Advances in Decision Sciences
Volume 2010 (2010), Article ID 810605, 19 pages
doi:10.1155/2010/810605
Research Article

Discretizing LTI Descriptor (Regular) Differential Input Systems with Consistent Initial Conditions

Athanasios D. Karageorgos,1 Athanasios A. Pantelous,2 and Grigoris I. Kalogeropoulos1

1Department of Mathematics, University of Athens, 15784 Athens, Greece
2Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK

Received 4 April 2009; Accepted 18 January 2010

Academic Editor: Graeme Wake

Copyright © 2010 Athanasios D. Karageorgos et al. This 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.

Abstract

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 ( 𝑘 𝑇 ) − 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 𝐸 ̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝑢 ( 𝑡 ) , ( 1 . 1 ) where matrices 𝐸 , 𝐴 ∈ ℳ ( 𝑛 × 𝑛 , 𝔽 ) (i.e., ℳ is the algebra of square matrices with elements in the field 𝔽 ) and 𝐵 ∈ ℳ ( 𝑛 × l , 𝔽 ) are constants; the state 𝑥 ( 𝑡 ) ∈ ℳ ( 𝑛 × 1 , 𝔽 ) has consistent initial conditions. We shall call 𝑥 0   a consistent initial condition for (1.1) at 𝑡 𝑜 if there is a solution for (1.1), which is defined on some interval [ 𝑡 𝑜 , 𝑡 𝑜 + 𝛾 ] , 𝛾 > 0 such that 𝑥 ( 𝑡 𝑜 ) = 𝑥 𝑜 ∈ ℳ ( 𝑛 × 1 , 𝔽 ) , the input 𝑢 ( 𝑡 ) ∈ ℳ ( 𝑙 × 1 , 𝔽 ) , and the d e t 𝐸 = 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, 𝑠 𝐸 − 𝐴 . Moreover, for the sake of simplicity, we set in the sequel ℳ 𝑛 = ℳ ( 𝑛 × 𝑛 ; 𝔽 ) and ℳ 𝑛 𝑚 = ℳ ( 𝑛 × 𝑚 ; 𝔽 ) .

Now, in what it follows, the pencil 𝑠 𝐸 − 𝐴 is regular, that is, d e t ( 𝑠 𝐸 − 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 [1–6], 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, 7–12], and so forth.

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

(i)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.(ii)Second, according to the authors’ knowledge, for the first time an upper bound for the error ‖ 𝑥 ( 𝑘 𝑇 ) − 𝑥 𝑘 ‖ , 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 𝑥 ( 𝑘 𝑇 ) , at time 𝑡 = 𝑘 𝑇 , and the relevant discrete-time points 𝑥 𝑘 .

This investigation is relevant to and it extends further the recent work proposed by Karampetakis; see [13] and Kalogeropoulos et al.; see [14]. 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 [16]. 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 [2–4, 17–22] et al. Thus, the class of strict equivalence is characterized by a uniquely defined element, known as a WCF, that is, 𝑠 𝐸 𝑤 − 𝐴 𝑤 . Consequently, when the pencil 𝑠 𝐸 − 𝐴 is regular, we have elementary divisors of the following type:

(i)e.d. of the type 𝑠 𝑑 , 𝑑 ∈ ℕ , are called zero finite elementary divisors (z. f.e.d.);(ii)e.d. of the type ( 𝑠 − 𝑎 ) 𝑐 , 𝑎 ≠ 0 , 𝑐 ∈ ℕ are called nonzero finite elementary divisors (nz. f.e.d.);(iii)e.d. of the type ̂ 𝑠 𝑞 are called infinite elementary divisors (i.e.d.).

Then, the Weierstrass canonical form of the regular pencil 𝑠 𝐸 − 𝐴 is defined by

𝑠 𝐸 𝑤 − 𝐴 𝑤 ≜ b l o c k d i a g 𝑠 𝐼 𝑝 − 𝐽 𝑝 , 𝑠 𝐻 𝑞 − 𝐼 𝑞 , ( 1 . 2 )

where the first normal Jordan type block 𝑠 𝐼 𝑝 − 𝐽 𝑝 is uniquely defined by the set of f.e.d.

𝑠 − 𝑎 1 𝑝 1 , … , 𝑠 − 𝑎 𝜈 𝑝 𝜈 , 𝜈 𝑗 = 1 𝑝 𝑗 = 𝑝 ( 1 . 3 )

of 𝑠 𝐹 − 𝐺 and has the form

𝑠 𝐼 𝑝 − 𝐽 𝑝 ≜ b l o c k d i a g 𝑠 𝐼 𝑝 1 − 𝐽 𝑝 1 𝑎 1 , … , 𝑠 𝐼 𝑝 𝜈 − 𝐽 𝑝 𝜈 𝑎 𝜈 . ( 1 . 4 )

And the 𝑞 blocks of the second uniquely defined block 𝑠 𝐻 𝑞 − 𝐼 𝑞 correspond to the i.e.d.

( ̂ 𝑠 ) 𝑞 1 , … , ( ̂ 𝑠 ) 𝑞 𝜎 , 𝜎 𝑗 = 1 𝑞 𝑗 = 𝑞 ( 1 . 5 )

of 𝑠 𝐸 − 𝐴 and has the form

𝑠 𝐻 𝑞 − 𝐼 𝑞 ≜ b l o c k d i a g 𝑠 𝐻 𝑞 1 − 𝐼 𝑞 1 , … , 𝑠 𝐻 𝑞 𝜎 − 𝐼 𝑞 𝜎 . ( 1 . 6 )

Thus the 𝐻 𝑞 is a nilpotent matrix of index 𝑞 ∗ = m a x { 𝑞 𝑗 ∶ 𝑗 = 1 , 2 , … , 𝜎 } , where 𝐻 𝑞 ∗ 𝑞 = 𝕆 and 𝐼 𝑝 𝑖 , 𝐽 𝑝 𝑖 ( 𝑎 𝑖 ) , 𝐻 𝑞 𝑖 are the matrices

𝐼 𝑝 𝑖 = ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ⎤ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑝 𝑖 × 𝑝 𝑖 , ℝ , 𝐻 𝑞 𝑖 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 0 1 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑞 𝑖 × 𝑞 𝑖 , ℝ , 𝐽 𝑝 𝑖 𝑎 𝑖 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑎 𝑖 1 0 ⋯ 0 0 𝑎 𝑖 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 𝑎 𝑖 1 0 0 0 0 𝑎 𝑖 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ∈ ℳ 𝑝 𝑖 × 𝑝 𝑖 , ℂ . ( 1 . 7 )

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

𝑥 ( 𝑡 ) = 𝑄 𝑦 ( 𝑡 ) , ( 1 . 8 ) 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 [17].

Theorem 1.1 (see [4, 23]). System (1.1) has the following solution: 𝑥 ( 𝑡 ) = 𝑄 𝑛 , 𝑝 𝑒 𝐽 𝑝 ( 𝑡 − 𝑡 𝑜 ) 𝑦 𝑝 𝑡 𝑜 + 𝑡 𝑡 𝑜 𝑒 𝐽 𝑝 ( 𝑡 − 𝑠 ) 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 − 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑡 ) , ( 1 . 9 ) where 𝑄 = [ 𝑄 𝑛 , 𝑝 𝑄 𝑛 , 𝑞 ] , 𝐵 = [ 𝐵 𝑝 , 𝑙 𝐵 𝑞 , 𝑙 ] , and 𝑦 ( 𝑡 𝑜 ) = [ 𝑦 𝑝 ( 𝑡 𝑜 ) 𝑦 𝑞 ( 𝑡 𝑜 ) ] = 𝑄 − 1 𝑥 ( 𝑡 𝑜 ) .

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

𝑥 ( 𝑡 ) = 𝑄 𝑛 , 𝑝 𝑒 𝐽 𝑝 ( 𝑡 − 𝑡 𝑜 ) 𝑦 𝑝 𝑡 𝑜 + 𝑄 𝑛 , 𝑞 𝑦 𝑞 𝑡 𝑜 + 𝑄 𝑛 , 𝑝 𝑡 𝑡 𝑜 𝑒 𝐽 𝑝 ( 𝑡 − 𝑠 ) 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 − 𝑄 𝑛 , 𝑞 𝑦 𝑞 𝑡 𝑜 − 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑡 ) = 𝑄 𝑛 , 𝑝 𝑄 𝑛 , 𝑞 𝑒 𝐽 𝑝 ( 𝑡 − 𝑡 𝑜 ) 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 ⎡ ⎢ ⎣ 𝑦 𝑝 𝑡 𝑜 𝑦 𝑞 𝑡 𝑜 ⎤ ⎥ ⎦ + 𝑄 𝑛 , 𝑝 𝑡 𝑡 𝑜 𝑒 𝐽 𝑝 ( 𝑡 − 𝑠 ) 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 − 𝑦 𝑞 𝑡 𝑜 − 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑡 ) . ( 1 . 1 0 )

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

⎡ ⎢ ⎣ 𝑦 𝑝 𝑡 𝑜 𝑦 𝑞 𝑡 𝑜 ⎤ ⎥ ⎦ = 𝑄 − 1 𝑥 𝑡 𝑜 , − 𝑦 𝑞 𝑡 𝑜 = 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) 𝑡 𝑜 . ( 1 . 1 1 )

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

𝑥 ( 𝑡 ) = 𝑄 𝑒 𝐽 𝑝 ( 𝑡 − 𝑡 𝑜 ) 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 𝑄 − 1 𝑥 𝑡 𝑜 + 𝑄 𝑛 , 𝑝 𝑡 𝑡 𝑜 𝑒 𝐽 𝑝 ( 𝑡 − 𝑠 ) 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) 𝑡 𝑜 − 𝑢 ( 𝑖 ) ( 𝑡 ) . ( 1 . 1 2 )

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

Φ 𝑡 , 𝑡 𝑜 = 𝑄 𝑒 𝐽 𝑝 ( 𝑡 − 𝑡 𝑜 ) 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 𝑄 − 1 . ( 1 . 1 3 )

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

𝑥 ( 𝑡 ) = Φ 𝑡 , 𝑡 𝑜 𝑥 𝑡 𝑜 + 𝑡 𝑡 𝑜 Φ ( 𝑡 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) 𝑡 𝑜 − 𝑢 ( 𝑖 ) ( 𝑡 ) , ( 1 . 1 4 ) where 𝑥 ( 𝑡 𝑜 ) ∈ 𝒯 belongs to the space of consistent initial conditions of system (1.1), that is, 𝒯 = 𝑥 ∈ ℳ ( 𝑛 × 1 , ℂ ) ∶ 𝑥 = 𝑄 𝑛 , 𝑝 𝐶 𝑝 − 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) 𝑡 𝑜 , w i t h 𝐶 𝑝 ∈ ℳ ( 𝑝 × 1 , ℂ ) . ( 1 . 1 5 )

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 [24], concerning the solution of regular systems in terms of the Laurent expansion terms of ( 𝑠 𝐸 − 𝐴 ) − 1 . The methodology of both papers, that is, [13] and the present, is equivalent, since they use the zero-order hold approximation.

Moreover, specifically for regular systems, Rachid, see [21], 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 𝑡 𝑜 = 0 , and the input function 𝑢 ( 𝑡 ) changes only through the time moments 𝑡 𝑘 = 𝑘 𝑇 for 𝑘 = 1 , 2 , … , that is,

𝑢 ( 𝑡 ) = 𝑢 ( 𝑘 𝑇 ) = 𝑢 𝑘 , ( 2 . 1 ) for every 𝑡 ∈ [ 𝑘 𝑇 , ( 𝑘 + 1 ) 𝑇 ) .

In more details, (2.1) implies that the input 𝑢 ( 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 𝑡 𝑘 = 𝑘 𝑇 until the next sampling period at time 𝑡 𝑘 + 1 = ( 𝑘 + 1 ) 𝑇 takes place.

Thus, the solution (1.14) is given by

𝑥 ( 𝑡 ) = Φ ( 𝑡 , 0 ) 𝑥 0 + 𝑡 0 Φ ( 𝑡 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑡 ) , ( 2 . 2 ) where 𝑥 0 = 𝑥 ( 0 ) ∈ 𝒯 .

Moreover, hereafter, we use the notation

𝑥 𝑘 ≡ 𝑥 ( 𝑘 𝑇 ) ∀ 𝑘 = 0 , 1 , 2 , … . ( 2 . 3 )

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) 𝑥 𝑘 = Φ ( 𝑘 𝑇 , 0 ) 𝑥 0 + 𝑘 − 1 𝑗 = 0 𝑇 0 Φ ( 𝑗 𝑇 + 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 𝑘 − 1 − 𝑗 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) . ( 2 . 4 )

Proof. First, we consider (2.2) at time moments 𝑡 = 𝑘 𝑇 and 𝑡 = ( 𝑘 + 1 ) 𝑇 .
Thus, we obtain for the time 𝑡 = 𝑘 𝑇
𝑥 𝑘 = Φ ( 𝑘 𝑇 , 0 ) 𝑥 0 + 𝑘 𝑇 0 Φ ( 𝑘 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) , ( 2 . 5 ) and for the time 𝑡 = ( 𝑘 + 1 ) 𝑇 𝑥 𝑘 + 1 = Φ ( ( 𝑘 + 1 ) 𝑇 , 0 ) 𝑥 0 + ( 𝑘 + 1 ) 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( ( 𝑘 + 1 ) 𝑇 ) . ( 2 . 6 ) The state-transition matrix (1.13) satisfies the following two properties: Φ ( 𝑇 , 0 ) ⋅ Φ ( 𝑘 𝑇 , 0 ) = Φ ( ( 𝑘 + 1 ) 𝑇 , 0 ) , ( 2 . 7 ) Φ ( 𝑇 , 0 ) ⋅ Φ ( 𝑘 𝑇 , 𝑠 ) = Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) . ( 2 . 8 ) 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 ) 𝑥 𝑘 = Φ ( ( 𝑘 + 1 ) 𝑇 , 0 ) 𝑥 0 + 𝑘 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + Φ ( 𝑇 , 0 ) 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) . ( 2 . 9 ) Consequently, by also using (2.6) and (2.9), we take the difference 𝑥 𝑘 + 1 − Φ ( 𝑇 , 0 ) 𝑥 𝑘 = ( 𝑘 + 1 ) 𝑇 𝑘 𝑇 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( ( 𝑘 + 1 ) 𝑇 ) − Φ ( 𝑇 , 0 ) 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) . ( 2 . 1 0 ) Afterwards, we should define 𝑄 − 1 = ⎡ ⎢ ⎣ 𝑄 𝑝 , 𝑛 𝑄 𝑝 , 𝑛 ⎤ ⎥ ⎦ , 𝑄 − 1 𝑄 = ⎡ ⎢ ⎣ 𝑄 𝑝 , 𝑛 𝑄 𝑞 , 𝑛 ⎤ ⎥ ⎦ 𝑄 = ⎡ ⎢ ⎣ 𝑄 𝑝 , 𝑛 𝑄 𝑞 , 𝑛 ⎤ ⎥ ⎦ 𝑄 𝑛 , 𝑝 𝑄 𝑛 , 𝑞 = ⎡ ⎢ ⎣ 𝑄 𝑝 , 𝑛 𝑄 𝑛 , 𝑝 𝑄 𝑝 , 𝑛 𝑄 𝑛 , 𝑞 𝑄 𝑞 , 𝑛 𝑄 𝑛 , 𝑝 𝑄 𝑞 , 𝑛 𝑄 𝑛 , 𝑞 ⎤ ⎥ ⎦ = 𝐼 𝑛 , ( 2 . 1 1 ) with 𝑄 𝑝 , 𝑛 𝑄 𝑛 , 𝑝 = 𝐼 𝑝 , 𝑄 𝑝 , 𝑛 𝑄 𝑛 , 𝑞 = 𝕆 𝑝 , 𝑞 , 𝑄 𝑞 , 𝑛 𝑄 𝑛 , 𝑝 = 𝕆 𝑞 , 𝑝 , 𝑄 𝑞 , 𝑛 𝑄 𝑛 , 𝑞 = 𝐼 𝑞 . ( 2 . 1 2 ) Thus, Φ ( 𝑇 , 0 ) 𝑄 𝑛 , 𝑞 = 𝑄 𝑒 𝐽 𝑝 𝑇 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 𝑄 − 1 𝑄 𝑛 , 𝑞 = 𝑄 𝑛 , 𝑝 𝑄 𝑛 , 𝑞 𝑒 𝐽 𝑝 𝑇 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 ⎡ ⎢ ⎣ 𝑄 𝑝 𝑛 𝑄 𝑛 𝑞 𝑄 𝑞 𝑛 𝑄 𝑛 𝑞 ⎤ ⎥ ⎦ = 𝑄 𝑛 , 𝑝 𝑄 𝑛 , 𝑞 𝑒 𝐽 𝑝 𝑇 𝕆 𝑝 , 𝑞 𝕆 𝑞 , 𝑝 𝐼 𝑞 𝕆 𝑝 , 𝑞 𝐼 𝑞 = 𝑄 𝑛 , 𝑝 , 𝑄 𝑛 , 𝑞 𝕆 𝑝 , 𝑞 𝐼 𝑞 = 𝑄 𝑛 , 𝑞 . ( 2 . 1 3 ) So Φ ( 𝑇 , 0 ) 𝑄 𝑛 , 𝑞 = 𝑄 𝑛 , 𝑞 . Consequently, (2.10) is transformed to (2.14) 𝑥 𝑘 + 1 = Φ ( 𝑇 , 0 ) 𝑥 𝑘 + ( 𝑘 + 1 ) 𝑇 𝑘 𝑇 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) − 𝑢 ( 𝑖 ) ( ( 𝑘 + 1 ) 𝑇 ) . ( 2 . 1 4 ) Note that by denoting 𝑠 = 𝑘 𝑇 + 𝑤 , the integral ∫ ( 𝑘 + 1 ) 𝑇 𝑘 𝑇 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 is given by ( 𝑘 + 1 ) 𝑇 𝑘 𝑇 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 = 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑘 𝑇 + 𝑤 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑘 𝑇 + 𝑤 ) 𝑑 𝑤 ( 2 . 1 ) = 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑘 𝑇 + 𝑤 ) 𝑑 𝑤 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 𝑘 . ( 2 . 1 5 ) Moreover, by using (2.7) and (2.8), we obtain 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑘 𝑇 + 𝑤 ) 𝑑 𝑤 = 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑘 𝑇 ) Φ ( 0 , 𝑤 ) 𝑑 𝑤 . ( 2 . 1 6 ) And considering again (2.8), we have Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑘 𝑇 ) = Φ ( 𝑇 , 0 ) ⋅ Φ ( 𝑘 𝑇 , 𝑘 𝑇 ) = Φ ( 𝑇 , 0 ) ( 2 . 1 7 ) because Φ ( 𝑘 𝑇 , 𝑘 𝑇 ) = 𝐼 𝑛 . Finally, we obtain 𝑇 0 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑘 𝑇 + 𝑤 ) 𝑑 𝑤 = 𝑇 0 Φ ( 𝑇 , 0 ) Φ ( 0 , 𝑤 ) 𝑑 𝑤 = 𝑇 0 Φ ( 𝑇 , 𝑤 ) 𝑑 𝑤 = 𝑇 0 Φ ( 𝑇 − 𝑤 , 0 ) 𝑑 𝑤 = 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 , ( 2 . 1 8 ) when we define 𝜆 = 𝑇 − 𝑤 . Thus, ( 𝑘 + 1 ) 𝑇 𝑘 𝑇 Φ ( ( 𝑘 + 1 ) 𝑇 , 𝑠 ) 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 ( 𝑠 ) 𝑑 𝑠 = 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 𝑘 . ( 2 . 1 9 ) Combining (2.14) and (2.19), we take the recursive formula (2.20) 𝑥 𝑘 + 1 = Φ ( 𝑇 , 0 ) 𝑥 𝑘 + 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 𝑘 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) − 𝑢 ( 𝑖 ) ( ( 𝑘 + 1 ) 𝑇 ) . ( 2 . 2 0 ) Then, the formula (2.4) is obtained by (2.20). Analytically,
(i) for 𝑘 = 0 , we take
𝑥 1 = Φ ( 𝑇 , 0 ) 𝑥 0 + 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 0 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑇 ) ; ( 2 . 2 1 )
(ii) for 𝑘 = 1 , we have
𝑥 2 = Φ ( 𝑇 , 0 ) 𝑥 1 + 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 1 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑇 ) − 𝑢 ( 𝑖 ) ( 2 𝑇 ) = Φ ( 𝑇 , 0 ) Φ ( 𝑇 , 0 ) 𝑥 0 + 𝑇 0 Φ ( 𝑇 , 0 ) Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 0 + Φ ( 𝑇 , 0 ) 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 𝑇 ) + 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 1 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 𝑇 ) − 𝑢 ( 𝑖 ) ( 2 𝑇 ) . ( 2 . 2 2 ) Since Φ ( 𝑇 , 0 ) 𝑄 𝑛 , 𝑞 = 𝑄 𝑛 , 𝑞 , we conclude to 𝑥 2 = Φ ( 2 𝑇 , 0 ) 𝑥 0 + 𝑇 0 Φ ( 𝑇 + 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 0 + 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 𝑝 𝐵 𝑝 , 𝑙 𝑢 1 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 2 𝑇 ) ; ( 2 . 2 3 )
(iii) for 𝑘 = 3 , we take
𝑥 3 = Φ ( 3 𝑇 , 0 ) 𝑥 0 + 𝑇 0 Φ ( 2 𝑇 + 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 0 + 𝑇 0 Φ ( 𝑇 + 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 ⋅ 𝐵 𝑝 , 𝑙 𝑢 1 + 𝑇 0 Φ ( 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 2 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑢 ( 𝑖 ) ( 0 ) − 𝑢 ( 𝑖 ) ( 3 𝑇 ) . ( 2 . 2 4 ) 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 𝑢 ( 𝑖 ) 𝑘 ≡ 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) with expressions of 𝑢 ( 𝑘 𝑇 ) . Thus, the following result is used, see [25],

𝑢 ( 𝑖 ) 𝑘 = 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) ≈ ∑ 𝑖 𝑗 = 0 ( − 1 ) 𝑗 𝑖 𝑗 𝑢 𝑘 − 𝑖 + 𝑗 𝑇 𝑖 . ( 2 . 2 5 )

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

𝑢 ( 𝑖 ) 𝑘 − 𝑢 ( 𝑖 ) 𝑘 + 1 ≈ ∑ 𝑖 + 1 𝑗 = 0 ( − 1 ) 𝑗 + 1 𝑖 + 1 𝑗 𝑢 𝑘 − 𝑖 + 𝑗 𝑇 𝑖 + 1 . ( 2 . 2 6 )

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 ]) 𝑥 𝑘 = Φ ( 𝑘 𝑇 , 0 ) 𝑥 0 + 𝑘 − 1 𝑗 = 0 𝑇 0 Φ ( 𝑗 𝑇 + 𝜆 , 0 ) 𝑑 𝜆 ⋅ 𝑄 𝑛 , 𝑝 𝐵 𝑝 , 𝑙 𝑢 𝑘 − 1 − 𝑗 + 𝑄 𝑛 , 𝑞 𝑞 ∗ − 1 𝑖 = 0 1 𝑇 𝑖 + 1 𝐻 𝑖 𝑞 𝐵 𝑞 , 𝑙 𝑖 + 1 𝑗 = 0 ( − 1 ) 𝑗 + 1 𝑖 + 1 𝑗 𝑢 𝑘 − 1 + 𝑗 . ( 2 . 2 7 )

Remark 2.3. Equation (2.20) reminds us with a new discrete-time system of the form 𝑥 𝑘 + 1 = 𝐴 𝑥 𝑘 + 𝐵 ( 𝜎 ) 𝑢 𝑘 , ( 2 . 2 8 ) where 𝐴 = Φ ( 𝑇 , 0 ) , whereas the differences 𝑢 ( 𝑖 ) ( 𝑘 𝑇 ) − 𝑢 ( 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 [13].

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 ‖ 𝑥 ( 𝑘 𝑇 ) − 𝑥 𝑘 ‖ , 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 = 𝑘 𝑇 , 𝑥 ( 𝑘 𝑇 ) and the relevant discrete-time points 𝑥 𝑘 , see (2.4), that is, we are interested in the difference ‖ 𝑥 ( 𝑘 𝑇 ) − 𝑥 𝑘