α<sup>i</sup>* <sup>−</sup> *<sup>α</sup>i*−1) for *<sup>i</sup>* <sup>=</sup> 2, . . . , 2*<sup>n</sup>* <sup>+</sup> 1.

0 1 0 0 0 ... 0 0 0 1 0 ... ... 0

0 0 0 0 ... ... 1

<sup>0</sup> <sup>0</sup> *<sup>a</sup>*′

0 ... <sup>−</sup>*a*′

<sup>2</sup> 0 ... 0 *<sup>a</sup>*′

2*n*−1 *a*′ 2*n*+1

<sup>2</sup>*<sup>n</sup>* ] .

 

<sup>0</sup> <sup>−</sup>*a*′ 3 *a*′ 2*n*+1

*C* = [ *a*′

<sup>2</sup>*x*3(*t*) + ... <sup>+</sup> <sup>0</sup>*x*2*<sup>n</sup>* <sup>+</sup> *<sup>a</sup>*′

<sup>1</sup>*x*2(*t*) <sup>−</sup> ... <sup>−</sup> *<sup>a</sup>*′

Discrete-Time Fractional-Order Systems: Modeling and Stability Issues

*x*(*t*) = *Ax*(*t*) + *Bu*(*t*) *x*(0) = *x*0, (15)

 ,

*y*(*t*) = *Cx*(*t*) + *Du*(*t*), (16)

 

0 0 . . . 1 *a*′ 2*n*+1  ,

, *B* =

<sup>2</sup>*n*−1*x*2*n*(*t*)],

<sup>2</sup>*nx*2*n*+1(*t*). (14)

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187

*Dα*<sup>1</sup> *x*1(*t*) = *x*<sup>2</sup> *D*(*α*2−*α*1)*x*2(*t*) = *x*<sup>3</sup>

*D*(*αi*−*αi*−1)*xi*(*t*) = *xi*+<sup>1</sup>

*D*(*α*2*<sup>n</sup>*−*α*2*n*−1)*x*2*n*(*t*) = *x*2*n*+<sup>1</sup>

*a*′ 2*n*+1

<sup>0</sup>*x*1(*t*) + <sup>0</sup>*x*2(*t*) + *<sup>a</sup>*′

The corresponding state-space group of equations is then derived

*D*[*γ*]

*D*[*γ*]

. . .

. . .

the expression of the system output being

*y*(*t*) = *a*′

The output equation takes the form

Matrices A, B, C and D are given by

*A* =

 

. . . . . . . . . . . . . . . . . . . . .

− <sup>1</sup> *a*′ 2*n*+1

−*a*′ 1 *a*′ 2*n*+1

*D*(*α*2*n*+1−*α*2*<sup>n</sup>*)*x*2*n*+1(*t*) = <sup>1</sup>

in which *α<sup>i</sup>* are the fractional orders that can be either commensurate or non-commensurate, with

$$
\alpha\_1 \le \alpha\_2 \le \alpha\_3 \le \dots \le \alpha\_{2n} \le \alpha\_{2n+1}.
$$

The procedure for obtaining a state-space representation from (9) is as follows: firstly, let us introduce an intermediate variable X (*s*) such that

$$G(s) = \frac{Y(s)}{\mathcal{X}(s)} \frac{\mathcal{X}(s)}{U(s)}$$

Next, we can write

$$G(s) = \frac{Y(s)}{\mathcal{X}(s)} \frac{\mathcal{X}(s)}{\mathcal{U}(s)} = \frac{(a\_0' + a\_2' s^{a\_2} + a\_4' s^{a\_4} + \dots + a\_{2n}' s^{a\_{2n}}) \mathcal{X}(s)}{(1 + a\_1' s^{a\_1} + a\_3' s^{a\_3} + \dots + a\_{2n+1}' s^{a\_{2n+1}}) \mathcal{X}(s)} \tag{10}$$

Let us put successively

*s*

$$\begin{aligned} \mathcal{X}\_1(s) &= \mathcal{X}(s) \\ \mathcal{X}\_2(s) &= s^{a\_1} \mathcal{X}(s) = s^{a\_1} \mathcal{X}\_1(s) \\ \mathcal{X}\_3(s) &= s^{a\_2} \mathcal{X}(s) = s^{(a\_2 - a\_1)} \mathcal{X}\_2(s) \\ &\vdots \\ \mathcal{X}\_{2n+1}(s) &= s^{a\_{2n}} \mathcal{X}(s) = s^{(a\_{2n} - a\_{2n-1})} \mathcal{X}\_{2n}(s). \end{aligned} \tag{11}$$

With (10) and relations (11), it is easy to build the following group of equations

$$\begin{aligned} \mathbf{s}^{a\_1} \mathcal{X}\_1(\mathbf{s}) &= \mathcal{X}\_2(\mathbf{s}) \\ \mathbf{s}^{(a\_{2n} - a\_1)} \mathcal{X}\_2(\mathbf{s}) &= \mathcal{X}\_3(\mathbf{s}) \\ &\vdots \\ \mathbf{s}^{(a\_i - a\_{i-1})} \mathcal{X}\_i(\mathbf{s}) &= \mathcal{X}\_{i+1}(\mathbf{s}) \\ &\vdots \\ \mathbf{s}^{(a\_{2n+1} - a\_{2n})} \mathcal{X}\_{2n+1}(\mathbf{s}) &= \frac{1}{a\_{2n+1}'} [\mathcal{U}(\mathbf{s}) - \mathcal{X}\_1(\mathbf{s}) - a\_1' \mathcal{X}\_2(\mathbf{s}) - \dots - a\_{2n-1}' \mathcal{X}\_{2n}(\mathbf{s})] \end{aligned} \tag{12}$$

The transposition in the time domain yields a state-space representation of System (9), with *xi*(*t*), *u*(*t*) and *y*(*t*) the respective inverse Laplace transforms of X*i*(*s*), *U*(*s*) and *Y*(*s*)

$$\begin{array}{ccccc}D^{a\_1}x\_1(t) & = & x\_2 & & & \\ D^{(a\_2-a\_1)}x\_2(t) & = & & x\_3 & & \\ & \vdots & & & & \\ D^{(a\_i-a\_{i-1})}x\_i(t) & = & & & & x\_{i+1} & \\ & \vdots & & & & & \\ \end{array} \tag{13}$$

$$\begin{array}{ll}D^{(a\_{2n}-a\_{2n-1})}\mathbf{x}\_{2n}(t) &=& \mathbf{x}\_{2n+1} \\ D^{(a\_{2n+1}-a\_{2n})}\mathbf{x}\_{2n+1}(t) &=& \frac{1}{a\_{2n+1}'} [\boldsymbol{\mu}(t) - \boldsymbol{\mathbf{x}}\_{1}(t) - a\_{1}'\mathbf{x}\_{2}(t) - \dots - a\_{2n-1}'\mathbf{x}\_{2n}(t)]\_{\boldsymbol{\nu}} \end{array}$$

the expression of the system output being

4 Advances in Discrete Time Systems

with

Next, we can write

Let us put successively

coefficients *ai* and *bj* denoted differently by *a*′

*<sup>G</sup>*(*s*) = *<sup>Y</sup>*(*s*)

introduce an intermediate variable X (*s*) such that

*<sup>G</sup>*(*s*) = *<sup>Y</sup>*(*s*)

*s*

*s* (*α*2−*α*1)

*s* (*αi*−*αi*−1)

(*α*2*n*+1−*α*2*<sup>n</sup>*)

*s*

X (*s*)

X (*s*) *<sup>U</sup>*(*s*) <sup>=</sup> (*a*′

*<sup>U</sup>*(*s*) <sup>=</sup> *<sup>a</sup>*′

To facilitate the development of the so-called construction procedure, (8) is rewritten with

<sup>2</sup>*sα*<sup>2</sup> <sup>+</sup> *<sup>a</sup>*′

<sup>1</sup>*sα*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*′

in which *α<sup>i</sup>* are the fractional orders that can be either commensurate or non-commensurate,

*α*<sup>1</sup> ≤ *α*<sup>2</sup> ≤ *α*<sup>3</sup> ≤ ... ≤ *α*2*<sup>n</sup>* ≤ *α*2*n*+1.

The procedure for obtaining a state-space representation from (9) is as follows: firstly, let us

X (*s*)

<sup>1</sup>*sα*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*′

*<sup>α</sup>*1X (*s*) = *s*

*<sup>α</sup>*2X (*s*) = *s*

*<sup>α</sup>*2*<sup>n</sup>*X (*s*) = *s*

[*U*(*s*) − X1(*s*) − *<sup>a</sup>*′

With (10) and relations (11), it is easy to build the following group of equations

<sup>2</sup>*sα*<sup>2</sup> <sup>+</sup> *<sup>a</sup>*′

X (*s*) *U*(*s*)

*<sup>G</sup>*(*s*) = *<sup>Y</sup>*(*s*)

<sup>0</sup> <sup>+</sup> *<sup>a</sup>*′

(1 + *a*′

X1(*s*) = X (*s*) X2(*s*) = *s*

X3(*s*) = *s*

. .

X2*n*+1(*s*) = *s*

*<sup>α</sup>*1X1(*s*) = X2(*s*)

X2(*s*) = X3(*s*) . .

X*i*(*s*) = X*i*+1(*s*) . . .

> *a*′ 2*n*+1

<sup>X</sup>2*n*+1(*s*) = <sup>1</sup>

<sup>0</sup> <sup>+</sup> *<sup>a</sup>*′

1 + *a*′

<sup>2</sup>*k*+<sup>1</sup> and *<sup>a</sup>*′

<sup>4</sup>*sα*<sup>4</sup> <sup>+</sup> ... <sup>+</sup> *<sup>a</sup>*′

<sup>4</sup>*sα*<sup>4</sup> <sup>+</sup> ... <sup>+</sup> *<sup>a</sup>*′

X2(*s*)

. (11)

X2*n*(*s*).

. (12)

<sup>1</sup>X2(*s*) <sup>−</sup> ... <sup>−</sup> *<sup>a</sup>*′

<sup>2</sup>*n*−1X2*n*(*s*)]

<sup>3</sup>*sα*<sup>3</sup> <sup>+</sup> ... <sup>+</sup> *<sup>a</sup>*′

*<sup>α</sup>*1X1(*s*)

(*α*2−*α*1)

(*α*2*<sup>n</sup>*−*α*2*n*−1)

<sup>3</sup>*sα*<sup>3</sup> <sup>+</sup> ... <sup>+</sup> *<sup>a</sup>*′

<sup>2</sup>*<sup>k</sup>* respectively [35]. This gives

<sup>2</sup>*nsα*2*<sup>n</sup>* )X (*s*)

<sup>2</sup>*n*+1*sα*2*n*+<sup>1</sup> )<sup>X</sup> (*s*) (10)

<sup>2</sup>*n*+1*sα*2*n*+<sup>1</sup> , (9)

2*nsα*2*<sup>n</sup>*

$$y(t) = a\_0' \mathbf{x}\_1(t) + \mathbf{0} \mathbf{x}\_2(t) + a\_2' \mathbf{x}\_3(t) + \dots + \mathbf{0} \mathbf{x}\_{2n} + a\_{2n}' \mathbf{x}\_{2n+1}(t). \tag{14}$$

The corresponding state-space group of equations is then derived

$$D^{[\gamma]}\mathbf{x}(t) = A\mathbf{x}(t) + Bu(t) \qquad \mathbf{x}(0) = \mathbf{x}\_0 \tag{15}$$

where *x*(*t*)=[*x*1(*t*) *x*2(*t*)... *x*2*n*+1(*t*)]*<sup>T</sup>* ∈ **R**2*n*+<sup>1</sup> is the column state vector and *x*(0) its initial value. Note that here *x*(0) can be assumed non null, instead of the assumption of null initial conditions necessary to define the transfer function model. Besides, we have

$$D^{[\gamma]}\mathfrak{x}(t) = \begin{bmatrix} D^{\gamma\_1}\mathfrak{x}\_1(t) \\ \vdots \\ D^{\gamma\_{2n+1}}\mathfrak{x}\_{2n+1}(t) \end{bmatrix},$$

in which *<sup>γ</sup><sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**⋆+, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>α</sup>*<sup>1</sup> for *<sup>i</sup>* <sup>=</sup> 1 and *<sup>γ</sup><sup>i</sup>* = (*α<sup>i</sup>* <sup>−</sup> *<sup>α</sup>i*−1) for *<sup>i</sup>* <sup>=</sup> 2, . . . , 2*<sup>n</sup>* <sup>+</sup> 1. The output equation takes the form

$$y(t) = \mathbb{C}x(t) + Du(t),\tag{16}$$

Matrices A, B, C and D are given by

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 1 & 0 & \dots & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \dots & \dots & 1 \\ -\frac{1}{d\_{2n+1}'} & \frac{-d\_1'}{d\_{2n+1}} & 0 & \frac{-d\_1'}{d\_{2n+1}'} & 0 & \dots & \frac{-d\_{2n-1}'}{d\_{2n+1}'} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ \frac{1}{d\_{2n+1}'} \end{bmatrix},$$

$$\mathbf{C} = \begin{bmatrix} a\_0' \ 0 \ a\_2' \ 0 \ \dots \ 0 \ a\_{2n}' \end{bmatrix}.$$

As for direct transmission matrix *D*, we have

$$D = \begin{bmatrix} \frac{a\_{2n}'}{a\_{2n+1}'} \end{bmatrix} \text{ if } \ a\_{2n} = \alpha\_{2n+1\prime} \text{ otherwise } \ D = [0].$$

The symbol *D* here should not be confused with the derivation operator *D*[.] .

**Remark 1.** *The dimension of the model, i.e., its number of state variables* 2*n* + 1 *is equal to the total number of non-null terms a*′ *l sα<sup>l</sup> present in the numerator and denominator of G*(*s*). *In the following,* 2*n* + 1 *is denoted nd.*

The controllability and observability properties as well as the stability have been well studied when the differentiation fractional-orders *γ<sup>i</sup>* are all equal to a unique value, say *α* ([43], [42]). In this case, (15) becomes

$$D^{[\mathfrak{a}]}\mathbf{x}(t) = \begin{bmatrix} D^{\mathfrak{a}}\mathbf{x}\_1 \\ D^{\mathfrak{a}}\mathbf{x}\_2 \\ \vdots \\ D^{\mathfrak{a}}\mathbf{x}\_{\mathfrak{n}\_d} \end{bmatrix} = A\mathbf{x}(t) + Bu(t), \qquad \mathbf{x}(0) = \mathbf{x}\_0 \tag{17}$$

is equal to *nd*. Besides, this system is observable if the rank of the observability matrix

 

is equal to *nd*. To our knowledge, no such results have been brought in the literature up to now for FOS with non commensurate fractional-order systems in continuous time. The next section of this chapter constitutes a contribution to the study of structural properties of non-commensurate FOS, using a different approach, that is discrete-time representation [34].

The development of a discrete-time model is based on the Grünwald-Letnikov's definition of fractional-order operators. This definition is another expression of (1), based on the generalization of the backward difference. The generalization of the integer-order difference

In what follows, we present a discretized representation of the continuous fractional-order state-space model of Equations (15) and (16), in which the fractional order can be

where *x*(*t*)=[*x*1(*t*) *x*2(*t*)... *xnd* (*t*)]*<sup>T</sup>* ∈ **R***nd* is the state vector. This latter model can be represented in discrete-time using a forward difference approximation of the derivative of order one of *x*(*t*) [32, 33]. For this purpose let us consider a sampling period *h*. Then for

*<sup>D</sup>*1*x*(*t*) <sup>≈</sup> <sup>∆</sup>1*x*((*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)*h*) = *<sup>x</sup>*((*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)*h*) <sup>−</sup> *<sup>x</sup>*(*kh*)

We start with applying a discretization of (17) with identical fractional orders. The fractional

*hα*

*x*(*t*) can be approximated by using the definition due to K. Grünwald

*k*+1 ∑ *j*=0

(−1)*<sup>j</sup>* �*α j* �

*kh* ≤ *t* < (*k* + 1)*h* the integer order first derivative *D*1*x*(*t*) can be approximated by

*<sup>h</sup> <sup>x</sup>*((*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)*h*) = <sup>1</sup>

*D*1*x*(*t*) = *Ax*(*t*) + *Bu*(*t*) (23)

∆1(*x*(*k* + 1)*h*) = *Ax*(*kh*) + *Bu*(*kh*) (25)

*<sup>h</sup>* . (24)

*x*(*k* + 1 − *j*)*h*) (26)

*C CA CA*<sup>2</sup> . . . *CAnd*−<sup>1</sup>  

Discrete-Time Fractional-Order Systems: Modeling and Stability Issues

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189

O =

to a non-integer-order (or fractional-order) difference is addressed in [33, 47].

**3.1. The discrete-time fractional-order difference operator**

Let us consider the state-space model with integer-order derivatives

**3. LTI discrete-time fractional-order modeling**

commensurate or non-commensurate.

We can then write

derivatives *D*[*α*]

*D*[*α*]

*<sup>x</sup>*(*t*) ≈ <sup>∆</sup>[*α*]

Applying the Laplace transform to (17), we have

$$X(s) = [s^{\alpha} \mathbb{I} - A]^{-1} [B \mathcal{U}(s) + \mathfrak{x}(0)]\_{\prime\prime}$$

Defining Φ(*t*) = L−1[*sα***I** − *A*] <sup>−</sup><sup>1</sup> as the corresponding state transition matrix, we obtain the state response given by

$$\mathbf{x}(t) = \Phi(t)\mathbf{x}(0) + \int\_0^t \Phi(t-\tau)B u(\tau)d\tau \tag{18}$$

It can be shown that Φ(*t*) can be expressed by

$$\Phi(t) = \sum\_{k=0}^{\infty} \frac{A^k t^{ka}}{\Gamma(1+ka)} \tag{19}$$

The left side term of this latter equation represents the Mittag-Leffler function [41] :

$$E\_{\alpha}(t) \stackrel{\triangle}{=} \sum\_{k=0}^{\infty} \frac{A^k t^{k\alpha}}{\Gamma(1+k\alpha)}\tag{20}$$

It has been established in [43] that the controllability and observability conditions of the continuous-time state-space representation of commensurate fractional-order systems are the same as in the integer-order case. Thus, System (17) is controllable if the rank of the controllability matrix

$$\mathcal{C} = \begin{bmatrix} \mathcal{B} & \mathcal{A}\mathcal{B} & \mathcal{A}^2 \mathcal{B} & \dots \mathcal{A}^{n\_d - 1} \mathcal{B} \end{bmatrix} \tag{21}$$

is equal to *nd*. Besides, this system is observable if the rank of the observability matrix

$$\mathcal{O} = \begin{bmatrix} \mathsf{C} \\ \mathsf{CA} \\ \mathsf{CA}^2 \\ \vdots \\ \mathsf{CA}^{n\_d - 1} \end{bmatrix} \tag{22}$$

is equal to *nd*. To our knowledge, no such results have been brought in the literature up to now for FOS with non commensurate fractional-order systems in continuous time. The next section of this chapter constitutes a contribution to the study of structural properties of non-commensurate FOS, using a different approach, that is discrete-time representation [34].

#### **3. LTI discrete-time fractional-order modeling**

The development of a discrete-time model is based on the Grünwald-Letnikov's definition of fractional-order operators. This definition is another expression of (1), based on the generalization of the backward difference. The generalization of the integer-order difference to a non-integer-order (or fractional-order) difference is addressed in [33, 47].

#### **3.1. The discrete-time fractional-order difference operator**

In what follows, we present a discretized representation of the continuous fractional-order state-space model of Equations (15) and (16), in which the fractional order can be commensurate or non-commensurate.

Let us consider the state-space model with integer-order derivatives

$$D^1\mathbf{x}(t) = A\mathbf{x}(t) + Bu(t) \tag{23}$$

where *x*(*t*)=[*x*1(*t*) *x*2(*t*)... *xnd* (*t*)]*<sup>T</sup>* ∈ **R***nd* is the state vector. This latter model can be represented in discrete-time using a forward difference approximation of the derivative of order one of *x*(*t*) [32, 33]. For this purpose let us consider a sampling period *h*. Then for *kh* ≤ *t* < (*k* + 1)*h* the integer order first derivative *D*1*x*(*t*) can be approximated by

$$D^1 \mathbf{x}(t) \approx \Delta^1 \mathbf{x}((k+1)h) = \frac{\mathbf{x}((k+1)h) - \mathbf{x}(kh)}{h}.\tag{24}$$

We can then write

6 Advances in Discrete Time Systems

*number of non-null terms a*′

In this case, (15) becomes

Defining Φ(*t*) = L−1[*sα***I** − *A*]

state response given by

controllability matrix

2*n* + 1 *is denoted nd.*

As for direct transmission matrix *D*, we have

*<sup>D</sup>* = [ *<sup>a</sup>*′

*l*

*x*(*t*) =

Applying the Laplace transform to (17), we have

It can be shown that Φ(*t*) can be expressed by

 

*Dαx*<sup>1</sup> *Dαx*<sup>2</sup> . . . *D<sup>α</sup>xnd*

*X*(*s*)=[*s*

*x*(*t*) = Φ(*t*)*x*(0) +

Φ(*t*) =

*Eα*(*t*)

The left side term of this latter equation represents the Mittag-Leffler function [41] :

 

*<sup>α</sup>***I** − *A*]

� *<sup>t</sup>* 0

∞ ∑ *k*=0

∞ ∑ *k*=0

It has been established in [43] that the controllability and observability conditions of the continuous-time state-space representation of commensurate fractional-order systems are the same as in the integer-order case. Thus, System (17) is controllable if the rank of the

*Akt kα*

*Akt kα*

<sup>−</sup>1[*BU*(*s*) + *x*(0)],

<sup>−</sup><sup>1</sup> as the corresponding state transition matrix, we obtain the

C = [*B AB A*2*B* ...*And*−1*B*] (21)

*D*[*α*]

2*n a*′ 2*n*+1

The symbol *D* here should not be confused with the derivation operator *D*[.]

] if *α*2*<sup>n</sup>* = *α*2*n*+1, otherwise *D* = [0].

*sα<sup>l</sup> present in the numerator and denominator of G*(*s*). *In the following,*

<sup>=</sup> *Ax*(*t*) + *Bu*(*t*), *<sup>x</sup>*(0) = *<sup>x</sup>*<sup>0</sup> (17)

Φ(*t* − *τ*)*Bu*(*τ*)*dτ* (18)

<sup>Γ</sup>(<sup>1</sup> <sup>+</sup> *<sup>k</sup>α*) (19)

<sup>Γ</sup>(<sup>1</sup> <sup>+</sup> *<sup>k</sup>α*) (20)

**Remark 1.** *The dimension of the model, i.e., its number of state variables* 2*n* + 1 *is equal to the total*

The controllability and observability properties as well as the stability have been well studied when the differentiation fractional-orders *γ<sup>i</sup>* are all equal to a unique value, say *α* ([43], [42]).

.

$$
\Delta^1(\mathbf{x}(k+1)h) = A\mathbf{x}(kh) + Bu(kh) \tag{25}
$$

We start with applying a discretization of (17) with identical fractional orders. The fractional derivatives *D*[*α*] *x*(*t*) can be approximated by using the definition due to K. Grünwald

$$D^{[a]}\mathbf{x}(t) \approx \,\,\Delta^{[a]}\_{\hbar}\mathbf{x}((k+1)\hbar) = \frac{1}{\hbar^a} \sum\_{j=0}^{k+1} (-1)^j \binom{a}{j} \mathbf{x}(k+1-j)\hbar) \tag{26}$$

In this equation, *<sup>α</sup>* ∈ **<sup>R</sup>**⋆<sup>+</sup> is the fractional order, *<sup>t</sup>* is the current time, *<sup>h</sup>* ∈ **<sup>R</sup>**⋆<sup>+</sup> is the sampling period or time increment. The term ( *α j* ) is calculated by the relation

$$\binom{a}{j} = \begin{cases} 1 & \text{for } j = 0\\ \frac{a(a-1)\ldots(a-j+1)}{j!} & \text{for } j > 0 \end{cases} \tag{27}$$

This leads to

where

non-commensurate) orders [53]

*x*(*k* + 1) = *A*0*x*(*k*) + *A*1*x*(*k* − 1) + *A*2*x*(*k* − 2) + ... + *Akx*(0) + *Bu*˜ (*k*) (34)

*k*+1 ∑ *j*=1

*hγ*1∆*γ*<sup>1</sup> *x*1(*k* + 1) . . . *<sup>h</sup>γnd* <sup>∆</sup>*γnd xnd* (*<sup>k</sup>* + <sup>1</sup>)

*Ajx*(*k* − *j* + 1)

 

Discrete-Time Fractional-Order Systems: Modeling and Stability Issues

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191

, *i* = 1, . . . , *nd* , (35)

, *i* = 1, . . . , *nd*}, *j* = 1, 2, . . . (36)

*<sup>i</sup>* = *hγ<sup>i</sup> Bi* where *Ai* and *Bi*

*Ajx*(*k* − *j*) + *Bu*˜ (*k*); *x*(0) = *x*<sup>0</sup> (37)

*y*(*k*) = *Cx*(*k*) (38)

*<sup>i</sup>* = *hγ<sup>i</sup> Ai* ; *B*˜

This description can be extended to FOS with different (commensurate or

*<sup>h</sup> x*(*k* + 1) = *Ax*(*k*) + *Bu*(*k*)

*<sup>h</sup> x*(*k* + 1) +

 

> �*γ<sup>i</sup>* 1 �

∆[*γ*]

*<sup>x</sup>*(*<sup>k</sup>* + <sup>1</sup>) = <sup>∆</sup>[*γ*]

∆[*γ*]

*Aj* = *diag*{−(−1)*j*+<sup>1</sup>

*x*(*k* + 1) =

In this case, *A*˜ and *B*˜ are calculated as follows *A*˜

commensurate or non-commensurate fractional-orders.

Using (35), we obtain the state equation

*<sup>h</sup> x*(*k* + 1) =

*A*<sup>0</sup> = *A*˜ + *diag*{

*k* ∑ *j*=0

denote the rows of *A* and *B* respectively. The corresponding output equation is

In this resulting discrete time state-space model, *Aj* is given by (32) and (33), in the case of a unique fractional-order while it is given by (35) and (36) in the case of different,

**Remark 2.** *The model defined by* (37) *and* (38) *can be viewed as a discrete-time model with time-delay in state. It has a varying number of steps of time-delays, equal to k, i.e., increasing with time. Instead,*

*the models addressed in [50–52] consider a finite constant number of steps of time-delays.*

in which *<sup>γ</sup><sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**⋆+, *<sup>i</sup>* <sup>=</sup> 1, 2, . . . denote any fractional orders. Here, we can write

� *γ<sup>i</sup> j* + 1 �

If we drop *h* (i.e., making *h* = 1), we find the discrete fractional-order difference operator ∆*<sup>α</sup>* as defined and used in [32, 33, 46]. Indeed, the choice of the sampling period *h* is very important when elaborating the discretized model of a given process. *h* is not necessarily small. It corresponds to an appropriate sampling rate, deduced from the dynamics (or frequency bandwidth) of the process, by using rules of thumbs and criteria, as exposed for example in [48, 49].

#### **3.2. The discrete-time fractional-order state-space model**

The previous results conduct to the linear discrete-time fractional-order state-space model

$$
\Delta^{[4]}\mathbf{x}(k+1) = A\mathbf{x}(k) + Bu(k); \qquad \qquad \mathbf{x}(0) = \mathbf{x}\_0 \tag{28}
$$

In this model, the differentiation order *α* is taken the same for all the state variables *xi*(*k*), *i* = 1, . . . , *nd*. Besides, from (27) we have

$$h^a \Delta^{[a]} \mathbf{x}(k+1) = \mathbf{x}(k+1) + \sum\_{j=1}^{k+1} (-1)^j \binom{n}{j} \mathbf{x}(k-j+1) \tag{29}$$

Substituting (30) into (29) yields

$$\mathbf{x}(k+1) = \mathbf{\tilde{A}x}(k) - \sum\_{j=1}^{k+1} (-1)^j \binom{n}{j} \mathbf{x}(k-j+1) + \mathbf{\tilde{B}u}(k) \tag{30}$$

with *A*˜ = *hαA* and *B*˜ = *hαB*. By setting *cj* = (−1)*<sup>j</sup>* ( *α j* ), (31) can be rewritten as follows

$$\mathbf{x}(k+1) = (\tilde{A} - c\_1 I\_{\mathbb{H}\_d})\mathbf{x}(k) - \sum\_{j=2}^{k+1} c\_j \mathbf{x}(k - j + 1) + \tilde{B}\mathbf{u}(k) \tag{31}$$

Let us put now

$$A\_0 = (\tilde{A} - c\_1 I\_{\mathbb{N}\_d}) \tag{32}$$

and, for all *j* > 0 :

$$A\_j = -\mathfrak{c}\_{j+1} I\_{n\_d} \tag{33}$$

This leads to

8 Advances in Discrete Time Systems

for example in [48, 49].

period or time increment. The term (

In this equation, *<sup>α</sup>* ∈ **<sup>R</sup>**⋆<sup>+</sup> is the fractional order, *<sup>t</sup>* is the current time, *<sup>h</sup>* ∈ **<sup>R</sup>**⋆<sup>+</sup> is the sampling

If we drop *h* (i.e., making *h* = 1), we find the discrete fractional-order difference operator ∆*<sup>α</sup>* as defined and used in [32, 33, 46]. Indeed, the choice of the sampling period *h* is very important when elaborating the discretized model of a given process. *h* is not necessarily small. It corresponds to an appropriate sampling rate, deduced from the dynamics (or frequency bandwidth) of the process, by using rules of thumbs and criteria, as exposed

The previous results conduct to the linear discrete-time fractional-order state-space model

In this model, the differentiation order *α* is taken the same for all the state variables *xi*(*k*),

*k*+1 ∑ *j*=1

(−1)*<sup>j</sup> α j* 

*x*(*k* + 1) = *Ax*(*k*) + *Bu*(*k*); *x*(0) = *x*<sup>0</sup> (28)

) is calculated by the relation

*<sup>j</sup>*! for *<sup>j</sup>* <sup>&</sup>gt; <sup>0</sup> (27)

*x*(*k* − *j* + 1) (29)

*x*(*k* − *j* + 1) + *Bu*˜ (*k*) (30)

*cjx*(*k* − *j* + 1) + *Bu*˜ (*k*) (31)

), (31) can be rewritten as follows

*A*<sup>0</sup> = (*A*˜ − *c*<sup>1</sup> *Ind* ) (32)

*Aj* = −*cj*+<sup>1</sup> *Ind* (33)

 1 for *j* = 0 *α*(*α*−1)...(*α*−*j*+1)

*α j*

*α j* =

**3.2. The discrete-time fractional-order state-space model**

*x*(*k* + 1) = *x*(*k* + 1) +

*k*+1 ∑ *j*=1

(−1)*<sup>j</sup> α j* 

> ( *α j*

*k*+1 ∑ *j*=2

∆[*α*]

*i* = 1, . . . , *nd*. Besides, from (27) we have

Substituting (30) into (29) yields

Let us put now

and, for all *j* > 0 :

*hα*∆[*α*]

*x*(*k* + 1) = *Ax*˜ (*k*) −

*x*(*k* + 1)=(*A*˜ − *c*<sup>1</sup> *Ind* )*x*(*k*) −

with *A*˜ = *hαA* and *B*˜ = *hαB*. By setting *cj* = (−1)*<sup>j</sup>*

$$\mathbf{x}(k+1) = A\_0 \mathbf{x}(k) + A\_1 \mathbf{x}(k-1) + A\_2 \mathbf{x}(k-2) + \dots + A\_k \mathbf{x}(0) + \bar{B} \mathbf{u}(k) \tag{34}$$

This description can be extended to FOS with different (commensurate or non-commensurate) orders [53]

$$
\Delta\_h^{\lfloor \gamma \rfloor} \mathfrak{x}(k+1) = A\mathfrak{x}(k) + B\mathfrak{u}(k)
$$

$$\mathbf{x}(k+1) = \boldsymbol{\Delta}\_{\boldsymbol{h}}^{[\gamma]} \mathbf{x}(k+1) + \sum\_{j=1}^{k+1} A\_j \mathbf{x}(k-j+1)$$

where

$$
\Delta\_h^{[\gamma]} \mathfrak{x}(k+1) = \begin{bmatrix}
\hbar^{\gamma\_1} \Delta^{\gamma\_1} \mathfrak{x}\_1(k+1) \\
\vdots \\
\hbar^{\gamma\_{\mathfrak{N}\_d}} \Delta^{\gamma\_{\mathfrak{N}\_d}} \mathfrak{x}\_{\mathfrak{N}\_d}(k+1)
\end{bmatrix}
$$

in which *<sup>γ</sup><sup>i</sup>* <sup>∈</sup> **<sup>R</sup>**⋆+, *<sup>i</sup>* <sup>=</sup> 1, 2, . . . denote any fractional orders. Here, we can write

$$A\_0 = \tilde{A} + \text{diag}\{\begin{pmatrix} \gamma\_i\\ 1 \end{pmatrix} \mid i = 1, \dots, n\_d \ \text{ }, \tag{35}$$

$$A\_j = \text{diag}\{- (-1)^{j+1} \binom{\gamma\_j}{j+1}, \ i = 1, \dots, n\_d \}, \ j = 1, 2, \dots \tag{36}$$

Using (35), we obtain the state equation

$$\mathbf{x}(k+1) = \sum\_{j=0}^{k} A\_j \mathbf{x}(k-j) + \mathsf{\tilde{B}} u(k); \quad \mathbf{x}(0) = \mathbf{x}\_0 \tag{37}$$

In this case, *A*˜ and *B*˜ are calculated as follows *A*˜ *<sup>i</sup>* = *hγ<sup>i</sup> Ai* ; *B*˜ *<sup>i</sup>* = *hγ<sup>i</sup> Bi* where *Ai* and *Bi* denote the rows of *A* and *B* respectively. The corresponding output equation is

$$y(k) = \mathbb{C}x(k)\tag{38}$$

In this resulting discrete time state-space model, *Aj* is given by (32) and (33), in the case of a unique fractional-order while it is given by (35) and (36) in the case of different, commensurate or non-commensurate fractional-orders.

**Remark 2.** *The model defined by* (37) *and* (38) *can be viewed as a discrete-time model with time-delay in state. It has a varying number of steps of time-delays, equal to k, i.e., increasing with time. Instead, the models addressed in [50–52] consider a finite constant number of steps of time-delays.*

Define *Gk* such that

$$\mathbf{G}\_{k} = \begin{cases} \ I\_{\mathbb{II}\_{d}} & \text{for } k = 0, \\\ \sum\_{j=0}^{k-1} A\_{j} \mathbf{G}\_{k-1-j} & \text{for } k \ge 1 \end{cases} \tag{39}$$

**4. Structural properties of LTI discrete-time FOS**

completely state reachable and controllable systems.

C*<sup>k</sup>* =

*3. The controllability Gramian, provided that* A0 *is non-singular*

*that transfers x*<sup>0</sup> = 0 *at k* = 0 *to xf* �= 0 *at k* = *K is given by*

W*r*(0, *k*) =

W*c*(0, *k*) = G−<sup>1</sup>

*k .*

We discuss here a fundamental question for dynamic systems modeled by (37) in the case of a non-commensurate fractional-order. This question is to determine whether it is possible to transfer the state of the system from a given initial state to any other state. We search below to extend two concepts of state reachability (or controllability from the origin) and controllability (or controllability to the origin) to the present case. We are interested in

Discrete-Time Fractional-Order Systems: Modeling and Stability Issues

**Definition 1.** *The linear discrete-time fractional-order system modeled by* (37) *is reachable if it is possible to find a control sequence such that an arbitrary state can be reached from the origin in a finite*

**Definition 2.** *The linear discrete-time fractional-order system modeled by* (37) *is controllable if it is possible to find a control sequence such that the origin can be reached from any initial state in a finite*

**Definition 3.** *For the linear discrete-time fractional-order system modeled by* (37) *we define the*

*k*−1 ∑ *j*=0

G0*B* G1*B* G2*B* ··· G*k*−1*B*

G*jBB<sup>T</sup>*G*<sup>T</sup>*

*<sup>k</sup>* <sup>W</sup>*r*(0, *<sup>k</sup>*)G−*<sup>T</sup>*

Note that *G*<sup>1</sup> = *A*<sup>0</sup> and the existence of *Wr*(0, 1) imposes *A*<sup>0</sup> to be non-singular. However, this is not that restrictive condition because a discrete model is often obtained by sampling a continuous one. Thus, in the remainder of this paper we assume that *A*<sup>0</sup> is non-singular. **Theorem 2.** *The linear discrete-time fractional-order system modeled by* (37) *is reachable if and only if there exists a finite time K such that rank*(C*K*) = *n or, equivalently, rank*(W*r*(0, *K*)) = *n.*

U*<sup>K</sup>* = [*uT*(*K* − 1) *uT*(*K* − 2) ... *uT*(0)]*<sup>T</sup>*

*K*W−<sup>1</sup>

U*<sup>K</sup>* = C*<sup>T</sup>*

(45)

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193

*<sup>j</sup>* , *k* ≥ 1 (46)

*<sup>k</sup>* , *k* ≥ 1 (47)

*<sup>r</sup>* (0, *K*)*xf* . (48)

**4.1. Reachability and controllability**

*time.*

*time.*

*following*

*1. The controllability matrix*

*2. The reachability Gramian*

*Furthermore, the input sequence*

*It is easy to show that* W*r*(0, *k*) = C*k*C*<sup>T</sup>*

**Theorem 1.** *The solution of* (37) *is given by*

$$\mathbf{x}(k) = \mathbf{G}\_k \mathbf{x}(0) + \sum\_{j=0}^{k-1} \mathbf{G}\_{k-1-j} \mathbf{B} u(j) \tag{40}$$

This theorem can be proved by induction [34]. The first part of the solution of (40) represents the free response of the system and the last part takes the role of the convolution sum corresponding to the forced response.

The corresponding transition matrix can be defined as

$$
\Phi(k,0) = \mathcal{G}\_{k\prime} \quad \Phi(0,0) = \mathcal{G}\_0 = I\_{\mathbb{N}\_d} \tag{41}
$$

**Remark 3.** *-* Φ(*k*, 0) *exhibits the particularity of being time-varying, in the sense that it is composed of a number of terms* A*<sup>j</sup> which grows along with k. This is due to the fractional-order feature of the model which takes into account all the past values of the states.*

By virtue of Equations (37) and (38), the discretization of this state-space realization is

$$\begin{aligned} \mathbf{x}(k+1) &= \sum\_{j=0}^{k} A\_j \mathbf{x}(k-j) + Bu(k); \quad \mathbf{x}(0) = \mathbf{x}\_0; \\\\ \mathbf{y}(k) &= \mathbf{C} \mathbf{x}(k) \end{aligned}$$

Here we have

$$A\_0 = A + \operatorname{diag} \{ \begin{pmatrix} n\_i \\ 1 \end{pmatrix} \mid i = 1, \dots, n\_d \} \tag{42}$$

$$A\_j = \text{diag}\{- (-1)^{j+1} \binom{n\_j}{j+1}, \ i = 1, \dots, n\_d \}, j = 1, 2, \dots \tag{43}$$

The output response for a given input sequence and initial conditions is given by

$$y(k) = \mathbf{C}G\_k x(0) + \sum\_{j=0}^{k-1} \mathbf{C} \mathbf{G}\_{k-1-j} \mathbf{B} u(j) \tag{44}$$
