# A New Fractional Order Hold and Its Capability in Frequency Response and Zero Placement

Authors: Farzad Hashemzadeh; Amin Rezaei Pish-Robat; Parviz Jabehdar-Maralani
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper introduces a new holder with application on digital control systems. This holder is a combination of fractional order hold (FROH) and zero-order hold (ZOH) that has the capability of both holders and a frequency response better than both of ZOH and FROH. For the stability of zeros of the sampled system two theorems are stated and proved with the assumption that the sampling period is very small. Also simulation results are studied to show the effectiveness of the proposed holder and better performance results in comparison with ZOH and FROH.


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## Body

## 1. Introduction

In digital control systems, the output of the system is sampled by an analog-to-digital converter (ADC) and then compared with digital reference inputs that are generated by the computer. The difference is given to the controller, and the output of the controller needs to be given to a holder block in order to give to the continuous time system. The holder block generates continuous time signal between sampling times, and the more common and usual holders are ZOH and FOH. The ZOH is the most common holder used as digital-to-analog converter (DAC) in industry. FROH is similar to first order hold (FOH) but has a variable slope [1], and due to an adjustable parameter, it has a suitable capability for placement of additional produced zeros [2]. Holders such as ZOH and FOH which are used in digital control systems sometimes generate additional non-minimum-phase zeros in z-domain [2]. In the design of a feedback control system, zeros play an important role, and most of the design methods based on pole-zero cancellation require that zeros of the system be stable. However, in discretizing continuous time systems with sample and hold, the stability of zeros of the sampled system is not necessarily preserved. Thus it is necessary to know the conditions under which we can obtain a minimum-phase sampled system. Depending on the relative degree of the system, FOH generates non-minimum-phase zeros and ZOH maps additional generated zeros to unit circle in z-plane. The number of the additional generated zeros depends on the relative degree of the system. Both FOH and ZOH cannot control the location of additional generated zeros. Based on disadvantages of ZOH and FOH, FROH was introduced to have non-minimum-phase system. Using adjustable slope of holder in FROH, very small sampling time guarantees that additional generated zeros of the sampled systems are minimum-phase in the system with relative degrees of 0, 1, and 2.Replacing ZOH with FROH in a digital control system, the ability to improve the stability of zeros of system is shown in [2]. Compared with ZOH, the stability of zeros of the sampled system can be improved by using FROH, and the conditions under which the FROH has better performance than ZOH have been clarified theoretically in [3]. Most of the researches on the properties of zeros of the sampled system are made under the assumption that ZOH, FOH, or the FROH is employed [4–6].In order to have minimum-phase additional generated zeros, when using FROH in the systems with relative degree one, the slope of the holder(β) should be β>-1 and in systems with relative degree two, the slope of the holder should be -1<β<0 [5].In this paper, a new holder to be applied to digital control systems is introduced; we named it mixed fractional order hold (MFROH). MFROH is a combination of FROH and ZOH that has the capabilities of both holders and has a frequency response better than ZOH and FROH. MFROH places additional generated zeros better than FROH and ZOH. The number of these additional zeros depends on the relative degree of the system, and hence it is difficult to design a digital controller for systems with large relative degrees [3]. MFROH has two adjustable parameters and has the same capability as FROH with respect to placement of additional generated zeros for different relative degrees and increases the range of acceptable slope of the holder, therefore it improves the capability of the holder in zero placements. At low frequencies, FROH is a better approximation to an ideal low pass filter than ZOH, but at high frequencies, ZOH is better than FROH. Because of the fact that MFROH is a combination of ZOH and FROH, it has the same behavior as FROH at low frequencies and the same behavior as ZOH at high frequencies, and hence by tuning its two parameters, we can have a better frequency response than ZOH and FROH. Therefore, with adjusting the parameters of MFROH, the amplitude of frequency response of MFROH would be very similar to ideal low-pass filter. MFROH has the capability that it can be converted to ZOH or FROH by tuning one of its parameters.In Section2 the structure of MFROH is introduced, and the frequency response of MFROH is given in Section 3. Stability studies of zeros in sampled system and two theorems for stability of zeros are stated and proved in Section 4. In Section 5 simulation results are studied, and conclusion is stated in Section 6.


## 2. The Structure of MFROH

MFROH is a line with slopeβ over the interval [kT,kT+ΔT] that coincides with the output at instant t=kT. Also it is a line with zero slope over the interval [kT+ΔT,kT+T] with the continuity preserving at t=kT+ΔT. A schematic of MFROH is shown in Figure 1 and is described by the following equations:
(1)u(t)={u(kT)+βu(kT)-u((k-1)T)T(t-kT)kT≤t≤(k+Δ)Tu(kT)+βu(kT)-u((k-1)T)T(ΔT)(k+Δ)T≤t≤(k+1)T0≤Δ≤1,
where T is the sampling time, β is the adjustable parameter of holder slope over the interval [kT,kT+ΔT], and Δ is the adjustable parameter describing the location of slope variation over interval [kT,kT+ΔT]. The impulse response of MFROH according to (1) is shown in Figure 2. According to (1) and Figure 2, the transfer function of MFROH is given by the following equation:
(2)Gβ,Δ=(1-e-sT)(1-Δβe-sT)s+β(1-e-sT)(1-e-ΔsT)Ts2=(1-e-sT)s{(1-Δβe-sT)+β(1-e-ΔsT)Ts}.Figure 1
Samples in MFROH.Figure 2
Impulse response of MFROH.In [7], implementation of FROH is discussed using a lot of ZOH. Given MFROH is the combination of ZOH and FROH, according to [7] MFROH can be implemented using ZOH blocks in practical application. The new proposed holder device has limits in frequency synthesis methods based on delta, bilinear, or tustins transformations.


## 3. Frequency Response of MFROH

In a discrete time system the high frequency terms of sampled signal that are generated in the sampling process must be attenuated before applying the signal to continuous time systems. In order to attenuate the high frequency terms of a sampled signal, it is usually fed to a hold circuit. Hold circuit is a low-pass filter that converts the sampled signal to a continuous-time signal. An ideal low-pass filter or an ideal holder converts sampled signal to continuous-time signal with no high frequency terms, but in the real world there is no such an ideal filter [8].


### 3.1. Frequency Response Specifications

Sampling process generates many additional terms in addition to the main term, and the ideal filter attenuates all these additional terms to zero and only passes the main term. Therefore, a practical filter is such a filter that passes the main term without any frequency distortion and attenuates all additional terms as much as possible. This means that the frequency response of practical filter must resemble to that of the ideal filter in Figure3 as much as possible.Figure 3
Frequency response of an ideal filter or an ideal holder.The amplitudes of frequency responses of ZOH and FROH for different values ofβ with sampling time T=0.01sec are shown in Figure 4.Figure 4
Amplitude of frequency responses of ZOH and FROH for different values ofβ.Regarding Figure4, for negative values of β and reducing β, the amplitude of frequency response at ω=ωs/2 is reduced dramatically, and it is not smooth and has large peaks after its first minimum, that is, for β=-1. The amplitude of frequency response increases after its first minimum in ω=ωs, and the filter passes undesirable signals generating from sampling around ω=ωs and leads to frequency aliasing. Therefore, the optimal value of β should be chosen among the positive values of β in FROH. By comparing the amplitudes of frequency responses of ZOH and FROH, it can be seen from Table 1 that for positive values of β at ω=ωs/2, the decrease in the ZOH is much more than the decrease in the FROH.Table 1
FROH's amplitude of the frequency responses for different values ofβ at ω=ωs/2.
β
-
1
-
0.4
0
0.4
1
|
G
h
(
(
j
ω
s
)
/
2
)
|
×
10
3
3.96
4.17
6.4
9.13
13.4In Figure5, the amplitudes of the frequency responses of ZOH and FROH for small values of β are shown. In low frequencies, the transfer function of FROH has a better approximation of ideal low-pass filter than the transfer function of ZOH, because at low frequencies the amplitude of the frequency response of ZOH is reduced much more than FROH, but at high frequencies ZOH is better.Figure 5
Amplitude of the frequency responses of ZOH and FROH for smallβ.The amplitudes of the frequency response of MFROH for different values ofβ and Δ are shown in Figure 6. According to Figure 6 and based on two adjustable parameters of MFROH, it is possible to adjust β and Δ so that the amplitude of the frequency response of MFROH becomes very similar to that of the ideal low-pass filter that is shown in Figure 3.Amplitude of the frequency response of MFROH for different values ofβ and Δ.
(a)
(b)


### 3.2. Adjusting Parameters of MFROH

In order to have the optimal values ofβ and Δ in the scense of frequency response of MFROH, a proper cost function of ideal low-pass filter must be defined. It is desirable that at low frequencies the amplitude of the frequency response of MFROH is flat and for frequencies higher than ω=ωs/2, it is falling sharply. Also it is preferred that the phase of frequency response of MFROH in passband is flat as much as possible. We assume that the Nyquist sampling condition is satisfied, and therefore passband defines in frequency interval [0,0.1ωs]. Optimal β and Δ are those that minimize the following cost function:
(3)(Δopt,βopt)≜{(Δ,β):J(Δ,β)is  minimum},
where J(Δ,β)=ω1J1(Δ,β)+ω2J2(Δ,β)+ω3J3(Δ,β). In weighted cost function, J1(Δ,β),  J2(Δ,β) and J3(Δ,β) are defined as follows:
(4)J1(Δ,β)=Max||G(jω)|-|G(j0)||,ω∈[0,0.1ωs],J2(Δ,β)=Max|∠G(jω)-∠G(j0)|,ω∈[0,0.1ωs],J3(Δ,β)=Max||G(jω)||,ω∈[ωs,∞).To obtain optimalβ and Δ, ω2 is set as 1/π and ω1 and ω3 are set to one.The result of optimization process is obtained asβopt=0.4 and Δopt=0.2 in which the genetic algorithm as a common optimization method is used. The amplitude and phase of the frequency response of MFROH with optimal β and Δ in comparison with frequency response of ZOH is shown in Figure 7.Amplitude and phase of the frequency response of ZOH and MFROH with optimalβ and Δ.
(a)
(b)As shown in Figure7, both amplitude and phase of frequency response of optimal MFROH in passband are flatter than those of ZOH.


## 3.1. Frequency Response Specifications

Sampling process generates many additional terms in addition to the main term, and the ideal filter attenuates all these additional terms to zero and only passes the main term. Therefore, a practical filter is such a filter that passes the main term without any frequency distortion and attenuates all additional terms as much as possible. This means that the frequency response of practical filter must resemble to that of the ideal filter in Figure3 as much as possible.Figure 3
Frequency response of an ideal filter or an ideal holder.The amplitudes of frequency responses of ZOH and FROH for different values ofβ with sampling time T=0.01sec are shown in Figure 4.Figure 4
Amplitude of frequency responses of ZOH and FROH for different values ofβ.Regarding Figure4, for negative values of β and reducing β, the amplitude of frequency response at ω=ωs/2 is reduced dramatically, and it is not smooth and has large peaks after its first minimum, that is, for β=-1. The amplitude of frequency response increases after its first minimum in ω=ωs, and the filter passes undesirable signals generating from sampling around ω=ωs and leads to frequency aliasing. Therefore, the optimal value of β should be chosen among the positive values of β in FROH. By comparing the amplitudes of frequency responses of ZOH and FROH, it can be seen from Table 1 that for positive values of β at ω=ωs/2, the decrease in the ZOH is much more than the decrease in the FROH.Table 1
FROH's amplitude of the frequency responses for different values ofβ at ω=ωs/2.
β
-
1
-
0.4
0
0.4
1
|
G
h
(
(
j
ω
s
)
/
2
)
|
×
10
3
3.96
4.17
6.4
9.13
13.4In Figure5, the amplitudes of the frequency responses of ZOH and FROH for small values of β are shown. In low frequencies, the transfer function of FROH has a better approximation of ideal low-pass filter than the transfer function of ZOH, because at low frequencies the amplitude of the frequency response of ZOH is reduced much more than FROH, but at high frequencies ZOH is better.Figure 5
Amplitude of the frequency responses of ZOH and FROH for smallβ.The amplitudes of the frequency response of MFROH for different values ofβ and Δ are shown in Figure 6. According to Figure 6 and based on two adjustable parameters of MFROH, it is possible to adjust β and Δ so that the amplitude of the frequency response of MFROH becomes very similar to that of the ideal low-pass filter that is shown in Figure 3.Amplitude of the frequency response of MFROH for different values ofβ and Δ.
(a)
(b)


## 3.2. Adjusting Parameters of MFROH

In order to have the optimal values ofβ and Δ in the scense of frequency response of MFROH, a proper cost function of ideal low-pass filter must be defined. It is desirable that at low frequencies the amplitude of the frequency response of MFROH is flat and for frequencies higher than ω=ωs/2, it is falling sharply. Also it is preferred that the phase of frequency response of MFROH in passband is flat as much as possible. We assume that the Nyquist sampling condition is satisfied, and therefore passband defines in frequency interval [0,0.1ωs]. Optimal β and Δ are those that minimize the following cost function:
(3)(Δopt,βopt)≜{(Δ,β):J(Δ,β)is  minimum},
where J(Δ,β)=ω1J1(Δ,β)+ω2J2(Δ,β)+ω3J3(Δ,β). In weighted cost function, J1(Δ,β),  J2(Δ,β) and J3(Δ,β) are defined as follows:
(4)J1(Δ,β)=Max||G(jω)|-|G(j0)||,ω∈[0,0.1ωs],J2(Δ,β)=Max|∠G(jω)-∠G(j0)|,ω∈[0,0.1ωs],J3(Δ,β)=Max||G(jω)||,ω∈[ωs,∞).To obtain optimalβ and Δ, ω2 is set as 1/π and ω1 and ω3 are set to one.The result of optimization process is obtained asβopt=0.4 and Δopt=0.2 in which the genetic algorithm as a common optimization method is used. The amplitude and phase of the frequency response of MFROH with optimal β and Δ in comparison with frequency response of ZOH is shown in Figure 7.Amplitude and phase of the frequency response of ZOH and MFROH with optimalβ and Δ.
(a)
(b)As shown in Figure7, both amplitude and phase of frequency response of optimal MFROH in passband are flatter than those of ZOH.


## 4. Stability of Zeros in Sampled System

About the importance of zeros of sampled system, it is better to consider that the stability properties of the discrete-time zeros do not influence the closed-loop stability, but, as shown in [9], unstable discretized zeros can lead to intersample ripple in some cases. Several techniques for control systems design are based on the cancellation of the system zeros by the controller. Unfortunately, such methods cannot be applied when the system has unstable zeros; see for example [10] and references therein. Therefore the study of the zeros of sampled system is important; it was studied for the ZOH devices in [11]. Also the stability properties of the FROH zeros (when T→0) were analyzed by Ishitobi [5].In this part of the paper, the stability properties of additional generated zeros of MFROH are analyzed. Applying MFROH, it is necessary to adjust parameters of MFROH so that the zeros of the sampled system lie inside the unit circle. However, selecting suitable values forβ and Δ for any given sampling period is quite a difficult task. In view of practical control engineering, the sampling period is usually small. Thus, in this paper, we concentrate on the case in which the sampling period T is small. For sufficient small sampling time T, the following theorem gives the stability of additional generated zeros of the sampled system in MFROH.Theorem 1.
Suppose thatGp(s) is a strictly proper nth-order transfer function and can be expressed as
(5)Gp(s)=K(s+b1)⋯(s+bm)(s+a1)⋯(s+an),K≠0,
where ai and bi∈C,  (i=1,…,m) and K≠0. Then the z-transform of the MFROH combined with the system (5) and for very small T is
(6)Hβ,Δ(z)=Z{Hβ,Δ(s)}=Z{Gβ,Δ(s)Gp(s)}=KTn-m(n-m)!(z-1)mEn-m(z,Δ,β)(z-1)nz,
where
(7)En-m(z,Δ,β)=(z-Δβ)Bn-m(z)+βn-m+1zBn-m+1(z)z-1-βn-m+1Bn-m+1(z,Δ)z-1,(8)Bp(z)=bp1zp-1+bp2zp-2+⋯+bpp,bpk=∑i=1k(-1)k-iip(p+lk-i),(k=1,…,p),(9)Bp(z,Δ)=bp0(Δ)zp+bp1(Δ)zp-1+⋯+bpp(Δ),bpk(Δ)=∑l=0p-k(-1)p-k-l(l+Δ)p(p+lp-k-l),(k=0,1,…,p),
and (nm)  indicates n!/(n-m)m!Note. Bn-m+1(z,Δ) and Bn-m+1(z) have term (z-1), and En-m+1(z,Δ) is written as (7) for simplicity.Before proving Theorem1, we need some preliminary lemmas.Lemma 2 (see [4]).
Thez-transform of [(1-e-sT)/s]Gp(s) for very small T is as follows:
(10)limT→0Z{1-e-sTsGp(s)}=KTn-m(n-m)!(z-1)mBn-m(z)(z-1)n.Lemma 3 (see [4]).
Thez-transform of [(1-e-sT)/s2]Gp(s) for very small T is as follows:
(11)limT→0Z{1-e-sTs2Gp(s)}=KTn-m+1(n-m+1)!(z-1)mBn-m+1(z)(z-1)n+1,
where Bn-m+1(z) is defined in (8).Lemma 4 (see [7]).
Thez-transform of [(1-e-sT)/s2]Gp(s)e-sΔT for very small T is as follows:
(12)limT→0Z{1-e-sTs2e-sΔTGp(s)}=KTn-m+1(n-m+1)!(z-1)mBn-m+1(z,Δ)(z-1)n+1z,
where Bn-m+1(z,Δ) is defined in (9).Proof of Theorem1.
Using the transfer function of MFROH in (2), the transfer function of the discrete system Hβ,Δ(s) can be expressed as follows:
(13)Hβ,Δ(s)={(1-e-sT)(1-Δβe-sT)s+β(1-e-sT)(1-e-ΔsT)Ts2}Gp(s)=(1-Δβe-sT)(1-e-sT)sGp(s)+βT(1-e-sT)sGp(s)s-βT(1-e-sT)sGp(s)se-sΔT.
The z-transform of Hβ,Δ(s) can be written as follows:
(14)Hβ,Δ(z)=Z{Hβ,Δ(s)}=Z{Gβ,Δ(s)Gp(s)}=(1-Δβz-1)Z{(1-e-sT)sGp(s)}+βTZ{(1-e-sT)s2Gp(s)}-βTZ{(1-e-sT)s2Gp(s)e-sΔT}.
Combining Lemmas 2, 3, and 4 with (14) and after some simplification we would have
(15)Hβ,Δ(z)=KTn-m(n-m)!(z-1)mEn-m(z,Δ,β)(z-1)nz.Theorem 5.
In this theorem one studies the additional generated zeros of the sampled system in MFROH in which, one assumes that the zeros of the transfer function of the system are minimum-phase in the s-plane.Case 1 (n-m=1).
The sufficient condition that all zeros ofHβ,Δ(z) in 0<T≪1 is minimum phase is that all zeros of Gp(s) is minimum phase and β and Δ satisfy the following condition:
(16)β>1Δ(Δ-2).
The boundary curve of (16) is as follows:
(17)β=1Δ(Δ-2).
In FROH for n-m=1, the sufficient condition for stable zeros of Hβ(z) with 0<T≪1 is that all zeros of Gp(s) are stable and -1<β [2]. In this method, the acceptable region of β to have stable zeros of sampled system is extended due to the additional parameter Δ. Therefore, with decreasing Δ, the lower bound of β increases; that is, for Δ=0.1, zeros of sampled system are stable for -5.26<β. For Δ=1 as we mentioned before, the behavior of MFROH switches to FROH, and we can see from [5] and (16) that the region of β is -1<β.Case 2 (n-m=2).
The sufficient condition that all zeros ofHβ,Δ(z) for 0<T≪1 is minimum phase is that all zeros of Gp(s) is minimum phase and the following conditions are satisfied:
(18)β>1Δ(Δ-2),β<0.
The boundary curve of (18) is as follows:
(19)β=1Δ(Δ-2),β=0.
In FROH for n-m=2, the sufficient condition for stable zeros of Hβ(z) with 0<T≪1 is that all zeros of Gp(s) are stable and -1<β<0 [2], whereas in this method, with additional parameter Δ we can extend the acceptable region of β for stability of zeros of sampled system. Note that by decreasing Δ, the range of acceptable β increases.Case 3 (n-m≥3).
At least one of the zeros ofHβ,Δ(z) will be unstable when 0<T≪1, but compared to FROH we can place the non-minimum-phase zeros by adjusting Δ.Proof of Theorem5.
From Theorem1, the additional generated zeros of sampled system in z-plane can be obtained as follows:
(20)En-m(z,Δ,β)=0.
For n-m=1, we have
(21)E1(z,Δ,β)=(-12βΔ2+βΔ+1)z+(12βΔ2-βΔ).
In order to have a stable zero for E1(z,Δ,β), (22) must be satisfied:
(22)|(1/2)βΔ2-βΔ-(1/2)βΔ2+βΔ+1|<1⟶-βΔ2+2βΔ+1>0⟶β>1Δ(Δ-2).
For n-m=2, we have
(23)E2(z,Δ,β)=(1-βΔ2+βΔ+13βΔ3)z2+(-23βΔ3+βΔ2+1)z+(13βΔ3-βΔ).
By applying the Jury stability criteria to (23), E2(z,Δ,β), the following constraints must be satisfied:
(24)(1)E2(1,Δ,β)>0,(2)E2(-1,Δ,β)>0,(3)|(13)βΔ3-βΔ|<|1-βΔ2+βΔ+(13)βΔ3|.
Combination of the above three constraints and after some simplification we have
(25)-βΔ2+2βΔ+1>0,(26)βΔ2(2Δ-3)>0.
In (26), since 0<Δ<1, hence, Δ2(2Δ-3)<0 will be satisfied for all Δ. Therefore, (25) and (26) will be simplified as (27)
(27)β>1Δ(Δ-2),β<0.
For n-m=3 we have
(28)E3(z,Δ,β)=(-14βΔ4+βΔ3-32βΔ2+βΔ+1)z3+(34βΔ4-2βΔ3+3βΔ+4)z2+(-34βΔ4+βΔ3+32βΔ2-3βΔ+1)z+(14βΔ4-βΔ).
By applying the Jury stability criteria to (28), E3(z,Δ,β), the following constraints must be satisfied:
(29)(1)E3(1,Δ,β)>0,(2)E3(-1,Δ,β)<0,(3)|(14)βΔ4-βΔ|<|-(14)βΔ4+βΔ3-(32)βΔ2+βΔ+1|.
Since all of the above conditions are not satisfied simultaneously, then E3(z,Δ,β) has at least one unstable zero.


## 5. Simulation Results

In this part, simulation results of the comparison among  the proposed holders, ZOH and FROH, are given. In Figure8, a continuous system is shown with unity feedback that the sampler and holder is in feedback path.Figure 8
A continuous system with a holder.The simulation result of applying ZOH, FROH, and MFROM in the holder box in Figure8 is shown in Figure 9. In Figure 9(a) step response of the closed-loop system is shown, and in Figure 9(b) a special part of the response is shown to show the difference of the holders. As shown from Figure 9, it is easy to see that the response of the system in MFROM is more similar to the continuous time response (no holder), and it is because of the fact that in MFROH there are two adjusting parameters that using them we can make the response more similar to the continuous time response of the system.(a) Step response of the closed-loop continuous system and digital control system when using MFROH, FROH, and ZOH. (b) Focusing on the step response to show the effect of different holders.
(a)
(b)


## 6. Conclusion

In this paper we introduced a new holder for digital control systems that has a better performance in terms of the frequency response than ZOH, FOH, and FROH and has better frequency response. This new holder has the capability of placing additional generated zeros similar to FROH. But the ranges of adjusting parameters in MFROH increase, and this is the advantage of MFROH when the holder parameters need to be varied. Simulation results shown in Section5 show the efficiency of the proposed holder in comparison with typical holders.



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*Source: 157292-2013-08-06.xml*