**4.1. Reachability and controllability**

10 Advances in Discrete Time Systems

*Gk* =

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

corresponding to the forced response.

The corresponding transition matrix can be defined as

*model which takes into account all the past values of the states.*

*x*(*k* + 1) =

*Aj* = *diag*{−(−1)*j*+<sup>1</sup>

*k* ∑ *j*=0

*A*<sup>0</sup> = *A* + *diag*{

∑*k*−1

*x*(*k*) = G*kx*(0) +

*Ind* for *k* = 0,

*k*−1 ∑ *j*=0

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

**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*

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

*α<sup>i</sup>* 1 

> *k*−1 ∑ *j*=0

 *α<sup>i</sup> j* + 1 

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

*y*(*k*) = *CGkx*(0) +

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

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

*<sup>j</sup>*=<sup>0</sup> *AjGk*<sup>−</sup>1−*<sup>j</sup>* for *<sup>k</sup>* <sup>≥</sup> <sup>1</sup> (39)

<sup>Φ</sup>(*k*, 0) = *Gk*, <sup>Φ</sup>(0, 0) = *<sup>G</sup>*<sup>0</sup> = *Ind* (41)

G*k*−1<sup>−</sup>*jBu*(*j*) (40)

, *i* = 1, . . . , *nd*} (42)

, *i* = 1, . . . , *nd*}, *j* = 1, 2, . . . (43)

*CGk*<sup>−</sup>1<sup>−</sup>*jBu*(*j*) (44)

Define *Gk* such that

Here we have

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 completely state reachable and controllable systems.

**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 time.*

**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 time.*

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

*1. The controllability matrix*

$$\mathcal{C}\_k = \begin{bmatrix} \mathbf{G}\_0 \mathbf{B} & \mathbf{G}\_1 \mathbf{B} & \mathbf{G}\_2 \mathbf{B} & \cdots & \mathbf{G}\_{k-1} \mathbf{B} \end{bmatrix} \tag{45}$$

*2. The reachability Gramian*

$$\mathbf{W}\_{r}(0,k) = \sum\_{j=0}^{k-1} \mathbf{G}\_{j} \mathbf{B} \mathbf{B}^{T} \mathbf{G}\_{j}^{T}, \quad k \ge 1 \tag{46}$$

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

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

$$\mathbf{W}\_{\mathcal{E}}(0,k) = \mathbf{G}\_{k}^{-1}\mathbf{W}\_{\mathcal{I}}(0,k)\mathbf{G}\_{k}^{-T}, \quad k \ge 1 \tag{47}$$

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. Furthermore, the input sequence*

$$\mathcal{U}\_{\mathbf{K}} = [\boldsymbol{\mu}^T(\mathbf{K} - \mathbf{1}) \quad \boldsymbol{\mu}^T(\mathbf{K} - \mathbf{2}) \quad \dots \quad \boldsymbol{\mu}^T(\mathbf{0})]^T$$

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

$$\mathcal{U}\_{\mathbf{K}} = \mathcal{C}\_{\mathbf{K}}^T \mathbf{W}\_r^{-1}(\mathbf{0}, \mathbf{K}) \mathbf{x}\_f. \tag{48}$$

**Remark 4.** *In the case of an integer order, it is well known that the rank of* C*<sup>k</sup> cannot increase for any k* ≥ *n. This results from the Cayley-Hamilton theorem. On the contrary, in the case of the linear discrete-time non-commensurate fractional-order system* (37)*, the rank of* C*<sup>k</sup> can increase for values of k* ≥ *n. In other words, it is possible to reach the final state xf in a number of steps greater than n. This is due to the nature of the elements* G*<sup>k</sup> which build up the controllability matrix* C*<sup>k</sup> and which exhibit the particularity of being time-varying, in the sense that they are composed of a number of terms* A*<sup>j</sup> that grows with k, as already mentioned in Remark* 2*. The full rank of* (C*k*) *can be reached in some Step k* = *K equal to, or greater than n.*

**Theorem 3.** *The linear discrete-time fractional-order system modeled by* (37) *is controllable if and only if there exists a finite time K such that rank*(W*c*(0, *K*)) = *n. Furthermore, an input sequence* U*<sup>K</sup>* = [*uT*(*K* − 1) *uT*(*K* − 2) ... *uT*(0)]*<sup>T</sup> that transfers x*<sup>0</sup> �= 0 *at k* = 0 *to xf* = 0 *at k* = *K is given by*

$$\mathcal{U}\_{\mathbf{K}} = -\mathcal{C}\_{\mathbf{K}}^T \mathbf{G}\_{\mathbf{K}}^{-T} \mathbf{W}\_{\mathbf{c}}^{-1} (\mathbf{0}, \mathbf{K}) \mathbf{x}\_0. \tag{49}$$

*with*

*and*

*previous result in [53].*

the following results

U˜

Y˜

 

*. . . . . . . .*

clear then that matrices *Gk* defined by (40) are polynomials in *A*0, i.e.,

<sup>0</sup> <sup>+</sup> *<sup>β</sup>*1*kAk*−<sup>1</sup>

<sup>0</sup> <sup>+</sup> *<sup>β</sup>*1*nAn*−<sup>1</sup>

*On the other hand, this system is controllable if and only if rank*(W*c*(0, *n*)) = *n.*

*Gk* = *A<sup>k</sup>*

*Gn* = *A<sup>n</sup>*

From the Cayley-Hamilton theorem, *A<sup>n</sup>*

M*<sup>K</sup>* =

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

*<sup>K</sup>* = [*yT*(0) *yT*(1) ... *yT*(*K* − 1)]*T*,

0 0 0 ... 0 0 *C*G0*B* 0 0 ... 0 0 *C*G1*B C*G0*B* 0 ... 0 0 *C*G2*B C*G1*B C*G0*B* ... 0 0

*C*G*K*−2*B C*G*K*−3*B C*G*K*−4*B* ... *C*G0*B* 0

**Remark 5.** *From the Cayley-Hamilton theorem, it is well known that for integer-order systems the rank of the observability matrix* O*<sup>k</sup> cannot increase in Step k* ≥ *n. Here too, it is remarkable that this is not true in the case of the discrete-time non-commensurate fractional-order system of* (37) *and* (38)*. Indeed, rank*(O*k*) *can increase for values k* ≥ *n. We can state that the observability of this type of systems can possibly be obtained in a number of steps greater than n. This is due to the same reasons as those exposed above in Remark* 3 *for controllability. In [53], the observability condition for the discrete-time fractional-order system as modeled by* (37)*, with non-commensurate order, is that the rank of* O*<sup>k</sup> should be equal to n at most in Step k* = *n. Our result shows that the full rank of* (O*k*) *can be reached in some Step k* = *K greater than n. This can be considered as an extension of the*

**Commensurate fractional-order case** In this section we address the particular case of commensurate FOS. The terms *Aj* are as expressed by *Aj* = −*cj*+1**I***n*, for all *j* > 0. It is

<sup>0</sup> <sup>+</sup> *<sup>β</sup>*2*kAk*−<sup>2</sup>

<sup>0</sup> <sup>+</sup> *<sup>β</sup>*2*nAn*−<sup>2</sup>

deduce that *Gk*<sup>+</sup>*n*, for all *k* ≥ 0 are linearly dependent on *Gn*−1, *Gn*−2, . . ., **I***n*. This implies

**Corollary 1.** *The linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *in the commensurate case is reachable if and only if rank*(C*n*) = *n or, equivalently, rank*(W*r*(0, *n*)) = *n*.

**Corollary 2.** *The linear discrete-time fractional-order system modeled by* (37) *and* (38) *in the commensurate case is observable if and only if rank*(O*n*) = *n or, equivalently, rank*(W*o*(0, *n*)) = *n*.

where the real coefficients *βjk* are calculated from the coefficients *cj*. In particular, we have

*.* ... *.*

<sup>0</sup> <sup>+</sup> ... <sup>+</sup> *<sup>β</sup>kk* **<sup>I</sup>***n*.

<sup>0</sup> <sup>+</sup> ... <sup>+</sup> *<sup>β</sup>nn* **<sup>I</sup>***n*.

<sup>0</sup> , *<sup>A</sup>n*−<sup>2</sup>

<sup>0</sup> , . . ., **<sup>I</sup>***n*. We

<sup>0</sup> is a linear combination of *<sup>A</sup>n*−<sup>1</sup>

*. . . . .*

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

 

. (53)

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195

#### **4.2. Observability**

In this section we aim at extending the concept of observability to the system of Equations (37) and (38), in the case of a non-commensurate fractional-order. We are interested in completely state observable systems.

**Definition 4.** *The linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *is observable at time k* = 0 *if and only if there exits some K* > 0 *such that the state x*<sup>0</sup> *at time k* = 0 *can be uniquely determined from the knowledge of uk, yk, k* ∈ [0, *K*]*.*

**Definition 5.** *For the linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *we define the following*

*1. The observability matrix*

$$\mathcal{O}\_k = \begin{bmatrix} \text{CG}\_0 \\ \text{CG}\_1 \\ \text{CG}\_2 \\ \vdots \\ \text{CG}\_{k-1} \cdot \end{bmatrix} \tag{50}$$

*2. The observability Gramian*

$$\mathbf{W}\_o(0,k) = \sum\_{j=0}^{k-1} \mathbf{G}\_j^T \mathbf{C}^T \mathbf{C} \mathbf{G}\_j. \tag{51}$$

*It is easy to show that* W*o*(0, *k*) = O*<sup>T</sup> <sup>k</sup>* O*k*.

**Theorem 4.** *The linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *is observable if and only if there exists a finite time K such that rank*(O*K*) = *n or, equivalently, rank*(W*o*(0, *K*)) = *n*. *Furthermore, the initial state x*<sup>0</sup> *at k* = 0 *is given by*

$$\mathbf{x}\_0 = \mathbf{W}\_o^{-1}(0, \mathbf{K}) \mathcal{O}\_K^T [\tilde{\mathcal{Y}}\_\mathbf{K} - \mathcal{M}\_\mathbf{K} \tilde{\mathcal{U}}\_\mathbf{K}] \tag{52}$$

*with*

12 Advances in Discrete Time Systems

*is given by*

**4.2. Observability**

(38) *we define the following*

*1. The observability matrix*

*2. The observability Gramian*

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

*in some Step k* = *K equal to, or greater than n.*

completely state observable systems.

**Remark 4.** *In the case of an integer order, it is well known that the rank of* C*<sup>k</sup> cannot increase for any k* ≥ *n. This results from the Cayley-Hamilton theorem. On the contrary, in the case of the linear discrete-time non-commensurate fractional-order system* (37)*, the rank of* C*<sup>k</sup> can increase for values of k* ≥ *n. In other words, it is possible to reach the final state xf in a number of steps greater than n. This is due to the nature of the elements* G*<sup>k</sup> which build up the controllability matrix* C*<sup>k</sup> and which exhibit the particularity of being time-varying, in the sense that they are composed of a number of terms* A*<sup>j</sup> that grows with k, as already mentioned in Remark* 2*. The full rank of* (C*k*) *can be reached*

**Theorem 3.** *The linear discrete-time fractional-order system modeled by* (37) *is controllable if and only if there exists a finite time K such that rank*(W*c*(0, *K*)) = *n. Furthermore, an input sequence* U*<sup>K</sup>* = [*uT*(*K* − 1) *uT*(*K* − 2) ... *uT*(0)]*<sup>T</sup> that transfers x*<sup>0</sup> �= 0 *at k* = 0 *to xf* = 0 *at k* = *K*

> *K*G−*<sup>T</sup> <sup>K</sup>* <sup>W</sup>−<sup>1</sup>

In this section we aim at extending the concept of observability to the system of Equations (37) and (38), in the case of a non-commensurate fractional-order. We are interested in

**Definition 4.** *The linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *is observable at time k* = 0 *if and only if there exits some K* > 0 *such that the state x*<sup>0</sup> *at time k* = 0

**Definition 5.** *For the linear discrete-time fractional-order system modeled by Equations* (37) *and*

 

> *k*−1 ∑ *j*=0 G*T*

**Theorem 4.** *The linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *is observable if and only if there exists a finite time K such that rank*(O*K*) = *n or, equivalently,*

> *K* � Y˜

*<sup>K</sup>* − M*K*U˜

*K*

*<sup>o</sup>* (0, *<sup>K</sup>*)O*<sup>T</sup>*

*C*G0 *C*G1 *C*G2 *. . . C*G*k*<sup>−</sup>1.  

O*<sup>k</sup>* =

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

*<sup>k</sup>* O*k*.

*rank*(W*o*(0, *K*)) = *n*. *Furthermore, the initial state x*<sup>0</sup> *at k* = 0 *is given by*

*<sup>x</sup>*<sup>0</sup> = <sup>W</sup>−<sup>1</sup>

*<sup>c</sup>* (0, *K*)*x*0. (49)

*<sup>j</sup> <sup>C</sup>TC*G*j*. (51)

� (52)

(50)

U*<sup>K</sup>* = −C*<sup>T</sup>*

*can be uniquely determined from the knowledge of uk, yk, k* ∈ [0, *K*]*.*

$$\begin{aligned} \tilde{\mathcal{U}}\_{\mathbf{K}} &= [\boldsymbol{u}^T(\mathbf{0}) \quad \boldsymbol{u}^T(\mathbf{1}) \quad \dots \quad \boldsymbol{u}^T(\mathbf{K}-\mathbf{1})]^T, \\\\ \tilde{\mathcal{V}}\_{\mathbf{K}} &= [\boldsymbol{y}^T(\mathbf{0}) \quad \boldsymbol{y}^T(\mathbf{1}) \quad \dots \quad \boldsymbol{y}^T(\mathbf{K}-\mathbf{1})]^T, \end{aligned}$$

*and*

$$\mathcal{M}\_K = \begin{bmatrix} 0 & 0 & 0 & \dots & 0 & 0 \\ \text{CG}\_0 B & 0 & 0 & \dots & 0 & 0 \\ \text{CG}\_1 B & \text{CG}\_0 B & 0 & \dots & 0 & 0 \\ \text{CG}\_2 B & \text{CG}\_1 B & \text{CG}\_0 B & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \dots & \vdots & \vdots \\ \text{CG}\_{K-2} B \text{ CG}\_{K-3} B \text{ CG}\_{K-4} B & \dots & \text{CG}\_0 B & 0 \end{bmatrix} . \tag{53}$$

**Remark 5.** *From the Cayley-Hamilton theorem, it is well known that for integer-order systems the rank of the observability matrix* O*<sup>k</sup> cannot increase in Step k* ≥ *n. Here too, it is remarkable that this is not true in the case of the discrete-time non-commensurate fractional-order system of* (37) *and* (38)*. Indeed, rank*(O*k*) *can increase for values k* ≥ *n. We can state that the observability of this type of systems can possibly be obtained in a number of steps greater than n. This is due to the same reasons as those exposed above in Remark* 3 *for controllability. In [53], the observability condition for the discrete-time fractional-order system as modeled by* (37)*, with non-commensurate order, is that the rank of* O*<sup>k</sup> should be equal to n at most in Step k* = *n. Our result shows that the full rank of* (O*k*) *can be reached in some Step k* = *K greater than n. This can be considered as an extension of the previous result in [53].*

**Commensurate fractional-order case** In this section we address the particular case of commensurate FOS. The terms *Aj* are as expressed by *Aj* = −*cj*+1**I***n*, for all *j* > 0. It is clear then that matrices *Gk* defined by (40) are polynomials in *A*0, i.e.,

$$G\_k = A\_0^k + \beta\_{1k} A\_0^{k-1} + \beta\_{2k} A\_0^{k-2} + \dots + \beta\_{k\iota} \mathbb{I}\_{\iota} \mathbb{I}\_{\iota}$$

where the real coefficients *βjk* are calculated from the coefficients *cj*. In particular, we have

$$G\_n = A\_0^n + \beta\_{1\_n} A\_0^{n-1} + \beta\_{2\_n} A\_0^{n-2} + \dots + \beta\_{n\_n} \mathbb{I}\_{n-1}$$

From the Cayley-Hamilton theorem, *A<sup>n</sup>* <sup>0</sup> is a linear combination of *<sup>A</sup>n*−<sup>1</sup> <sup>0</sup> , *<sup>A</sup>n*−<sup>2</sup> <sup>0</sup> , . . ., **<sup>I</sup>***n*. We deduce that *Gk*<sup>+</sup>*n*, for all *k* ≥ 0 are linearly dependent on *Gn*−1, *Gn*−2, . . ., **I***n*. This implies the following results

**Corollary 1.** *The linear discrete-time fractional-order system modeled by Equations* (37) *and* (38) *in the commensurate case is reachable if and only if rank*(C*n*) = *n or, equivalently, rank*(W*r*(0, *n*)) = *n*. *On the other hand, this system is controllable if and only if rank*(W*c*(0, *n*)) = *n.*

**Corollary 2.** *The linear discrete-time fractional-order system modeled by* (37) *and* (38) *in the commensurate case is observable if and only if rank*(O*n*) = *n or, equivalently, rank*(W*o*(0, *n*)) = *n*.

**Remark 6.** *We therefore observe that the controllability and observability criteria for the commensurate fractional-order case are similar to those of the integer-order case, in the sense that if a state cannot be reached in n steps, then it is not reachable at all and that if an initial state cannot be deduced from n steps of input-output data, then it is not observable at all. The result put forward in [53] which states that a necessary and sufficient condition for the discrete-time fractional-order system as modeled in* (37) *and* (38) *to be observable is that the rank of* O*<sup>k</sup> should be equal to n at most in Step k* = *n is true only in the case of commensurate FOS.*

In order to study the asymptotic stability property of this system, we consider its unforced

**Definition 6.** *System* (54) *is asymptotically stable if for each k* ≥ 1 *and any initial condition x*0*, the*

 *n* ∑ *i*=1 *x*2 *<sup>i</sup>* (*k*),

*Ajx*(*k* − *j*), *x*(0) = *x*0. (54)

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

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197

*<sup>k</sup>*→<sup>∞</sup> � *<sup>x</sup>*(*k*) � <sup>=</sup> 0. (55)

*x*(*k*) = *Gkx*0, *k* ≥ 1, *G*<sup>0</sup> = **I***n*. (56)

*<sup>k</sup>* ) = *σmax*(*Gk*),

� *Gk* �≤ 1, *k* ≥ 1. (57)

*k* ∑ *j*=0

*lim*

� *x*(*k*) �=

*x*(*k* + 1) =

version, i.e.,

*following equality is verified*

We use the 2-norm of the state vector *x*(*k*), i.e.,

where *xi*(*k*) are the components of *x*(*k*).

The 2-norm of the transition matrix is

respectively.

S acts on **x** as follows

The solution to (54) for given initial conditions *x*<sup>0</sup> is

It follows that System (54) is asymptotically stable if and only if

� *Gk* �= *λ*

values for *k* < 0, assuming the system to be causal. Thus, we have

1 2 *max*(*GkG<sup>T</sup>*

where *λmax* and *σmax* refer to the maximal eigenvalue and the maximal singular value of *Gk*

Let us define the backward shift operator S as follows ([55], [56]). We consider an infinite sequence of samples of vector *x*, denoted **x**, starting from *k* = 0 up to infinity, and with null

**x** = {. . . , 0, 0, *x*(0), *x*(1), *x*(2),..., *x*(*N*),...}.

S**x** = S{. . . , 0, 0, *x*(0), *x*(1), *x*(2) ,..., *x*(*k*), *x*(*k* + 1),...}

= {. . . , 0, 0, 0, 0, *x*(0), *x*(1) , *x*(2),..., *x*(*k* − 1),...}.

**Figure 1.** Computed values of the output sequence

#### **4.3. Stability analysis**

In this section, we investigate the property of stability of linear discrete-time FOS. We know that the study of the stability of FOS is limited to the result of the "'argument principle"', applicable to the sole category of commensurate-order systems. As far as non-commensurate orders are considered in the new models introduced, other adequate tools are required. In this context, new results concerning asymptotic stability are developed below. Besides, we introduce the concept of practical stability and we establish some mathematical conditions to check this property.

**Asymptotic stability** Let us consider the state-space model given by (37) and (38)

$$\mathbf{x}(k+1) = \sum\_{j=0}^{k} A\_j \mathbf{x}(k-j) + B\boldsymbol{u}(k), \quad \mathbf{x}(0) = \mathbf{x}\_{0\prime}$$

$$y(k) = \mathsf{Cx}(k)\_{\prime}$$

Its solution, given by (40) is

$$\varkappa(k) = G\_k \varkappa\_0 + \sum\_{j=0}^{k-1} G\_{k-1-j} B \nu(j).$$

In order to study the asymptotic stability property of this system, we consider its unforced version, i.e.,

$$\mathbf{x}(k+1) = \sum\_{j=0}^{k} A\_j \mathbf{x}(k-j), \quad \mathbf{x}(0) = \mathbf{x}\_0. \tag{54}$$

**Definition 6.** *System* (54) *is asymptotically stable if for each k* ≥ 1 *and any initial condition x*0*, the following equality is verified*

$$\lim\_{k \to \infty} \parallel \mathfrak{x}(k) \parallel = 0. \tag{55}$$

We use the 2-norm of the state vector *x*(*k*), i.e.,

14 Advances in Discrete Time Systems

*Step k* = *n is true only in the case of commensurate FOS.*

amplitude

*x*(*k* + 1) =

*k* ∑ *j*=0

*x*(*k*) = *Gkx*<sup>0</sup> +

**Figure 1.** Computed values of the output sequence

**4.3. Stability analysis**

to check this property.

Its solution, given by (40) is

**Remark 6.** *We therefore observe that the controllability and observability criteria for the commensurate fractional-order case are similar to those of the integer-order case, in the sense that if a state cannot be reached in n steps, then it is not reachable at all and that if an initial state cannot be deduced from n steps of input-output data, then it is not observable at all. The result put forward in [53] which states that a necessary and sufficient condition for the discrete-time fractional-order system as modeled in* (37) *and* (38) *to be observable is that the rank of* O*<sup>k</sup> should be equal to n at most in*

time (s)

In this section, we investigate the property of stability of linear discrete-time FOS. We know that the study of the stability of FOS is limited to the result of the "'argument principle"', applicable to the sole category of commensurate-order systems. As far as non-commensurate orders are considered in the new models introduced, other adequate tools are required. In this context, new results concerning asymptotic stability are developed below. Besides, we introduce the concept of practical stability and we establish some mathematical conditions

**Asymptotic stability** Let us consider the state-space model given by (37) and (38)

*y*(*k*) = *Cx*(*k*),

*k*−1 ∑ *j*=0

*Ajx*(*k* − *j*) + *Bu*(*k*), *x*(0) = *x*0,

*Gk*<sup>−</sup>1<sup>−</sup>*jBu*(*j*).

$$\|\|\mathbf{x}(k)\|\| = \sqrt{\sum\_{i=1}^{n} \mathbf{x}\_i^2(k)}.$$

where *xi*(*k*) are the components of *x*(*k*). The solution to (54) for given initial conditions *x*<sup>0</sup> is

$$\mathbf{x}(k) = \mathbf{G}\_k \mathbf{x}\_0, \quad k \ge 1, \ \mathbf{G}\_0 = \mathbf{I}\_{\mathbb{N}}.\tag{56}$$

It follows that System (54) is asymptotically stable if and only if

$$\|\|\: G\_k \mid\| \le 1, \quad k \ge 1. \tag{57}$$

The 2-norm of the transition matrix is

$$\parallel G\_k \parallel = \lambda\_{\max}^{\frac{1}{2}}(G\_k G\_k^T) = \sigma\_{\max}(G\_k)\_{\prime}$$

where *λmax* and *σmax* refer to the maximal eigenvalue and the maximal singular value of *Gk* respectively.

Let us define the backward shift operator S as follows ([55], [56]). We consider an infinite sequence of samples of vector *x*, denoted **x**, starting from *k* = 0 up to infinity, and with null values for *k* < 0, assuming the system to be causal. Thus, we have

$$\mathbf{x} = \{\ldots, 0, 0, \mathbf{x}(0), \mathbf{x}(1), \mathbf{x}(2), \ldots, \mathbf{x}(N), \ldots\}.$$

S acts on **x** as follows

$$\begin{split} \mathcal{S}\mathbf{x} &= \mathcal{S}\{\ldots, 0, 0, \mathbf{x}(0), \mathbf{x}(1), \boxed{\mathbf{x}(2)}, \ldots, \mathbf{x}(k), \mathbf{x}(k+1), \ldots\} \\ &= \{\ldots, 0, 0, 0, 0, \mathbf{0}, \mathbf{x}(0), \boxed{\mathbf{x}(1)}, \mathbf{x}(2), \ldots, \mathbf{x}(k-1), \ldots\}. \end{split}$$

We then define the column sequence

$$
\widetilde{\mathbf{x}} = \begin{bmatrix} \vdots \\ 0 \\ \mathbf{x}(0) \\ \mathbf{x}(1) \\ \vdots \\ \mathbf{x}(k) \\ \vdots \end{bmatrix}
$$

Similarly, we can use this representation to rewrite (37) and its output equation

$$y(k) = \mathsf{Cx}(k)$$

in the equivalent form

$$
\widetilde{\mathbf{x}} = \widetilde{\mathbf{S}} \widetilde{\mathbf{A}} \widetilde{\mathbf{x}} + \widetilde{\mathbf{S}} \widetilde{\mathbf{B}} \widetilde{\mathbf{u}}\_{\prime} \tag{58}
$$

$$
\widetilde{\mathbf{y}} = \widetilde{\mathbf{C}} \widetilde{\mathbf{x}}.\tag{59}
$$

**<sup>C</sup>**� <sup>=</sup>

**Theorem 5.** *Putting* <sup>A</sup>*<sup>s</sup>* <sup>=</sup> **<sup>S</sup>**�**A**�*, the system*

*in which the norm definition is*

*<sup>s</sup>*[*I*,*J*] *denoting the* [*I*, *J*]

*as*

*with* A*<sup>i</sup>*

as follows

Besides, we can write

 

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

*is asymptotically stable if and only if ρ*(A*s*) ≤ 1, *where ρ is the spectral radius of operator* A*s, defined*

*i*→∞ �A*<sup>i</sup> <sup>s</sup>*�∗ 1

*s*.

The proof of this theorem is in [36]. In practice, a finite-time observation of the system is desirable. For this purpose, we consider the concept of practical stability ([54], [57]) defined

**Definition 7.** *System* (54) *is practically stable in a finite-time horizon Lh* > 0 *if for each* 1 ≤ *k* ≤ *Lh*

From solution (56), System (54) is practically stable if and only if � *Gk* �≤ *M* for 1 ≤ *k* ≤ *L*.

*G*<sup>1</sup> 0 0 ... ... 0 *G*<sup>2</sup> 0 0 ... ... 0 *G*<sup>3</sup> 0 0 ... ... 0

*GLh* 0 0 ... ... 0

× × × ×××

<sup>0</sup> 0 ... ... 0

<sup>0</sup> 0 ... 0

�A*<sup>i</sup> s*[*I*,*J*] �,

*ρ*(A*s*) = *lim*

*<sup>s</sup>*�∗ = *sup* [*I*,*J*]

�A*<sup>i</sup>*

*th block matrix of* A*<sup>i</sup>*

*and any initial condition x*<sup>0</sup> *the following inequality is verified*

A*<sup>L</sup> <sup>s</sup>* =  

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

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

× *ALh*+<sup>1</sup>

× × *ALh*+<sup>2</sup>

*where M is a strictly positive finite given number.*

*C* 0 0 0 ... 0 *C* 0 0 ... 0 0 *C* 0 ... 000 *C* ...  

.

�**<sup>x</sup>** <sup>=</sup> **<sup>S</sup>**�**A**��**<sup>x</sup>** <sup>=</sup> <sup>A</sup>*s*�**<sup>x</sup>** (60)

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� *x*(*k*) �≤ *M* � *x*<sup>0</sup> �, (62)

 

*<sup>i</sup>* , (61)

In this representation, the expressions of the different components are

$$
\widetilde{\mathbf{u}} = \begin{bmatrix} \vdots \\ 0 \\ u(0) \\ u(1) \\ \vdots \\ u(k) \\ \vdots \end{bmatrix}, \quad \widetilde{\mathbf{y}} = \begin{bmatrix} \vdots \\ 0 \\ y(0) \\ y(1) \\ \vdots \\ y(k) \\ y(k) \\ \vdots \end{bmatrix}, \quad \widetilde{\mathbf{S}} = \begin{bmatrix} 0 & 0 & 0 & \dots & \dots \\ \mathbb{I}\_n & 0 & \dots & \dots & \dots \\ 0 & \mathbb{I}\_n & \dots & \dots & \dots \\ 0 & 0 & \mathbb{I}\_n & \dots & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{bmatrix}.
$$

and

$$
\widetilde{\mathbf{A}} = \begin{bmatrix} A\_0 & 0 & 0 & 0 & \dots \\ A\_1 \ A\_0 & 0 & 0 & \dots \\ A\_2 \ A\_1 \ A\_0 & 0 & \dots \\ A\_3 \ A\_2 \ A\_1 & A\_0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{bmatrix}, \quad \widetilde{\mathbf{B}} = \begin{bmatrix} B & 0 & 0 & 0 & \dots \\ 0 & B & 0 & 0 & \dots \\ 0 & 0 & B & 0 & \dots \\ 0 & 0 & 0 & B & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{bmatrix}
$$

and

$$
\tilde{\mathbf{C}} = \begin{bmatrix}
\mathbf{C} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \dots \\
\mathbf{0} & \mathbf{C} & \mathbf{0} & \mathbf{0} & \dots \\
\mathbf{0} & \mathbf{0} & \mathbf{C} & \mathbf{0} & \dots \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C} & \dots \\
\vdots & \vdots & \vdots & \vdots & \vdots
\end{bmatrix}.
$$

**Theorem 5.** *Putting* <sup>A</sup>*<sup>s</sup>* <sup>=</sup> **<sup>S</sup>**�**A**�*, the system*

$$
\widetilde{\mathbf{x}} = \overline{\mathbf{S}} \mathbf{A} \widetilde{\mathbf{x}} = \mathcal{A}\_{\sf s} \widetilde{\mathbf{x}} \tag{60}
$$

*is asymptotically stable if and only if ρ*(A*s*) ≤ 1, *where ρ is the spectral radius of operator* A*s, defined as*

$$\rho(\mathcal{A}\_{\mathbf{s}}) = \lim\_{\bar{i} \to \infty} \|\mathcal{A}\_{\mathbf{s}}^{\bar{i}}\|\_{\ast} \,^{\frac{1}{\bar{\mathbf{1}}}} \,. \tag{61}$$

*in which the norm definition is*

16 Advances in Discrete Time Systems

in the equivalent form

and

and

**u**� =

 

**<sup>A</sup>**� <sup>=</sup>

 

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

. . . 0 *u*(0) *u*(1) . . . *u*(*k*) . . .

 

We then define the column sequence

�**x** =

Similarly, we can use this representation to rewrite (37) and its output equation

In this representation, the expressions of the different components are

 

. . . 0 *y*(0) *y*(1) . . . *y*(*k*) . . .

 

> 

, **<sup>B</sup>**� <sup>=</sup>

, **<sup>S</sup>**� <sup>=</sup>

 

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

 

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

, �**<sup>y</sup>** <sup>=</sup>

*A*<sup>0</sup> 0 0 0... *A*<sup>1</sup> *A*<sup>0</sup> 0 0... *A*<sup>2</sup> *A*<sup>1</sup> *A*<sup>0</sup> 0... *A*<sup>3</sup> *A*<sup>2</sup> *A*<sup>1</sup> *A*<sup>0</sup> ...

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

�**<sup>x</sup>** <sup>=</sup> **<sup>S</sup>**�**A**��**<sup>x</sup>** <sup>+</sup> **<sup>S</sup>**�**B**�**u**�, (58)

0 0 0 ...... **I***<sup>n</sup>* 0 ... ...... 0 **I***<sup>n</sup>* ... ...... 0 0 **I***<sup>n</sup>* ......

*B* 0 0 0 ... 0 *B* 0 0 ... 0 0 *B* 0 ... 000 *B* ...    

�**<sup>y</sup>** <sup>=</sup> **<sup>C</sup>**��**x**. (59)

 

. . . 0 *x*(0) *x*(1) . . . *x*(*k*) . . .

 

$$||\mathcal{A}\_{\mathbf{s}}^{i}||\_{\*} = \sup\_{\left[I,J\right]} ||\mathcal{A}\_{\mathbf{s}\left[I,J\right]}^{i}||\_{J}$$

*with* A*<sup>i</sup> <sup>s</sup>*[*I*,*J*] *denoting the* [*I*, *J*] *th block matrix of* A*<sup>i</sup> s*.

The proof of this theorem is in [36]. In practice, a finite-time observation of the system is desirable. For this purpose, we consider the concept of practical stability ([54], [57]) defined as follows

**Definition 7.** *System* (54) *is practically stable in a finite-time horizon Lh* > 0 *if for each* 1 ≤ *k* ≤ *Lh and any initial condition x*<sup>0</sup> *the following inequality is verified*

$$\|\|\ x(k)\|\| \le M \parallel \ x\_0 \parallel \tag{62}$$

*where M is a strictly positive finite given number.*

From solution (56), System (54) is practically stable if and only if � *Gk* �≤ *M* for 1 ≤ *k* ≤ *L*. Besides, we can write

$$\mathcal{A}\_{\mathbf{s}}^{L} = \begin{bmatrix} G\_1 & 0 & 0 & \dots & \dots & 0 \\ G\_2 & 0 & 0 & \dots & \dots & 0 \\ G\_3 & 0 & 0 & \dots & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ G\_{L\_h} & 0 & 0 & \dots & \dots & 0 \\ \times & A\_0^{L\_h+1} & 0 & \dots & \dots & 0 \\ \times & \times & A\_0^{L\_h+2} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ \times & \times & \times & \times & \times & \times \end{bmatrix}$$

Define the block matrix

$$\mathcal{A}\_{\mathop{\bf c}}^{L\_{\hbar}}{}\_{L\_{\hbar}} = \begin{bmatrix} G\_1 & 0 & 0 & \dots & \dots & 0 \\ G\_2 & 0 & 0 & \dots & \dots & 0 \\ G\_3 & 0 & 0 & \dots & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ G\_{L\_{\hbar}} & 0 & 0 & \dots & \dots & 0 \\ \end{bmatrix}.$$

The Dirichlet boundary condition is stated as follows [59]: we consider the control action that maintains a temperature *u*(*t*) fixed at the end point *χ* = *L* and that a temperature measurement can be taken inside the rod, at a fixed abscissa *χ*<sup>0</sup> (see Figure 2). Therefore, the

As for the temperature measured at a fixed point *χ*0, this represents the observation, that is

*y*(*t*) = *T*(*χ*0, *t*).

Let us denote the Laplace transforms of *T*(*χ*, *t*) and *u*(*t*) respectively by *T*(*χ*,*s*) and *u*(*s*). The assumption of null initial conditions is made here, i.e., the rod is left to cool down before experiment. This conducts us to an easier computing process. Indeed non null initial conditions in the fractional case, as exposed for example in [31], conduct to computations

*ξ*

*<sup>u</sup>*(*s*) <sup>=</sup> *<sup>T</sup>*(*χ*0,*s*)

exp(*z*) <sup>≈</sup> <sup>∑</sup>*<sup>P</sup>*

*<sup>T</sup>*(*L*,*s*) <sup>=</sup>

This expression establishes that the described device exhibits the feature of fractional-order model. It is the case for various other types of diffusion, say chemical diffusion processes, neutron flux, and also for dielectric relaxation, electrode polarization, transmission lines, etc.. The four exp(.) terms composing the last part of (67) are expressed by using a Padé approximation of order 2. Let us recall here that, for instance, for exp(.) with given argument

*k*=0

The determination of *H*(*χ*0,*s*) is made for an aluminium rod, with a set of numerical values

∑*P k*=0

(2*P*−*k*)! *k*!(*P*−*k*)!

(2*P*−*k*)! *k*!(*P*−*k*)! (*z*)*<sup>k</sup>*

(−*z*)*<sup>k</sup>* .

**5.2. Fractional-order state-space model of the aluminium rod**

that are more complicated to handle. Equation (63) and (64) become

*d*2*T*(*χ*,*s*) *<sup>d</sup>χ*<sup>2</sup> <sup>=</sup> *<sup>s</sup>*

The corresponding solution leads to the transfer function

*z*, a Padé approximation of order *P* yields

most of them used in [30]. These are

• rod length *L* = 0.4 m.

*<sup>H</sup>*(*χ*0,*s*) = *<sup>y</sup>*(*s*)

*T*(0, *t*) = 0. *T*(*L*, *t*) = *u*(*t*). (64)

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*T*(*χ*,*s*) = 0, with (65)

. (67)

*T*(0,*s*)) = 0, *T*(*L*,*s*) = *u*(*s*). (66)

*sinh χ*0 *s ξ* 

*sinh L s ξ*

boundary conditions are

From the above, we can state that System (54) is practically stable if and only if � A*Lh s Lh* �∗≤ *M*. These results are illustrated by a numerical example in [36].

**Remark 7.** *The state-space representations elaborated in the previous sections for the continuous time and the discrete time were considered for a SISO process. They are easily extended to Multiple Input, Multiple Output (MIMO) systems, with the use of a vector of inputs u instead of a scalar input, and a vector of outputs y instead of a scalar output. MIMO systems are treated with a same procedure. This enables the study of their structural properties and permits to deal with control issues of such systems either in time domain or frequency domain, such as MIMO* H∞ *problem for example in[58].*

### **5. Practical example**

#### **5.1. Fractional-order modeling of a thermal system**

We consider a thermal system consisting in the regulation of the temperature profile of an aluminium rod with constant thermal conductivity *κ* and diffusivity *ξ*, as described in [30, 59]. A possible version is shown in Figure 2. Temperature *T*(*χ*, *t*) is evaluated at abscissa *χ*

**Figure 2.** The metallic rod thermal system.

along the rod axis. The rod length is *L*, and the origin is set at the non heated end. Measuring points of *T*(*χ*, *t*), by means of probes, are provided at different distances. To build a model of this system, we use an approximation by considering that the temperature diffusion for this type of geometry is governed by the following partial differential equation (PDE)

$$\frac{\partial T(\chi, t)}{\partial t} = \xi \frac{\partial^2 T(\chi, t)}{\partial \chi^2}, \quad \chi \in (0, L), \quad t \ge 0 \tag{63}$$

The Dirichlet boundary condition is stated as follows [59]: we consider the control action that maintains a temperature *u*(*t*) fixed at the end point *χ* = *L* and that a temperature measurement can be taken inside the rod, at a fixed abscissa *χ*<sup>0</sup> (see Figure 2). Therefore, the boundary conditions are

$$T(0,t) = 0.\qquad T(L,t) = \mu(t).\tag{64}$$

As for the temperature measured at a fixed point *χ*0, this represents the observation, that is

$$y(t) = T(\chi\_{0\prime}t).$$

#### **5.2. Fractional-order state-space model of the aluminium rod**

Let us denote the Laplace transforms of *T*(*χ*, *t*) and *u*(*t*) respectively by *T*(*χ*,*s*) and *u*(*s*). The assumption of null initial conditions is made here, i.e., the rod is left to cool down before experiment. This conducts us to an easier computing process. Indeed non null initial conditions in the fractional case, as exposed for example in [31], conduct to computations that are more complicated to handle. Equation (63) and (64) become

$$\frac{d^2\overline{T}(\chi\_\prime s)}{d\chi^2} = \frac{s}{\overline{\xi}}\overline{T}(\chi\_\prime s) = 0,\quad\text{with}\tag{65}$$

$$
\overline{T}(0, \mathbf{s})\rangle = 0, \qquad \overline{T}(L, \mathbf{s}) = \overline{\mathfrak{u}}(\mathbf{s}).\tag{66}
$$

The corresponding solution leads to the transfer function

$$H(\chi\_{0\prime}s) = \frac{\overline{y}(s)}{\overline{u}(s)} = \frac{\overline{T}(\chi\_{0\prime}s)}{\overline{T}(L\_{\prime}s)} = \frac{\sinh\left(\chi\_{0}\sqrt{\frac{s}{\xi}}\right)}{\sinh\left(L\sqrt{\frac{s}{\xi}}\right)}.\tag{67}$$

This expression establishes that the described device exhibits the feature of fractional-order model. It is the case for various other types of diffusion, say chemical diffusion processes, neutron flux, and also for dielectric relaxation, electrode polarization, transmission lines, etc.. The four exp(.) terms composing the last part of (67) are expressed by using a Padé approximation of order 2. Let us recall here that, for instance, for exp(.) with given argument *z*, a Padé approximation of order *P* yields

$$\exp(z) \approx \frac{\sum\_{k=0}^{P} \frac{(2P-k)!}{k!(P-k)!} (z)^k}{\sum\_{k=0}^{P} \frac{(2P-k)!}{k!(P-k)!} (-z)^k}.$$

The determination of *H*(*χ*0,*s*) is made for an aluminium rod, with a set of numerical values most of them used in [30]. These are

• rod length *L* = 0.4 m.

18 Advances in Discrete Time Systems

Define the block matrix

**5. Practical example**

**Figure 2.** The metallic rod thermal system.

A*Lh s Lh* =

*M*. These results are illustrated by a numerical example in [36].

**5.1. Fractional-order modeling of a thermal system**

*∂T*(*χ*, *t*) *<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>ξ</sup>*  

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

From the above, we can state that System (54) is practically stable if and only if � A*Lh*

**Remark 7.** *The state-space representations elaborated in the previous sections for the continuous time and the discrete time were considered for a SISO process. They are easily extended to Multiple Input, Multiple Output (MIMO) systems, with the use of a vector of inputs u instead of a scalar input, and a vector of outputs y instead of a scalar output. MIMO systems are treated with a same procedure. This enables the study of their structural properties and permits to deal with control issues of such systems either in time domain or frequency domain, such as MIMO* H∞ *problem for example in[58].*

We consider a thermal system consisting in the regulation of the temperature profile of an aluminium rod with constant thermal conductivity *κ* and diffusivity *ξ*, as described in [30, 59]. A possible version is shown in Figure 2. Temperature *T*(*χ*, *t*) is evaluated at abscissa *χ*

along the rod axis. The rod length is *L*, and the origin is set at the non heated end. Measuring points of *T*(*χ*, *t*), by means of probes, are provided at different distances. To build a model of this system, we use an approximation by considering that the temperature diffusion for this type of geometry is governed by the following partial differential equation (PDE)

*∂χ*<sup>2</sup> , *<sup>χ</sup>* <sup>∈</sup> (0, *<sup>L</sup>*), *<sup>t</sup>* <sup>≥</sup> <sup>0</sup> (63)

*∂*2*T*(*χ*, *t*)

*G*<sup>1</sup> 0 0 ... ... 0 *G*<sup>2</sup> 0 0 ... ... 0 *G*<sup>3</sup> 0 0 ... ... 0  

.

*s Lh* �∗≤

*GLh* 0 0 ... ... 0

• abscissa of the measuring point, set here at *χ*<sup>0</sup> = <sup>2</sup>∗*<sup>L</sup>* <sup>3</sup> m.

• rod thermal conductivity *κ* = 237 W/m.K, and diffusivity *ξ* = 9975.10−<sup>8</sup> m2/s. With this set of practical values, we obtain the following transfer function

$$H(\chi\_0, s) = \frac{0.6667 + 22.2501s^{\frac{1}{2}} + 188.1246s - 1982.7s^{\frac{3}{2}} + 5293.9s^2}{1 + 33.3751s^{\frac{1}{2}} + 430.7064s + 1982.7s^{\frac{3}{2}} + 3529.3s^2}. \tag{68}$$

physical coherence, the result is not satisfactory, since the necessary input sequence yielded is *u*(*k*)=[−278.2396 − 99.607 − 36.5687 − 18.3551], that is, with (all) negative elements. Related to this feature, the values of the model output matrix *C* is found to be made of all negative elements. A further insight should be therefore directed towards the modeling process, initiated with the representation of the diffusive phenomena by PDE equations. Next, a more detailed evaluation of the precision when achieving the Padé approximation would be expectable. It remains that one major improvement to expect is the effective introduction in the modeling of fractional-order systems of non null initial conditions [31]. Taking them into account in the computations is a way to make the representations closer to the reality of the phenomena. The case treated here, which aims to describe the diffusion of a given fixed temperature at one end of the rod, implies to deal with realistic operation conditions. For example, an interesting case to investigate would be to consider the rod set initially at ambient temperature, say currently around 300 *oK*, as a non null initial condition, and to track the diffusion of a given input temperature with a more elaborated model.

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**Figure 3.** Output temperature evolution over 3000 points (Sampling period: 1s).

**Figure 4.** Output temperature evolution, starting from zero initial condition (Detail).

Next we apply to (68) the procedure exposed above from (8) to (17), so as to obtain the corresponding fractional-order state-space model. We find a state-space model with commensurate fractional orders, of the form (17). In our case, it can be reduced to four states, by eliminating redundant states through a variable change. In the resulting model, there exists a direct transmission matrix *D*, what can be foreseen when considering the monomials of highest degree in *s* of numerator and denominator of (68).

The computed set of matrices *A*, *B*, *C*, *D* of the resulting fractional-order state-space model for the rod, in continuous time, is therefore

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -0.0003 & -0.0095 & -0.1220 & -0.5618 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ 0.0002833 \end{bmatrix},$$

$$C = \begin{bmatrix} -0.8333 \ -27.8126 \ -457.9350 \ -4956.8 \end{bmatrix}, \quad D = [1.5].$$

#### **5.3. Analysis of the discrete-time state-space model of the rod**

We convert the continuous-time state space model described by the set of matrices (69) into a discrete-time version. This is done by choosing the sampling period *h*, making *h* = 1s. The sampling period value taken in [30] for the same aluminium rod is somewhat smaller, that is 0.5s. We set the input variable *u*(*t*) = *T*(*L*, *t*) = 320 *oK*. In other words, the temperature at abscissa *χ* = *L* is maintained constant. The observed output *y*(*t*) = *T*(*χ*0, *t*) is the temperature diffusing at a measuring point at *χ*0. The response obtained is shown in Figure 3. It indicates that the output temperature starts increasing from null initial condition set at *T*(0, *t*) = 0 *oK*, which is an assumption for deriving the model used. In the interval of the experiment over 3000 steps, it diffuses and tends to a value approaching the input fixed value. A detailed plot represented in Figure 4 shows at the beginning an increase of *y*(*t*), followed by a brief fall under zero. This type of behaviour at first sight is similar to non-minimum phase system behaviour. However, its interpretation as a physical phenomenon appears to be not realistic, considering the underlying process in study.

Besides, as an example of study of the structural properties of the model proposed, we have led a reachability test, following the theoretical developments of previous Section 4. Starting from a zero initial state *x*(0)=[0000] ′ , we set a final state to be reached as *xf* = [−0.13 − 0.13 − 0.13 − 0.13] ′ . The computation achieved yields the result that *xf* is exactly reached in four steps, i.e. in a number of steps equal to the rank of the controllability matrix, as depicted in Figure 5. Nevertheless, from the point of view of physical coherence, the result is not satisfactory, since the necessary input sequence yielded is *u*(*k*)=[−278.2396 − 99.607 − 36.5687 − 18.3551], that is, with (all) negative elements. Related to this feature, the values of the model output matrix *C* is found to be made of all negative elements. A further insight should be therefore directed towards the modeling process, initiated with the representation of the diffusive phenomena by PDE equations. Next, a more detailed evaluation of the precision when achieving the Padé approximation would be expectable. It remains that one major improvement to expect is the effective introduction in the modeling of fractional-order systems of non null initial conditions [31]. Taking them into account in the computations is a way to make the representations closer to the reality of the phenomena. The case treated here, which aims to describe the diffusion of a given fixed temperature at one end of the rod, implies to deal with realistic operation conditions. For example, an interesting case to investigate would be to consider the rod set initially at ambient temperature, say currently around 300 *oK*, as a non null initial condition, and to track the diffusion of a given input temperature with a more elaborated model.

**Figure 3.** Output temperature evolution over 3000 points (Sampling period: 1s).

20 Advances in Discrete Time Systems

• abscissa of the measuring point, set here at *χ*<sup>0</sup> = <sup>2</sup>∗*<sup>L</sup>*

*<sup>H</sup>*(*χ*0,*s*) = 0.6667 <sup>+</sup> 22.2501*<sup>s</sup>*

• rod thermal conductivity *κ* = 237 W/m.K, and diffusivity *ξ* = 9975.10−<sup>8</sup> m2/s.

1

Next we apply to (68) the procedure exposed above from (8) to (17), so as to obtain the corresponding fractional-order state-space model. We find a state-space model with commensurate fractional orders, of the form (17). In our case, it can be reduced to four states, by eliminating redundant states through a variable change. In the resulting model, there exists a direct transmission matrix *D*, what can be foreseen when considering the monomials

The computed set of matrices *A*, *B*, *C*, *D* of the resulting fractional-order state-space model

1

With this set of practical values, we obtain the following transfer function

1 + 33.3751*s*

of highest degree in *s* of numerator and denominator of (68).

0100 0010 0001 −0.0003 −0.0095 −0.1220 −0.5618

**5.3. Analysis of the discrete-time state-space model of the rod**

be not realistic, considering the underlying process in study.

Starting from a zero initial state *x*(0)=[0000]

as *xf* = [−0.13 − 0.13 − 0.13 − 0.13]

−0.8333 −27.8126 −457.9350 −4956.8 �

We convert the continuous-time state space model described by the set of matrices (69) into a discrete-time version. This is done by choosing the sampling period *h*, making *h* = 1s. The sampling period value taken in [30] for the same aluminium rod is somewhat smaller, that is 0.5s. We set the input variable *u*(*t*) = *T*(*L*, *t*) = 320 *oK*. In other words, the temperature at abscissa *χ* = *L* is maintained constant. The observed output *y*(*t*) = *T*(*χ*0, *t*) is the temperature diffusing at a measuring point at *χ*0. The response obtained is shown in Figure 3. It indicates that the output temperature starts increasing from null initial condition set at *T*(0, *t*) = 0 *oK*, which is an assumption for deriving the model used. In the interval of the experiment over 3000 steps, it diffuses and tends to a value approaching the input fixed value. A detailed plot represented in Figure 4 shows at the beginning an increase of *y*(*t*), followed by a brief fall under zero. This type of behaviour at first sight is similar to non-minimum phase system behaviour. However, its interpretation as a physical phenomenon appears to

Besides, as an example of study of the structural properties of the model proposed, we have led a reachability test, following the theoretical developments of previous Section 4.

that *xf* is exactly reached in four steps, i.e. in a number of steps equal to the rank of the controllability matrix, as depicted in Figure 5. Nevertheless, from the point of view of

′

′

for the rod, in continuous time, is therefore

 

*C* = �

*A* =

<sup>3</sup> m.

3

 

0 0 0.0002833

, *D* = [1.5].

, we set a final state to be reached

. The computation achieved yields the result

 ,

(69)

3

<sup>2</sup> + 5293.9*s*<sup>2</sup>

<sup>2</sup> <sup>+</sup> 3529.3*s*<sup>2</sup> . (68)

<sup>2</sup> + 188.1246*s* − 1982.7*s*

 , *<sup>B</sup>* <sup>=</sup>

<sup>2</sup> + 430.7064*s* + 1982.7*s*

**Figure 4.** Output temperature evolution, starting from zero initial condition (Detail).

In the closed loop, we have

 

in which

*x*(1) *x*(2) *x*(3) . . . *x*(*k* + 1) . . .

 

**A** �**<sup>F</sup>** <sup>=</sup>

Further, we decompose **A**

 

=

 

Then, putting *AFj* = *Aj* + *BFj*, we can write

*x*(*k* + 1) =

*x*(*k* + 1) =

This leads to the following relation between the two column sequences

. .

using the backward shift S, Equation (58) is rewritten as follows

. .

�**<sup>F</sup>** as follows

. .

. .

Thus, successively, we build the recurrence beginning with

*k* ∑ *j*=0

> *k* ∑ *j*=0

*x*(1) = *AF*0*x*(0)=(*A*<sup>0</sup> + *BF*0)*x*(0),

*x*(2) = *AF*0*x*(1) + *AF*1*x*(0)=(*A*<sup>0</sup> + *BF*0)*x*(1)+(*A*<sup>1</sup> + *BF*1)*x*(0),

*x*(3) = *AF*0*x*(2) + *AF*1*x*(1)*AF*2*x*(0)=(*A*<sup>0</sup> + *BF*0)*x*(2)+(*A*<sup>1</sup> + *BF*1)*x*(1)+(*A*<sup>2</sup> + *BF*2)*x*(0),...

(*A*<sup>0</sup> + *BF*0) 0 0 0... (*A*<sup>1</sup> + *BF*1) (*A*<sup>0</sup> + *BF*0) 0 0... (*A*<sup>2</sup> + *BF*2) (*A*<sup>1</sup> + *BF*1) (*A*<sup>0</sup> + *BF*0) 0...

. .

. . . . . . . .

.

Similarly to the notations used in (37), i.e., considering an infinite column sequence �**x**, and

�**<sup>x</sup>** <sup>=</sup> **<sup>S</sup>**�**<sup>A</sup>**

(*A*<sup>0</sup> + *BF*0) 0 0 0... (*A*<sup>1</sup> + *BF*1) (*A*<sup>0</sup> + *BF*0) 0 0... (*A*<sup>2</sup> + *BF*2) (*A*<sup>1</sup> + *BF*1) (*A*<sup>0</sup> + *BF*0) 0...

. .

. . . . . . . .

.

(*Aj* + *BFj*)*x*(*k* − *j*). (71)

http://dx.doi.org/10.5772/51983

205

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

*AFjx*(*k* − *j*). (72)

 

�**F**�**x**, (74)

    *x*(0) *x*(1) *x*(2) . . . *x*(*k*) . . .

 

(73)

(75)

**Figure 5.** Output temperature evolution during control sequence.

#### **6. Stabilization issue**

In this section, we deal with the problem of stabilization. For this purpose, we consider the system given by (37) and (38), which we aim to stabilize, by handling state equation (37)

$$\mathbf{x}(k+1) = \sum\_{j=0}^{k} A\_j \mathbf{x}(k-j) + B\boldsymbol{u}(k), \quad \mathbf{x}(0) = \mathbf{x}\_0.$$

Let us consider the control *u* as a state-feedback control, in a classical state-feedback control loop, with a null reference input (*r* = 0), as illustrated in Figure 6. We assume that

**Figure 6.** State-feedback control loop.

$$\mu(k) = \sum\_{j=0}^{k} F\_{\hat{\jmath}} \mathbf{x}(k-j),\tag{70}$$

That is

$$\mu(k) = F\_0 \mathbf{x}(k) + F\_1 \mathbf{x}(k-1) + F\_2 \mathbf{x}(k-2) + \dots + F\_k \mathbf{x}(0).$$

In the closed loop, we have

22 Advances in Discrete Time Systems

**6. Stabilization issue**

**Figure 5.** Output temperature evolution during control sequence.

*x*(*k* + 1) =

**Figure 6.** State-feedback control loop.

That is

*k* ∑ *j*=0

In this section, we deal with the problem of stabilization. For this purpose, we consider the system given by (37) and (38), which we aim to stabilize, by handling state equation (37)

Let us consider the control *u* as a state-feedback control, in a classical state-feedback control

loop, with a null reference input (*r* = 0), as illustrated in Figure 6. We assume that

*u*(*k*) =

*k* ∑ *j*=0

*u*(*k*) = *F*0*x*(*k*) + *F*1*x*(*k* − 1) + *F*2*x*(*k* − 2) + ... + *Fkx*(0).

*Ajx*(*k* − *j*) + *Bu*(*k*), *x*(0) = *x*0.

*Fjx*(*k* − *j*), (70)

$$\mathbf{x}(k+1) = \sum\_{j=0}^{k} (A\_j + BF\_j)\mathbf{x}(k-j). \tag{71}$$

Then, putting *AFj* = *Aj* + *BFj*, we can write

$$\mathbf{x}(k+1) = \sum\_{j=0}^{k} A\_{Fj} \mathbf{x}(k-j). \tag{72}$$

Thus, successively, we build the recurrence beginning with

$$\mathbf{x}(1) = A\_{F0}\mathbf{x}(0) = (A\_0 + BF\_0)\mathbf{x}(0),$$

$$\mathbf{x}(2) = A\_{F0}\mathbf{x}(1) + A\_{F1}\mathbf{x}(0) = (A\_0 + BF\_0)\mathbf{x}(1) + (A\_1 + BF\_1)\mathbf{x}(0),$$

$$\mathbf{x}(\mathbf{3}) = A\_{F0}\mathbf{x}(2) + A\_{F1}\mathbf{x}(1)\\A\_{F2}\mathbf{x}(0) = (A\_0 + BF\_0)\mathbf{x}(2) + (A\_1 + BF\_1)\mathbf{x}(1) + (A\_2 + BF\_2)\mathbf{x}(0), \dots$$

This leads to the following relation between the two column sequences

$$
\begin{bmatrix}
\mathbf{x}(1) \\
\mathbf{x}(2) \\
\mathbf{x}(3) \\
\vdots \\
\mathbf{x}(k+1) \\
\vdots
\end{bmatrix} = \begin{bmatrix}
(A\_0 + BF\_0) & 0 & 0 & 0 & \dots \\
(A\_1 + BF\_1) \ (A\_0 + BF\_0) & 0 & 0 & \dots \\
(A\_2 + BF\_2) \ (A\_1 + BF\_1) \ (A\_0 + BF\_0) & 0 \dots \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\end{bmatrix} \begin{bmatrix}
\mathbf{x}(0) \\
\mathbf{x}(1) \\
\mathbf{x}(2) \\
\vdots \\
\mathbf{x}(k) \\
\vdots \\
\vdots \\
\end{bmatrix} \tag{73}
$$

Similarly to the notations used in (37), i.e., considering an infinite column sequence �**x**, and using the backward shift S, Equation (58) is rewritten as follows

$$
\tilde{\mathbf{x}} = \overline{\mathbf{S}} \overleftarrow{\mathbf{A}\_F} \tilde{\mathbf{x}} \tag{74}
$$

in which

$$
\widetilde{\mathbf{A}\_{\mathbf{F}}} = \begin{bmatrix}
(A\_0 + BF\_0) & 0 & 0 & 0 & \dots \\
(A\_1 + BF\_1) \ (A\_0 + BF\_0) & 0 & 0 & \dots \\
(A\_2 + BF\_2) \ (A\_1 + BF\_1) \ (A\_0 + BF\_0) & 0 & \dots \\
\vdots & \vdots & \vdots & \vdots & \vdots
\end{bmatrix} \tag{75}
$$

Further, we decompose **A** �**<sup>F</sup>** as follows

$$
\widetilde{\mathbf{A\_F}} = \begin{bmatrix} A\_0 & 0 & 0 & 0 & \dots \\ A\_1 \ A\_0 & 0 & 0 & \dots \\ A\_2 \ A\_1 \ A\_0 & 0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{bmatrix} + \begin{bmatrix} B \ 0 \ 0 \ 0 \dots \\ 0 \ B \ 0 \ 0 \dots \\ 0 \ 0 \ B \ 0 \dots \\ \vdots \ \vdots \vdots & \vdots & \vdots \end{bmatrix} + \begin{bmatrix} F\_0 \ 0 \ 0 \ 0 \dots \\ F\_1 \ F\_0 \ 0 \ 0 \dots \\ F\_2 \ F\_1 \ F\_0 \ 0 \dots \\ \vdots \vdots \vdots & \vdots & \vdots \end{bmatrix} \tag{76}
$$

Equation (58) becomes

$$
\widetilde{\mathbf{x}} = \overline{\mathbf{S}} \overline{\mathbf{A}}\_{\mathbf{F}} \widetilde{\mathbf{x}} = \overline{\mathbf{S}} \overline{\mathbf{A}} \widetilde{\mathbf{x}} + \overline{\mathbf{S}} \overline{\mathbf{B}} \overline{\mathbf{F}} \widetilde{\mathbf{x}},\tag{77}
$$

**Remark 8.** *The state feedback* (70) *uses the entire memory of the state variable. In practice, this could be undesirable and computationally unwieldy. It is preferable to design a controller with short-memory [45]. Let be Lh the restricted length of the memory (or horizon). The practical state feedback is*

*Fjx*(*k* − *j*), *for k* ≥ *Lh*,

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

http://dx.doi.org/10.5772/51983

207

*Fjx*(*k* − *j*), *for k* < *Lh*

*Stability of the short-memory state feedback controller could be investigated with the same development*

We have reviewed some tools for modeling and analysis of FOS in discrete time, introducing state-space representation for both commensurate and non commensurate fractional orders. These latter new approaches of modeling and analysis of such systems have revealed new

Our contribution concerns the analysis of the controllability and the observability of linear discrete-time FOS. We have introduced a new formalism and established testable sufficient conditions for guaranteeing the controllability and the observability. Some aspects of controllability and observability of such systems had not been treated before. Let us recall the remarkable point that, in the case of the linear discrete-time non-commensurate FOS, the rank of the controllability matrix can increase for values greater than the dimension of the system. In other words, it is possible to reach the final state in a number of steps greater than this dimension number. They are expected to give birth to further investigations and applications. With the use of a new formalism, an approach to analysis of asymptotic stability

The modeling of a practical system has been treated, which points out theoretical

Finally, the preliminary results presented in this chapter enabled us to make first steps into

1 Laboratoire de Conception et Conduite des Systèmes de Production (L2CSP), Faculty of Electrical Engineering and Computer Science, Mouloud Mammeri University of Tizi-Ouzou,

2 Department of Electrical/Electronics & Computer Engineering, University of Sharjah,

investigation on stabilization and practical stabilization of linear discrete-time FOS.

*u*(*k*) =

*u*(*k*) =

properties, not shown in continuous time representations.

and practical stability of discrete-time FOS has been proposed.

assumptions to match with real conditions.

Saïd Djennoune1 and Maâmar Bettayeb<sup>2</sup>

*Lh* ∑ *j*=0

*k* ∑ *j*=0

*formulated as:*

*given in this section.*

**7. Conclusion**

**Author details**

Tizi-Ouzou, Algeria

United Arab Emirates

Saïd Guermah1,

*and*

where

$$
\widetilde{\mathbf{F}} = \begin{bmatrix} F\_0 & 0 & 0 & 0 & \dots \\ F\_1 & F\_0 & 0 & 0 & \dots \\ F\_2 & F\_1 & F\_0 & 0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{bmatrix} \tag{78}
$$

The reachability is reformulated here under the operator-theoretic formulation (Equations (58) and (59)) of the discrete-time fractional-order system.

**Definition 8.** *The pair* (**A**�, **<sup>B</sup>**�) *is K-step reachable if the operator reachability Gramian defined by*

$$
\bar{\mathbf{W}}\_{\mathbf{r}}^{\overline{\mathbf{P}}} = \tilde{\mathbf{C}}\_{p}^{T} \tilde{\mathbf{C}}\_{p\prime} \tag{79}
$$

*where*

$$
\tilde{\mathbf{C}}\_p = [\tilde{\mathbf{B}} \quad \tilde{\mathbf{A}} \tilde{\mathbf{B}} \quad \dots \tilde{\mathbf{A}}^{p-1} \tilde{\mathbf{A}}]\_\prime
$$

*is invertible.*

Similarly to Equation (39), let us now define the transition matrix form *Gk* such that

$$G\_{Fk} = \begin{cases} \mathbb{I}\_{\mathbb{II}\_d} & \text{for } k = 0, \\ \sum\_{j=0}^{k-1} A\_{Fj} G\_{F(k-1-j)} & \text{for } k \ge 1. \end{cases} \tag{80}$$

We now consider the problem of choosing the state feedback operator gain � **F** that stabilizes System (77), i.e., which ensures that

$$\|\|G\_{Fk}\|\| \le 1 \quad \text{for } k \ge 1.$$

The following statement is derived from Theorem 5 established above, and Lemma 2 in [60]. **Corollary 3.** *Suppose that* (**A**�, **<sup>B</sup>**�) *is K-reachable and let*

$$\widetilde{\mathbf{F}} = -\widetilde{\mathbf{B}}^T \widetilde{\mathbf{A}}^T (\widetilde{\mathbf{W}}\_r^p)^{-1} \widetilde{\mathbf{A}}^{p+1},$$

*then the closed-loop system* (77) *is asymptotically stable, that is <sup>ρ</sup>*(**A**� *Fs*) <sup>&</sup>lt; 1, *where* **<sup>A</sup>**� *Fs* <sup>=</sup> **<sup>S</sup>**�**A**� *<sup>F</sup>*

**Remark 8.** *The state feedback* (70) *uses the entire memory of the state variable. In practice, this could be undesirable and computationally unwieldy. It is preferable to design a controller with short-memory [45]. Let be Lh the restricted length of the memory (or horizon). The practical state feedback is formulated as:*

$$u(k) = \sum\_{j=0}^{L\_h} F\_j \mathbf{x}(k - j), \qquad \text{for} \quad k \ge L\_{h\nu}$$

*and*

24 Advances in Discrete Time Systems

**A** �**<sup>F</sup>** <sup>=</sup>

Equation (58) becomes

where

*where*

*is invertible.*

 

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

*A*<sup>0</sup> 0 0 0... *A*<sup>1</sup> *A*<sup>0</sup> 0 0... *A*<sup>2</sup> *A*<sup>1</sup> *A*<sup>0</sup> 0...  +

�**<sup>x</sup>** <sup>=</sup> **<sup>S</sup>**�**<sup>A</sup>**

 

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

� **F** =

(58) and (59)) of the discrete-time fractional-order system.

*GFk* =

**Corollary 3.** *Suppose that* (**A**�, **<sup>B</sup>**�) *is K-reachable and let*

System (77), i.e., which ensures that

 

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

*B* 0 0 0... 0 *B* 0 0... 0 0 *B* 0...

�**F**�**<sup>x</sup>** <sup>=</sup> **<sup>S</sup>**�**A**��**<sup>x</sup>** <sup>+</sup> **<sup>S</sup>**�**B**��

*F*<sup>0</sup> 0 0 0... *F*<sup>1</sup> *F*<sup>0</sup> 0 0... *F*<sup>2</sup> *F*<sup>1</sup> *F*<sup>0</sup> 0...

The reachability is reformulated here under the operator-theoretic formulation (Equations

**Definition 8.** *The pair* (**A**�, **<sup>B</sup>**�) *is K-step reachable if the operator reachability Gramian defined by*

**<sup>C</sup>**� *<sup>p</sup>* = [**B**� **<sup>A</sup>**�**B**� ... **<sup>A</sup>**� *<sup>p</sup>*−1**A**�],

� **I***nd* for *k* = 0,

� *GFk* � ≤ 1 for *k* ≥ 1.

The following statement is derived from Theorem 5 established above, and Lemma 2 in [60].

*<sup>r</sup>* )−1**A**� *<sup>p</sup>*<sup>+</sup>1,

**<sup>F</sup>**� <sup>=</sup> <sup>−</sup>**B**�*T***A**� *<sup>T</sup>*(�**W***<sup>p</sup>*

*then the closed-loop system* (77) *is asymptotically stable, that is <sup>ρ</sup>*(**A**� *Fs*) <sup>&</sup>lt; 1, *where* **<sup>A</sup>**� *Fs* <sup>=</sup> **<sup>S</sup>**�**A**� *<sup>F</sup>*

**W** �**p <sup>r</sup>** <sup>=</sup> **<sup>C</sup>**�*<sup>T</sup>*

Similarly to Equation (39), let us now define the transition matrix form *Gk* such that

∑*k*−1

We now consider the problem of choosing the state feedback operator gain �

 +

   

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

*F*<sup>0</sup> 0 0 0... *F*<sup>1</sup> *F*<sup>0</sup> 0 0... *F*<sup>2</sup> *F*<sup>1</sup> *F*<sup>0</sup> 0...  

**<sup>F</sup>**�**x**, (77)

. (78)

*<sup>p</sup>* **<sup>C</sup>**� *<sup>p</sup>*, (79)

*<sup>j</sup>*=<sup>0</sup> *AFjGF*(*k*−1−*j*) for *<sup>k</sup>* <sup>≥</sup> 1. (80)

**F** that stabilizes

(76)

$$\mu(k) = \sum\_{j=0}^{k} F\_j \mathfrak{x}(k-j)\_{\prime} \qquad \text{for} \quad k < L\_h$$

*Stability of the short-memory state feedback controller could be investigated with the same development given in this section.*

#### **7. Conclusion**

We have reviewed some tools for modeling and analysis of FOS in discrete time, introducing state-space representation for both commensurate and non commensurate fractional orders. These latter new approaches of modeling and analysis of such systems have revealed new properties, not shown in continuous time representations.

Our contribution concerns the analysis of the controllability and the observability of linear discrete-time FOS. We have introduced a new formalism and established testable sufficient conditions for guaranteeing the controllability and the observability. Some aspects of controllability and observability of such systems had not been treated before. Let us recall the remarkable point that, in the case of the linear discrete-time non-commensurate FOS, the rank of the controllability matrix can increase for values greater than the dimension of the system. In other words, it is possible to reach the final state in a number of steps greater than this dimension number. They are expected to give birth to further investigations and applications. With the use of a new formalism, an approach to analysis of asymptotic stability and practical stability of discrete-time FOS has been proposed.

The modeling of a practical system has been treated, which points out theoretical assumptions to match with real conditions.

Finally, the preliminary results presented in this chapter enabled us to make first steps into investigation on stabilization and practical stabilization of linear discrete-time FOS.
