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IMA Journal of Mathematical Control and Information
, Volume Advance Article – Feb 27, 2017

16 pages

/lp/ou_press/influence-of-discretization-step-on-positivity-of-a-certain-class-of-vjRYy7R2Bn

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnx005
- Publisher site
- See Article on Publisher Site

Abstract A method is proposed in this article to study the influence of discretization step on positivity for a certain class of fractional continuous discrete-time linear system introduced in Kaczorek (2011a, B. Pol. Acad. Sci-Tech. Sci.,59, 575—579). The relation between the value of the sampling step and positivity of this class of system is proposed, i.e., under which conditions the two-dimensional discrete-time linear system obtained by discretization from the two-dimensional continuous discrete-time linear system will be also positive if the two-dimensional continuous discrete-time linear system is positive. Numerical examples are introduced to illustrate the proposed results. 1. Introduction In recent years there has been a growing interest in two-dimensional fractional systems that are subjected to positivity constraints on their dynamical variables. Such positive systems must have state variables never negative, when given a positive initial state. These systems have been studied by many authors (Farina & Rinaldi, 2000; Kaczorek, 2002) in different applications and arise naturally in industrial process involving chemical reactors, heat exchangers and distillation columns, circuits, storage systems, compartmental systems, etc. In the last three decades, many efforts have been made to develop fractional systems in different fields of research. Fractional calculus is a generalization of ordinary derivation and integration to non-integer orders. Actually, we can find many papers and books devoted to its theoretical and application aspects, see for example the works of Oustaloup (1999), Podlubny (1999), Sabatier et al. (2007), Monje et al. (2010), Kaczorek (2011b) and Baleanu et al. (2012). A new class of fractional positive systems is considered in this article. A variety of fractional models having positive fractional linear systems behaviour can be found in engineering, biology, management science, medicine, fluid dynamic, robotic, aeronautic and control. Some other applications of positive fractional order systems was proposed in Kaczorek (2011b) where an overview of the state of the art in fractional positive linear systems theory is given. The problem of positivity of the two-dimensional fractional linear systems was considered by Kaczorek (2011b) and the notion of two-dimensional positive fractional continuous-discrete time system and positivity constraints have been introduced in Kaczorek (2011a). The problem of discretization was investigated for the one-dimensional continuous-time linear system in Kaczorek (2013a) and later in Kaczorek (2013b) for the one-dimensional continuous-time fractional linear systems. In this article, we consider the class of two-dimensional fractional continuous-discrete-time linear systems which were considered in Kaczorek (2011a) and the aim is to extend results as shown in Bouagada (2004) to positive fractional continuous-discrete time linear systems. Necessary and sufficient conditions on the sampling step are established to conserve the positivity. The article is organized as follow. In Section 2, the discretization of the two-dimensional continuous-discrete fractional linear system is proposed. The corresponding two-dimensional discrete fractional linear system is derived and then solved. In Section 3, new necessary and sufficient positivity conditions are introduced. In Section 4, the main result is introduced and discussions about those results are proposed. Finally, in Section 5, some numerical examples are provided to illustrate the effectiveness of the approach by some simulation results. The following notations are used in this article. Let $$\mathbb{R}^{n\times m}_{+}$$ be the set of non-negative real $$n\times m$$ matrices and $$\mathbb{R}^{n\times 1}= \mathbb{R}^{n}$$. The nonnegative integers set will be denoted by $$\mathbb{Z}_{+}$$, the set of the strictly positive integers will be denoted $$\mathbb{Z}^{\ast}_{+}=\left\{1,2,3,...\right\}$$ and the $$n\times n$$ identity matrix will be denoted by $$I_{n}$$. The notation $$i=\overline{p,q}$$ with $$p,q\in \mathbb{Z}_{+}$$ and $$p\leq q$$ means all integer $$i \in \left\{p,p+1,p+2,\cdots,q-1,q\right\}$$. 2. Preliminaries We first recall here some definitions and results taken from Monje et al. (2010) and Kaczorek (2011a) which we shall need later on. Necessary and sufficient positivity conditions were derived in Kaczorek (2011a). The discretization is based on the Grunwald–Letnikov definition for the approximation of the fractional derivatives. After that we will derive the solution of the discretized system. Definition 2.1 The Caputo fractional derivative for $$n-1<\alpha<n, \ n\in \mathbb{Z}^{\ast}_{+}$$ is defined by the following formula Dαf(t)=dαf(t)dtα=1Γ(1−α)∫0tf˙(τ)(t−τ)αdτ, (2.1) where $$\dot{f}(\tau) = df(\tau)/d\tau$$ and $${\it{\Gamma}}(x)= \int^{\alpha}_{0} e^{-t} t^{x-1} dt$$, $$Re(x)>0$$ is the Euler gamma function. Let us consider the two-dimensional continuous-discrete time fractional linear system introduced in Kaczorek (2011a) and described for $$\ 0<\alpha<1 $$, $$t\in \mathbb{R}$$ and $$i\in \mathbb{Z_{+}}=\left\{0,1,2,...\right\}$$ by the equations dαx(t,i+1)dtα=A0x(t,i)+A1x(t,i+1)+Bu(t,i) (2.2) y(t,i)=Cx(t,i)+Du(t,i), (2.3) where, $$x(t,i) \in \mathbb{R}^{n}$$, $$y(t,i) \in \mathbb{R}^{p}$$ and $$u(t,i) \in \mathbb{R}^{m}$$ are respectively the state, input and output vectors, and the matrices $$A_{0}, A_{1}\in \mathbb{R}^{n \times n}$$, $$B \in \mathbb{R}^{n \times m}$$, $$C \in \mathbb{R}^{p \times n}$$ and $$D \in \mathbb{R}^{p \times m}$$. The states $$x(t,0) \in \mathbb{R}^{n}$$ for all $$t\in \mathbb{R}\ $$ and $$x(0,i) \in \mathbb{R}^{n}$$ for all $$i\in \mathbb{Z_{+}}$$ are the boundary conditions. We assume that the continuous variable $$t$$ and the discrete variable $$i$$ are independent. Note that we treat here a particular case of the general system given in Kaczorek (2011a), which depends only on one fractional derivative $$ d^{\alpha} x(t,i+1)/dt^{\alpha}$$. Definition 2.2 A matrix $$A=(a_{ij})_{i,j=\overline{1,n}} \in \mathbb{R}^{n\times n}$$ is called a Metzler matrix if all its off-diagonal entries are nonnegative; i.e., : $$a_{ij}\geq0$$ for $$i\neq j$$. Theorem 2.1 (see Kaczorek, 2011a) The systems (2.2) and (2.3) are positive if and only if: $$A_{0} \in \mathbb{R}^{n\times n}_{+}$$, $$ B\in \mathbb{R}^{n\times m}_{+}$$, $$C \in \mathbb{R}^{p\times n}_{+}$$, $$D \in \mathbb{R}^{p\times m}_{+}$$, $$A_{1}$$ is a Metzler matrix. In the hope to discretize the continuous variable $$t$$ of the systems (2.2) and (2.3), we introduce the Grunwald–Letnikov definition of fractional derivative approximation. In general, as shown in Monje et al. (2010), we define the generalized approximation of the fractional order differential operator. Definition 2.3 Let $$h>0$$ be the sampling step and $$ \alpha$$ be the fractional order satisfying $$n-1<\alpha<n$$ with $$n \in \mathbb{Z^{*}_{+}}$$. The generalized approximation of the fractional order differentiation operator for a given one-dimensional function $$x$$ is defined for all steps $$h$$ and all $$t= kh$$, $$ k \in \mathbb{Z_{+}}$$ by the formula dαx(t)dtα=Dαx(kh)=1hα∑p=0k+1cα(p) x((k+1−p)h), (2.4) where cα(p)=(−1)p(αp) (2.5) with (αp)={1ifp=0α(α−1)(α−2)⋯(α−p+1)p!ifp>0 (2.6) it follows from equation (2.4) that Dαx(kh)=h−α[x((k+1)h)−αx(kh)+∑p=2k+1cα(p) x((k+1−p)h)] (2.7) and x((k+1)h)=hαDαx(kh)+αx(kh)−∑p=2k+1cα(p) x((k+1−p)h). (2.8) Remark 2.1 In our case $$n=1$$. Note that the relation (2.7) is a generalization of the standard derivative in the case of $$\alpha=1$$. 2.1. Discretization of the two-dimensional continuous-discrete time fractional linear system The formulas (2.4) and (2.8) can be extended to the two-dimensional functions and can be applied to the systems (2.2) and (2.3). Thus, dαx(t,i+1)dtα=Dαx(kh,i+1)=h−α∑p=0k+1cα(p) x((k+1−p)h,i+1) (2.9) and x((k+1)h,i+1)=hαDαx(kh,i+1)+αx(kh,i+1)−∑p=2k+1cα(p) x((k+1−p)h,i+1). (2.10) From now on, for notational convenience, the sampling step $$h$$ is removed from the discretized two-dimensional sequences, i.e., $$ x(kh,i) = x(k,i)$$, $$ y(kh,i) = y(k,i)$$ and $$u(kh,i) = u(k,i)$$ for all $$k,i \in \mathbb{Z}_{+}$$. Theorem 2.2 Let’s consider $$h>0$$. The fractional two-dimensional continuous-discrete time systems (2.2) and (2.3) where $$0<\alpha<1$$ is dicretized to the corresponding $$\alpha$$-order fractional two-dimensional discrete-time system defined by the equations, x(k+1,i+1)=A^0x(k,i)+A^1x(k,i+1)+B^u(k,i)−∑p=2k+1cα(p) x(k+1−p,i+1) (2.11) y(k,i)=Cx(k,i)+Du(k,i) (2.12) for all $$k,i\in \mathbb{Z}_{+}$$. with the matrices $$\hat{A}_{j}$$, for $$\ j=0,1$$ and $$\hat{B}$$ are defined by the relations $$\hat{A}_{0} = h^{\alpha}A_{0}$$, $$\hat{A}_{1} = h^{\alpha}A_{1} + \alpha I_{n}$$, $$\hat{B} = h^{\alpha}B$$ with the boundary conditions $$x(k,0) \in \mathbb{R}^{n}$$ for all $$k\in \mathbb{Z_{+}}\ $$ and $$x(0,i) \in \mathbb{R}^{n}$$ for all $$i\in \mathbb{Z_{+}}$$. Proof. Let’s consider $$h>0$$ and the equation (2.2) for $$t= kh, \ k=0,1,2,\cdots$$. This then yields Dαx(k,i+1)=dαx(k,i+1)dtα=A0x(k,i)+A1x(k,i+1)+Bu(k,i). (2.13) Substituting relation (2.13) in the relation (2.10) we obtain x(k+1,i+1) =hα[A0x(k,i)+A1x(k,i+1)+Bu(k,i)] +αx(k,i+1)−∑p=2k+1cα(p) x(k+1−p,i+1), (2.14) which leads to relation (2.11). □ Remark 2.2 It is important to know that the two-dimensional linear system (2.11) and (2.12) is a fractional system for the first direction and standard for the second one. The two-dimensional discrete-time (2.11) and (2.12) is a specific model and has not been discussed in the standard literature. For this reason it is essential, to calculate the solution and study the positivity criteria. 2.2. Solution of the system First, we will relate some basic notions and properties concerning the Z-transform of a two-dimensional function $$x(k,i)$$ used to solve the equation (2.11). Definition 2.4 The Z-transform of a discrete bidimensional function $$x(k,i)$$ is the bidimensional function $$X(z_{1},z_{2})$$ defined by the formula X(z1,z2)=Z[x(k,i)]=∑k=0∞∑i=0∞x(k,i)z1−kz2−i, (2.15) where $$z_{1},z_{2} \in \mathbb{C}$$. For more details about definition and properties of the bidimensional Z-transform, see Jayaraman et al. (2011). We now derive the solution of the equation (2.11). Theorem 2.3 The solution of the equation (2.11) with boundary conditions x(k,0)∈Rn and x(0,i)∈Rnfork,i∈Z+ (2.16) takes the following form x(k,i)=∑e=0k∑f=0iTk−e−1,i−f−1B^u(e,f)+∑e=1k[Tk−e−1,i−1A^0−∑p=2k−ecα(k−e−p)Tp,i]]x(e,0) +∑f=1i[Tk−1,i−f−1A^1−∑p=2k−1cα(k−p)Tp,f]x(0,f)+[Tk−1,i−1A^0−∑p=0k−1cα(k−p)Tk−p,i]x(0,0), (2.17) where the matrices $$T_{ef}$$ are defined by the following Tef={ Inif e=f=0A^0Te−1,f−1+A^1Te−1,f−∑p=2k+1cα(p)Te−p,fife+f>00n (zero matrix)ife<0 and/or f<0. (2.18) Based on the literature (see Monje et al., 2010; Kaczorek, 2011b), the matrices $$T_{ef}$$ are called the transition matrices. Proof. Applying the Z-transform properties on both members of equation (2.11) yields, z1z2[X(z1,z2)−X(z1,0)−X(0,z2)+x(0,0)]=A^0X(z1,z2)+A^1z2[X(z1,z2)−X(z1,0)] −∑p=2k+1cα(p)z1−p+1z2[X(z1,z2)−X(z1,0)]+B^U(z1,z2), (2.19) where $$X(z_{1},z_{2})=Z\left[x(k,i)\right]$$ and $$ U(z_{1},z_{2})= Z\left[u(k,i)\right]$$. Multiplying the both members of the relation (2.19) by the value $$ z^{-1}_{1}z^{-1}_{2}$$ we obtain GX(z1,z2)=X(0,z2)−x(0,0)+z1−1z2−1B^U(z1,z2) +[In−z1−1A^1+∑p=2k+1cα(p)z1−pIn]X(z1,0), (2.20) where the matrix $$G$$ is defined by the formula G=[In−z1−1z2−1A^0−z1−1A^1+∑p=2k+1cα(p)z1−pIn]. (2.21) The matrix $$G$$ is invertible i.e., det(G)=∑p=02k+2∑q=02k+2apqz1−pz2−q≠0 for some $$z_{1},z_{2} \in \mathbb{C}$$, where $$a_{pq}$$ for $$1\leq p\leq2k+2, \ 1\leq q\leq2k+2$$ are real coefficients and depend on the matrices $$\hat{A}_{0}, \hat{A}_{1}$$. In this case, the inverse of the polynomial matrix $$G$$ can be expressed in the form of the following sum G−1=∑e=0∞∑f=0∞Tefz1−ez2−f. (2.22) Therefore, from the relation (2.20) we obtain X(z1,z2) =G−1[X(0,z2)−x(0,0)+[In−z1−1A^1+∑p=2k+1cα(p)z1−pIn]X(z1,0) +z1−1z2−1B^U(z1,z2)]. (2.23) The calculation of the Z-transform $$X(z_{1},z_{2})$$ of the solution of the equation (2.11) is on the basis of the value of the matrix $$G^{-1}$$, and consequently on the transition matrices $$T_{ef}$$. Since $$G.G^{-1}=G^{-1}.G= I_{n}$$, it yields [∑e=0∞∑f=0∞Tefz1−ez2−f][In−z1−1z2−1A^0−z1−1A^1+∑p=2k+1cα(p)z1−pIn]=In (2.24) hence ∑e=0∞∑f=0∞[Tef−A^0Te−1,f−1−A^1Te−1,f+∑p=2k+1cα(p)Te−p,f]=In. (2.25) Comparing the coefficients at the same powers of $$z_{1}$$ and $$z_{2}$$ of (2.15) leads to the relation (2.18). The transition matrices are then defined by the recursive sequence (2.18), and we can remark the matrix identities $$T_{10}=\hat{A}_{1}$$ and $$T_{01}=0_{n}$$. Substituting relation (2.22) in (2.20), we obtain X(z1,z2) =(∑e=0∞∑f=0∞Tefz1−ez2−f)[In−z1−1A^1+∑p=2k+1cα(p)Tefz1−p−ez2−f]X(z1,0) +(∑e=0∞∑f=0∞Tefz1−ez2−f)[X(0,z2)−x(0,0)+z1−1z2−1B^U(z1,z2)]. (2.26) Applying the inverse Z-transform and the convolution theorem to the relation (2.26) we obtain the solution (2.17) of the equation (2.11). □ See Jayaraman et al. (2011) for more details about definition and properties of the inverse Z-transform. 3. Positivity criteria Our goal is to study the influence of the value of discretization step on the positivity of the system (2.11) and (2.12), for this reason, necessary and sufficient conditions on positivity of the system (2.11) and (2.12) will be derived. Definition 3.1 The two-dimensional $$\alpha$$-order fractional discrete-time system (2.11) and (2.12) is positive, if all the states and outputs are positive i.e., $$x(k,i) \in \mathbb{R}^{n}_{+}$$, $$y(k,i) \in \mathbb{R}^{p}_{+}$$, $$k,i\in \mathbb{Z}_{+}$$, for all boundary conditions $$x(k,0) \in \mathbb{R}^{n}_{+}$$, $$x(0,i) \in \mathbb{R}^{n}_{+}$$, $$k,i\in \mathbb{Z}_{+}$$, and all the entries $$u(k,i) \in \mathbb{R}^{n}_{+}$$,$$k,i\in \mathbb{Z}_{+}$$. In the following lemma, we recall a known result, which we prove by another technique. lemma 3.1 Let’s consider $$0<\alpha<1$$, then $$c_{\alpha}(p)<0$$, for all $$p\geq1$$. Proof. Let $$0<\alpha\ <1$$ and $$p\geq1$$, then cα(p)=(−1)p(αp) with (αp)=α(α−1)(α−2)⋯(α−p+1)p! so, we have $$p$$ factors in the product $$ \alpha(\alpha-1)(\alpha-2)\cdots (\alpha-p+1)$$ and $$\alpha$$ is the only positive factor. Then, only two cases are possible if the integer $$p$$ is even then $$(p-1)$$ is odd, so $$(-1)^{p}>0$$ and $$\underbrace{(\alpha-1)(\alpha-2)\cdots (\alpha-p+1)}_{(p-1) \ \mathrm{negative} \ \mathrm{values}}<0$$ if the integer $$p$$ is odd then $$(p-1)$$ is even, $$(-1)^{p}<0$$ and $$\underbrace{(\alpha-1)(\alpha-2)\cdots (\alpha-p+1)}_{(p-1) \ \mathrm{negative} \ \mathrm{values}}>0$$. Consequently we deduce that in all cases the values of $$(-1)^{p}$$ and the product $$(\alpha-1)(\alpha-2)\cdots (\alpha-p+1)$$ are of opposite signs. □ Remark 3.1 Note that, this lemma has been proved by induction in many papers. Theorem 3.1 The $$\alpha$$-order two-dimensional fractional discrete-time system (2.11) and (2.12) is positive if only if the matrices $$\hat{A}_{0}, \hat{A}_{1} \in \mathbb{R}^{n\times n}_{+}, \hat{B} \in \mathbb{R}^{n\times m}_{+} , C \in \mathbb{R}^{p\times n}_{+}$$ and $$ D \in \mathbb{R}^{p\times m}_{+}$$ Proof. a) Sufficient condition: From Lemma (3.1), we have $$-c_{\alpha}\left(p\right)>0$$. If the matrices verify $$\hat{A}_{0}, \hat{A}_{1} \in \mathbb{R}^{n\times n}_{+}, \hat{B} \in \mathbb{R}^{n\times m}_{+} , C \in \mathbb{R}^{p\times n}_{+}$$ and $$ D \in \mathbb{R}^{p\times m}_{+}$$, and if $$u(k,i) \in \mathbb{R}^{+}$$ with all positive boundary conditions, then so based on relations (2.17) and (2.18) we conclude that the transition matrices $$T_{ef}$$ are positive and hence the solution $$x(k,i)$$ is positive for all positive boundary conditions. b) Necessary condition: The idea follows from the result of Kaczorek (2008). Suppose that the system (2.11) and (2.12) is positive and let us prove the positivity of the matrices $$\hat{A}_{0},\hat{A}_{1},\hat{B},C,D$$. Assuming that $$x(0,0)=e_{ni}$$, $$i=\overline{1,n}$$, where $$e_{ni}$$ is the $$i^{th}$$ column of the identity matrix $$I_{n}$$, ie.: $$e_{ni} = (0,0,\cdots,1,\cdots,0)^{T}$$. Also suppose that $$x(1,0)=0$$ and $$u(0,0)=0$$. From the equation (2.11) we obtain $$x(1,1)= \hat{A}_{0} x(0,0) = \hat{A}_{0} e_{ni} = \hat{A}_{0i}$$, such that $$\hat{A}_{0i}$$ is the $$i{th}$$ column of the matrix $$\hat{A}_{0}$$. So, $$\hat{A}_{0i}=x(1,1) \in \mathbb{R}^{n}_{+}$$. Continue stepping along the columns we deduce that $$\hat{A}_{0} \in \mathbb{R}^{n}_{+}$$. By the same analogy Assuming that $$x(0,1)=e_{ni}, \ i=\overline{1,n}$$, and that $$x(0,0)=0$$ and $$u(0,0)=0$$. Substituting in the equation (2.11) we obtain $$x(1,1)= \hat{A}_{1} x(0,1) = \hat{A}_{1} e_{ni} = \hat{A}_{1i}$$, such that $$\hat{A}_{1i}$$ is the $$i^{th}$$ column of the matrix $$\hat{A}_{1}$$. Hence, $$\hat{A}_{1i}=x(1,1) \in \mathbb{R}^{n}_{+}$$. Consequently the matrix $$\hat{A}_{1}\in \mathbb{R}^{n}_{+}$$. Assuming That $$x(0,0)=x(0,1)=0$$ and that $$u(0,0)=e_{ni}, \ i=\overline{1,m}$$. Substituting in the equation (2.11) we obtain $$x(1,1)= \hat{B} u(0,0) = \hat{B} e_{ni} = \hat{B}_{i}$$ where $$\hat{B}_{i}$$ is the $$i{th}$$ column of the matrix $$\hat{B}$$. Since $$x(1,1)$$ is positive $$\hat{B}_{i} \in \mathbb{R}^{m}_{+}$$. Consequently the matrix $$\hat{B}$$ is positive because of positivity of all its columns $$\hat{B}_{i}, \ i=\overline{1,m}$$. By the same manner one can prove the positivity of the matrices $$C$$ and $$D$$ by using the equation (2.12). □ 4. Results and discussions In this section, we want to know conditions to guarantee that the two-dimensional discrete-time fractional system (2.11) and (2.12) obtained by discretization remains positive when the two-dimensional continuous-discrete fractional system (2.2) and (2.3) is supposed positive. We regard also the influence of the sampling step on the positivity of the system (2.11) and (2.12). 4.1. Influence of discretization step on positivity Here we will expose our main theorem. Theorem 4.1 Let’s consider $$h>0$$ be the discretization step. Assume that the conditions of the theorem 2.1 are satisfied. Then we have one of the following cases: If the matrix $$A_{1}$$ is a Metzler positive matrix, then the system (2.11) and (2.12) remains positive for all sampling step $$h>0$$. If the matrix $$A_{1}$$ is a Metzler nonpositive, then the system (2.11) and (2.12) remains positive if and only if 0<h≤(αmaxi|aii(1)|)1α, i=1,n¯, (4.1) where $$a^{(1)}_{ii}$$, for $$i= \overline{1,n}$$, are the strictly negative diagonal entries of the matrix $$A_{1}$$. Proof. Let $$h>0$$ and assume that the fractional system (2.2)-(2.3) is positive, i.e., $$A_{0} \in \mathbb{R}^{n\times n}_{+}$$, $$ B\in \mathbb{R}^{n\times m}_{+}$$, $$C \in \mathbb{R}^{p\times n}_{+}$$, $$D \in \mathbb{R}^{p\times m}_{+}$$, $$A_{1}$$ is a Metzler matrix. Since $$0<\alpha<1$$ and $$h>0$$ we obtain $$\hat{A}_{0} = h^{\alpha}{A}_{0} \in \mathbb{R}^{n\times n}_{+}$$, $$ \hat{B} = h^{\alpha}{B}\in \mathbb{R}^{n\times m}_{+}$$. Thus, by applying Theorem 3.1, the $$\alpha$$-order fractional discrete-time system (2.11) and (2.12) remains positive if and only if the matrix $$\hat{A}_{1} \in \mathbb{R}^{n\times n}_{+}$$. We have $$\hat{A}_{1} = h^{\alpha}{A}_{1} + \alpha I_{n} $$ with $$A_{1}$$ is a Metzler matrix. Then the inequality $$\hat{A}_{1} \geq 0$$ is equivalent to the comparison $${A}_{1}\geq \frac{-\alpha}{h^{\alpha}} I_{n}$$. Then two cases have to be considered, If $$A_{1}$$ is a positive matrix, then the last relation is obvious. If $$A_{1}$$ is nonpositive matrix; which means that at least one diagonal entry of the matrix $$A_{1}$$ is strictly negative; hence the matrix $$\hat{A}_{1}$$ is not necessarily positive. Necessary condition: Since all of off diagonal entries of the matrix $${A}_{1}$$ are positive, and those of the matrix $$\frac{-\alpha}{h^{\alpha}} I_{n}$$ are all null, it will be clear that, the last inequality will be based on the comparison between diagonal entries of the matrix $${A}_{1}$$ and the value $$\frac{-\alpha}{h^{\alpha}}$$. Let $$a^{(1)}_{ii}$$ be the diagonal entries of the matrix $${A}_{1}$$. (a) it is easy to show that if all $$a^{(1)}_{ii}\geq0$$ then $$a^{(1)}_{ii}\geq0> \frac{-\alpha}{h^{\alpha}}$$. (b) for $$a^{(1)}_{ii}<0$$ then $$ 0 > a^{(1)}_{ii}\geq \frac{-\alpha}{h^{\alpha}}$$ thus hα≤−αaii(1) hence 0<h≤(αmaxi|aii(1)|)1α. Sufficient condition: Assuming that the matrix $$A_{1}$$ is a Metzler matrix with at least one diagonal strictly negative entry, and we will prove that the matrix $$\hat{A}_{1}$$ is positive. It is clear that since $$ a^{(1)}_{ij}\geq 0$$ for $$i\neq j$$, the off diagonal entries of the matrix $$\hat{A}_{1}=h^{\alpha}{A}_{1} + \alpha I_{n}$$ are all positive. Also, if $$ a^{(1)}_{ii}\geq 0$$, the diagonal entries of the matrix $$\hat{A}_{1}$$ are positive. For the strictly negative diagonal entries of the matrix $$A_{1}$$ verifying 0<h≤(αmaxi|aii(1)|)1α we have for any diagonal entry $$a^{(1)}_{ii}<0$$ 0<hα≤α|aii(1)| thus aii(1)+αhα≥0 we conclude that all diagonal entries of the matrix $$\hat{A}_{1}=h^{\alpha}{A}_{1} + \alpha I_{n}$$ are positive. □ Remark 4.1 Note that, one can substitute the condition of the first case in the Theorem 4.1 by the condition `If the matrix $$A_{1}$$ is positive’. Because obviously, all positive matrices are Metzler matrices. 4.2. Discussions Note that from the Theorem 4.1 and for the nonpositive Metzler matrix $$A_{1}$$, the choice of the sampling step depends on the value of the fractional derivation order $$\alpha$$ and the scalar $$\mathrm{max}_{i}\left|a^{(1)}_{ii}\right|$$, such that $$a^{(1)}_{ii}$$ are the strictly negative diagonal entries of the matrix $$A_{1}$$. We denote the confidence interval $$I_{\alpha,M}= ]0,\left(\frac{\alpha}{M}\right)^{\frac{1}{\alpha}}]$$ where $$M= \mathrm {max}_{i}\left|a^{(1)}_{ii}\right|$$ for $$i=\overline{1,n}$$. Then we deduce that to preserve positivity of the system (2.11) and (2.12), the sampling step $$h$$ must verify $$h \in I_{\alpha,M}$$. The length of the confidence interval $$I_{\alpha,M}$$ may increase or decrease, depending on the strictly positive values of $$\alpha$$ and $$M$$. The following corollary is then deduced. Corollary 4.1 Consider the system (2.2) and (2.3), and let the matrix $$A_{1}$$ be a nonpositive Metzler matrix, then Lmax={e1MeifM<e−11MifM≥e−1, (4.2) where $$L_{max}= max_{\alpha}\left\{l(I_{\alpha,M}), \mathrm{for} \ 0<\alpha<1 \right\}$$, and $$l(I_{\alpha,M})$$ denotes the length of the interval $$I_{\alpha,M}$$ Proof. Let the length of the interval $$I_{\alpha,M}$$ be denoted by the function $$l(I_{\alpha,M})=l_{M}(\alpha)$$ of the variable $$\alpha \in ]0,1[$$. Obviously the length of $$I_{\alpha,M}$$ is defined by the relation lM(α)=(αM)1α, 0<α<1 (4.3) and after derivation, it yields $$l{'}_{M}(\alpha) = \left[\frac{1-ln\left(\frac{\alpha}{M}\right)}{\alpha^{2}}\right] l_{M}(\alpha)$$. We obtain one of the following cases, if $$0<M<\frac{1}{\mathrm{e}}$$ then $$0<l_{M}(\alpha)\leq \mathrm{exp}(\frac{1}{M \mathrm{e}})$$. if $$M \geq \frac{1}{\mathrm{e}}$$ then $$0< l_{M}(\alpha) <\frac{1}{M}$$. □ Remark 4.2 The Corollary 4.1 shows that the length of the confidence interval will increase in the case where $$0<M< \frac{1}{\mathrm{e}}\approx 0.3679$$, and the choice of the sampling step will not be more constrained. Contrarily where $$M \geq 0.3679$$ this choice will be done in a smallest interval. In both cases, since $$lim_{\alpha\rightarrow 0}l_{M}(\alpha)=0$$ the vanishing values of the fractional derivation order $$\alpha$$ leads to a vanishing values of the sampling step $$h$$. 5. Numerical examples In this section, we present some numerical simulations that illustrate the theoretical results derived in the previous section. Example 5.1 Consider the system (2.2) and (2.3) for $$\alpha=0.5$$ with the matrices A0=[ 10 01], A1=[ −10 0−0.9], B=[ 1 0.5],C=[ 23] and D=0. The input $$u(t,j)=1$$ for $$t\geq0$$ and $$j \in \mathbb{Z}_{+}$$ with boundary conditions $x(t,0)=\left[ \begin{array}{c} \ 0.1 \\ \ 0.1 \end{array} \right]$ for $$t\geq0$$ and $x(0,j)=\left[ \begin{array}{c} \ 0.2 \\ \ 0.1 \end{array} \right]$ for $$j \in \mathbb{Z}_{+}$$. Applying Theorem 2.1, then the system (2.2) and (2.3) is positive. From the Theorem 4.1, since the matrix $$A_{1}$$ is a Metzler nonpositive matrix, then to preserve the positivity after discretization, we shall choice a discretization step $$h$$ verifying the relation (4.1); that leads to 0<h≤(0.5maxi|aii(1)|)10.5 then $$0< h\leq 0.25$$. The matrix $$\hat{A}_{1}$$ is defined $$\hat{A}_{1}= h^{0.5}A_{1}+ 0.5 I_{2}$$ then A^1=[ −h0.5+0.50 h0.52h0.5+0.5]. The matrices $$\hat{A}_{0}, \hat{B}$$ are obviously positive. Let $$\epsilon> 0$$. For $$h=h_{1}=0.25+\epsilon>0$$, the entries $$h_{1}^{0.5}$$ and $$2h_{1}^{0.5}+0.5$$ are positive except $$-h_{1}^{0.5} +0.5$$; because since $$\epsilon>0$$, we have $$-(\epsilon +0.25)^{0.5}< -0.5$$ therefore $$-h_{1}^{0.5} +0.5 < 0$$. Thus, the two-dimensional discrete-time fractional system (2.11) and (2.12) obtained after discretization is not positive. For $$h=0.72$$, see Fig. 1. Fig. 1. View largeDownload slide State vector of the system from Example 5.1.2 with $$h=0.72.$$ Fig. 1. View largeDownload slide State vector of the system from Example 5.1.2 with $$h=0.72.$$ For $$h=h_{2}=0.25-\epsilon>0$$ then the entries $$-h_{2}^{0.5} +0.5$$, $$h_{2}^{0.5}$$ and $$2h_{2}^{0.5}+0.5$$ are positive because since $$-\epsilon <0$$ we have $$ -(-\epsilon +0.25)^{0.5} > -0.5$$ hence $$-h_{2}^{0.5} +0.5 > 0.$$ So, the matrix $$\hat{A}_{1} \in \mathbb{R}^{2\times 2}_{+}$$, therefore the system (2.11) and (2.12) is a positive system. For $$h=0.2$$, see Fig.2. Fig. 2. View largeDownload slide State vector of the system from Example 5.1.2 with $$h=0.72.$$ Fig. 2. View largeDownload slide State vector of the system from Example 5.1.2 with $$h=0.72.$$ Example 5.2 Consider the system (2.2) and (2.3) for $$0<\alpha< 1$$ and the matrices A0=[ 0.20 1.20.51], A1=[ −0.460 2.12.02],B=[ 0.22 1.02], C=[ 0.220.33] and D=0. We have $$M =0.46 \geq \frac{1}{\mathrm{e}}$$. Figure 3 shows that the variation of the length of confidence interval $$I_{\alpha,0.46}$$ depends on the value of the fractional derivation order $$\alpha \in ]0,1[$$. Fig. 3. View largeDownload slide Variation of $$l_{0.46}$$, the length of confidence interval as a function of the derivation order $$\alpha.$$ Fig. 3. View largeDownload slide Variation of $$l_{0.46}$$, the length of confidence interval as a function of the derivation order $$\alpha.$$ The function $$l_{M}=l_{0.46}$$ is defined by the following relation: l0.46(α)=(α0.46)1α. Hence, the maximum value of $$h$$ in the confidence interval $$I_{\alpha,0.46}$$ is 2.174 which corresponds to the length equal to $$L_{max}=2.174$$ at $$\alpha \approx 1$$. From Fig. 3, we can conclude that for the vanishing values of the fractional order derivative $$\alpha$$ verifying $$0< \alpha< 0.2$$ the sampling step $$h$$ must take a vanishing values too ie: $$h<0.0155$$, which may affect the precision and the problem complexity. The problem will be posed for the fractional models with a smaller fractional order derivative. Example 5.3 Consider the system (2.2)-(2.3) for $$0 < \alpha< 1$$ and the following matrices A0=[ 0.20 0.31.23], A1=[ −0.20 0.1−0.15],B=[ 0.52 1.11], C=[ 1.420.53] and D=0. We have $$ M =0.2 <\frac{1}{\mathrm{e}}$$. The corresponding function $$l_{M}=l_{0.2}$$ is defined by the following relation l0.2(α)=(α0.2)1α. Figure 4 shows the variation of the length of confidence interval $$I_{\alpha,0.2}$$ depending on the value of the fractional order derivative $$\alpha \in ]0,1[$$. Fig. 4. View largeDownload slide Variation of $$l_{0.2}$$, the length of confidence interval as a function of the derivation order $$\alpha.$$ Fig. 4. View largeDownload slide Variation of $$l_{0.2}$$, the length of confidence interval as a function of the derivation order $$\alpha.$$ Hence, the maximum value for sampling step $$h$$ is 6.28 which corresponds to $$L_{max}=l_{0.2}(\alpha)=exp(0.2\mathrm{e})\approx 6.29$$ at $$\alpha=0.2 \mathrm{e} \approx 0.542$$. Thus, we can see from Fig. 4 that for the vanishing fractional order $$\alpha$$ verifying $$0<\alpha< 0.11$$, the sampling step $$h$$ should not exceed the value 0.005 to guarantee preservation of positivity, however for values of $$\alpha\geq 0.11$$, it should take an acceptable values. 6. Conclusion In this article, an analysis of the influence of the value of discretization step on positivity of fractional two-dimensional discrete-time linear system obtained by discretization from two-dimensional continuous-discrete linear system is investigated. Necessary and sufficient conditions are then derived and illustrative numerical examples are also given to illustrate the applicability of the proposed approach. An open problem is an extension of the results to the two-dimensional continuous fractional linear system. Another open problem is to study influence of the discretization step on the asymptotic stability which will be considered in a separate paper. References Baleanu D. , Machado J. A. T. & Luo A. C. J. ( 2012 ) Fractional Dynamics and Control. Springer New York Dordrecht Heidelberg London : Springer Science & Business Media. Bouagada D. ( 2004 ) Influence of the value of discretization step on positivity of 2D linear continuous-discrete systems. Far East J. Math. Sci. , 15 107 – 112 . Farina L. & Rinaldi S. ( 2000 ) Positive Linear Systems . Theory and Applications. New York : J. Wiley. Jayaraman S. , Esakkirajan S. & Veerakumar T. ( 2011 ) Digital Image Processing. New Delhi : Tata Mc Graw-Hill Education Private Limited. Kaczorek T. ( 2002 ) Positive 1D and 2D Systems . Springer-Verlag London Berlin Heidelberg. Kaczorek T. ( 2008 ) Positive fractional continuous-time systems and their reachability. Int. J. Appl. Math. Comput. Sci. , 18 , 223 – 228 . CrossRef Search ADS Kaczorek T. ( 2011a ) Positive fractional 2D hybrid linear systems. B. Pol. Acad. Sci-Tech. Sci. 59 , 575 – 579 . Kaczorek T. ( 2011b ) Selected Problems of Fractional Systems Theory. Berlin Heidelberg : Springer. Kaczorek T. ( 2013a ) Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems. Archives of Electrical Engineering , 62 , 345 – 355 . Kaczorek T. ( 2013b ) Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems. Int. J. Appl. Math. Comput. Sci. , 23 , 501 – 506 . Kurek J. ( 1985 ) The general state-space model for a two-dimensional. IEEE Trans. Autom. Contr. AC , 30 , 600 – 602 . Google Scholar CrossRef Search ADS Monje C. A. , Chen Y. , Vinagre B. M. , Xue D. & Feliu V. ( 2010 ) Fractional-Order Systems and Controls, Fundamentals and Applications . London Dordrecht Heidelberg New York : Springer. Oustaloup A. ( 1999 ) La commande crone. Paris : Hermes. Podlubny I. ( 1999 ) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. New york : Academic Press. Sabatier J. , Agrawal O. P. & Machado J. A. T. ( 2007 ) Advances in Fractional Calculus . Theoretical developments and Applications in Physics and Engineering. Springer Dordrecht : The Netherlands. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

IMA Journal of Mathematical Control and Information – Oxford University Press

**Published: ** Feb 27, 2017

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