Improving the Asymptotic Properties of Discrete System Zeros in Fractional-Order Hold Case

Abstract

Remarkable improvements in the asymptotic properties of discrete system zeros may be achieved by properly adjusted fractional-order hold (FROH) circuit. This paper analyzes asymptotic properties of the limiting zeros, as the sampling period 𝑇 tends to zero, of the sampled-data models on the basis of the normal form representation of the continuous-time systems with FROH. Moreover, when the relative degree of the continuous-time system is equal to one or two, an approximate expression of the limiting zeros for the sampled-data system with FROH is also given as power series with respect to a sampling period up to the third-order term. And, further, the corresponding stability conditions of the sampling zeros are discussed for fast sampling rates. The ideas of the paper here provide a more accurate approximation for asymptotic zeros, and certain known achievements on asymptotic behavior of limiting zeros are shown to be particular cases of the results presented.

1. Introduction

Zeros, along with poles, are fundamental characteristics of linear time-invariant systems, and the stability of zeros is one of the most important issues in the model matching and adaptive control problems. When a continuous-time system is discretized by the use of a sampler and a hold, the mapping between the discrete-time poles and their continuous-time counterparts is very simple; namely, stability of poles is reserved. There is unfortunately no simple transformation between the discrete-time zeros and their continuous-time ones because the zeros of discrete-time systems depend on sampling period𝑇[1]. More precisely, it is generally impossible to tranform a continuous-time system with zeros in the left-half plane to a discrete-time system with zeros inside the unit circle. That is to say, the stability of zeros is not necessarily preserved except in special cases. Therefore, one of the special cases (i.e., the limiting case) is that the sampling period𝑇tends to zero which has attracted considerable attention from the engineering point of view.

Perhaps the first attempt to study discrete system zeros was given by Åström and coworkers [1], who describe the asymptotic behavior of the discrete-time zeros for fast sampling rate when the original continuous-time plant is discretized with zero-order hold (ZOH). In this case, the discretized zeros are further called limiting zeros which are composed of the intrinsic zeros and sampling zeros [2]. The former ones have counterparts in the underlying continuous-time system and go to unity [3] while the latter ones, which have no continuous-time counterparts and are generated in the sampling process, go toward roots of a certain polynomial [4, 5] determined by relative degree of the continuous-time system.

In much of discussion about the properties of discrete-time zeros, ZOH has been mainly employed as a hold circuit since it is used most commonly in practice [1, 3, 6–10]. Taking into account the fact that the type of hold circuit used critically influences the position of zeros, it is an interesting problem to investigate the zeros in the case of various holds. Hagiwara et al. [4] have carried out a comparative study and demonstrated that a first-order hold (FOH) provides no advantage over ZOH as far as the stability of zeros of the resulting discrete-time systems is concerned. Passino and Antsaklis [11] have considered the fractional-order hold (FROH) as an alternative to the ZOH and shown that it can locate the zeros of discrete-time system inside the unit circle by some examples while ZOH fails to do so. In the very motivating work by Ishitobi [12], the properties of limiting zeros with FROH have been analyzed, and the corresponding pulse-transfer function has been also derived.

Moreover, Ishitobi has definitely presented the relationship between the relative degree and discretized zeros behavior when the continuous-time systems have the relative degree up to five for sufficiently small sampling periods. Further, Bàrcena et al. [13, 14] and Liang et al. [15, 16] have extended Ishitobi’s results [12] from different angles and methods by investigating the limiting zeros in the case of a FROH. In addition, the limiting FROH zeros [12] have been also extended by Blachuta [17], who describes the accuracy of the asymptotic results for both the intrinsic and sampling zeros in terms of Bernoulli numbers and parameters of the continuous-time transfer function for sufficiently small sampling periods𝑇.

In FROH case, the intrinsic zeros are located inside (resp., outside) the unit circle for small sampling periods when the corresponding continuous-time zeros lie strictly in the left-half plane (resp., right). For sampling zeros, at least one of the zeros lies strictly outside the unit circle if the relative degree of a continuous-time transfer function is greater than or equal to three [12, 18]. This fact indicates that even though all the zeros of such a continuous-time system are stable, the corresponding discrete-time system has at least one unstable zero in the limiting case as the sampling period tends to zero. Thus, attention is here focused on continuous-time systems with relative degree less than or equal to two. More specifically, the corresponding discrete-time plants have one or two sampling zero(s) in the case of a FROH when the relative degree of a continuous-time transfer function is one or two. However, in these cases, the sampling zeros are located just on the unit circle, that is, in the marginal case of the stability. More importantly, it is a valuable research topic to find the criteria which guarantee that stable discretized zeros are obtained. Thus, the asymptotic behavior of the sampling zeros is an interesting issue as we explore the stability properties of the sampling zeros by analyzing the asymptotic properties as the sampling period tends to zero.

The objective of this paper is to analyze the improved asymptotic properties of the limiting zeros for discrete-time models by using a new kind of method. More precisely, we give an approximate expression of limiting zeros for the sampled-data system on the basis of the normal form representation of continuous-time system with FROH as power series with respect to a sampling period up to the third-order term when the relative degree of the continuous-time system is one or two. Our results include also the finding of how close limiting zeros are to the actual intrinsic and sampling ones, irrespectively of whether they are stable or not. The approach used could be referred to as an extension of that of [12, 17, 18], and one of the principal contribution in this paper, in particular, would consequently propose an analytical method to obtain the FROH zeros as stable as possible, or with improved asymptotic properties even when unstable, for a given continuous-time plant. Finally, we further discuss the stability of the sampling zeros for sufficiently small sampling periods, and some interesting examples are given to validate the main results.

2. Sampled-Data Models with FROH

Consider an𝑛th continuous-time system with relative degree one or two described by a transfer function 𝐺(𝑠)𝑁(𝑠)=𝐾𝐷(𝑠),𝐾≠0, (1) where 𝑁(𝑠)=𝑠^𝑚+𝑏_𝑚−1𝑠^𝑚−1+𝑏_𝑚−2𝑠^𝑚−2+⋯+𝑏₀,𝐷(𝑠)𝑚=𝑛−1or𝑛−2,=𝑠^𝑛+𝑎_𝑛−1𝑠^𝑛−1+𝑎_𝑛−2𝑠^𝑛−2+⋯+𝑎₀. (2) The paper treats systems with relative degree one or two because at least one of the limiting zeros is unstable when the relative degree is greater than or equal to three though it is slightly a limitation.

2.1. Case of Relative Degree One (𝑚=𝑛−1)

The normal form of (1) with the relative degree one,𝑚=𝑛−1is represented with an input𝑢and an output𝑦[19, 20] as ̇̇𝜉=−𝑑𝜉+𝐾𝑢−𝜔,𝜼=𝑃𝜼+𝐪𝜉,𝑦=𝜉, (3) where 𝜼=𝜼₁⋯𝜼_𝑛−1^𝑇,𝜔=𝐜^𝑇𝜼,𝐜=𝑟₀𝑟₁⋯𝑟_𝑛−2^𝑇,⋱𝑃=01𝑂𝑂1−𝑏₀⋯−𝑏_𝑛−20⋮01,𝐪=, (4) and the scalars𝑑and 𝑟_𝑖 (𝑖=0,…,𝑛−2) are obtained from 𝐷(𝑠)(𝑠)𝑁(𝑠)(𝑠),𝑄(𝑠)𝑅(𝑠)=𝑄+𝑅=𝑠+𝑑,=𝑟_𝑛−2𝑠^𝑛−2+𝑟_𝑛−3𝑠^𝑛−3+⋯+𝑟₀, (5) where 𝑑=𝑎_𝑛−1−𝑏_𝑛−2,𝑟_𝑖=𝑎_𝑖−𝑏_𝑖−1−𝑏_𝑖𝑑,𝑖=0,…,𝑛−2. (6)

When the FROH signal reconstruction method is considered, the input is described by 𝑣(𝑡)()𝑢())𝑇)=𝑢𝑘𝑇+𝛽𝑘𝑇−𝑢((𝑘−1𝑇),()(𝑡−𝑘𝑇𝑘𝑇≤𝑡<𝑘+1𝑇,𝑘=0,1,…, (7) where𝛽is a real design parameter and𝑇is a sampling period [11, 12, 18]. It is obvious that FROH is reduced to ZOH for𝛽=0while it becomes the FOH for𝛽=1. The signal reconstruction of a FROH with𝛽=−0.5is shown in Figure 1.

Figure 1 

The signal reconstruction of a fractional-order hold with𝛽=−0.5.

Suppose𝑢(𝑡)=𝑣(𝑡), and when a FROH is applied, we have 𝑢̇(𝑡)𝑢())𝑇)=𝛽𝑘𝑇−𝑢((𝑘−1𝑇,𝑢̈(𝑡)=⋯=0. (8)

Furthermore, (3) leads to the derivatives of the output 𝑦̇=−𝑑𝜉+𝐾𝑢−𝐜^𝑇̈𝑑𝜼,𝑦=²−𝐜^𝑇𝐪𝜉−𝑑𝐾𝑢+𝑑𝐜^𝑇−𝐜^𝑇𝑃̇𝑦𝜼+𝐾𝑢,₍₃₎=−𝑑³+2𝑑𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪𝜉+𝑑²−𝐜^𝑇𝐪+𝐾𝑢𝑑𝐜^𝑇𝑃−𝐜^𝑇𝑃²−𝑑²𝐜^𝑇+𝐜^𝑇𝐪𝐜^𝑇̇𝑦𝜼−𝑑𝐾𝑢,₍₄₎=𝑑⁴−3𝑑²𝐜^𝑇𝐪+2𝑑𝐜^𝑇𝑃𝐪−𝐜^𝑇𝑃²𝐪+𝐜^𝑇𝐪²𝜉+−𝑑³+2𝑑𝐜^𝑇𝐪−𝐜^𝑇+𝑑𝑃𝐪𝐾𝑢₃𝐜^𝑇−2𝑑𝐜^𝑇𝐪𝐜^𝑇+𝐜^𝑇𝑃𝐪𝐜^𝑇+𝑑𝐜^𝑇𝑃²−𝐜^𝑇𝑃³−𝑑²𝐜^𝑇𝑃+𝐜^𝑇𝐪𝐜^𝑇𝑃𝜼+𝑑²−𝐜^𝑇𝐪𝐾𝑢̇ (9)(10)(11)(12) which are expressed by 𝜉, 𝜼, and 𝐾𝑢. Further, the derivatives of 𝜼 are also represented by 𝜉, 𝜼, and 𝐾𝑢 as ̇̈()𝑃𝜼=𝑃𝜼+𝐪𝜉,𝜼=𝑃𝐪−𝐪𝑑𝜉+²−𝐪𝐜^𝑇𝜼𝜼+𝐪𝐾𝑢,₍₃₎=𝑃²𝐪−𝐪𝐜^𝑇𝐪−𝑃𝐪𝑑+𝐪𝑑²()+𝑃𝜉+𝑃𝐪−𝐪𝑑𝐾𝑢₃−𝐪𝐜^𝑇𝑃−𝑃𝐪𝐜^𝑇+𝐪𝐜^𝑇𝑑𝜼+𝐪𝐾𝑢̇. (13)(14)(15) Hence, by substituting (9)–(15) into the right-hand side of 𝑦_𝑘+1=^∞_𝑖=0𝑇^𝑖𝑦𝑖!^𝑘(𝑖),𝜼_𝑘+1=^∞_𝑖=0𝑇^𝑖𝜼𝑖!^𝑘(𝑖) (16) and defining the state variables𝑥_𝑘=[𝑦_𝑘,𝜼^𝑇_𝑘]^𝑇, where the subscript𝑘denotes𝑡=𝑘𝑇, the discrete-time state equations are definitely obtained. It is easy to show that zeros of a discrete-time system for a transfer function (1) are derived from (16).

Now, by applying the explicit expressions of𝑦_𝑘,𝑦̇_𝑘,…,𝑦_𝑘⁽⁴⁾and𝜼_𝑘,…,𝜼_𝑘⁽³⁾, the zeros of (16) are analyzed as follows: 𝑦_𝑘+1=⁴_𝑖=0𝑇^𝑖𝑦𝑖!^𝑘(𝑖)+𝑂𝑇⁵=𝑑1−𝑑𝑇+²−𝑟_𝑛−22𝑇²+2𝑑𝑟_𝑛−2−𝑑³−𝐜^𝑇𝑃𝐪6𝑇³+𝑑⁴−3𝑑²𝑟_𝑛−2+2𝑑𝐜^𝑇𝑃𝐪𝑟²_𝑛−2−𝐜^𝑇𝑃²𝐪𝑇24⁴𝑦_𝑘+𝛽1+2𝑑𝑇−2+𝑑𝛽6𝑇²+4𝑑²−4𝑟_𝑛−2−𝑟_𝑛−2𝛽+𝑑²𝛽𝑇24³+2𝑑𝑟_𝑛−2−𝑑³−𝐜^𝑇𝑃𝐪𝑇24⁴𝐾𝑢_𝑘+−𝛽2𝑇+𝑑𝛽6𝑇²−𝑑²−𝑟_𝑛−2𝛽𝑇24³𝐾𝑢_𝑘−1+−𝐜^𝑇𝑇+𝑑𝐜^𝑇−𝐜^𝑇𝑃2𝑇²+𝑑𝐜^𝑇𝑃−𝐜^𝑇𝑃²+𝑟_𝑛−2−𝑑²𝐜^𝑇6𝑇³+𝑑³−2𝑑𝑟_𝑛−2𝐜^𝑇+𝐜^𝑇𝑃𝐪𝐜^𝑇+𝑑𝐜^𝑇𝑃²−𝐜^𝑇𝑃³−𝑑²𝐜^𝑇𝑃+𝑟_𝑛−2𝐜^𝑇𝑃×(24)⁻¹𝑇⁴𝜼_𝑘+𝑂𝑇⁵,𝜼_𝑘+1=³_𝑖=0𝑇^𝑖𝜼𝑖!^𝑘(𝑖)+𝑂𝑇⁴=𝐪𝑇+𝑃𝐪−𝐪𝑑2𝑇²+𝑃²𝐪−𝑟_𝑛−2𝐪+𝐪𝑑²−𝑃𝐪𝑑6𝑇³𝑦_𝑘+𝐪2+𝐪𝛽6𝑇²𝑃𝐪−𝐪𝑑6𝑇³𝐾𝑢_𝑘−𝐪𝛽6𝑇²𝐾𝑢_𝑘−1+𝑃𝐼+𝑃𝑇+²−𝐪𝐜^𝑇2𝑇²+𝑃³−𝐪𝐜^𝑇𝑃−𝑃𝐪𝐜^𝑇+𝑑𝐪𝐜^𝑇6𝑇³𝜼_𝑘+𝑂𝑇⁴. (17)(18) The reason why the explicit expressions of𝑦_𝑘,𝑦̇_𝑘,…,𝑦_𝑘⁽⁴⁾and𝜼_𝑘,…,𝜼_𝑘⁽³⁾are used is to obtain the approximate expansion of the limiting zeros for the discrete-time system with the order𝑇³.

2.2. Case of Relative Degree Two (𝑚=𝑛−2)

The normal form of (1) with the relative degree two,𝑚=𝑛−2is represented [19, 20] as ̇01𝝃=0100𝝃+𝐾𝑢−𝜔−𝑑₀𝜉₁−𝑑₁𝜉₂,𝜼̇=𝑃𝜼+𝐪𝜉₁,𝑦=10𝝃, (19) where 𝝃=𝜉₁𝜉₂^𝑇,𝜼=𝜼₁⋯𝜼_𝑛−2^𝑇,𝜔=𝐜^𝑇𝜼,𝐜=𝐜₀𝐜₁⋯𝐜_𝑛−3^𝑇,⋱𝑃=01𝑂𝑂1−𝑏₀⋯−𝑏_𝑛−4−𝑏_𝑛−30⋮01,𝐪=, (20) and the scalars 𝑑_𝑖 (𝑖=0,1) and 𝐜_𝑖 (𝑖=0,…,𝑛−3) are obtained from 𝐷(𝑠)(𝑠)𝑁(𝑠)(𝑠),𝑄(𝑠)=𝑄+𝑅=𝑠²+𝑑₁𝑠+𝑑₀,𝑅(𝑠)=𝐜_𝑛−3𝑠^𝑛−3+⋯+𝐜₀, (21) where 𝑑₀=𝑎_𝑛−2−𝑏_𝑛−4−𝑏_𝑛−3𝑑₁,𝑑₁=𝑎_𝑛−1−𝑏_𝑛−3,𝐜_𝑖=𝑎_𝑖−𝑏_𝑖−2−𝑏_𝑖−1𝑑₁−𝑏_𝑖𝑑₀,𝑖=0,…,𝑛−3. (22) When a FROH is used, the normal form (19) yields the derivatives of the output 𝑦̈=𝐾𝑢−𝑑₀𝜉₁−𝑑₁𝜉₂−𝐜^𝑇𝑦𝜼,₍₃₎=𝑑₀𝑑₁−𝐜^𝑇𝐪𝜉₁+𝑑²₁−𝑑₀𝜉₂−𝑑₁+𝑑𝐾𝑢₁𝐜^𝑇−𝐜^𝑇𝑃̇𝑦𝜼+𝐾𝑢,₍₄₎=𝑑²₀−𝑑₀𝑑²₁+𝑑₁𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪𝜉₁+−𝐜^𝑇𝐪+2𝑑₀𝑑₁−𝑑³₁𝜉₂+𝑑²₁−𝑑₀𝐾𝑢−𝑑₁𝐾𝑢̇+−𝑑²₁−𝑑₀𝐜^𝑇+𝑑₁𝐜^𝑇𝑃−𝐜^𝑇𝑃²𝑦𝜼,₍₅₎=𝑑₀𝑑³₁−𝑑²₁𝐜^𝑇𝐪+2𝐜^𝑇𝐪𝑑₀−2𝑑²₀𝑑₁−𝐜^𝑇𝑃²𝐪+𝑑₁𝐜^𝑇𝑃𝐪𝜉₁+𝑑²₀−3𝑑₀𝑑²₁+2𝑑₁𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪+𝑑⁴₁𝜉₂+−𝑑³₁+2𝑑₀𝑑₁−𝐜^𝑇𝐪+𝐜𝐾𝑢_𝑇𝐪𝐜^𝑇−2𝑑₀𝑑₁𝐜^𝑇+𝑑³₁𝐜^𝑇−𝑑²₁−𝑑₀𝐜^𝑇𝑃+𝑑₁𝐜^𝑇𝑃²−𝐜^𝑇𝑃³𝜼+𝑑²₁−𝑑₀𝐾𝑢̇. (23)(24)(25)(26)

Further, the derivatives of 𝜼 are also represented as 𝜼̇=𝑃𝜼+𝐪𝜉₁,𝜼̈=𝑃𝐪𝜉₁+𝐪𝜉₂+𝑃²𝜼𝜼,₍₃₎=𝑃²𝐪−𝐪𝑑₀𝜉₁+𝑃𝐪−𝐪𝑑₁𝜉₂+𝑃+𝐪𝐾𝑢₃−𝐪𝐜^𝑇𝜼𝜼,₍₄₎=−𝑃𝐪𝑑₀+𝐪𝑑₀𝑑₁+𝑃³𝐪−𝐪𝐜^𝑇𝐪𝜉₁+−𝐪𝑑₀+𝑃²𝐪−𝑃𝐪𝑑₁+𝐪𝑑²₁𝜉₂+𝑃𝐪−𝐪𝑑₁𝑢̇+𝐾𝑢+𝐪𝐾−𝑃𝐪𝐜^𝑇+𝐪𝑑₁𝐜^𝑇+𝑃⁴−𝐪𝐜^𝑇𝑃𝜼. (27)(28)(29)(30) Hence, by substituting (23)–(30) into the right-hand side of (16) and 𝑦̇_𝑘+1=^∞_𝑖=0𝑇^𝑖𝑦𝑖!^𝑘(𝑖+1) (31) and defining the state variables 𝑥_𝑘=[𝑦_𝑘,𝑦̇_𝑘,𝜼^𝑇_𝑘]^𝑇, the discrete-time state equations are obtained.

Now, by using the explicit expressions of 𝑦_𝑘,𝑦^′_𝑘,…,𝑦_𝑘⁽⁵⁾ and 𝜼_𝑘,…,𝜼_𝑘⁽⁴⁾, the zeros of the discrete-time system (16) and (31) are analyzed as follows: 𝑦_𝑘+1=⁵_𝑖=0𝑇^𝑖𝑦𝑖!^𝑘(𝑖)+𝑂𝑇⁶=𝑑1−⁰2𝑇²+𝑑₀𝑑₁−𝐜^𝑇𝐪6𝑇³+𝑑²₀−𝑑₀𝑑²₁+𝑑₁𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪𝑇24⁴−2𝐜^𝑇𝐪𝑑₀−2𝑑²₀𝑑₁+𝑑₀𝑑³₁−𝑑²₁𝐜^𝑇𝐪−𝑑₁𝐜^𝑇𝑃𝐪+𝐜^𝑇𝑃²𝐪×(120)⁻¹𝑇⁵𝑦_𝑘+𝑑𝑇−¹2𝑇²+𝑑²₁−𝑑₀6𝑇³+2𝑑₀𝑑₁−𝑑³₁−𝐜^𝑇𝐪𝑇24⁴+𝑑²₀−3𝑑₀𝑑²₁+2𝑑₁𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪+𝑑⁴₁𝑇120⁵𝑦̇_𝑘+12+𝛽6𝑇²−4𝑑₁+𝑑₁𝛽𝑇24³+5𝑑²₁−5𝑑₀+𝑑²₁−𝑑₀𝛽𝑇120⁴+2𝑑₀𝑑₁−𝐜^𝑇𝐪−𝑑³₁𝑇120⁵𝐾𝑢_𝑘+−𝛽6𝑇²+𝑑₁𝛽𝑇24³−𝑑²₁−𝑑₀𝛽𝑇120⁴𝐾𝑢_𝑘−1+−𝐜^𝑇2𝑇²+𝑑₁𝐜^𝑇−𝐜^𝑇𝑃6𝑇³+𝑑₁𝐜^𝑇𝑃−𝑑²₁−𝑑₀𝐜^𝑇−𝐜^𝑇𝑃²𝑇24⁴+𝐜^𝑇𝐪𝐜^𝑇−2𝑑₀𝑑₁𝐜^𝑇+𝑑³₁𝐜^𝑇𝑑₁𝐜^𝑇𝑃²−𝑑²₁−𝑑₀𝐜^𝑇𝑃−𝐜^𝑇𝑃³×(120)⁻¹𝑇⁵𝜼_𝑘+𝑂𝑇⁶,𝑦̇_𝑘+1=⁴_𝑖=0𝑇^𝑖𝑦𝑖!^𝑘(𝑖+1)+𝑂𝑇⁵=−𝑑₀𝑑𝑇+⁰𝑑₁−𝐜^𝑇𝐪2𝑇²+𝑑²₀−𝑑₀𝑑²₁+𝑑₁𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪6𝑇³+𝐜^𝑇𝑃²𝐪−𝑑₁𝐜^𝑇𝑃𝐪−2𝐜^𝑇𝐪𝑑₀+2𝑑²₀𝑑₁−𝑑₀𝑑³₁+𝑑²₁𝐜^𝑇𝐪×(24)⁻¹𝑇⁴𝑦_𝑘+1−𝑑₁𝑑𝑇+²¹−𝑑₀2𝑇²+2𝑑₀𝑑₁−𝑑³₁−𝐜^𝑇𝐪6𝑇³+𝑑²₀−3𝑑₀𝑑²₁+2𝑑₁𝐜^𝑇𝐪−𝐜^𝑇𝑃𝐪+𝑑⁴₁𝑇24⁴𝑦̇_𝑘+𝛽1+2𝑇−3𝑑₁+𝑑₁𝛽6𝑇²+4𝑑²₁−4𝑑₀+𝑑²₁−𝑑₀𝛽𝑇24³+2𝑑₀𝑑₁−𝐜^𝑇𝐪−𝑑³₁𝑇24⁴𝐾𝑢_𝑘+−𝛽2𝑑𝑇+¹𝛽6𝑇²−𝑑²₁−𝑑₀𝛽𝑇24³𝐾𝑢_𝑘−1+−𝐜^𝑇𝑑𝑇+¹𝐜^𝑇−𝐜^𝑇𝑃2𝑇²+𝑑₁𝐜^𝑇𝑃−𝐜^𝑇𝑃²−𝑑²₁−𝑑₀𝐜^𝑇6𝑇³+𝐜^𝑇𝐪𝐜^𝑇−2𝑑₀𝑑₁𝐜^𝑇+𝑑³₁𝐜^𝑇−𝑑²₁−𝑑₀𝐜^𝑇𝑃−𝐜^𝑇𝑃³+𝑑₁𝐜^𝑇𝑃²×(24)⁻¹𝑇⁴𝜼_𝑘+𝑂𝑇⁵,𝜼_𝑘+1=⁴_𝑖=0𝑇^𝑖𝜼𝑖!^𝑘(𝑖)+𝑂𝑇⁵=𝐪𝑇+𝑃𝐪2𝑇²+𝑃²𝐪−𝐪𝑑₀6𝑇³+𝑃³𝐪−𝐪𝐜^𝑇𝐪−𝑃𝐪𝑑₀+𝐪𝑑₀𝑑₁𝑇24⁴𝑦_𝑘+𝐪2𝑇²+𝑃𝐪−𝐪𝑑₁6𝑇³+𝑃²𝐪−𝐪𝑑₀𝑇24⁴+−𝑃𝐪𝑑₁+𝐪𝑑²₁𝑇24⁴𝑦̇_𝑘−𝐪𝛽𝑇24³𝐾𝑢_𝑘−1+𝐪6+𝐪𝛽𝑇24³+𝑃𝐪−𝐪𝑑₁𝑇24⁴𝐾𝑢_𝑘+𝑃𝐼+𝑃𝑇+²2𝑇²+𝑃³−𝐪𝐜^𝑇6𝑇³+𝑃⁴−𝑃𝐪𝐜^𝑇−𝐪𝐜^𝑇𝑃−𝑃𝐪𝐜^𝑇𝑇24⁴𝜼_𝑘+𝑂𝑇⁵. (32)(33)(34) Similarly, the reason why the explicit expressions of𝑦_𝑘,𝑦̇_𝑘,…,𝑦_𝑘⁽⁵⁾and𝜼_𝑘,…,𝜼_𝑘⁽⁴⁾are used is to obtain the approximate expansion of the limiting zeros for the discrete-time system with the order𝑇³when the relative degree of continuous-time systems is two.

3. Main Results

In the following, a more accurate approximate model of the sampled-data system is considered by neglecting the higher order terms, and the approximate expression of the limiting zeros is further calculated in this section.

3.1. Case of Relative Degree One (𝑚=𝑛−1)

An approximate expression of limiting zeros for the discrete-time system is derived from (17) and (18). The first result is given by the following theorem.

Theorem 1. The zeros of a discrete-time system corresponding to the continuous-time transfer function (1) with FROH are given for𝑇≪1approximately by the roots of 𝛽−1−2+3𝑑+𝑑𝛽6𝑇−4𝑑²−𝑟_𝑛−2+𝑑²−𝑟_𝑛−2𝛽𝑇24²−2𝑑𝑟_𝑛−2−𝑑³−𝐜^𝑇𝑃𝐪−5𝐜^𝑇𝑃𝐪𝛽+2𝐜^𝑇𝐪𝑑𝛽𝑇24³𝑧+𝛽2−𝑑𝛽6𝑑𝑇+²−𝑟_𝑛−2𝛽𝑇24²−5𝐜^𝑇𝑃𝐪𝛽−2𝐜^𝑇𝐪𝑑𝛽𝑇24³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+2𝑃³+𝐪𝐜^𝑇𝑃𝑇12³=0. (35)

Proof. The limiting zeros of the discrete-time system (16) are equivalent to zeros in (17) and (18), which are given by substituting𝑦_𝑘=𝑦_𝑘+1=0into (17) and (18) as follows: 𝑀₁𝐾𝑈_𝑘−1𝐻𝐾𝑈_𝑘=𝟎_𝑛, (36) where𝑈_𝑘−1,𝐻, and𝑈_𝑘are the𝑧-transforms of𝑢_𝑘−1,𝜼_𝑘, and𝑢_𝑘, respectively, and the matrix𝑀₁is defined by 𝑀₁=𝑚₁₁𝐦^𝑇₁₂𝑚₁₃−𝑧𝟎^𝑇1𝐦₃₁𝑀₃₂𝐦₃₃, (37) with 𝑚₁₁=𝑇𝑚₁₁+𝑂𝑇⁴,𝐦^𝑇₁₂=𝑇𝐦^𝑇₁₂+𝑂𝑇⁵,𝑚₁₃=𝑇𝑚₁₃+𝑂𝑇⁵,𝑚₁₁𝛽=−2+𝑑𝛽6𝑑𝑇−²−𝑟_𝑛−2𝛽𝑇24²,𝐦^𝑇₁₂=−𝐜^𝑇+𝑑𝐜^𝑇−𝐜^𝑇𝑃2𝑇+𝑑𝐜^𝑇𝑃−𝐜^𝑇𝑃²6𝑇²+𝑟_𝑛−2−𝑑²𝐜^𝑇6𝑇²+𝑑³−2𝑑𝑟_𝑛−2𝐜^𝑇𝑇24³+𝐜^𝑇𝑃𝐪𝐜^𝑇+𝑑𝐜^𝑇𝑃²−𝐜^𝑇𝑃³𝑇24³+−𝑑²𝐜^𝑇𝑃+𝑟_𝑛−2𝐜^𝑇𝑃𝑇24³,𝑚₁₃=𝛽1+2−𝑑2+𝑑𝛽6𝑇+𝑑²−𝑟_𝑛−26+𝑑²−𝑟_𝑛−2𝛽𝑇24²+−𝑑³+2𝑑𝑟_𝑛−2−𝐜^𝑇𝑃𝐪𝑇24³,𝐦₃₁=−𝐪𝛽6𝑇²+𝑂𝑇⁴,𝑀₃₂=()𝑃−𝑧+1𝐼+𝑃𝑇+²−𝐪𝐜^𝑇2𝑇²+𝑑𝐪𝐜^𝑇−𝑃𝐪𝐜^𝑇+𝑃³−𝐪𝐜^𝑇𝑃6𝑇³+𝑂𝑇⁴,𝐦₃₃=𝐪2+𝐪𝛽6𝑇²+𝑃𝐪−𝐪𝑑6𝑇³+𝑂𝑇⁴. (38) Thus, the zeros are derived from 𝑀₁=0. (39) From the relationship 𝑀=𝑀₁100010101, (40) it is obvious that the condition|𝑀|=0is equivalent to|𝑀₁|=0.
Expanding the result along the second row leads to the following equation: |𝑀|()𝐦=−−𝑧+1^𝑇12𝑚₁₃𝑀₃₂𝐦₃₃−𝑚₁₁+𝑚₁₃𝐦^𝑇₁₂𝐦₃₁+𝐦₃₃𝑀₃₂=𝑇−𝑧𝐴₁+(𝑧−1)𝐴₂=0, (41) where 𝐴₁=𝑑1−2𝑑𝑇+²−𝑟_𝑛−26𝑇²+2𝑑𝑟_𝑛−2−𝑑³−𝐜^𝑇𝑝𝐪𝑇24³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+2𝑃³+𝐪𝐜^𝑇𝑃𝑇12³=Δ₁×Δ₂,𝐴₂=−𝛽2+𝑑𝛽6𝑑𝑇−²−𝑟_𝑛−2𝛽𝑇24²+5𝐜^𝑇𝑝𝐪𝛽−2𝐜^𝑇𝐪𝑑𝛽𝑇24³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+2𝑃³+𝐪𝐜^𝑇𝑃𝑇12³=Δ₁×Δ₂. (42) Then, |𝑀|=Δ₁−Δ₁𝑧−Δ₁×Δ₂. (43) Hence, the approximate values of limiting zeros of the discrete-time system are obtained as the roots of (35).

Remark 2. Equation (35) implies that an approximation of the sampling zero is expressed as 𝛽−1−2+𝑑2+𝑑𝛽6𝑇−𝑑²−𝑟_𝑛−26+𝑑²−𝑟_𝑛−2𝛽𝑇24²+2𝑑𝑟_𝑛−2−𝑑³−𝐜^𝑇𝑃𝐪−5𝐜^𝑇𝑃𝐪𝛽+2𝐜^𝑇𝐪𝑑𝛽𝑇24³𝑧+𝛽2−𝑑𝛽6𝑑𝑇+²−𝑟_𝑛−2𝛽𝑇24²−5𝐜^𝑇𝑃𝐪𝛽−2𝐜^𝑇𝐪𝑑𝛽𝑇24³=0, (44) and the approximate values of the intrinsic zeros are derived from )𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+2𝑃³+𝐪𝐜^𝑇𝑃𝑇12³=0. (45)

Remark 3. Theorem 1 is applicable to also the case of multiple zeros of the continuous-time system (1) with FROH and further gives approximate values with higher order of accuracy than those of the previous result [17].

Remark 4. An insightful observation in Theorem 1 is that it has a form of a correction to the asymptotic result of Ishitobi [12, 18] in the form of a power term of𝑇. Similarly, the following result (Theorem 6) both the intrinsic zeros and sampling zeros is also clarified in a more precise manner than Ishitobi’s result [12, 17, 18] when the relative degree of continuous-time systems is two.

Remark 5. On the basis of the approach in [21], it is immediate to derive the asymptotic condition of the limiting zeros in the case of a FROH with relative degree one: 𝑀₁≈𝛽1+2−𝑑2+𝑑𝛽6𝑇+4𝑑²−4𝑟_𝑛−2+𝑑²−𝑟_𝑛−2𝛽𝑇24²+2𝑑𝑟_𝑛−2−𝑑³−𝐜^𝑇𝑃𝐪𝑇24³×2−𝑧+1−+2+𝛽𝑑𝛽32+𝛽²𝑇+𝛽6𝑟_𝑛−2+3𝑟_𝑛−2𝛽−𝛽𝑑²182+𝛽³𝑇²+𝛽1014𝑑𝛽𝑟_𝑛−2−348𝑑³+348𝑑𝑟_𝑛−2+𝛽1188𝑑𝛽²𝑟_𝑛−2−1026𝑑³𝛽−1193𝑑³𝛽²+𝛽36𝐜^𝑇𝑃𝐪𝛽+36𝐜^𝑇𝑃𝐪+9𝐜^𝑇𝑃𝐪𝛽²×1082+𝛽⁴⁻¹𝑇³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+2𝑃³+𝐪𝐜^𝑇𝑃𝑇12³. (46) When the relative degree of continuous-time systems is one and the continuous-time input is generated by a FROH, further research is needed to establish connections between (46) and (35) of Theorem 1 in this paper, wherein the idea (35) has more decent effect than the literature [21] in terms of techniques in studying the discrete system zeros.

3.2. Case of Relative Degree Two (𝑚=𝑛−2)

Next, we present asymptotic properties of limiting zeros of discrete-time control system in the case of a FROH as power series with respect to a sampling period up to the third-order term when the relative degree of the continuous-time system is two. An approximate expression, in fact, of zeros of a discrete-time system is derived from (32)−(34), and the other results of this paper are given by the following Theorem.

Theorem 6. The zeros of a discrete-time system for the continuous-time transfer function (1) with FROH are given for𝑇≪1approximately by the roots of 12+𝛽6−4𝑑₁+𝑑₁𝛽24𝑇+5𝑑²₁−5𝑑₀𝑇120²+𝛽𝑑²₁−𝛽𝑑₀𝑇120²+2𝑑₀𝑑₁−𝐜_𝑛−3−𝑑³₁𝑇120³𝑧²+3+𝛽6−2𝑑₁+𝑑₁𝛽6𝑇+15𝑑²₁−5𝑑₀+8𝛽𝑑²₁−3𝛽𝑑₀𝑇120²+𝑑₀𝑑₁+𝐜40^𝑛−3−𝑑120³¹+𝑑30⁰𝑑₁𝛽+𝐜180_𝑛−3𝛽−𝑑72³¹𝛽𝑇80³𝑧−𝛽3+5𝑑₁𝛽24𝑇+4𝑑₀𝛽−9𝑑²₁𝛽𝑇120²+9𝑑³₁−4𝑑₀𝑑₁𝛽−10𝐜_𝑛−3𝛽𝑇720³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³=0. (47)

Proof. Zeros of the discrete-time system (16) and (31), equivalent to (32)–(34), are given by substituting𝑦_𝑘=𝑦_𝑘+1=0into (32)–(34) as follows: 𝑀₂𝑌_𝑑𝐾𝑈_𝑘−1𝐻𝐾𝑈_𝑘=𝟎_𝑛, (48) where𝑌_𝑑is the𝑧-transforms of𝑦̇_𝑘and the matrix𝑀₂is defined by 𝑀₂=𝑚₁₁𝑚₁₂𝐦^𝑇₁₃𝑚₁₄𝑚₂₁𝑚₂₂𝐦^𝑇₂₃𝑚₂₄0−𝑧𝟎^𝑇1𝐦₄₁𝐦₄₂𝑀₄₃𝐦₄₄, (49) with 𝑚₁₁=𝑇𝑚₁₁+𝑂𝑇⁵,𝑚₁₂=𝑇𝑚₁₂+𝑂𝑇⁵,𝐦^𝑇₁₃=𝑇𝐦^𝑇₁₃+𝑂𝑇⁵,𝑚₁₄=𝑇𝑚₁₄+𝑂𝑇⁵,𝑚₁₁𝑑=1−¹2𝑑𝑇+²¹−𝑑₀6𝑇²+−𝑑³₁+2𝑑₀𝑑₁−𝐜_𝑛−3𝑇24³,𝑚₁₂𝛽=−6𝑑𝑇+¹𝛽𝑇24²−𝑑²₁−𝑑₀𝛽𝑇120³𝐦^𝑇₁₃𝐜=−^𝑇2𝑑𝑇+¹𝐜^𝑇−𝐜^𝑇𝑃6𝑇²+−𝑑²₁−𝑑₀𝐜^𝑇+𝑑₁𝐜^𝑇𝑃−𝐜^𝑇𝑃²𝑇24³,𝑚₁₄=12+𝛽6−𝑑𝑇+¹6−𝑑₁𝛽𝑇24²+𝑑²₁−𝑑₀+𝑑24²¹−𝑑₀𝛽𝑇120³,𝑚₂₁=−𝑧+1−𝑑₁𝑑𝑇+²¹−𝑑₀2𝑇²+−𝑑³₁+2𝑑₀𝑑₁−𝐜_𝑛−36𝑇³+𝑂𝑇⁴,𝑚₂₂𝛽=−2𝑑𝑇+¹𝛽6𝑇²−𝑑²₁−𝑑₀𝛽𝑇24³+𝑂𝑇⁴,𝐦^𝑇₂₃=−𝐜^𝑇𝑑𝑇+¹𝐜^𝑇−𝐜^𝑇𝑃2𝑇²+𝑑₁𝐜^𝑇𝑃−𝑑²₁−𝑑₀𝐜^𝑇−𝐜^𝑇𝑃²6𝑇³+𝑂𝑇⁴,𝑚₂₄=𝛽1+2−𝑑𝑇+¹2−𝑑₁𝛽6𝑇²+𝑑²₁−𝑑₀6+𝑑²₁−𝑑₀𝛽𝑇24³+𝑂𝑇⁴,𝐦₄₁=𝐪2𝑇²+𝑃𝐪−𝐪𝑑₁6𝑇³+𝑂𝑇⁴,𝐦₄₂=−𝐪𝛽𝑇24³+𝑂𝑇⁴,𝑀₄₃=()𝑃−𝑧+1𝐼+𝑃𝑇+²2𝑇²+𝑃³−𝐪𝐜^𝑇6𝑇³+𝑂𝑇⁴,𝐦₄₄=𝐪6+𝐪𝛽𝑇24³+𝑂𝑇⁴. (50)
Thus, the zeros are derived from 𝑀₂=0. (51)
From the relationship 𝑀=𝑀₂1000010000100101, (52) it is obvious that the condition|𝑀|=0is equivalent to|𝑀₂|=0.
Expanding the result along the third row leads to the following equation: 𝑀()𝑚=−−𝑧+1¹¹𝐦^𝑇₁₃𝑚₁₄𝑚₂₁𝐦^𝑇₂₃𝑚₂₄𝐦₄₁𝑀₄₃𝐦₄₄−𝑚₁₁𝑚₁₂+𝑚₁₄𝐦^𝑇₁₃𝑚₂₁𝑚₂₂+𝑚₂₄𝐦^𝑇₂₃𝐦₄₁𝐦₄₂+𝐦₄₄𝑀₄₃=𝑇−𝑧𝐴₁+(−𝑧+1)𝐴₂=0, (53) where 𝐴₁=12−𝑑₁6𝑑𝑇+²¹−𝑑₀𝑇24²+−𝐜_𝑛−3+2𝑑₀𝑑₁−𝑑³₁𝑇120³×𝑑−𝑧−1+¹3𝑑𝑇−²¹𝑇18²+𝑑³₁+3𝑑₀𝑑₁−9𝐜_𝑛−3𝑇270³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³=Δ₁×Δ₂×Δ₃,𝐴₂=𝛽6𝑑𝑇−¹𝛽𝑇24²+𝑑²₁−𝑑₀𝛽𝑇120³×−𝑧−2+3𝑑₁4𝑇+−13𝑑²₁+𝑑80⁰𝑇10²+7𝑑₀𝑑₁−2403𝑑³₁−𝐜960^𝑛−3𝑇12³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³=Δ₁×Δ₂×Δ₃. (54)(55) Equations (54) and (55) will be calculated in the appendix. Then, |𝑀|=−Δ₁×Δ₂−Δ₁×Δ₂𝑧+Δ₁×Δ₂×Δ₃. (56) Hence, the approximate values of the zeros of the discrete-time system are obtained as the roots of (47).

Remark 7. Equation (47) implies that the approximations of the sampling zeros are expressed as 12+𝛽6−𝑑₁6+𝑑₁𝛽24𝑇+5𝑑²₁−5𝑑₀+𝛽𝑑²₁−𝛽𝑑₀𝑇120²+2𝑑₀𝑑₁−𝐜_𝑛−3−𝑑³₁𝑇120³𝑧²+3+𝛽6−2𝑑₁+𝑑₁𝛽6𝑇+15𝑑²₁−5𝑑₀+8𝛽𝑑²₁−3𝛽𝑑₀𝑇120²+𝑑₀𝑑₁+𝐜40^𝑛−3−𝑑120³¹+𝑑30⁰𝑑₁𝛽+𝐜180_𝑛−3𝛽−𝑑72³¹𝛽𝑇80³𝑧−𝛽3+5𝑑₁𝛽24𝑇+4𝑑₀𝛽−9𝑑²₁𝛽𝑇120²+9𝑑³₁−4𝑑₀𝑑₁𝛽−10𝐜_𝑛−3𝛽𝑇720³=0, (57) and the approximate values of the intrinsic zeros are derived from )𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³=0. (58)

Remark 8. When FROH is implemented in practice, an approximate fractional-order hold (AFROH) using ZOH would be convenient practical solution. The basic idea of AFROH is that, at each sampling interval, the output of FROH is approximated by staircase waveforms that can be generated by ZOH [15, 22] (see Figure 2). Therefore, an asymptotic expression of the limiting zeros in AFROH case is derived similarly.

Figure 2 

Output of the approximate fractional-order hold with𝑁=2and𝛽=−0.5.

In the particular case when the sampling period tends to zero, it is immediate to obtain the following Corollary although a similar result is also obtained by Ishitobi [12, 18].

Corollary 9. One has the following cases.
Case a. Assume that the relative degree of a continuous-time system is one. If−1<𝛽(resp.,𝛽<−1), then the sampling zero of the sampled-data model is stable (resp., unstable) in the case of a FROH when the sampling period tends to zero.
Case b. Assume that the relative degree of a continuous-time system is two. If−1<𝛽<0(resp.,𝛽≤−1or𝛽>0), then the sampling zeros of the sampled-data model are stable (resp., unstable) in the case of a FROH when the sampling period tends to zero.

Proof. One has the following cases.
Case a. For𝑛−𝑚=1, we have from (35) 𝐴₁=𝛽𝑧;𝛽−1−2𝛽𝑧+2. (59) Simple straightforward calculation will verify that the root of𝐴₁(𝑧;𝛽)=0is stable if−1<𝛽 and is unstable if𝛽≤−1. In addition, when𝛽=−1, there remains the possibility that the sampling zero corresponding to𝐴₁(𝑧;𝛽), which approaches𝑧=−1, lies inside the open unit disc as𝑇→0.
Case b. For𝑛−𝑚=2, the polynomial𝐴₂(𝑧;𝛽)is represented from (47) as 𝐴₂=1𝑧;𝛽2+𝛽6𝑧²+12+𝛽6𝛽𝑧−3. (60) When we perform the bilinear transformation𝑧=(𝜔+1)/(𝜔−1)on the above equation, the polynomial is written as 𝐴₂𝑧;𝛽=3𝜔²+31+𝛽𝜔−𝛽=0. (61) It is clear that the two roots of (61) lie in the open left half of𝜔-plane if−1<𝛽<0, and at least one of them stays in the closed right-half plane if𝛽≤−1or𝛽≥0. In particular, only one of the sampling zeros approaches −1 at𝛽=0. Namely, the stability of the sampling zeros is marginal, that is, in the case of a ZOH [4, 23].

Remark 10. When the FROH signal reconstruction device is used, the parameter𝛽which is the device adjustable gain (generalised gain) is the major factor that decides the stability properties of sampling zeros of sampled-data systems with FROH. In other words, the appropriate𝛽is determined to obtain the FROH that provides sampling zeros as stable as possible, or with improved stability properties even when being unstable, for a given continuous-time plant.

Remark 11. If the relative degree of a continuous-time transfer function is two and the sum of the zeros is less than or equal to the sum of the poles, the limiting zeros of the sampled system with FROH of−1<𝛽<0stay definitely inside the unit circle while those with ZOH may lie outside or on the unit circle. Therefore, the FROH with−1<𝛽<0will produce all stable sampling zeros for a wider class of continuous-time plants than that of the ZOH.

4. Simulation Examples

This section presents three interesting examples to show the stability of sampling zeros with FROH by improved asymptotic properties. It has also shown that the stability of zeros will be improved by using FROH instead of ZOH. Both kinds of zeros are calculated by applying MATLAB, and in the simulation figures (Figures 3, 4, 5, 6, 7, 8, and 9), the solid line and dotted line indicate the exact values and approximate values, respectively.

Figure 3 

Intrinsic zero of sampled-data model with relative degree one.

Figure 4 

Intrinsic zero of sampled-data model with relative degree one.

Figure 5 

Sampling zero of sampled-data model with relative degree one and𝛽=−1/2.

Figure 6 

Sampling zero of sampled-data model with relative degree one and𝛽=1.

Figure 7 

Intrinsic zero of sampled-data model with relative degree two.

Figure 8 

Sampling zero of sampled-data model with relative degree two and𝛽=−1/2.

Figure 9 

Sampling zero of sampled-data model with relative degree two and𝛽=−1/2.

Example 1. Consider the following transfer function with the relative degree one [21]: 𝐺(𝑠)=(𝑠+2)²𝑠().𝑠+1)(𝑠−2 (62) The approximate values (35) and the exact values of zeros of the sampled-data system for the transfer function (62) are shown in Tables 1–4 and corresponding figures, where the intrinsic zeros are shown in Table 1 and the sampling zero is respectively shown in Tables 2, 3, and 5 owing to the difference of the parameter 𝛽. Equation (35) gives good approximation also for the case of a continuous-time transfer function with FROH.

Table 1 

Intrinsic zeros of the sampled-data system with relative degree one.

Table 2 

Sampling zero of the sampled-data system with relative degree one and 𝛽=−1/2.

Table 3 

Sampling zero of the sampled-data system with relative degree one and 𝛽=1.

Table 4 

Sampling zero of the sampled-data system with relative degree one.

Table 5 

Sampling zero of the sampled-data system with relative degree one and 𝛽=−2.

When the continuous-time systems have relative degree one, a discrete-time system corresponding to a continuous-time transfer function (62) has two intrinsic zeros and one sampling zero in the case of a FROH. In particular, the values of the intrinsic zeros with FROH are approximately equal to those with ZOH owing to the parameter𝛽(see also Remark 13). Further, the stability of sampling zero with FROH depends on the parameter𝛽. When−1<𝛽(resp.,𝛽<−1), the sampling zero of the sampled-data model is stable (resp., unstable) in the case of a FROH for small sampling periods (see Tables 2–4).

Case a  (𝛽=−1/2). See Table 2 and Figure 5.

Case b  (𝛽=1). See Table 3 and Figure 6.

Case c  (𝛽=−2). See Table 5. From the foregoing analysis, it is obvious that the limiting zeros of the sampled-data system with FROH of−1<𝛽are located inside the unit circle. In addition, (46) gives good approximation and the sampling zero lies inside the unit circle for small sampling periods at𝛽=−1/2and 𝛽=1 (see Table 4). Furthermore, it can be seen from the corresponding Tables 2–4 that (35) can offer a more accurate approximation than that of (46) in terms of the stable sampling zero of discrete-time model.

Example 2. Consider a transfer function with the relative degree two 𝐺(𝑠)=𝑠+7().𝑠+1)(𝑠+2)(𝑠+3 (63)
On the basis of the results in [21, 24], the stability condition of sampling zeros with ZOH is dissatisfied since𝑑₁=𝑎₂−𝑏₀=−1<0. However, the stability of the sampling zeros will be preserved in the case of a FROH when−1<𝛽<0. The approximate values (47) and the exact values of zeros of the sampled-data system for the transfer function (63) are shown in Tables 6–9, where the intrinsic zero is shown in Table 6 and the sampling zeros are shown in Tables 7, 8, and 9. Equation (47) gives good approximation and the sampling zeros lie inside the unit circle for small sampling periods with FROH, while ZOH fails to do so.

Table 6 

Intrinsic zero of the sampled-data system with relative degree two.

Table 7 

Sampling zeros of the sampled-data system with relative degree two and 𝛽=−1/2.

Table 8 

Sampling zeros of the sampled-data system with relative degree two and 𝛽=1.

Table 9 

Sampling zeros of the sampled-data system with relative degree two and 𝛽=−2.

Remark 12. From Examples 1 and 2, it can be obviously seen that FROH is reduced to FOH for𝛽=1. The limiting zeros for sufficiently small𝑇in the case of a FOH are stable with relative degree one while it is unstable with relative degree two. Thus, a FOH provides no advantage over ZOH and FROH with the stability of the limiting zeros [4].

Remark 13. When the FROH signal reconstruction device is used, the parameter𝛽, so called the device adjustable gain (generalised gain), is also a factor which affects the intrinsic zeros of sampled-data systems by numerically verifying in the case of a FROH. More precisely, it only affects the distribution of intrinsic zeros while the stability of intrinsic zeros is still preserved for different values of𝛽. See also the literature by De la Sen [25], who has similar conclusion by applying different technique.
Next, we display the improvement of the asymptotic properties of discrete system zeros with FROH through an example of an electronic circuit in the remainder of this section.

Example 3. Consider an electric circuit shown in Figure 10 [15], where𝑅_𝑖 (𝑖=1,…,4) and 𝐶_𝑗 (𝑗=1,…,3) represent resistance and condenser, respectively.

Figure 10 

The electric circuit plant.

The transfer function with the voltage𝑒_𝑖(𝑡)as an input and with the voltage𝑒₀(𝑡)as an output is given by 𝐺(𝑠)=∑¹_𝑘=0𝑏_𝑘𝑠^𝑘∑³_ℓ=0𝑎_ℓ𝑠^ℓ=𝐾𝑠+𝑏₀𝑠³+𝑎₂𝑠²+𝑎₁𝑠+𝑎₀, (64) where 𝑏𝐾=¹𝑎₃,𝑏₀=𝑏₀𝑏₁,𝑎₀=𝑎₀𝑎₃,𝑎₁=𝑎₁𝑎₃,𝑎₂=𝑎₂𝑎₃. (65) It is easy to see that the relative degree of transfer function (64) is two. Here, when the parameters are set as𝑅₁=𝑅₂=1 [kΩ],𝑅₃=5 [kΩ],𝑅₄=13/70 [kΩ], and𝐶₁=𝐶₂=𝐶₃=1 [μF], the corresponding discrete-time system with ZOH has an unstable sampling zero for the sufficiently small sampling periods according to the𝑎₂−𝑏₀<0[21, 24]. In fact, the absolute value of the sampling zero of the discrete-time system with ZOH is 1.006418 for𝑇=0.001. The magnitudes of limiting zeros of the corresponding discrete-time system with FROH are shown in Figure 11 for the sampling period𝑇=0.001. All the limiting zeros stay inside the unit circle for−1<𝛽<0(see also Figure 11). The stability condition can be achieved by means of a suitable choice of the parameter𝛽of the improving asymptotic properties.

Figure 11 

The magnitudes of zeros of the sampled-data models with FROH for 𝑇=0.001 s.

Remark 14. From Example 3, it has been shown that the limiting zeros of the sampled-data models with FROH can be located inside the stability region by analyzing the improved asymptotic properties while ZOH fails to do so. In addition, the limiting zeros with AFROH are also stable in some cases due to the same advantages of the FROH and AFROH cases.

5. Conclusions

This paper has analyzed the improved asymptotic behavior of limiting zeros for the discrete-time system by using Taylor expansion and the FROH signal reconstruction device. When the normal form representation of continuous-time system with relative degree one or two is discretized, we have given an approximate expression of limiting zeros as power series expansions with respect to a sampling period up to the third-order term. Furthermore, the stability of the sampling zeros is also discussed as the sampling period tends to zero. Finally, it has been shown that FROH provides advantage over ZOH with stability of the limiting zeros of sampled-data systems. The idea of this paper is a further extension of the previous results. For a future study, an extension of the approach to multivariable systems is left.

Appendix

Calculation of (54) and (55). Denote 𝐴₁=𝑚₁₁𝑚₁₂+𝑚₁₄𝐦^𝑇₁₃𝑚₂₁𝑚₂₂+𝑚₂₄𝐦^𝑇₂₃𝐦₄₁𝐦₄₂+𝐦₄₄𝑀₄₃,𝐴₂=𝑚₁₁−𝑚₁₂𝐦^𝑇₁₃𝑚₂₁−𝑚₂₂𝐦^𝑇₂₃𝐦₄₁−𝐦₄₂𝑀₄₃. (A.1)

It is immediate to obtain the value of|𝐴₁|from [21].

Here, consider a matrix𝑀_𝛼which is defined by neglecting the higher order terms𝑂(⋅)with respect to𝑇in the matrix𝐴₂since the interests lie in the case of small sampling periods𝑇.

Multiplying𝑀_𝛼by 𝐿₁=1𝑇ℓ₁𝟎^𝑇_𝑛−2ℓ₂𝟎^𝑇_𝑛𝐼_𝑛^𝑇, (A.2) where ℓ₁1=−𝑚₀𝛽2−𝑑₁𝛽6𝑑𝑇+²¹−𝑑₀𝛽𝑇24²,ℓ₂=−𝛽𝑇²24𝑚₀,𝑚₀=𝛽6−𝑑₁𝛽𝑑24𝑇+²¹−𝑑₀𝛽𝑇120² (A.3) from the left-hand side leads to 𝐿₁𝑀_𝛼=𝑚₁₁−𝑚₁₂𝐦^𝑇₁₃𝑚₂₁0𝐦^𝑇₂₃𝐦₄₁𝟎_𝑛−1𝑀₄₃, (A.4) where 𝑚₂₁=−𝑧−2+3𝑑₁4𝑇+8𝑑₀−13𝑑²₁𝑇80²+28𝑑₀𝑑₁−3𝑑³₁−40𝐜_𝑛−3𝑇960³,𝐦^𝑇₂₃=𝐜^𝑇2𝑑𝑇−¹𝐜^𝑇8𝑇²+9𝑑²₁𝐜^𝑇−4𝑑₀𝐜^𝑇−20𝐜^𝑇𝑃²𝑇480³,𝐦₄₁=𝐪4𝑇²+8𝑃𝐪−5𝑑₁𝐪𝑇48³,𝑀₄₃=()𝑃1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³. (A.5) Noting here that 𝛿=𝑚₂₁×𝑀₄₃−𝐦^𝑇₂₃×𝐦₄₁=−𝑧−2+3𝑑₁4𝑇+8𝑑₀−13𝑑²₁𝑇80²+28𝑑₀𝑑₁−3𝑑³₁−40𝐜_𝑛−3𝑇960³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³+𝐪𝐜^𝑇(𝑧−1)𝑇24³=−𝑧−2+3𝑑₁4𝑇+−13𝑑²₁+𝑑80⁰𝑇10²+7𝑑₀𝑑₁−2403𝑑³₁−𝐜960^𝑛−3𝑇12³×)𝑃(1−𝑧𝐼+𝑃𝑇+²2𝑇²+𝑃³6𝑇³. (A.6) Then, 𝐴₂=−𝑚₁₂𝛿. (A.7) As a result, the calculation is completed.

Acknowledgments

This work is partially supported by the National Basic Research Program of China (“973” Grant no. 2013CB328903), the National Natural Science Foundation of China (no. 60574003), the Natural Science Foundation Project of CQ CSTC (no. cstc2012jjA40026), and Project no. CDJXS12170006 supported by the Fundamental Research Funds for the Central Universities. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation of the paper.