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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

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

Now, in what it follows, the pencil

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 [

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

Second, according to the authors’ knowledge, for the first time an upper bound for the error

This investigation is relevant to and it extends further the recent work proposed by Karampetakis; see [

Recently, in [

For reasons of convenience, some basic concepts and definitions from matrix pencil theory are introduced; see for more details [

Then, the Weierstrass canonical form of the regular pencil

where the first normal Jordan type block

of

And the

of

Thus the

Now, considering [

System (

However, (

In order the system (

Consequently, we obtain the desired expression for (

Moreover, by definition, the state-transition matrix of the autonomous linear descriptor differential system, that is,

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

In this section, we provide a computationally efficient method for the analytic discretization of LTI descriptor regular differential systems with

Moreover, specifically for regular systems, Rachid, see [

However, in the present paper, a different to the above research works discretization technique for LTI descriptor differential

First, we denote

In more details, (

Thus, the solution (

Moreover, hereafter, we use the notation

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

An analytic formula for the discretized solution (

First, we consider (

Thus, we obtain for the time

(i) for

(ii) for

(iii) for

Now, in order to complete the discretization process of the solution (

Profoundly, based on (

The discrete-time solution which represents (

Equation (

The induction method (show that it is true for

In the next section, according to the authors’ knowledge, an upper bound for the error

In this section, we provide an analytic expression that penalizes our choice for the sampling period

Following the results of the previous sections, we obtain (

The solution (

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

The equality (

Consider that

The equality (

We define that

Thus, we obtain

Now, we consider Lemmas

The equality (

Making the transformation

Since

Following the results of Lemma

Consequently, (

For the calculation of the

Now, for the calculation of the upper bound of the difference (

The inequality (

Considering (_{p}

The inequality (

We known that

Moreover,

The inequality (

It is known that

Now, the whole discussion of this section is completed with the statement of Theorem

The upper bound for the error

Now, with the upper bound derived by (

Consider that the sampling period

Consequently, we have seen that if the sampling period

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

Finally, the results of this paper can be further enriched by the analytic determination of an appropriate sampling period

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.