Mathematical Approaches in Advanced Control Theories 2013View this Special Issue
Research Article | Open Access
Improving the Asymptotic Properties of Discrete System Zeros in Fractional-Order Hold Case
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 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.
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 period. 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 periodtends 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 , 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 . The former ones have counterparts in the underlying continuous-time system and go to unity  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.  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  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 , 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  from different angles and methods by investigating the limiting zeros in the case of a FROH. In addition, the limiting FROH zeros  have been also extended by Blachuta , 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 periods.
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 anth continuous-time system with relative degree one or two described by a transfer function where 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 ()
When the FROH signal reconstruction method is considered, the input is described by whereis a real design parameter andis a sampling period [11, 12, 18]. It is obvious that FROH is reduced to ZOH forwhile it becomes the FOH for. The signal reconstruction of a FROH withis shown in Figure 1.
Suppose, and when a FROH is applied, we have
Furthermore, (3) leads to the derivatives of the output which are expressed by , , and . Further, the derivatives of are also represented by , , and as Hence, by substituting (9)–(15) into the right-hand side of and defining the state variables, where the subscriptdenotes, 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 ofand, the zeros of (16) are analyzed as follows: The reason why the explicit expressions ofandare used is to obtain the approximate expansion of the limiting zeros for the discrete-time system with the order.
2.2. Case of Relative Degree Two ()
The normal form of (1) with the relative degree two,is represented [19, 20] as where and the scalars () and () are obtained from where When a FROH is used, the normal form (19) yields the derivatives of the output
Further, the derivatives of are also represented as Hence, by substituting (23)–(30) into the right-hand side of (16) and and defining the state variables , the discrete-time state equations are obtained.
Now, by using the explicit expressions of and , the zeros of the discrete-time system (16) and (31) are analyzed as follows: Similarly, the reason why the explicit expressions ofandare used is to obtain the approximate expansion of the limiting zeros for the discrete-time system with the orderwhen 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 ()
Theorem 1. The zeros of a discrete-time system corresponding to the continuous-time transfer function (1) with FROH are given forapproximately by the roots of
Proof. The limiting zeros of the discrete-time system (16) are equivalent to zeros in (17) and (18), which are given by substitutinginto (17) and (18) as follows:
where,, andare the-transforms of,, and, respectively, and the matrixis defined by
Thus, the zeros are derived from
From the relationship
it is obvious that the conditionis equivalent to.
Expanding the result along the second row leads to the following equation: where Then, 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 and the approximate values of the intrinsic zeros are derived from
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 .
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 of. 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 , it is immediate to derive the asymptotic condition of the limiting zeros in the case of a FROH with relative degree one: 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  in terms of techniques in studying the discrete system zeros.
3.2. Case of Relative Degree Two ()
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 forapproximately by the roots of
Proof. Zeros of the discrete-time system (16) and (31), equivalent to (32)–(34), are given by substitutinginto (32)–(34) as follows:
whereis the-transforms ofand the matrixis defined by
Thus, the zeros are derived from
From the relationship it is obvious that the conditionis equivalent to.
Expanding the result along the third row leads to the following equation: where Equations (54) and (55) will be calculated in the appendix. Then, 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 and the approximate values of the intrinsic zeros are derived from
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.
Corollary 9. One has the following cases.
Case a. Assume that the relative degree of a continuous-time system is one. If(resp.,), 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(resp.,or), 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. For, we have from (35) Simple straightforward calculation will verify that the root ofis stable if and is unstable if. In addition, when, there remains the possibility that the sampling zero corresponding to, which approaches, lies inside the open unit disc as.
Case b. For, the polynomialis represented from (47) as When we perform the bilinear transformationon the above equation, the polynomial is written as It is clear that the two roots of (61) lie in the open left half of-plane if, and at least one of them stays in the closed right-half plane ifor. In particular, only one of the sampling zeros approaches −1 at. 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 parameterwhich 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 appropriateis 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 ofstay definitely inside the unit circle while those with ZOH may lie outside or on the unit circle. Therefore, the FROH withwill 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.
Example 1. Consider the following transfer function with the relative degree one : 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 . Equation (35) gives good approximation also for the case of a continuous-time transfer function with FROH.
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 parameter(see also Remark 13). Further, the stability of sampling zero with FROH depends on the parameter. When(resp.,), 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 c . See Table 5. From the foregoing analysis, it is obvious that the limiting zeros of the sampled-data system with FROH ofare located inside the unit circle. In addition, (46) gives good approximation and the sampling zero lies inside the unit circle for small sampling periods atand (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
On the basis of the results in [21, 24], the stability condition of sampling zeros with ZOH is dissatisfied since. However, the stability of the sampling zeros will be preserved in the case of a FROH when. 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.