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A Spectral Analysis of Feedback Regulation near

and beyond Nyquist Frequency

Dan Wang, Xu Chen†

Abstract—A fundamental challenge in sampled-data control

arises when a continuous-time plant is subject to disturbances

that possess signiﬁcant frequency components beyond the Nyquist

frequency of the feedback sensor. Such intrinsic difﬁculties create

formidable barriers for fast high-performance controls in modern

and emerging technologies such as additive manufacturing and

vision servo, where the update speed of sensors is low compared to

the dynamics of the plant. This paper analyzes spectral properties

of closed-loop signals under such scenarios, with a focus on

mechatronic systems. We propose a spectral analysis method that

provides new understanding of the time- and frequency-domain

sampled-data performance. Along the course of uncovering

spectral details in such beyond-Nyquist controls, we also report

a fundamental understanding on the infeasibility of single-rate

high-gain feedback to reject disturbances not only beyond but

also below the Nyquist frequency. New metrics and tools are then

proposed to systematically quantify the limit of performance.

Validation and practical implications of the limitations are

provided with experimental case studies performed on a precision

mirror galvanometer platform for laser scanning.

Index Terms—Nyquist frequency, feedback regulation, high-

gain control, sampled-data control

I. INT ROD UC TI ON

MANY modern manufacturing systems are increasingly

subjected to the challenge of limited sensing in the

design of control systems. For instance, in hard disk drive

systems, the sampling speed of the closed loop is limited

by the amount of physical servo sectors [1], [2]. In selective

laser sintering additive manufacturing, infrared thermography

cameras are expected to feedback more than 100,000 frames

of data every second, which is currently unattainable in a

real-time control framework [3], [4]. Similar scenarios also

appear in many other systems, such as vision-guided high-

speed controls [5], [6] and chemical processes. This paper

studies performance of the control system in this important

problem space.

The focused feedback system here is a sampled-data one

with its fast continuous dynamics controlled by a slow-

sampled data feedback. To better motivate the research, we

brieﬂy review the existing metrics of sampled-data perfor-

mance. Let a plant Pc(s)be controlled by a digital controller

C(z)under a sampling time Ts(in seconds). It is a standard re-

sult from digital control theory that single-rate high-gain con-

trol (|C(ejΩoTs)|=∞) can asymptotically reject disturbances

at frequency Ωoin the sampled output. However, for the

actual continuous-time output, the situation is more involved.

Based on sampled-data control [7]–[11], periodic sampling

The authors are with the Department of Mechanical Engineering, Uni-

versity of Connecticut, Storrs, CT, 06269, USA (emails: {dan.wang,

xchen}@uconn.edu). †: corresponding author.

at Tspartitions the continuous-time frequency into inﬁnite

regions of [2kπ/Ts,2(k+1)π/Ts)where k=0,±1,±2,. . . ,

and a continuous-time disturbance yields a fundamental mode

plus an inﬁnite number of shifted replicas in the partitioned

regions. Due to the sampled-data architecture, the conventional

concept of frequency responses does not apply to evaluate

the full system performance here [7]–[12]. Three variations

are introduced: (i) the fundamental transfer function (FTF)

[11], (ii) the performance frequency gain (PFG) [13], [14],

and (iii) the robust frequency gain (RFG) [7]. FTF reveals

partial information of the full intersample behavior because

it focuses only on the fundamental mode. PFG studies the

overall sampled-data behavior within certain frequency regions

by employing an input-to-output power gain function [15].

RFG forms a metric for robustness by maximizing the input-

to-output power ratio over all possible combinations of the

magnitudes and phases of the input [16].

Although a sizable literature has studied the generalized

frequency responses in sampled-data control, analyses and

evaluations for the case with beyond-Nyquist disturbances

have not been sufﬁciently developed. For instance, under a

beyond-Nyquist disturbance, PFG and RFG only provide a

scalar value as an indicator of the regulation performance.

The distribution and closed-loop impact of each sampling-

induced alias mode remain not well understood. This can

be problematic for control practitioners since it is hard to

distinguish whether a spectral peak in the observed output

comes from below- or beyond-Nyquist disturbance sources.

As will be shown, the spectral effects of high-gain control on

beyond-Nyquist disturbances differ greatly from those below

π/Ts. This research uncovers the spectral details and, by doing

so, reveals the infeasibility of sub-Nyquist high-gain servo

design to reject beyond-Nyquist disturbances in mechatronic

systems that have low-pass type of dynamics. In particular, we

present and validate the existence of an upper frequency bound

for rejecting disturbances even below the Nyquist frequency.

This bound implies a fundamental limitation for high-gain

feedback control of sampled-data systems. We provide tools

to analyze the limitation and guidance to implement the tools

in practical problems. Theoretical analyses in this paper are

veriﬁed by both simulation and experimentation on a laser

scanning platform in additive manufacturing.

The main contributions of the paper are:

•building a full spectral analysis method to evaluate the

intersample behavior for beyond-Nyquist disturbances in

sampled-data control;

•applying the proposed method to analyze single-rate high-

gain control and discovering the existence of a principal

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

2

sampled-data bandwidth Bpbelow the Nyquist frequency;

•verifying numerically and experimentally the theoretical

results in additive manufacturing.

A preliminary version of the ﬁndings was presented in [17]. In

this paper, we substantially expand the research with new theo-

retical results and experimental veriﬁcations. In the remainder

of the paper, Section II reviews several basics of sampled-

data control; the main spectral analysis method is provided

in Section III; Sections IV and V provide the numerical and

experimental veriﬁcations of the algorithm, respectively, after

which Section VI concludes the paper.

Notations:x[n]and xc(t)denote, respectively, a discrete

sequence and a continuous-time signal. X(ejω)denotes the

discrete-time Fourier transform (DTFT) of x[n].Xc(jΩ)is the

Fourier transform of xc(t).ω=ΩTs, and Ωis in rad/s.

ℜ(c)denotes the real part of a complex number c∈C. For

a sampled-data system with measurements collected every Ts

sec, single-rate control refers to digital control implemented

at the same sampling time of Ts.

II. PR EL IM INA RI ES

Consider the sampled-data control system in Fig. 1,

where the solid and the dashed lines represent, respectively,

continuous- and discrete-time signal ﬂows. The main elements

in the block diagram include the continuous-time plant Pc(s),

the analog-to-digital converter (ADC) that samples the con-

tinuous output at Ts, the discrete-time controller C(z), and

the signal holder H. In this paper, we focus on the case

where His a zero-order hold (ZOH). The developed tools

and analytic framework can be applied to generalized sample

hold functions.

dc

+

//Huc//Pc(s)yc0

+//◦yc

//ADC

−

yd//

C(z)

ud◦

e

oo

Fig. 1: Block diagram of a sampled-data control system.

Some basic properties and assumptions of sampled-data

control are reviewed ﬁrst for setting up the problem.

It is assumed that 1) Pc(s) = P0(s)e−sτwhere τ≥0; P0(s)

and C(z)both are LTI, proper, and rational; 2) the coefﬁcients

of all transfer functions are real; 3) the closed loop satisﬁes

the non-pathological sampling condition [18].

Under assumption 3), the closed-loop sampled-data system

is stable if and only if the discrete-time closed loop, consisting

of C(z)and the ZOH equivalent of Pc(s), is stable [19], [20].

Lemma 1. [21] If Xc(jΩ)exists, the sampling process

converting xc(t)to x[n] = xc(nTs)gives

Xejω=1

Ts

∞

∑

k=−∞

Xc(j(ω

Ts

−2π

Ts

k)).(1)

Following conventions, we refer to Xc(jω/Ts) (k=0)as

the fundamental mode and the other terms (k6=0)in the right

side of (1) as the shifted replicas.

Because of (1), after dcpasses the ADC and enters the

feedback loop, yc(t)contains a fundamental mode plus an

inﬁnite number of aliases:

Lemma 2. [22] If dc(t) = ejΩotand the sampling time is Ts

in Fig. 1, then the Fourier transform of the continuous-time

plant output yc(t)is

Yc(jΩ) = 2π1−1

Ts

Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)δ(Ω−Ωo)

−2π

Ts

Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)

∞

∑

k=−∞,k6=0

δ(Ω−Ωo−2π

Ts

k),(2)

where δ(Ω−Ωo)denotes a shifted Dirac delta impulse,

H(jΩ) = (1−e−jΩTs)/(jΩ)is the Fourier transform of the

ZOH, and Sd(ejΩoTs)is the frequency response of the discrete-

time sensitivity function Sd(z) = 1/(1+Pd(z)C(z)), where

Pd(z), the ZOH equivalent of Pc(s), has the DTFT

Pd(ejΩoTs) = 1

Ts

∞

∑

k=−∞

Pc(j(Ωo+2π

Ts

k))H(j(Ωo+2π

Ts

k)).(3)

In practice, the pure analog output yc(t)is infeasible to

collect and store on digital computers. As an alternative, a fast

signal sampled at T0

sis used to approximate the continuous-

time output with T0

s=Ts/F(F>1 and F∈Z). The problem

then reduces to a multirate (MR) sampled-data control one, as

shown in Fig. 2, where the dotted and dashed lines represent

the fast and slow signals sampled by T0

sand Ts, respectively.

To reveal the performance of the fast-sampled output ydh, we

adopt the PFG metric [13], which considers the power ratio

between the input disturbance d[k] = d(kT 0

s)and the output

ydh [k] = ydh(kT 0

s):

Deﬁnition 1. Let d[k]∈nd[k]:d[k] = cejΩk T 0

s,kck2<∞obe

applied to an MR system in Fig. 2. The PFG P(ejΩT0

s)is

deﬁned as

P(ejΩT0

s),sup

d6=0

kydh [k]kp

kd[k]kp

,(4)

where k·kprepresents the discrete-time signal power

kd[k]kp,v

u

u

tlim

N→∞

1

2N+1

N

∑

k=−N

kd[k]k2,(5)

and k·kdenotes the Euclidean vector norm.

H

+

‐

+

′

′

′

Fig. 2: Block diagram of multirate sampled-data analysis.

III. SPE CT RA L ANALYSIS OF BEYON D-NY QU IS T

REG UL ATIO N PROB LE MS

To better motivate the analysis, consider two fast-sampled

outputs ydh in Figs. 3 and 4 collected from experimentation

on the mirror galvanometer system in Section V. The outputs

are fast sampled at T0

s=Ts/Fwith F=4. The Nyquist

frequency is ΩN=5kHz. The disturbance frequencies are

below ΩNat 3kHz and beyond ΩNat 7kHz, respectively.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

3

Under a classic PID control design, the two single-harmonic

excitations generate aliased modes at multiple frequencies

(bottom plots of Figs. 3 and 4). When classic single-rate

high-gain control [23] is applied to the feedback system,

distinct differences show in the output spectra (top plots of

Figs. 3 and 4). Furthermore, all the spectral spikes are not

fully attenuated despite the zero steady-state Ts-sampled output

(Figs. 7a and 8a). How do the results happen? What is the

governing mechanics of the beyond-Nyquist compensation?

How would the spectral distribution change with respect to

the excitation frequency?

0

0.05

high-gain control on

02468101214161820

Frequency(kHz)

0

0.02

0.04

0.06

0.08 high-gain control off

X: 7000

Y: 0.05446

X: 3000

Y: 0.05397

X: 3000

Y: 0.02361

X: 7000

Y: 0.02772

1Γ

Ω

Γ

Ω

1Γ

∗Ω

Γ

∗Ω

Fig. 3: Fast Fourier transform of ydh(t)with input disturbance

frequency at 1.4ΩN.

0

0.02

0.04

0.06

0.08

high-gain control on

0 2 4 6 8 10 12 14 16 18 20

Frequency(kHz)

0

0.05

0.1

high-gain control off

X: 7000

Y: 0.02331

X: 3000

Y: 0.02333

X: 3000

Y: 0.07651

X: 7000

Y: 0.01086

1Γ

ΩΓ

Ω

1Γ

∗ΩΓ

∗Ω

Fig. 4: Fast Fourier transform of ydh(t)with input disturbance

frequency at 0.6ΩN.

To decipher the characteristics of the individual frequency

spikes, we propose a spectral analysis method integrating the

principles of loop shaping, the limiting conditions of high-gain

control, and the PFG. For a generalized sampled-data control

system in Fig. 1, to determine the magnitudes of the individual

spectral spikes, we deﬁne the characteristic feedback loop gain

Γk(Ωo),Pc(j(Ωo+2π

Tsk))H(j(Ωo+2π

Tsk))

TsPd(ejΩoTs)Td(ejΩoTs),(6)

where

Td(ejΩoTs),Pd(ejΩoTs)C(ejΩoTs)

1+Pd(ejΩoTs)C(ejΩoTs)=Pd(ejΩoTs)C(ejΩoTs)Sd(ejΩoTs).(7)

After substituting (6) into (2) and recalling that FejΩ0t=

2πδ(Ω−Ω0), the steady-state continuous-time output is sim-

pliﬁed to

yc(t) = [1−Γ0(Ω0)]ejΩ0t−

∞

∑

k=−∞,k6=0

Γk(Ω0)ej(Ω0+2π

Tsk)t.(8)

Fact 1. Based on (3) and (6), it is immediate that

∞

∑

k=−∞

Γk(Ωo) = Td(ejΩoTs).(9)

For the case of real-valued disturbances in practice, let

dc(t) = cos(Ωot+φ). Recall cos(Ω0t+φ) = ℜ(ej(Ω0t+φ)),

F{ℜ(x(t))}= [X(−jΩ) + X(jΩ)]/2, and δ(−Ω−Ωo) =

δ(Ω+Ωo). Laplace transform to the real part of (8) gives

Yc(jΩ) = πejφ(1−Γ0(Ωo))δ(Ω−Ωo)

+πe−jφ(1−Γ0(−Ωo))δ(Ω+Ωo)

−πejφ∞

∑

k=−∞,k6=0

Γk(Ω0)δ(Ω−Ωo−2π

Ts

k)

−πe−jφ∞

∑

k=−∞,k6=0

Γ−k(−Ω0)δ(Ω+Ωo+2π

Ts

k).(10)

By the deﬁnition in (6), Γk(Ωo)is conjugate

symmetric, namely, Γ−k(−Ω0) = Γk(Ω0). Thus in (10),

the gains for two fundamental modes, |1−Γ0(Ω0)|and

|1−Γ0(−Ω0)|=1−Γ0(Ω0), are equal, and the gains for

their related aliased harmonics, |Γk(Ω0)|and |Γ−k(−Ω0)|,

are also equal. The collective effect of these modes governs

the dynamics of the output.

It is noteworthy that simultaneously rejecting all modes

of Yc(jΩ)in (10) is unattainable. Similar to the feedback

limitation on simultaneously rejecting disturbances and sensor

noises, the gains for the fundamental modes and the aliases

cannot be reduced at the same time. For example, letting

C(ejΩoTs) = 0 in (7) yields Γk(Ωo) = 0 for any k, namely,

a zero gain for each harmonic |Γk(Ω0)|and a unit gain for

the fundamental mode |1−Γ0(Ω0)|in (10). Thus, perfect

“rejection” of the aliased harmonics yields no attenuation of

the fundamental disturbance.

To understand the differences in the top plots of Figs. 3

and 4, we explore the shape of the mode gain Γk(Ωo)under

high-gain control.

Deﬁnition 2. Under ideal single-rate high-gain control, the

new characteristic feedback loop gain is

Γ∗

k(Ωo),lim

|C(ejΩoTs)|→∞

Γk(Ωo) = Pc(j(Ωo+2π

Tsk))H(j(Ωo+2π

Tsk))

TsPd(ejΩoTs).(11)

Fact 2. From the deﬁnition of Pd(ejΩoTs)in (3), it is immediate

that the summation of Γ∗

k(Ωo)over kis

∞

∑

k=−∞

Γ∗

k(Ω0) = 1,∀Ω0.(12)

(12) will be revisited in Section III-B. Similar to

Γk(Ωo),Γ∗

k(Ωo)is also conjugate symmetric: 1−Γ∗

0(Ωo)=

1−Γ∗

0(−Ωo);Γ∗

k(Ωo)=Γ∗

−k(−Ω0).

A. Characteristic feedback loop gains Γk(Ωo)and Γ∗

k(Ωo)

In this subsection, the properties of the characteristic feed-

back loop gains are discussed. From (6) and (11), we obtain

that Γk(Ωo) = Γ∗

k(Ωo)Td(ejΩoTs). Since Td(z)is typically a

low-pass ﬁlter whose bandwidth BTis commonly 10%-20% of

the Nyquist frequency [22], we have |Γ∗

k(Ωo)|>|Γk(Ωo)|for

most frequencies. Furthermore, we can obtain the following

characteristics:

1) If Ωo+2kπ/Ts∈[0,BT), then the low-pass H(j(Ωo+

2kπ/Ts))/Ts≈1 in (11). For mechatronic systems

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

4

where the plant usually has high gains at low fre-

quencies, Pc(j(Ωo+2kπ/Ts))H(j(Ωo+2kπ/Ts))/Ts≈

Pd(ej(Ωo+2kπ/Ts)Ts) = Pd(ejΩoTs), and |Td(ejΩoTs)|≈1,

yielding both Γk(Ωo)and Γ∗

k(Ωo)to be approximately

1. Thus, |1−Γk(Ωo)|and |1−Γ∗

k(Ωo)|are both small.

In particular, since Pd(1) = Pc(0)[24] and H(0)/Ts=1,

we have 1−Γ∗

0(0) = 0.

2) If Ωo+2kπ/Ts∈[BT,π/Ts), then |Td(ejΩoTs)|<1, and

thus |Γ∗

k(Ωo)|>|Γk(Ωo)|. For most frequencies in this

region, |Γ∗

k(Ωo)| ≈ 1, and |1−Γ∗

k(Ωo)| 1.

3) If Ωo+2kπ/Ts∈[π/Ts,2π/Ts), the low-pass ZOH

|H(j(Ω+2kπ/Ts))|reduces quickly outside its ap-

proximate bandwidth π/Ts. Although high-gain control

still makes |Γ∗

k(Ωo)|>|Γk(Ωo)|, the overall magni-

tudes |Γ∗

k(Ωo)|and |Γk(Ωo)|are very small. Thereby,

1−Γ∗

k(Ωo)and |1−Γk(Ωo)|both approximate 1.

Interestingly, Γ∗

k(Ωo)has high gains at the Nyquist frequency

and its odd multiplications. To see this point, we analyze the

property of |Pd(ejπ

TsTs)|=|Pd(−1)|in (11). It is well known

that all continuous-time systems with relative degree larger

than or equal to two have limiting nonminimum-phase zeros

in their ZOH equivalent [24]. In particular, real unstable zeros

appear in Pdat high frequencies for small values of Ts. As a

result, |Pd(−1)|in (11) is small or even zero, yielding a large

Γ∗

k(π

Ts). More speciﬁcally, we have the following result:

Lemma 3. If Pc(s) = 1/snand nis a positive even integer,

then Γ∗

k(π

Ts) = ∞.

Proof: See the Appendix.

Lemma 3 illustrates a danger of designing single-rate high-

gain controllers near the Nyquist frequency. With the limiting

case of Γ∗

k(π

Ts)and 1 −Γ∗

k(π

Ts)both being inﬁnity, a continuity

analysis gives that Γ∗

k(Ω)and 1−Γ∗

k(Ω)have very high gains

near the Nyquist frequency. Correspondingly, from (10), the

continuous-time output is signiﬁcantly ampliﬁed. It is also

worth pointing out that the special case of Pc(s) = 1/s2

is common in precision motion control (e.g., in hard disk

drives [25] and in wafer scanners used in semiconductor

manufacturing).

Fig. 5 illustrates the magnitudes of Γ(∗)

kand 1 −Γ(∗)

kin a

motion-control example in Section IV. The Nyquist frequency

is indicated by the vertical line at π/Ts. The shapes of the

curves match well with the above analysis. As an analysis tool,

Fig. 5 reveals several fundamental performance limitations of

single-rate high-gain control:

•First, based on the top plot in Fig. 5, unless at very

low frequencies (below BT) where Γk(Ω)≈Γ∗

k(Ω), the

aliased harmonics are all ampliﬁed by single-rate high-

gain control.

•Second, high-gain control in C(z)only provides enhanced

rejection of the fundamental disturbance mode below the

intersection frequency of |1−Γ∗

0(Ω)|and |1−Γ0(Ω)|(Bc

in Fig. 5). In addition, the achievable maximum attenua-

tion—indicated by the magnitude |1−Γ∗

0(Ω)|—decreases

with increasing frequency. For common servo design with

low-pass type of complementary sensitivity functions Td,

the ﬁrst two points suggest that single-rate high-gain

control cannot reject continuous-time disturbances near

and above Nyquist frequency.

•Third, for Ωo∈(π/Ts,2π/Ts),|Γ∗

k(Ωo)|>|Γk(Ωo)|, and

|1−Γ∗

0(Ωo)|&|1−Γ0(Ωo)| ≈ 1. In this interval, under

single-rate high-gain control, Ωobeing closer to π/Ts

causes larger servo degradation, which is different from

classic servo control where disturbances at lower frequen-

cies are commonly easier to be attenuated.

BTπ/Ts2π/Ts

Bc

Fig. 5: Magnitude responses of Γk(Ω),Γ∗

k(Ω), 1 −Γk(Ω)

and 1 −Γ∗

k(Ω)as a function of Ω+2πk/Ts, where Γ∗

k(Ωo)

and Γk(Ωo)denote the characteristic feedback loop gain with

and without high-gain control respectively. The ﬁrst three

vertical lines indicate, respectively, the Nyquist frequency, the

sampling frequency and 3

/2Ts.

Remark 1. For implementation, it is noteworthy that with

the low-pass dynamics in ZOH, the ﬁrst few frequency modes

in (10) are usually dominant in magnitude. In Fig. 5, after

3·2π

Ts, the magnitudes of Γk(Ω)and Γ∗

k(Ω)are relatively

insigniﬁcant, and 1−Γ(∗)

k(Ω)is practically equal to 1.

B. Typical spectrum of yc(t)in sampled-data control

In this subsection, we extend the analysis and study the full

beyond-Nyquist spectra of the output signals.

Let Ωo∈(π/Ts,2π/Ts)and Ω0

o=2π/Ts−Ωo∈(0,π/Ts).

Consider two different disturbances dc(t),cos(Ωot)and

˜

dc(t),cos(Ω0

ot), respectively, at above and below the Nyquist

frequency. The Fourier transforms of the continuous-time

disturbances are

Dc(jΩ) = πδ(Ω−Ωo) + πδ(Ω+Ωo),

˜

Dc(jΩ) = πδ(Ω−Ωo−2π

Ts

) + πδ(Ω+Ωo−2π

Ts

).

From (1), the sampled disturbance spectra and hence yd[k]

are the same. However, the spectra of yc(t)are fundamentally

different for the two types of disturbances, as illustrated in

Figs. 6a and 6c. One important difference is the location

of the fundamental mode (Ωofor dcand 2π−Ωofor d0

c).

For Ωobeing above Nyquist frequency, the magnitude of the

fundamental mode |1−Γ0(Ωo)|is close to 1 (cf. Fig. 5). The

dominant aliased mode Γ1(−Ωo)occurs at 2π/Ts−Ωobelow

the Nyquist frequency (see Fig. 6a). With single-rate high-gain

control at Ωo, the magnitude of Γk(Ωo)increases towards the

limiting case Γ∗

k(Ωo). In particular, Γ1(−Ωo)increases towards

Γ∗

1(−Ωo)≈1 (Fig. 6b). Meanwhile, |1−Γk(Ωo)|stays close

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

5

to 1 or is even increased. Collectively, dc(t)is ampliﬁed by

single-rate high-gain control.

On the other hand, for Ω0

obelow the Nyquist frequency,

the fundamental mode 1 −Γ0(Ω0

o)can be effectively reduced

(from the dashed line to the solid line in the bottom plot of

Fig. 5). The aliased modes Γk(Ω0

o)still increase to Γ∗

k(Ω0

o).

However, |Γ∗

k(Ω0

o)|remains small in the top plot of Fig. 5

since the lowest frequency of the alias is already beyond the

Nyquist frequency (at 2π/Ts−Ω0

o). Thus, ˜

dc(t)can be reduced

by single-rate high-gain control.

The graphical tool is justiﬁed by the experimental results in

Figs. 3 and 4. In Fig. 3, the fundamental mode occurs at 7000

Hz, and the ampliﬁed mode at 3000 Hz corresponds to the alias

mode below the Nyquist frequency. In Fig. 4, the frequencies

of the two modes are switched. We can now distinguish that

Fig. 3 describes the trend of the case in Figs. 6a and 6b while

Fig. 4 matches the results in Figs. 6c and 6d.

Ω

Yc

πe±jφ

| ||| | |||

1−Γ0(Ωo)1−Γ0(−Ωo)

Γ1(−Ωo)

Γ−1(Ωo)

Γ1(Ωo)

Γ−1(−Ωo)

Γ2(−Ωo)

Γ−2(Ωo)......

π

Ts

2π

Ts

−π

Ts

−2π

Ts

3π

Ts

4π

Ts

−3π

Ts

−4π

Ts

(a) dc(t) = cos(Ωot)with the baseline control

Ω

| ||| | |||

1−Γ∗

0(Ωo)1−Γ∗

0(−Ωo)

Γ∗

1(−Ωo)

Γ∗

−1(Ωo)

Γ∗

1(Ωo)

Γ∗

−1(−Ωo)

Γ∗

2(−Ωo)

Γ∗

−2(Ωo)......

π

Ts

2π

Ts

−π

Ts

−2π

Ts

3π

Ts

4π

Ts

−3π

Ts

−4π

Ts

(b) dc(t) = cos(Ωot)with enhanced discrete-time high-gain control at Ωo

Ω

| ||| | |||

1−Γ0(Ω0

o)1−Γ0(−Ω0

o)

Γ1(−Ω0

o)

Γ−1(Ω0

o)Γ1(Ω0

o)

Γ−1(Ω0

o)Γ2(−Ω0

o)

Γ−2(Ω0

o)

......

π

Ts

2π

Ts

−π

Ts

−2π

Ts

3π

Ts

4π

Ts

−3π

Ts

−4π

Ts

(c) ˜

dc(t) = cos(Ω0

ot) = cos((2π/Ts−Ωo)t)with the baseline control

Ω

| ||| | |||

1−Γ∗

0(Ω0

o)1−Γ∗

0(−Ω0

o)

Γ∗

1(−Ω0

o)

Γ∗

−1(Ω0

o)

Γ∗

1(Ω0

o)

Γ∗

−1(Ω0

o)Γ∗

2(−Ω0

o)

Γ∗

−2(Ω0

o)

......

π

Ts

2π

Ts

−π

Ts

−2π

Ts

3π

Ts

4π

Ts

−3π

Ts

−4π

Ts

(d) ˜

dc(t) = cos(Ω0

ot)with enhanced discrete-time high-gain control at Ωo

Fig. 6: Illustration of the spectrum of yc(t)in sampled-data

control when π/Ts<Ωo<2π/Ts. Dashed spikes: δ(Ω+Ω0)

and its aliases; solid spikes: δ(Ω−Ω0)and its aliases.

Next we show how to connect the frequency-domain results

with the time-domain observations. With sub-Nyquist high-

gain control, the Ts-sampled disturbances dc(t)and ˜

dc(t)can

be perfectly rejected from the sampled output, as shown in the

corresponding time-domain responses of Figs. 7a and 8a. The

disturbance rejection may conventionally suggest null gains in

the spectrum below the Nyquist frequency, which is, however,

neither the case for dc(t)or ˜

dc(t). In fact, Fig. 6b contains

signiﬁcant components at 2π/Ts−Ωo. And Fig. 7b shows the

hidden ampliﬁcation of the disturbances. To connect the spec-

tral distribution with the zero steady-state Ts-sampled output,

an important piece is the effect of the sampling operation in

the frequency domain. Take the case of dc(t)as example. After

yc(t)is sampled at Ts, each solid spike in Fig. 6b creates an

alias at Ωo(cf. Lemma 1). Based on (10), the magnitude of the

discrete-time spectral peak at Ω0is a normalized version of

[1−Γ∗

o(Ωo)] −Γ∗

−1(Ωo)−Γ∗

1(Ωo)−Γ∗

−2(Ωo)−Γ∗

2(Ωo)−... ,

which equals 0 from (12). For the case where the disturbance

is beyond Nyquist frequency in Fig. 6b, because there is little

control over 1 −Γ0(Ωo), and Γ±k(Ωo)(k6=0) is ampliﬁed,

the aliasing effect cancels the fundamental component after

sampling. Fig. 6d, on the other hand, achieves zero Ts-sampled

output by reducing the magnitude of 1 −Γ0(Ω0

o).

1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

Time(sec)

-0.05

0

0.05

Slow-sampled

output (V)

(a) yd(t)with the sampling time of Ts.

1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

Time(sec)

-0.1

-0.05

0

0.05

0.1

Fast-sampled output (V)

(b) ydh (t)with the sampling time of T0

s.

Fig. 7: Plant output with the input disturbance at 1.4ΩN(The

solid and dashed lines represent the cases with single-rate

high-gain control on and off, respectively.)

1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

Time(sec)

-0.05

0

0.05

Slow-sampled

output (V)

(a) yd(t)with the sampling time of Ts.

1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

Time(sec)

-0.1

-0.05

0

0.05

0.1

Fast-sampled output (V)

(b) ydh (t)with the sampling time of T0

s.

Fig. 8: Plant output with the input disturbance at 0.6ΩN(The

solid and dashed lines represent the cases with single-rate

high-gain control on and off, respectively.)

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

6

C. Performance frequency gain and the fundamental mode

With the understanding of individual mode shapes, we

can better relate the spectral responses to the time-domain

data in sampled-data control and explain the beyond-Nyquist

disturbance rejection. This section connects the analysis of

the individual modes with the PFG metric. An important

observation is that under single-rate high-gain control, PFG

also has a high gain near the Nyquist frequency.

Recall the transformation of a sampled-data system into an

MR one by fast sampling in Fig. 2. The fast and slow signals

are sampled by T0

sand Ts=FT 0

s, respectively. Let D(ejΩT0

s)

denote the DTFT of d[k]. Analogous to the derivation of (2),

the DTFT of the fast-sampled output ydh[k][15] is

Ydh(ejΩT0

s) = 1−1

FPdh(ejΩT0

s)H(ejΩT0

s)Td(ejΩTs)/Pd(ejΩTs)D(ejΩT0

s)

+1

FPdh(ejΩT0

s)H(ejΩT0

s)Td(ejΩTs)/Pd(ejΩTs)

F−1

∑

k=1

D(ej(ΩT0

s−2πk

F)),(13)

where Pdand Pdh represent the ZOH plant models under the

sampling time of Tsand T0

s, respectively, and the transfer

function of the ZOH interpolator is

H(z) =

F−1

∑

k=0

z−k=(F z =1

1−z−F

1−z−1z6=1.(14)

Based on (13), the MR characteristic feedback loop gain is

deﬁned as

Γk(Ωo) = Pdh(ej(ΩoT0

s+2πk

F))H(ej(ΩoT0

s+2πk

F))Td(ejΩ0Ts)

FPd(ejΩ0Ts),(15)

and the limiting case with single-rate high-gain control is

Γ∗

k(Ωo) = Pdh(ej(ΩoT0

s+2πk

F))H(ej(ΩoT0

s+2πk

F))

FPd(ejΩ0Ts).(16)

Lemma 4. For the MR system in Fig. 2, the modiﬁed PFG

under single-rate high-gain control at Ω0is

Ph(ejΩ0T0

s) = v

u

u

t

1−Γ∗

0(Ωo)

2+

F−1

∑

k=1

Γ∗

k(Ωo)

2.(17)

The derivation is similar to the one introduced in [15] and is

omitted here. Lemma 4 connects the input-output power ratio

with the gains of the individual signal modes. PFG evaluates

the overall effect of the intersample behavior and how a

sampled-data control system attenuates or ampliﬁes input

disturbances in certain frequencies, whereas the characteristic

feedback loop gains look into each individual mode in the

spectra of the continuous-time (and fast-sampled) outputs.

Note that independent of the baseline controller, the mod-

iﬁed PFG is a property of the plant itself since Γ∗

k(Ωo)

depends on Pd h,H,F, and Pdalone. In addition, the modiﬁed

PFG is a pointwise quantity that focuses on the limiting

case where ideal high-gain control is applied at one value of

Ω0, that is, Td(ejΩoTs) = 1. This pointwise high-gain control

can be achieved with tools such as special Youla-Kucera

parameterizations, disturbance observers, and peak ﬁlters [23],

[26], [27]. To introduce Td(ejΩoTs) = 1 at different values of

Ω0, the high-gain controller would need to be retuned or be

adaptive. When the customized high-gain control is turned

off, the high-gain controller is replaced by a regular servo

algorithm (e.g. PID and lead-lag compensation), and therefore

Γ∗

0(Ωo)and Γ∗

k(Ωo)are replaced by Γ0(Ωo)and Γk(Ωo)in

(15). The modiﬁed PFG then describes the performance of a

baseline LTI controller.

For a typical plant dynamic in Section V, Ph(ejΩT0

s)is

calculated and plotted in Fig. 9.

ܤ

Fig. 9: Performance frequency gain under high-gain control.

Deﬁnition 3. The intersection frequency between the curve

expressed by (17) and the line of Ph(ejΩT0

s) = 0 dB is called

the principal sampled-data bandwidth Bp.

Lemma 5. For general mechatronic systems, Bpis smaller

than the Nyquist frequency.

Proof: See the Appendix.

Implications: Similar to the analyses of the discrete-time

sensitivity function in digital control, the proposed PFG analy-

sis gives an important threshold frequency Bpin sampled-data

control. For disturbance frequencies below Bp, the power of

the fast-sampled output signal is smaller than that of the input

disturbance. In other words, sub-Nyquist high-gain control is

efﬁcient for rejecting disturbances with frequencies below Bp.

However, for beyond-Bpdisturbances with Ph>0 dB, single-

rate high-gain control exacerbates the servo performance.

Remark 2. In practice, disturbances can also enter from the

input of the plant in Fig. 2. In this case, the input disturbance di

is related to din Fig. 2 by D(ejΩT0

s) = Di(ejΩT0

s)Pd h(ejΩT0

s).

We can analogously deﬁne and compute the modiﬁed input

PFGs

P0

b(ejΩ0T0

s) = sup

di6=0

kydh kp

kdikp

=

Pdh(ejΩ0T0

s)sk1−Γ0(Ωo)k2+

F−1

∑

k=1

kΓk(Ωo)k2,(18)

and

P0

h(ejΩ0T0

s) = lim

Td(ejΩ0T0

s)→1

P0

b(ejΩ0T0

s) = |Pdh|sk1−Γ∗

0(Ωo)k2+

F−1

∑

k=1

Γ∗

k(Ωo)

2.(19)

The modiﬁed input PFG can be veriﬁed by the time-domain

deﬁnition in (18), that is, dividing output signal power by input

signal power. Dividing the modiﬁed input PFG P0

h(ejΩ0T0

s)

by Pdh(ejΩ0T0

s), we can then generate the modiﬁed PFG

Ph(ejΩ0T0

s).

Before presenting the numerical and experimental results,

we brieﬂy summarize the application steps of the proposed

spectral analysis method:

1) Determine Γk(Ωo), the characteristic feedback loop

gains, by (6) and (15). In addition, determine Γ∗

k(Ωo),

the limiting cases with single-rate high-gain control, by

(11) and (16).

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

7

2) Plot the magnitude responses of Γk(Ω),Γ∗

k(Ω), 1 −

Γk(Ω), and 1 −Γ∗

k(Ω)to look into individual spec-

tral spikes. Note that these are hybrid functions of

continuous- and discrete-time frequency responses.

3) Calculate and plot the modiﬁed PFG based on (17).

4) Identify the principal sampled-data bandwidth Bp, as

shown in Fig. 9.

5) Run simulation and experimentation to get the time-

and frequency-domain results with below- and beyond-

Bpdisturbance input. Numerically compute the modiﬁed

PFG from the input-to-output power ratio in Deﬁnition

1. The results should verify the location of Bpand the

trend of the individual spectral spikes.

IV. NUM ER IC AL VE RI FIC ATIO N

Consider a plant Pc(s) = 3.74488 ×109/(s2+565.5s+

319775.2)with an input delay of 10 µs. Let the sampling time

be Ts=1/2640 sec. The baseline controller is a PID controller

C(z) = kp+ki/(z−1) + kd(z−1)/zwith kp=7.51 ×10−5,

ki=3.00×10−5, and kd=3.60×10−4. Such a design provides

a bandwidth at 92 Hz that complies with the rule-of-thumb of

around 10% of the Nyquist frequency. yc(t)is fast-sampled

at T0

s=Ts/20 to approximate the continuous-time output

in Fig. 1. Single-frequency vibrations below and above the

Nyquist frequency are introduced to the plant. The narrow-

band disturbance observer (DOB) [23] is applied on top of the

PID controller. Such a design provides perfect compensation

of above- and below-Nyquist sinusoidal signals in the sampled

output yd[k].

Figs. 10 and 11 present the time- and frequency-domain

computation results, which verify the limitation of single-rate

high-gain control for beyond-Nyquist disturbance rejection.

The results match with the prediction in Fig. 6 that single-

rate high-gain control ampliﬁes beyond-Nyquist disturbances.

When the disturbance occurs at 2376 Hz (1.8ΩN), the inter-

sample signal is signiﬁcantly ampliﬁed in Fig. 10b, although

high-gain control yields zero sampled-output at steady state

(Fig. 10a). The ampliﬁcation is also evident in the frequency

domain (Fig. 11). Single-rate high-gain control barely changes

the fundamental component at 2376 Hz but greatly ampliﬁes

the aliased component at 264 Hz.

Fig. 12 veriﬁes the case with regular below-Nyquist distur-

bances. The Ts-sampled output also reaches zero at steady state

and is omitted here. With the fundamental mode at 924 Hz

(below the Nyquist frequency), single-rate high-gain control

can attenuate this spectral spike. As theoretically predicted

by Fig. 6d, the aliased harmonics are, however, all ampliﬁed.

Therefore, the actual continuous-time output contains inter-

sample ripples.

V. EX PE RI ME NTAL VERIFICATION

Experiments are conducted on a galvo scanner platform

(Fig. 13), a key component in laser-based additive manufac-

turing. Typically, a galvo scanner is composed of mirrors,

galvanometers, and control systems. The mirrors are actuated

to reﬂect the input laser beam to generate a scanning trajectory

at high speed with high precision. The angular rotation of the

mirrors are measured by encoders.

0.01 0.02 0.03 0.04 0.05 0.06

Time (sec)

-10

0

10

Normalized output

(a) yc(t)sampled at Ts.

0.081 0.082 0.083 0.084 0.085 0.086 0.087 0.088 0.089 0.09

Time (sec)

-10

0

10

Normalized output

(b) yc(t)sampled at Ts/20.

Fig. 10: Plant output with the input disturbance at 1.8ΩN(The

solid and dashed lines represent the cases with single-rate

high-gain control on and off, respectively.)

0

5

spectrum amplitude

high-gain control on

0 5 10 15 20 25

Frequency (kHz)

0

5

10 high-gain control off

Fig. 11: Fast Fourier transform of yc(t)sampled at Ts/20.

To form a baseline servo system, a built-in PID-type con-

troller C0(z)is embedded in the motor driver. C0(z)and the

actual plant P0(z)are treated as the new plant Pdh(z)in this

study. Fig. 14 shows the frequency response of the measured

and identiﬁed Pd h(z). The DOB [23] with C(z) = 1 in Fig. 15 is

implemented on a dSPACE DS1104 processor board to enable

high-gain control at selective frequencies. Transfer functions

inside the DOB block are all implemented at a sampling

time of Ts=0.1ms. Thus the Nyquist frequency ΩNequals

5 kHz. The fundamental sampling time used to measure ydh

is T0

s=0.025ms. That is, the fast sampling is conducted at

T0

s=T/Fwith F=4 for diagnosis of the beyond-Nyquist

performance. A single-harmonic disturbance with magnitude

0.1V and frequency ωo=2πΩoT0

s(Ωoin Hz) is introduced to

the system. In addition, the system is subjected to broadband

random disturbances at a magnitude of about 20 mV.

Fig. 16 illustrates the theoretically computed input PFGs

using (18) for the baseline controller and (19) for the cus-

tomized high-gain controller. Experimental data of P0

h(ejΩ0T0

s)

is obtained by following the time-domain deﬁnition in (18) for

each value of Ωo. From Fig. 16, the three experimental PFGs

of P0

h(ejΩ0T0

s)at 3 kHz (0.6ΩN), 4 kHz (0.8Ω), and 7 kHz

(1.4ΩN) match the theoretical computations very well.

As stated in Remark 2, Ph(ejΩ0T0

s)is obtained by means

of dividing P0

h(ejΩ0T0

s)by Pdh(ejΩ0T0

s). Three groups of

validations for Ph(ejΩ0T0

s)are shown in Table I and Fig. 17.

The results show that the mismatch between the experimental

and theoretical values is very small, and thus the modiﬁed PFG

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

8

0.081 0.082 0.083 0.084 0.085 0.086 0.087 0.088 0.089 0.09

Time (sec)

-5

0

5

Normalized output

(a) yc(t)sampled at Ts/20. (The solid and dashed lines represent the

cases with single-rate high-gain control on and off, respectively.)

0

5high-gain control on

0 5 10 15 20 25

Frequency (kHz)

0

5

spectrum amplitude

high-gain control off

(b) Fast Fourier transform of yc(t)sampled at Ts/20.

Fig. 12: Plant output with the input disturbance at 0.7ΩN

Laser source Galvo scanner

Power supply

Monitor

PC server with

dSPACE and

Matlab

White screen

Servo driver

Fig. 13: Schematic of the hardware platform.

is an efﬁcient tool for evaluating the intersample behavior. One

principal reason for the mismatch is that in Deﬁnition 1, PFG

is evaluated according to N→∞, while only a ﬁnite duration

of the signal can be reached in experiments.

Disturbance Group Experimental Average Theoretical

frequency PFGs (dB) PFGs

3 kHz

G1 -1.189

-1.033 dB -2.847 dBG2 -0.961

G3 -0.949

4 kHz

G1 5.481

5.913 dB 4.135 dBG2 5.999

G3 6.258

7 kHz

G1 6.023

6.242 dB 6.374 dBG2 6.357

G3 6.345

TABLE I: Experimental results of the modiﬁed PFG.

We have already seen the different cases of time-domain

responses in Figs. 7 and 8. Additionally, Fig. 19 veriﬁes

the performance limitation for disturbances even below the

Nyquist frequency. From the slow-sampled data in Figs. 7a, 8a,

and 19a, the single-rate DOB is successful in “compensating”

the sampled output. However, similar as the case in the

previous numerical study, the hidden performance loss for the

case with beyond- and near-Nyquist disturbances is obvious

from Figs. 7b and 19b.

Fortunately, these performance differences can be predicted

by the modiﬁed PFG in Fig. 17 and the characteristic feedback

loop gains in Fig. 18. The PFG plots predict that high-gain

control results in decreased output power for the disturbance

at 3kHz and increased output energy for the disturbances

-100

-50

0

50

Magnitude (dB)

measured system

identified system

102103104

Frequency (Hz)

-200

0

200

Phase (deg)

Fig. 14: Bode plot of Pd h(z)sampled at T0

s.

C

z

Q

z

1

++

+

+

‐

yd

ydh

u

DOB

0

0

H

H

+

+yd

e‐

ud

ucyc

yc0

dc

C

z

Q

z

1

++

+

+

d

yd

yc

H

1

++

+

+

+

‐

y(k)

e

d(k)

r(k)

+

+

0

Q

z

1

++

+

+

+

‐

d

yd

yc

e

DOB

0

0

H

+

‐

y(k)

ed(k)

r(k)

+

+

y(t)u(t)

u(k)

∆

ZOH G(s)

Gd(z)

H

+

+yd

e‐

ud

udh ydh

deq

H

+yd

e‐

ud

udh ydh

+

d

deq

Fig. 15: Block diagram with a disturbance observer (DOB).

at 4kHz and 7 kHz. Fig. 18 additionally reveals that both

the fundamental and aliased mode gains are increased when

disturbances occur at 4kHz and 7 kHz.

The experimental result in Fig. 3 veriﬁes that for the

input disturbance with Ωo=1.4ΩN=7 kHz, the fundamental

component at 7 kHz and the aliased harmonic at 3 kHz

are ampliﬁed when customized single-rate high-gain control

is turned on. For the case with Ωo=0.8ΩN=4 kHz, the

magnitude of the fundamental mode at 4 kHz barely changes,

but the aliased harmonic at 6 kHz is ampliﬁed by sub-

Nyquist high-gain control, resulting in the overall ampliﬁca-

tion. For Ωo=0.6ΩN=3 kHz, 1−Γ∗

0(Ωo)<|1−Γ0(Ωo)|,

and |Γ∗

1(−Ωo)|>|Γ1(−Ωo)|; although the aliased mode at 7

kHz is slightly ampliﬁed, the attenuation of the fundamental

mode at 3 kHz is signiﬁcant (Fig. 4), resulting in the overall

attenuation.

In summary, we experimentally veriﬁed that the character-

istic feedback loop gains, along with the modiﬁed PFG, are

reliable tools for analyzing servo performance in sampled-data

control. Single-rate high-gain control is observed to amplify all

beyond-Nyquist and even some below-Nyquist disturbances.

VI. CO NC LU SI ON A ND DI SC USSIONS

In this paper, the problem of sampled-data regulation control

against structured disturbances around and beyond the Nyquist

frequency is analyzed. It is shown that the conventional sub-

Nyquist single-rate high-gain control is infeasible to attenuate

disturbances near and beyond the Nyquist frequency. We

discover an intersection frequency deﬁned as the principal

sampled-data bandwidth Bp. Only for below-Bpdisturbances

can single-rate high-gain control be effective in disturbance

rejection. A spectral analysis is further proposed to look into

individual spectral modes. The proposed characteristic feed-

back loop gains are combined with the performance frequency

gain to evaluate the overall sampled-data performance. The

results imply that the rejection of beyond-Nyquist vibration

disturbances must rely on tools that can facilitate the inter-

sample attenuation, such as customized multirate control [28],

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

9

ܤ

Fig. 16: Input performance frequency gains (PFGs) with high-

gain control on and off.

𝐵𝑝

Fig. 17: Theoretical and experimental PFGs.

sampled-data internal model principle [29], sampled-data LQG

[30], sampled-data all-stabilizing control [31], and sampled-

data H∞theory (see [32] and the references therein). For

potential future work, the proposed study may have synergy

with the analysis and design of nonlinear systems in the

frequency domain (e.g., [33]–[35]).

VII. APP EN DI X: PROO FS

Proof of Lemma 3

Proof: Recall L−1{Pc(s)/s}=tn/(n!)by the inverse

Laplace transform. Thus, the ZOH equivalent of Pc(s)is

Pd(z) = (1−z−1)Z(kTs)n

n!=Tn

s(1−z−1)

n!

∞

∑

k=0

knz−k.(20)

We adopt two fundamental functions in number theory to eval-

uate Pd(z)at z=−1. Notice that ∑∞

k=0kn(−1)−k=−η(−n) =

−(1−21+n)ζ(−n), where η(n)is the Dirichlet Eta Function

and ζ(n)is the Riemann Zeta Function [36] deﬁned by

ζ(n),(∑∞

k=1

1

kn,ℜ{n}>1

(1−21−n)−1∑∞

k=1

(−1)k−1

kn,ℜ{n}>0.

ζ(n)is furthermore extended to the whole complex plane by

analytic continuation and satisﬁes the functional equation [37],

[38, Chapter 2]

ζ(−n) = 2−nπ−n−1sin−πn

2Γ(1+n)ζ(1+n),(21)

where Γ(n)is the Gamma function and n∈C. From the factor

sin−πn

2,ζ(−n)has zeros at all positive even integers of n

(called the "trivial zeros"). Hence after substitution into (20),

Pd(−1) = 0 when nis a positive even integer. By deﬁnition,

Pc(j(Ωo+2π

Ts

k))H(j(Ωo+2π

Ts

k)) = 1−e−j(Ωo+2π

Tsk)Ts

jn+1(Ωo+2π

Tsk)n+1,

which is ﬁnite when Ωo=π/Ts. Hence Lemma 3 holds.

Remark 3. A numerical evaluation of the zeta function is

available at [39]. The ZOH equivalents of 1/snfor nup to

-200

-100

0

Magnitude (dB)

k

*( )

k( )

0 5 10 15 20 25 30

+2 k/Ts (kHz)

-100

-50

0

Magnitude (dB)

1- k

*( )

1- k( )

Fig. 18: Magnitude responses of Γk(Ω),Γ∗

k(Ω), 1−Γk(Ω)and

1−Γ∗

k(Ω)as a function of Ω+2πk/Ts. The ﬁrst three vertical

lines indicate, respectively, the Nyquist frequency (5 kHz), the

sampling frequency and 3

/2Ts.

1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

Time(sec)

-0.05

0

0.05

Slow-sampled

output (V)

(a) yd(t)with the sampling time of Ts.

1.0002 1.0004 1.0006 1.0008 1.001 1.0012 1.0014 1.0016 1.0018 1.002

Time(sec)

-0.1

-0.05

0

0.05

0.1

Fast-sampled output (V)

(b) ydh (t)with the sampling time of T0

s.

0

0.02

0.04

0.0

6

high-gain control on

0 2 4 6 8 101214161820

Frequency(kHz)

0

0.05

high-gain control off

X: 4000

Y: 0.04633

X: 6000

Y: 0.04634

X: 4000

Y: 0.04488 X: 6000

Y: 0.007473

1Γ

ΩΓ

Ω

1Γ

∗ΩΓ

∗Ω

(c) Fast Fourier transform of ydh(t).

Fig. 19: Plant output with the input disturbance at 0.8ΩN(For

(a) and (b), the solid and dashed lines represent the cases with

single-rate high-gain control on and off, respectively.)

8 are numerically evaluated in [24]. There, a zero at −1 is

evident for n=2,4,6,8.

Proof of Lemma 5

Proof: Take an inertia system Pc(s) = 1/s2in mo-

tion control as example.1We show that Ph(ej·0·T0

s)<1 and

Ph(ejΩNT0

s)>1. If PFG is a continuous function of Ωofor

Ωo∈(0,ΩN),Ph(ejΩ0T0

s)then must cross over the 0dB line at

least once below the Nyquist frequency ΩN. In other words,

under the metric of PFG, it is inevitable that some band-limited

1Without loss of generality, the gain of Pc(s)is normalized to unity. Non-

unity gains are canceled in the computation of Γk(Ωo)and Γ∗

k(Ωo)in (16).

The final version of record is available at http://dx.doi.org/10.1109/TMECH.2018.2795960

10

disturbances below the Nyquist frequency are ampliﬁed by

single-rate high-gain control.

With Pc(s) = 1/s2, we have Pd(z) = T2

s(z+1)/2(z−1)2

and Pd h(z) = T02

s(z+1)/2(z−1)2. When Ω0=0, (17)

yields Ph(ej·0·T0

s) = q

1−Γ∗

0(0)

2+∑F−1

k=1

Γ∗

k(0)

2. Based

on the deﬁnition in (16) as well as the expressions of Pd(z)

and Pd h(z), we get Γ∗

k(0) = 1 when k=0 (by applying

L’Hospital’s Rule twice), and Γ∗

k(0) = 0 when k6=0. Therefore,

Ph(ej·0·T0

s) = 0(<1).

Next consider the case of Ωo=ΩN=π/Ts. Evaluating the

frequency responses of Pd(z),Pdh(z), and H(z)in (14) yields

Γ∗

0(ΩN) = 8e jπ/F(ejπ/F+1)

F3[ejπ/F−1]3(ejπ+1).(22)

For F>2, Γ∗

0(ΩN)→∞since ejπ+1=0. Hence, in (17),

Ph(ejΩNT0

s)→∞>1.

With Ph(ej·0·T0

s)<1 and Ph(ejΩNT0

s)>1, we thus have Bp<

ΩNbased on a continuity analysis.

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