ADSAdvances in Decision Sciences1532-76121173-9126Hindawi Publishing Corporation81060510.1155/2010/810605810605Research ArticleDiscretizing LTI Descriptor (Regular) Differential Input Systems with Consistent Initial ConditionsKarageorgosAthanasios D.1PantelousAthanasios A.2KalogeropoulosGrigoris I.1WakeGraeme1Department of MathematicsUniversity of Athens, 15784 AthensGreeceuoa.gr2Department of Mathematical SciencesUniversity of Liverpool, Liverpool L69 3BXUKliv.ac.uk201001022010201004042009180120102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A technique for discretizing efficiently the solution of a Linear descriptor (regular) differential input system with consistent initial conditions, and Time-Invariant coefficients (LTI) is introduced and fully discussed. Additionally, an upper bound for the error x¯(kT)x¯k that derives from the procedure of discretization is also provided. Practically speaking, we are interested in such kind of systems, since they are inherent in many physical, economical and engineering phenomena.

1. Introduction: Preliminary Results

During the discretization (or sampling) process, we should replace the original continuous-time systems with finite sequences of values at specified discrete-time points. This important process is commonly used whenever the differential systems involve digital inputs, and by having numerical data, the sampling operation and the quantization are necessary. Additionally, the discretization (or sampling) process is occurred whenever significant measurements for the system are obtained in an intermittent fashion. For instance, we can consider a radar tracking system, where there is information about the azimuth and the elevation, which is obtained as the antenna of the radar rotates. Consequently, the scanning operation of the radar produces many important sampled data.

In our approach, we consider the LTI descriptor (or generalized) differential input systems of type Eẋ̲(t)=Ax̲(t)+Bu̲(t), where matrices E,A(n×n,𝔽) (i.e., is the algebra of square matrices with elements in the field 𝔽) and B(n×l,𝔽) are constants; the state x̲(t)(n×1,𝔽) has consistent initial conditions. We shall call x̲0  a consistent initial condition for (1.1) at to if there is a solution for (1.1), which is defined on some interval [to,to+γ], γ>0 such that x̲(to)=x̲o(n×1,𝔽), the input u̲(t)(l×1,𝔽), and the detE=0 are related to the matrix pencil theory, since its algebraic, geometric, and dynamic properties stem from the structure of the associated pencil, that is, sE-A. Moreover, for the sake of simplicity, we set in the sequel n=(n×n;𝔽) and nm=(n×m;𝔽).

Now, in what it follows, the pencil sE-A is regular, that is, det(sE-A)0.

Practically speaking, descriptor (or generalized) regular (or singular) differential systems constitute a more general class than linear state space systems do. Considering applications, these kinds of systems appear in the modelling procedure of many physical, engineering, mechanical, actuarial, and financial problems. For instance, in engineering, in electrical networks, and in constrained mechanics, the reader may consult , and so forth. In Economics, the famous Leontief input-output singular dynamic model is well known; see for instance some of the numerous references [2, 3, 712], and so forth.

In this paper, we provide two main research directions that are being summarized briefly below.

First, we want to provide a computationally efficient method for discretizing LTI descriptor regular differential systems with input signals and consistent initial conditions. The consistency of the initial conditions is necessary, because it eliminates completely the possibility to appear as a distributional expression for the solution of system (1.1), that is, Dirac delta functions and its derivatives.

Second, according to the authors’ knowledge, for the first time an upper bound for the error x̲(kT)-x̲k, which is derived during the discretization process, is finally obtained. Consequently, an analytic expression that penalizes our choice for the sampling period T is provided through the notion of the ·-norm of the difference between the continuous-time solution x̲(kT), at time t=kT, and the relevant discrete-time points x̲k.

This investigation is relevant to and it extends further the recent work proposed by Karampetakis; see  and Kalogeropoulos et al.; see . Concerning the mathematical tools, only the Weierstrass canonical form (WCF) and some fundamental elements of matrix pencil theory are required.

Recently, in [15, 16], several numerical issues of the WCF of a regular matrix pencil are presented and discussed. Thus, briefly speaking, in these research papers, two important computational tools are considered: (a) the QZ algorithm to specify the required root range of the pencil and (b) the updating technique to compute the index of annihilation; see . The proposed updating technique takes advantages of the already computed rank of the sequences of matrices that appears during our procedure reducing significantly the required floating-point operations. The algorithms are implemented in a numerical stable manner, giving efficient results.

For reasons of convenience, some basic concepts and definitions from matrix pencil theory are introduced; see for more details [24, 1722] et al. Thus, the class of strict equivalence is characterized by a uniquely defined element, known as a WCF, that is, sEw-Aw. Consequently, when the pencil sE-A is regular, we have elementary divisors of the following type:

e.d. of the type sd, d, are called zero finite elementary divisors (z. f.e.d.);

e.d. of the type (s-a)c, a0, c are called nonzero finite elementary divisors (nz. f.e.d.);

e.d. of the type ŝq are called infinite elementary divisors (i.e.d.).

Then, the Weierstrass canonical form of the regular pencil sE-A is defined by

sEw-Awblockdiag{sIp-Jp,sHq-Iq},

where the first normal Jordan type block sIp-Jp is uniquely defined by the set of f.e.d.

(s-a1)p1,,(s-aν)pν,j=1νpj=p

of sF-G and has the form

sIp-Jpblockdiag{sIp1-Jp1(a1),,sIpν-Jpν(aν)}.

And the q blocks of the second uniquely defined block sHq-Iq correspond to the i.e.d.

(ŝ)q1,,(ŝ)qσ,j=1σqj=q

of sE-A and has the form

sHq-Iqblockdiag{sHq1-Iq1,,sHqσ-Iqσ}.

Thus the Hq is a nilpotent matrix of index q*=max{qj:j=1,2,,σ}, where Hqq*=𝕆 and Ipi,Jpi(ai),Hqi are the matrices

Ipi=(pi×pi,),Hqi=(qi×qi,),Jpi(ai)=[ai1000ai10000ai10000ai](pi×pi,).

Now, considering [4, 17, 18], and the transformation

x̲(t)=Qy̲(t), we obtain the following result. (Note that matrix Q is not unique. However, different matrices Q are mutually related.) We use the above transformation, (1.8), since we want to create the WCF; see .

Theorem 1.1 (see [<xref ref-type="bibr" rid="B3">4</xref>, <xref ref-type="bibr" rid="B5">23</xref>]).

System (1.1) has the following solution: x̲(t)=Qn,p{eJp(t-to)y̲p(to)+toteJp(t-s)Bp,lu̲(s)ds}-Qn,qi=0q*-1HqiBq,lu̲(i)(t), where Q=[Qn,pQn,q], B=[Bp,lBq,l], and y̲(to)=[y̲p(to)y̲q(to)]=Q-1x̲(to).

However, (1.9) should be transposed to (1.12), which is practically more useful. Thus, we obtain

x̲(t)=Qn,peJp(t-to)y̲p(to)+Qn,qy̲q(to)+Qn,ptoteJp(t-s)Bp,lu̲(s)ds-Qn,qy̲q(to)-Qn,qi=0q*-1HqiBq,lu̲(i)(t)=[Qn,pQn,q][eJp(t-to)𝕆p,q𝕆q,pIq][y̲p(to)y̲q(to)]+Qn,ptoteJp(t-s)Bp,lu̲(s)ds+Qn,q(-y̲q(to)-i=0q*-1HqiBq,lu̲(i)(t)).

In order the system (1.1) to obtain consistent initial conditions, we should consider that

[y̲p(to)y̲q(to)]=Q-1x̲(to),-y̲q(to)=i=0q*-1HqiBq,lu̲(i)(to).

Consequently, we obtain the desired expression for (1.9), that is,

x̲(t)=Q[eJp(t-to)𝕆p,q𝕆q,pIq]Q-1x̲(to)+Qn,ptoteJp(t-s)Bp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(to)-u̲(i)(t)).

Moreover, by definition, the state-transition matrix of the autonomous linear descriptor differential system, that is, Eẋ̲(t)=Ax̲(t), is given by

Φ(t,to)=Q[eJp(t-to)𝕆p,q𝕆q,pIq]Q-1.

Finally, after some simple algebraic calculations, we obtain another more elegance form for the solution of system (1.1), that is,

x̲(t)=Φ(t,to)x̲(to)+totΦ(t,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(to)-u̲(i)(t)), where x̲(to)𝒯 belongs to the space of consistent initial conditions of system (1.1), that is, 𝒯={x̲(n×1,):x̲=Qn,pC̲p-Qn,qi=0q*-1HqiBq,lu̲(i)(to),withC̲p(p×1,)}.

2. Discretisation of Nonhomogeneous LTI Descriptor Regular Differential System

In this section, we provide a computationally efficient method for the analytic discretization of LTI descriptor regular differential systems with input signal and consistent initial conditions. In the vast literature of descriptor systems, Karampetakis has proposed two discretization methods for regular systems without (see [13, Section 3]) and with inputs (see [13, Section 4]), which are based on the research work by Koumboulis and Mertzios, see , concerning the solution of regular systems in terms of the Laurent expansion terms of (sE-A)-1. The methodology of both papers, that is,  and the present, is equivalent, since they use the zero-order hold approximation.

Moreover, specifically for regular systems, Rachid, see , has proposed a different method, which discretizes the exact solution using Euler approximation techniques.

However, in the present paper, a different to the above research works discretization technique for LTI descriptor differential input systems is presented, which extends further the research work [8, 14].

First, we denote T>0 as the constant sampling period. Without loss of generality and because of the complex notations that follow, we assume that to=0, and the input function u̲(t) changes only through the time moments tk=kT for k=1,2,, that is,

u̲(t)=u̲(kT)=u̲k, for every t[kT,(k+1)T).

In more details, (2.1) implies that the input u̲(t) is being inserted into a mechanism of zero-order change after the end of each sampling process. Thus, the input remains constant from time tk=kT until the next sampling period at time tk+1=(k+1)T takes place.

Thus, the solution (1.14) is given by

x̲(t)=Φ(t,0)x̲0+0tΦ(t,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(t)), where x̲0=x̲(0)𝒯.

Moreover, hereafter, we use the notation

x̲kx̲(kT)k=0,1,2,.

The following theorem provides us with an analytic formula based on (2.2).

Theorem 2.1.

An analytic formula for the discretized solution (2.2) of system (1.1) is given by (2.4) x̲k=Φ(kT,0)x̲0+j=0k-10TΦ(jT+λ,0)dλ·Qn,pBp,lu̲k-1-j+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(kT)).

Proof.

First, we consider (2.2) at time moments t=kT and t=(k+1)T.

Thus, we obtain for the time t=kT

x̲k=Φ(kT,0)x̲0+0kTΦ(kT,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(kT)), and for the time t=(k+1)Tx̲k+1=Φ((k+1)T,0)x̲0+0(k+1)TΦ((k+1)T,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)((k+1)T)). The state-transition matrix (1.13) satisfies the following two properties: Φ(T,0)·Φ(kT,0)=Φ((k+1)T,0),Φ(T,0)·Φ(kT,s)=Φ((k+1)T,s). Now, we multiply (2.5) from the left with Φ(T,0), and we consider (2.7) and (2.8). Thus, we obtain Φ(T,0)x̲k=Φ((k+1)T,0)x̲0+0kTΦ((k+1)T,s)Qn,pBp,lu̲(s)ds+Φ(T,0)Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(kT)). Consequently, by also using (2.6) and (2.9), we take the difference x̲k+1-Φ(T,0)x̲k=  kT  (k+1)TΦ((k+1)T,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u(i)(0)-u(i)((k+1)T))-Φ(T,0)Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(kT)). Afterwards, we should define Q-1=[Q̃p,nQ̃p,n],Q-1Q=[Q̃p,nQ̃q,n]Q=[Q̃p,nQ̃q,n][Qn,pQn,q]=[Q̃p,nQn,pQ̃p,nQn,qQ̃q,nQn,pQ̃q,nQn,q]=In, with Q̃p,nQn,p=Ip,Q̃p,nQn,q=𝕆p,q,Q̃q,nQn,p=𝕆q,p,Q̃q,nQn,q=Iq. Thus, Φ(T,0)Qn,q=Q[eJpT𝕆p,q𝕆q,pIq]Q-1Qn,q=[Qn,pQn,q][eJpT𝕆p,q𝕆q,pIq][Q̃pnQnqQ̃qnQnq]=[Qn,pQn,q][eJpT𝕆p,q𝕆q,pIq][𝕆p,qIq]=[Qn,p,Qn,q][𝕆p,qIq]=Qn,q. So Φ(T,0)Qn,q=Qn,q. Consequently, (2.10) is transformed to (2.14) x̲k+1=Φ(T,0)x̲k+kT(k+1)TΦ((k+1)T,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(kT)-u̲(i)((k+1)T)). Note that by denoting s=kT+w, the integral kT(k+1)TΦ((k+1)T,s)Qn,pBp,lu̲(s)ds is given by kT(k+1)TΦ((k+1)T,s)Qn,pBp,lu̲(s)ds=0TΦ((k+1)T,kT+w)Qn,pBp,lu̲(kT+w)dw=(2.1)0TΦ((k+1)T,kT+w)dw·Qn,pBp,lu̲k. Moreover, by using (2.7) and (2.8), we obtain 0TΦ((k+1)T,kT+w)dw=0TΦ((k+1)T,kT)Φ(0,w)dw. And considering again (2.8), we have Φ((k+1)T,kT)=Φ(T,0)·Φ(kT,kT)=Φ(T,0) because Φ(kT,kT)=In. Finally, we obtain 0TΦ((k+1)T,kT+w)dw=0TΦ(T,0)Φ(0,w)dw=0TΦ(T,w)dw=0TΦ(T-w,0)dw=0TΦ(λ,0)dλ, when we define λ=T-w. Thus, kT(k+1)TΦ((k+1)T,s)Qn,pBp,lu̲(s)ds=0TΦ(λ,0)dλ·Qn,pBp,lu̲k. Combining (2.14) and (2.19), we take the recursive formula (2.20) x̲k+1=Φ(T,0)x̲k+0TΦ(λ,0)dλ·Qn,pBp,lu̲k+Qn,qi=0q*-1HqiBq,l(u̲(i)(kT)-u̲(i)((k+1)T)). Then, the formula (2.4) is obtained by (2.20). Analytically,

(i) for k=0, we take

x̲1=Φ(T,0)x̲0+0TΦ(λ,0)dλ·Qn,pBp,lu̲0+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(T));

(ii) for k=1, we have

x̲2=Φ(T,0)x̲1+0TΦ(λ,0)dλ·Qn,pBp,lu̲1+Qn,qi=0q*-1HqiBq,l(u̲(i)(T)-u̲(i)(2T))=Φ(T,0)Φ(T,0)x̲0+0TΦ(T,0)Φ(λ,0)dλ·Qn,pBp,lu̲0+Φ(T,0)Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(T))+0TΦ(λ,0)dλ·Qn,pBp,lu̲1+Qn,qi=0q*-1HqiBq,l(u̲(i)(T)-u̲(i)(2T)). Since Φ(T,0)Qn,q=Qn,q, we conclude to x̲2=Φ(2T,0)x̲0+0TΦ(T+λ,0)dλ·Qn,pBp,lu̲0+0TΦ(λ,0)dλ·QnpBp,lu̲1+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(2T));

(iii) for k=3, we take

x̲3=Φ(3T,0)x̲0+0TΦ(2T+λ,0)dλ·Qn,pBp,lu̲0+0TΦ(T+λ,0)dλ·Qn,p·Bp,lu̲1+0TΦ(λ,0)dλ·Qn,pBp,lu̲2+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(3T)). Continuing as above, we finally obtain the analytic formula (2.4).

Now, in order to complete the discretization process of the solution (2.2) of the system (1.1), we should replace the ith-order derivative u̲k(i)u̲(i)(kT) with expressions of u̲(kT). Thus, the following result is used, see ,

u̲k(i)=u̲(i)(kT)j=0i(-1)j(ij)u̲k-i+jTi.

Profoundly, based on (2.25), the following difference can be easily proved.

u̲k(i)-u̲k+1(i)j=0i+1(-1)j+1(i+1j)u̲k-i+jTi+1.

Remark 2.2.

The discrete-time solution which represents (2.2) of the continuous-time system (1.1) is given by (2.27), (see also [13, Theorem 3]) x̲k=Φ(kT,0)x̲0+j=0k-10TΦ(jT+λ,0)dλ·Qn,pBp,lu̲k-1-j+Qn,qi=0q*-11Ti+1HqiBq,lj=0i+1(-1)j+1(i+1j)u̲k-1+j.

Remark 2.3.

Equation (2.20) reminds us with a new discrete-time system of the form x̲k+1=Ax̲k+B(σ)u̲k, where A=Φ(T,0), whereas the differences u̲(i)(kT)-u̲(i)((k+1)T) can be replaced by (2.26) in order to get the matrix B(σ). This approach is equivalent to Theorem 3.7 in .

Remark 2.4.

The induction method (show that it is true for n = 0; assume that it is true for n=k; try to prove that it is true for n=k+1) can prove (2.4). However, we strongly believe that our approach seems to be more natural, since (2.4) is analytically constructed.

In the next section, according to the authors’ knowledge, an upper bound for the error x̲(kT)-x̲k, which is derived during the discretization process, is provided for the first time.

3. Error Analysis for the Discretization (or Sampling) Process

In this section, we provide an analytic expression that penalizes our choice for the sampling period T through the notion of the Euclidean ·-norm of the difference between the continuous-time solution (1.13), at time t=kT, x̲(kT) and the relevant discrete-time points x̲k, see (2.4), that is, we are interested in the difference x̲(kT)-x̲k.

Following the results of the previous sections, we obtain (3.2). Analytically, we firstly note that the comparison of the solutions is provided at time moment t=kT.

The solution (1.13) at time moment t=kT for k=1,2,3,, is given by (3.1)

x̲(kT)=Φ(kT,0)x̲0+0kTΦ(kT,s)Qn,pBp,lu̲(s)ds+Qn,qi=0q*-1HqiBq,l(u̲(i)(0)-u̲(i)(kT)) Combining (2.4) and (3.1), we obtain the following important expression:

x̲(kT)-x̲k=0kTΦ(kT,s)Qn,pBp,lu̲(s)ds-j=0k-10TΦ(jT+λ,0)dλ·Qn,pBp,lu̲k-1-j (note that it is not important to consider the completely discretized equation (2.27) instead of (2.4), since the second part with the ith-order derivatives is eliminated).

In order to compare the two solutions at a fixed time moments, the following lemmas are stated and straightforwardly proved.

Lemma 3.1.

The equality (3.3) holds 0kTΦ(kT,s)Qn,pBp,lu̲(s)ds=j=0k-1jT(j+1)TΦ(kT,s)ds·Qn,pBp,lu̲j.

Proof.

Consider that 0kTΦ(kT,s)Qn,pBp,lu̲(s)ds=0TΦ(kT,s)Qn,pBp,lu̲(s)ds+T2TΦ(kT,s)Qn,pBp,lu̲(s)ds++(k-1)TkTΦ(kT,s)Qn,pBp,lu̲(s)ds. Since (2.1) holds, we take 0kTΦ(kT,s)Qn,pBp,lu̲(s)ds=0TΦ(kT,s)Qn,pBp,lu̲0ds+T2TΦ(kT,s)Qn,pBp,lu̲1ds++(k-1)TkTΦ(kT,s)Qn,pBp,lu̲k-1ds=0TΦ(kT,s)ds·Qn,pBp,lu̲0+T2TΦ(kT,s)ds·Qn,pBp,lu̲1++(k-1)TkTΦ(kT,s)ds·Qn,pBp,lu̲k-1. Thus, (3.3) derives.

Lemma 3.2.

The equality (3.6) holds jT(j+1)TΦ(kT,s)ds=0TΦ((k-j)T-λ,0)dλ.

Proof.

We define that s=jT+λ; if s=jT, then λ=0; if s=(j+1)T, then λ=T.

Thus, we obtain

jT(j+1)TΦ(kT,s)ds=0TΦ(kT,jT+λ)dλ=0TΦ((k-j)T-λ,0)dλ since it can easily be proven that Φ(kT,jT+λ)=Φ((k-j)T-λ,0).

Now, we consider Lemmas 3.1 and 3.2, and (3.2) is transformed to (3.9)

x̲(kT)-x̲k=j=0k-10TΦ((k-j)T-λ,0)dλ·Qn,pBp,lu̲j-j=0k-10TΦ(jT+λ,0)dλ·Qn,pBp,lu̲k-1-j.

Lemma 3.3.

The equality (3.10) holds j=0k-10TΦ(jT+λ,0)dλ·Qn,pBp,lu̲k-1-j=j=0k-10TΦ((k-1-j)T+λ,0)dλ·Qn,pBp,lu̲j.

Proof.

Making the transformation k-1-j=i, we obtain j=k-1-i.

Since j=0,1,,k-1, then i=k-1,k-2,,0. So, we obtain

j=0k-10TΦ(jT+λ,0)dλ·Qn,pBp,lu̲k-1-j=i=0k-1Φ((k-1-i)T+λ,0)dλ·Qn,pBp,lu̲i. Now, if we also define that j=i, (3.10) holds.

Following the results of Lemma 3.3, (3.9) is obtained the following expression: x̲(kT)-x̲k=j=0k-10T(Φ((k-j)T-λ,0)-Φ((k-1-j)T+λ,0))dλ·Qn,pBp,lu̲j. Moreover, it is true that

Φ((k-j)T-λ,0)-Φ((k-j)T-T+λ,0)=Φ((k-j)T,0)(Φ(-λ,0)-Φ(λ-T,0)).

Consequently, (3.12) is given by

x̲(kT)-x̲k=j=0k-1Φ((k-j)T,0)·0T(Φ(-λ,0)-Φ(λ-T,0))dλ·Qn,pBp,lu̲j, which is practically more significant in what follows.

For the calculation of the upper bound of the difference x̲(kT)-x̲k, we use the Euclidean ·-norm for vectors. Thus, we take

x̲(kT)-x̲k=j=0k-1Φ((k-j)T,0)·0T(Φ(-λ,0)-Φ(λ-T,0))dλ·Qn,pBp,lu̲jj=0k-1Φ((k-j)T,0)·0TΦ(-λ,0)-Φ(λ-T,0)dλQn,p·Bp,l·u̲j.

Now, for the calculation of the upper bound of the difference (3.15), one more lemma and some propositions should be considered.

Lemma 3.4.

The inequality (3.16) holds eJpti=1ν[eαit(m=0pi-1((tmm!)2(pi-m)))1/2].

Proof.

Considering (1.7), the block matrix Jp is given by Jp=[Jp1(α1)Jp2(α2)Jpν(αν)], where α1,α2,,αν are nonzero finite elementary divisors of matrix pencil sE-A; consult Section 1 for more details. Moreover, Jpi(αi)=[αi1αiαi]pi×pifor  every  i=1,2,,v. Then eJpt=[eJp1(α1)teJp2(α2)teJpν(αν)t]with    p1+p2++pν=p. Considering the well-known property x+yx+y, it is derived that eJpteJp1(α1)t++eJpν(αν)t=i=1νeJpi(αi)t. Note that eJpi(αi)t=eαit[1tt22!t(pi-1)(pi-1)!01tt(pi-2)(pi-2)!t1]. Consequently, by using some basic properties of the Euclidean norm, we take eJpi(αi)t=[(eαit)2pi+(teαit)2(pi-1)+(t22!eαit)2(pi-2)++(pi-(pi-1))(tpi-1(pi-1)!eαit)2]1/2=[e2αitm=0pi-1((tmm!)2(pi-m))]1/2=eαit[m=0pi-1((tmm!)2(pi-m))]1/2. Thus, we conclude to the following result: eJpi(αi)t=eαit[m=0pi-1(tmm!)2(pi-m)]1/2, for every i=1,2,,ν. Hence, inequality (3.16) holds.

Proposition 3.5.

The inequality (3.24) holds Φ((k-j)T,0)Q·Q-1((i=1νeαiT(m=0pi-1(Tmm!)2(pi-m))1/2)k-j+q).

Proof.

We known that Φ((k-j)T,0)=Q[eJp(k-j)T𝕆p,q𝕆q,pIq]Q-1. Then, using also the well-known property of the Euclidean norm x+yx+y, we obtain Φ((k-j)T,0)Q(eJp(k-j)T+Iq)Q-1=Q·Q-1(eJp(k-j)T+q), since we have Iq=12+12++12q=q.

Moreover,

eJp(k-j)T=eJpTeJpTeJpT(eJpT)k-j. Considering, now, Lemma 3.4, we have eJp(κ-j)T(i=1ν[eαiT(m=0pi-1((Tmm!)2(pi-m)))1/2])k-j. Thus, the statement of Proposition 3.5 derives.

Proposition 3.6.

The inequality (3.29) holds Φ(-λ,0)-Φ(λ-T,0)Q·Q-1·{i=1ν  [e-αiλ(m=0pi-1((-1)mλmm!)2(pi-m))1/2+eαi(λ-T)(m=0pi-1((λ-T)mm!)2(pi-m))1/2]}

Proof.

It is known that Φ(-λ,0)-Φ(λ-T,0)=Q[e-Jpλ𝕆p,q𝕆q,pIq]Q-1-Q[eJp(λ-T)𝕆p,q𝕆q,pIq]Q-1=Q[e-Jpλ-eJp(λ-T)𝕆p,q𝕆q,p𝕆p]Q-1. Then, the norm is Φ(-λ,0)-Φ(λ-T,0)Q·e-Jpλ-eJp(λ-T)·Q-1Q·Q-1(e-Jpλ+eJp(λ-T)). Considering also Lemma 3.4, (3.29) derives.

Now, the whole discussion of this section is completed with the statement of Theorem 3.7 whose proof is based on Lemmas 3.1 and 3.2 and Propositions 3.5 and 3.6.

Theorem 3.7.

The upper bound for the error x̲(kT)-x̲k, which is derived during the discretization process, is given by x̲(kT)-x̲kj=0k-1[{(i=1ν[eαiT(m=0pi-1(Tmm!)2(pi-m))1/2])k-j+q}·0T{i=1ν[e-αiλ(m=0pi-1((-1)m·λmm!)2(pi-m))1/2+eαi(λ-T)(m=0pi-1((λ-T)mm!)2(pi-m))1/2]}dλ·u̲j]·Q2·Q-12·Qn,p·Bp,l.

Now, with the upper bound derived by (3.32), we have an analytic expression that penalizes our choice for the sampling period T. Although, the above expression is very complicated with a lot of parameters involved, it provides an insightful overview of the error obtained during the discretization (or sampling) process. The following Remark evaluates our upper bound. According to the existing literature, see for instance  and references therein, when the sampling period T tends to zero, we are waiting to obtain x̲(kT)-x̲k0; that is, the continuous-time solution at time t=kT is exactly the same with the discrete-time points x̲k.

Remark 3.8.

Consider that the sampling period T tends to zero, that is, T0. In the integral, the function consists exponential and polynomial, continuous functions with respect to λ. Thus, it is well known from the basic Calculus that limx00xf(t)dt=limx0(F(x)-F(0))=limx0F(x)-F(0)=F(0)-F(0)=0. Moreover, in our case since T0, we obtain limT00T{i=1ν[e-αiλ(m=0pi-1((-1)m·λmm!)2(pi-m))1/2+eαi(λ-T)(m=0pi-1((λ-T)mm!)2(pi-m))1/2]}dλ=0. Then limT0j=0k-1[{(i=1ν[eαiT(m=0pi-1(Tmm!)2(pi-m))1/2])k-j+q}·0T{i=1ν[e-αiλ(m=0pi-1((-1)m·λmm!)2(pi-m))1/2+eαi(λ-T)(m=0pi-1((λ-T)mm!)2(pi-m))1/2i=1ν[eαiT(m=0pi-1(Tmm!)2(pi-m))12]]}dλ·u̲j]·Q2·Q-12·Qn,p·Bp,l=0 since ν,k,pi for every i=1,,ν are finite.

Consequently, we have seen that if the sampling period T tends to zero, that is, T0, then the solution of the discretized system (1.1), that is, (2.4) (or (2.27)), is an efficient approximation of the relative solution of continuous-time system, that is, (1.13).

4. Conclusion

In this paper, we have presented and fully discussed a technique for discretizing efficiently the solution of a linear descriptor (regular) differential input system with consistent initial conditions, and time-invariant coefficients. Additionally, according to the authors’ knowledge, for the first time an upper bound for the error x̲(kT)-x̲k that derives from the procedure of discretization is provided. This error expression penalizes our choice for the sampling period T through the notion of the ·-norm of the difference between the continuous-time solution x̲(kT), at time t=kT, and the relevant discrete-time points x̲k.

Finally, the results of this paper can be further enriched by the analytic determination of an appropriate sampling period interval, that is, T[a,b], which is a future plan of research. An optimal upper bound is also under consideration. Moreover, we are also interested in investigating the case that the non-consistent initial conditions are considered, since distributional solutions are derived.

Acknowledgments

The project is cofunded by the European Social Fund and National Resources—THALIS I. The authors are very grateful to the anonymous referees for their comments, which improved the quality of the paper.

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