ArticlePDF Available

A Spectral Analysis of Feedback Regulation Near and Beyond Nyquist Frequency

Article

A Spectral Analysis of Feedback Regulation Near and Beyond Nyquist Frequency

Abstract and Figures

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.
Content may be subject to copyright.
1
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(ejoTs)|=) can asymptotically reject disturbances
at frequency oin 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(j)is the
Fourier transform of xc(t).ω=Ts, and is in rad/s.
(c)denotes the real part of a complex number cC. 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)esτwhere τ0; 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(j)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) = ejotand the sampling time is Ts
in Fig. 1, then the Fourier transform of the continuous-time
plant output yc(t)is
Yc(j) = 2π11
Ts
Pc(j)H(j)Sd(ejTs)C(ejTs)δ(o)
2π
Ts
Pc(j)H(j)Sd(ejTs)C(ejTs)
k=,k6=0
δ(o2π
Ts
k),(2)
where δ(o)denotes a shifted Dirac delta impulse,
H(j) = (1ejTs)/(j)is the Fourier transform of the
ZOH, and Sd(ejoTs)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(ejoTs) = 1
Ts
k=
Pc(j(o+2π
Ts
k))H(j(o+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 FZ). 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] = cejk T 0
s,kck2<obe
applied to an MR system in Fig. 2. The PFG P(ejT0
s)is
defined as
P(ejT0
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 N=5kHz. The disturbance frequencies are
below Nat 3kHz and beyond Nat 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
1Γ
Ω
Γ
Ω
1Γ
Ω
Γ
Ω
Fig. 3: Fast Fourier transform of ydh(t)with input disturbance
frequency at 1.4N.
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
1Γ
ΩΓ
Ω
1Γ
ΩΓ
Ω
Fig. 4: Fast Fourier transform of ydh(t)with input disturbance
frequency at 0.6N.
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(o),Pc(j(o+2π
Tsk))H(j(o+2π
Tsk))
TsPd(ejoTs)Td(ejoTs),(6)
where
Td(ejoTs),Pd(ejoTs)C(ejoTs)
1+Pd(ejoTs)C(ejoTs)=Pd(ejoTs)C(ejoTs)Sd(ejoTs).(7)
After substituting (6) into (2) and recalling that Fej0t=
2πδ(0), the steady-state continuous-time output is sim-
plified to
yc(t) = [1Γ0(0)]ej0t
k=,k6=0
Γk(0)ej(0+2π
Tsk)t.(8)
Fact 1. Based on (3) and (6), it is immediate that
k=
Γk(o) = Td(ejoTs).(9)
For the case of real-valued disturbances in practice, let
dc(t) = cos(ot+φ). Recall cos(0t+φ) = (ej(0t+φ)),
F{(x(t))}= [X(j) + X(j)]/2, and δ(o) =
δ(+o). Laplace transform to the real part of (8) gives
Yc(j) = πejφ(1Γ0(o))δ(o)
+πejφ(1Γ0(o))δ(+o)
πejφ
k=,k6=0
Γk(0)δ(o2π
Ts
k)
πejφ
k=,k6=0
Γk(0)δ(+o+2π
Ts
k).(10)
By the definition in (6), Γk(o)is conjugate
symmetric, namely, Γk(0) = Γk(0). Thus in (10),
the gains for two fundamental modes, |1Γ0(0)|and
|1Γ0(0)|=1Γ0(0), are equal, and the gains for
their related aliased harmonics, |Γk(0)|and |Γk(0)|,
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(j)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(ejoTs) = 0 in (7) yields Γk(o) = 0 for any k, namely,
a zero gain for each harmonic |Γk(0)|and a unit gain for
the fundamental mode |1Γ0(0)|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(o)under
high-gain control.
Definition 2. Under ideal single-rate high-gain control, the
new characteristic feedback loop gain is
Γ
k(o),lim
|C(ejoTs)|→
Γk(o) = Pc(j(o+2π
Tsk))H(j(o+2π
Tsk))
TsPd(ejoTs).(11)
Fact 2. From the definition of Pd(ejoTs)in (3), it is immediate
that the summation of Γ
k(o)over kis
k=
Γ
k(0) = 1,0.(12)
(12) will be revisited in Section III-B. Similar to
Γk(o),Γ
k(o)is also conjugate symmetric: 1Γ
0(o)=
1Γ
0(o);Γ
k(o)=Γ
k(0).
A. Characteristic feedback loop gains Γk(o)and Γ
k(o)
In this subsection, the properties of the characteristic feed-
back loop gains are discussed. From (6) and (11), we obtain
that Γk(o) = Γ
k(o)Td(ejoTs). Since Td(z)is typically a
low-pass filter whose bandwidth BTis commonly 10%-20% of
the Nyquist frequency [22], we have |Γ
k(o)|>|Γk(o)|for
most frequencies. Furthermore, we can obtain the following
characteristics:
1) If o+2kπ/Ts[0,BT), then the low-pass H(j(o+
2kπ/Ts))/Ts1 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(o+2kπ/Ts))H(j(o+2kπ/Ts))/Ts
Pd(ej(o+2kπ/Ts)Ts) = Pd(ejoTs), and |Td(ejoTs)|1,
yielding both Γk(o)and Γ
k(o)to be approximately
1. Thus, |1Γk(o)|and |1Γ
k(o)|are both small.
In particular, since Pd(1) = Pc(0)[24] and H(0)/Ts=1,
we have 1Γ
0(0) = 0.
2) If o+2kπ/Ts[BT,π/Ts), then |Td(ejoTs)|<1, and
thus |Γ
k(o)|>|Γk(o)|. For most frequencies in this
region, |Γ
k(o)| ≈ 1, and |1Γ
k(o)|  1.
3) If o+2kπ/Ts[π/Ts,2π/Ts), the low-pass ZOH
|H(j(+2kπ/Ts))|reduces quickly outside its ap-
proximate bandwidth π/Ts. Although high-gain control
still makes |Γ
k(o)|>|Γk(o)|, the overall magni-
tudes |Γ
k(o)|and |Γk(o)|are very small. Thereby,
1Γ
k(o)and |1Γk(o)|both approximate 1.
Interestingly, Γ
k(o)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()and 1Γ
k()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()Γ
k(), 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 |1Γ
0()|and |1Γ0()|(Bc
in Fig. 5). In addition, the achievable maximum attenua-
tion—indicated by the magnitude |1Γ
0()|—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 o(π/Ts,2π/Ts),|Γ
k(o)|>|Γk(o)|, and
|1Γ
0(o)|&|1Γ0(o)| ≈ 1. In this interval, under
single-rate high-gain control, obeing 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(),Γ
k(), 1 Γk()
and 1 Γ
k()as a function of +2πk/Ts, where Γ
k(o)
and Γk(o)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()and Γ
k()are relatively
insignificant, and 1Γ()
k()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 o(π/Ts,2π/Ts)and 0
o=2π/Tso(0,π/Ts).
Consider two different disturbances dc(t),cos(ot)and
˜
dc(t),cos(0
ot), respectively, at above and below the Nyquist
frequency. The Fourier transforms of the continuous-time
disturbances are
Dc(j) = πδ(o) + πδ(+o),
˜
Dc(j) = πδ(o2π
Ts
) + πδ(+o2π
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 (ofor dcand 2πofor d0
c).
For obeing above Nyquist frequency, the magnitude of the
fundamental mode |1Γ0(o)|is close to 1 (cf. Fig. 5). The
dominant aliased mode Γ1(o)occurs at 2π/Tsobelow
the Nyquist frequency (see Fig. 6a). With single-rate high-gain
control at o, the magnitude of Γk(o)increases towards the
limiting case Γ
k(o). In particular, Γ1(o)increases towards
Γ
1(o)1 (Fig. 6b). Meanwhile, |1Γk(o)|stays close
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.
5
to 1 or is even increased. Collectively, dc(t)is amplified by
single-rate high-gain control.
On the other hand, for 0
obelow the Nyquist frequency,
the fundamental mode 1 Γ0(0
o)can be effectively reduced
(from the dashed line to the solid line in the bottom plot of
Fig. 5). The aliased modes Γk(0
o)still increase to Γ
k(0
o).
However, |Γ
k(0
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/Ts0
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.
Yc
πe±jφ
| ||| | |||
1Γ0(o)1Γ0(o)
Γ1(o)
Γ1(o)
Γ1(o)
Γ1(o)
Γ2(o)
Γ2(o)......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(a) dc(t) = cos(ot)with the baseline control
| ||| | |||
1Γ
0(o)1Γ
0(o)
Γ
1(o)
Γ
1(o)
Γ
1(o)
Γ
1(o)
Γ
2(o)
Γ
2(o)......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(b) dc(t) = cos(ot)with enhanced discrete-time high-gain control at o
| ||| | |||
1Γ0(0
o)1Γ0(0
o)
Γ1(0
o)
Γ1(0
o)Γ1(0
o)
Γ1(0
o)Γ2(0
o)
Γ2(0
o)
......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(c) ˜
dc(t) = cos(0
ot) = cos((2π/Tso)t)with the baseline control
| ||| | |||
1Γ
0(0
o)1Γ
0(0
o)
Γ
1(0
o)
Γ
1(0
o)
Γ
1(0
o)
Γ
1(0
o)Γ
2(0
o)
Γ
2(0
o)
......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(d) ˜
dc(t) = cos(0
ot)with enhanced discrete-time high-gain control at o
Fig. 6: Illustration of the spectrum of yc(t)in sampled-data
control when π/Ts<o<2π/Ts. Dashed spikes: δ(+0)
and its aliases; solid spikes: δ(0)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/Tso. 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 o(cf. Lemma 1). Based on (10), the magnitude of the
discrete-time spectral peak at 0is a normalized version of
[1Γ
o(o)] Γ
1(o)Γ
1(o)Γ
2(o)Γ
2(o)... ,
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(o), and Γ±k(o)(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(0
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 1.4N(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 0.6N(The
solid and dashed lines represent the cases with single-rate
high-gain control on and off, 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.
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(ejT0
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(ejT0
s) = 11
FPdh(ejT0
s)H(ejT0
s)Td(ejTs)/Pd(ejTs)D(ejT0
s)
+1
FPdh(ejT0
s)H(ejT0
s)Td(ejTs)/Pd(ejTs)
F1
k=1
D(ej(T0
s2πk
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) =
F1
k=0
zk=(F z =1
1zF
1z1z6=1.(14)
Based on (13), the MR characteristic feedback loop gain is
defined as
Γk(o) = Pdh(ej(oT0
s+2πk
F))H(ej(oT0
s+2πk
F))Td(ej0Ts)
FPd(ej0Ts),(15)
and the limiting case with single-rate high-gain control is
Γ
k(o) = Pdh(ej(oT0
s+2πk
F))H(ej(oT0
s+2πk
F))
FPd(ej0Ts).(16)
Lemma 4. For the MR system in Fig. 2, the modified PFG
under single-rate high-gain control at 0is
Ph(ej0T0
s) = v
u
u
t
1Γ
0(o)
2+
F1
k=1
Γ
k(o)
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(o)
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
0, that is, Td(ejoTs) = 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(ejoTs) = 1 at different values of
0, 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(o)and Γ
k(o)are replaced by Γ0(o)and Γk(o)in
(15). The modified PFG then describes the performance of a
baseline LTI controller.
For a typical plant dynamic in Section V, Ph(ejT0
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(ejT0
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(ejT0
s) = Di(ejT0
s)Pd h(ejT0
s).
We can analogously define and compute the modified input
PFGs
P0
b(ej0T0
s) = sup
di6=0
kydh kp
kdikp
=
Pdh(ej0T0
s)sk1Γ0(o)k2+
F1
k=1
kΓk(o)k2,(18)
and
P0
h(ej0T0
s) = lim
Td(ej0T0
s)1
P0
b(ej0T0
s) = |Pdh|sk1Γ
0(o)k2+
F1
k=1
Γ
k(o)
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(ej0T0
s)
by Pdh(ej0T0
s), we can then generate the modified PFG
Ph(ej0T0
s).
Before presenting the numerical and experimental results,
we briefly summarize the application steps of the proposed
spectral analysis method:
1) Determine Γk(o), the characteristic feedback loop
gains, by (6) and (15). In addition, determine Γ
k(o),
the limiting cases with single-rate high-gain control, by
(11) and (16).
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.
7
2) Plot the magnitude responses of Γk(),Γ
k(), 1
Γk(), and 1 Γ
k()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/(z1) + kd(z1)/zwith kp=7.51 ×105,
ki=3.00×105, and kd=3.60×104. 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 (1.8N), 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 1.8N(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 Nequals
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(oin 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(ej0T0
s)
is obtained by following the time-domain definition in (18) for
each value of o. From Fig. 16, the three experimental PFGs
of P0
h(ej0T0
s)at 3 kHz (0.6N), 4 kHz (0.8), and 7 kHz
(1.4N) match the theoretical computations very well.
As stated in Remark 2, Ph(ej0T0
s)is obtained by means
of dividing P0
h(ej0T0
s)by Pdh(ej0T0
s). Three groups of
validations for Ph(ej0T0
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
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.
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 0.7N
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 o=1.4N=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 o=0.8N=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 o=0.6N=3 kHz, 1Γ
0(o)<|1Γ0(o)|,
and |Γ
1(o)|>|Γ1(o)|; 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],
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.
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 Htheory (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 L1{Pc(s)/s}=tn/(n!)by the inverse
Laplace transform. Thus, the ZOH equivalent of Pc(s)is
Pd(z) = (1z1)Z(kTs)n
n!=Tn
s(1z1)
n!
k=0
knzk.(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) =
(121+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,{n}>1
(121n)1
k=1
(1)k1
kn,{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) = 2nπn1sinπn
2Γ(1+n)ζ(1+n),(21)
where Γ(n)is the Gamma function and nC. From the factor
sinπn
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(o+2π
Ts
k))H(j(o+2π
Ts
k)) = 1ej(o+2π
Tsk)Ts
jn+1(o+2π
Tsk)n+1,
which is finite when o=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(),Γ
k(), 1Γk()and
1Γ
k()as a function of +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
1Γ
ΩΓ
Ω
1Γ
ΩΓ
Ω
(c) Fast Fourier transform of ydh(t).
Fig. 19: Plant output with the input disturbance at 0.8N(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(ejNT0
s)>1. If PFG is a continuous function of ofor
o(0,N),Ph(ej0T0
s)then must cross over the 0dB line at
least once below the Nyquist frequency N. 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(o)and Γ
k(o)in (16).
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.
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(z1)2
and Pd h(z) = T02
s(z+1)/2(z1)2. When 0=0, (17)
yields Ph(ej·0·T0
s) = q
1Γ
0(0)
2+F1
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 o=N=π/Ts. Evaluating the
frequency responses of Pd(z),Pdh(z), and H(z)in (14) yields
Γ
0(N) = 8e jπ/F(ejπ/F+1)
F3[ejπ/F1]3(ejπ+1).(22)
For F>2, Γ
0(N)since ejπ+1=0. Hence, in (17),
Ph(ejNT0
s)>1.
With Ph(ej·0·T0
s)<1 and Ph(ejNT0
s)>1, we thus have Bp<
Nbased on a continuity analysis.
REF ER EN CE S
[1] T. Atsumi and W. C. Messner, “Compensating for zoh-induced residual
vibrations in head-positioning control of hard disk drives,IEEE/ASME
Transactions on Mechatronics, vol. 19, no. 1, pp. 258–268, Feb 2014.
[2] Q.-W. Jia, “Intersample ripple-free multirate control with application to
a hard disk drive servo,IEEE/ASME Transactions on Mechatronics,
vol. 10, no. 3, pp. 341–345, June 2005.
[3] S. Berumen, F. Bechmann, S. Lindner, J.-P. Kruth, and T. Craeghs,
“Quality control of laser- and powder bed-based Additive Manufacturing
(AM) technologies,” Physics Procedia, 2010.
[4] W. E. Frazier, “Metal additive manufacturing: A review,” J. Mater. Eng.
Perform., vol. 23, no. 6, pp. 1917–1928, 2014.
[5] J. Tani, S. Mishra, and J. T. Wen, “Identification of fast-rate systems
using slow-rate image sensor measurements,” IEEE/ASME Transactions
on Mechatronics, vol. 19, no. 4, pp. 1343–1351, Aug 2014.
[6] Y. C. Chang, B. Berry-Pusey, R. Yasin, N. Vu, B. Maraglia, A. X. Chatzi-
ioannou, and T. C. Tsao, “An automated mouse tail vascular access
system by vision and pressure feedback,” IEEE/ASME Transactions on
Mechatronics, vol. 20, no. 4, pp. 1616–1623, Aug 2015.
[7] Y. Yamamoto and P. Khargonekar, “Frequency response of sampled-data
systems,” IEEE Transactions on Automatic Control, vol. 41, no. 2, pp.
166–176, 1996.
[8] M. Araki, Y. Ito, and T. Hagiwara, “Frequency response of sampled-data
systems,” Automatica, vol. 32, no. 4, pp. 483 – 497, 1996.
[9] T. Hagiwara, M. Suyama, and M. Araki, “Upper and lower bounds of the
frequency response gain of sampled-data systems,” Automatica, vol. 37,
no. 9, pp. 1363 – 1370, 2001.
[10] Y. Yamamoto and M. Araki, “Frequency responses for sampled-data
systems – their equivalence and relationships,Linear Algebra and its
Applications, vol. 205-206, pp. 1319–1339, 1994.
[11] G. C. Goodwin and M. Salgado, “Frequency domain sensitivity functions
for continuous time systems under sampled data control,” Automatica,
vol. 30, no. 8, pp. 1263–1270, 1994.
[12] J. S. Freudenberg, R. H. Middleton, and J. H. Braslavsky, “Inherent de-
sign limitations for linear sampled-data feedback systems,” International
Journal of Control, vol. 61, no. 6, pp. 1387–1421, 1995.
[13] O. Lindgarde and B. Lennartson, “Performance and robust frequency
response for multirate sampled-data systems,” in American Control
Conference, 1997. Proceedings of the 1997, vol. 6. IEEE, 1997, pp.
3877–3881.
[14] M. W. Cantoni and K. Glover, “Frequency-domain analysis of linear
periodic operators with application to sampled-data control design,”
Proceedings of IEEE Conference on Decision and Control, vol. 5, pp.
4318–4323 vol.5, 1997.
[15] T. Oomen, M. van de Wal, and O. Bosgra, “Design framework for high-
performance optimal sampled-data control with application to a wafer
stage,” International Journal of Control, vol. 80, no. 6, pp. 919–934,
2007.
[16] O. Lindgärde, Frequency analysis of sampled-data systems applied to a
lime slaking process. Chalmers University of Technology„ 1999.
[17] D. Wang, M. Tomizuka, and X. Chen, “Spectral distribution and impli-
cations of feedback regulation beyond nyquist frequency,” in Flexible
Automation (ISFA), International Symposium on. IEEE, 2016, pp. 23–
30.
[18] R. E. Kalman, Y. C. Ho, and K. S. Narendra, “Controllability of linear
dynamical systems,” Contributions to differential equations, vol. 1, no. 2,
pp. 189–213, 1963.
[19] B. A. Francis and T. Georgiou, “Stability theory for linear time-invariant
plants with periodic digital controllers,” IEEE Transactions on Automatic
Control, vol. 33, no. 9, pp. 820–832, 1988.
[20] R. Middleton and J. Freudenberg, “Non-pathological sampling for
generalized sampled-data hold functions,” Automatica, vol. 31, no. 2,
pp. 315 – 319, 1995.
[21] M. Hayes, Statistical digital signal processing and modeling. Wiley-
India, 2009.
[22] K. J. Astrom and B. Wittenmark, Computer-Controlled Systems: Theory
and Design, 3rd ed. Upper Saddle River, N.J: Prentice Hall, Nov. 1996.
[23] X. Chen and M. Tomizuka, “A minimum parameter adaptive approach
for rejecting multiple narrow-band disturbances with application to hard
disk drives,IEEE Transactions on Control Systems Technology, vol. 20,
no. 2, pp. 408–415, March 2012.
[24] K. Astrom, P. Hagander, and J. Sternby, “Zeros of sampled systems,
Automatica, vol. 20, no. 1, pp. 31 – 38, 1984.
[25] A. Al Mamun, G. Guo, and C. Bi, Hard disk drive: mechatronics and
control. CRC Press, Taylor & Francis Group, London, 2007.
[26] I. D. Landau, A. C. Silva, T.-B. Airimitoaie, G. Buche, and M. Noe,
“Benchmark on adaptive regulationrejection of unknown/time-
varying multiple narrow band disturbances,European Journal of Con-
trol, vol. 19, no. 4, pp. 237 – 252, 2013.
[27] L. Sievers and A. von Flotow, “Comparison and extensions of control
methods for narrow-band disturbance rejection,IEEE Transactions on
Signal Processing, vol. 40, no. 10, pp. 2377–2391, 1992.
[28] X. Chen and H. Xiao, “Multirate forward-model disturbance observer
for feedback regulation beyond Nyquist frequency,” Systems & Control
Letters, vol. 94, pp. 181–188, August 2016.
[29] H. Fujioka and S. Hara, “Output regulation for sampled-data feedback
control systems: Internal model principle and Hservo controller
synthesis,” Journal of the Chinese Institute of Engineers, vol. 33, no. 3,
pp. 335–346, Mar. 2011.
[30] T. Chen and B. A. Francis, “H2-optimal sampled-data control,IEEE
Transactions on Automatic Control, vol. 36, no. 4, pp. 387–397, Apr.
1991.
[31] R. Ravi, P. P. Khargonekar, K. D. Minto, and C. N. Nett, “Controller
parametrization for time-varying multirate plants,” Automatic Control,
IEEE Transactions on, vol. 35, no. 11, pp. 1259–1262, Nov. 1990.
[32] S. Lall and G. Dullerud, “An LMI solution to the robust synthesis
problem for multi-rate sampled-data systems,” Automatica, vol. 37,
no. 12, pp. 1909–1922, Dec. 2001.
[33] X. Jing and Z. Lang, Frequency domain analysis and design of nonlinear
systems based on Volterra series expansion: a parametric characteristic
approach. Springer, 2015.
[34] Z. Q. Lang, S. A. Billings, R. Yue, and J. Li, “Output frequency response
function of nonlinear volterra systems,” Automatica, vol. 43, no. 5, pp.
805–816, 2007.
[35] Z. Xiao and X. Jing, “Frequency-domain analysis and design of linear
feedback of nonlinear systems and applications in vehicle suspensions,”
IEEE/ASME transactions on mechatronics, vol. 21, no. 1, pp. 506–517,
2016.
[36] I. S. Gradshtein, Table of Integrals, Series, and Products. Academic
Press, Jan 1980, vol. 1.
[37] E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann
zeta-function. Oxford University Press, 1986.
[38] H. Edwards, Riemann’s zeta function, ser. Pure and Applied Mathemat-
ics. Elsevier Science, 1974.
[39] J. Sondow and E. W. Weisstein, “Riemann zeta func-
tion,” from MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/RiemannZetaFunction.html, accessed
Dec. 5 2016.
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.
... In addition, when iM = jN holds for some i ∈ [0, t 1 ] and j ∈ [0, t 2 ], the corresponding columns e k+iM and e k+ jN in matrix M k are identical (see, e.g., the 7th and 9th columns of M k in (19)), yielding redundant pairs of variables in (4) (say there are n d number of them). Then, the number of independent variables becomes l + t 1 + t 2 + 2 − n d and the necessary condition (22) reduces to (20). To more quantitatively define n d , we recall that a signal batch could provide at most L/M measurements from sensor S 1 and L/N measurements from sensor S 2 , hence the number of prior batches used in the recovery that contain measurements from S 1 (denoted as n d,M ) or S 2 (denoted as n d,N ) are ...
... Application: Beyond-Nyquist Disturbances Rejection An immediate result of slowly sampled data in a feedback system is that the controlled process will not be able to reject fast disturbances, or more specifically, signals beyond the Nyquist frequency of the feedback sensor. Our preliminary study [22] has reported by simulation and experimentation that a well-designed classic high-gain control could amplify instead of attenuating the actual disturbance when its main spectral components are near or beyond the Nyquist frequency of the sensor. However, with the proposed model-based information recovery technique, rejecting beyond Nyquist disturbances using classic high-gain control becomes possible, as we shall see from an example below. ...
Article
Full-text available
This paper considers the real-time recovery of a fast time series by using sparsely sampled measurements from sensors whose sampling speeds are prohibitively slow originally. Specifically, when the fast signal is an autoregressive process, we propose an online information recovery algorithm that reconstructs the dense underlying temporal dynamics fully by systematically modulating two slow sensors, and by exploiting a model-based fusion of the sparsely collected data. We provide the design of collaborative sensing and model-based information recovery algorithm, impacts of parameter choosing and model singularity, and methods to reduce computational complexity and increase prediction robustness. The proposed method is experimentally verified in an optical beam steering platform for additive manufacturing. Application to a closed-loop disturbance rejection problem reveals the feasibility to eliminate fast disturbance signals with the slow and not fully aligned sensor pair in real time, and in particular, the rejection of narrow-band disturbances whose frequencies are much higher than the Nyquist frequencies of the sensors.
... F OURIER transform infrared (FTIR) imaging spectrometers serve as one of the most valuable tools for rapid identification of materials [1], [2], which can be applied in analytical process in biomedicine [3], [4], wines evaluation [5], etc. However, FTIR spectra are unavoidably contaminated by band overlap and random noise during the acquisition process (see Fig. 1), which severely degrades the quality of the spectrum data and limits the precision of the subsequent processing tasks, such as feature extraction [6], spectrum identification [7], [8], temperature feedback control [4], target detection [9], [10], etc. Typically, band overlap will be induced by proper functioning of an experimental apparatus, which can be viewed as a combination of several factors, such as slit function, grating response, circuit response, etc. ...
Patent
Fourier transform infrared (FTIR) imaging spectrometers are often corrupted by the problems of band overlap and random noise during the infrared spectrum acquisition process. In this paper, we present a novel blind reconstruction method with wavelet transform regularizations for infrared spectrum obtained from the aging instrument. Inspired by the finding that the wavelet coefficient distribution of the clean spectrum is sparser than that of the degraded one, a blind reconstruction model for infrared spectrum is proposed in this paper to regularize the distribution of the degraded spectrum by total variation regularization. This method outperforms when suppressing random noise and preserving the spectral structure details. In addition, an effective optimization scheme is introduced in overcoming the issue of formulated optimization. The instrument response function and desired spectrum can be simultaneously estimated through the proposed method which can efficiently mitigate the effects caused by instrument degradation. Finally, extensive experiments on synthetic and real noisy infrared spectra are carried out to demonstrate the superiority of the proposed method over the existing state-of-the-art ones. Thus, the high-resolution spectrum data will promote the applications of FTIR imaging spectrometer in the mechatronics industry.
... An immediate result of slowly sampled data in a feedback system is that the controlled process will not be able to reject fast disturbances, or more specifically, signals beyond the Nyquist frequency. Our preliminary study [14,15] have reported by simulation and experimentation that a well-designed classic highgain control could amplify instead of attenuate the actual disturbance when its main spectral components are near or beyond the Nyquist frequency of the sensor. However, with the proposed model-based information recovery technique, rejecting beyond Nyquist disturbances using classic high-gain control becomes possible, as we shall see from an example below. ...
Preprint
Full-text available
This paper considers the real-time recovery of a fast time series (e.g., updated every T seconds) by using sparsely sampled measurements from two sensors whose sampling intervals are much larger than T (e.g., MT and NT, where M and N are integers). Specifically, when the fast signal is an autoregressive process, we propose an online information recovery algorithm that reconstructs the dense underlying temporal dynamics fully, by systematically modulating the sensor speeds MT and NT, and by exploiting a model-based fusion of the sparsely collected data. We provide the collaborative sensing design, parametric analysis and optimization of the algorithm. Application to a closed-loop disturbance rejection problem reveals the feasibility to annihilate fast disturbance signals with the slow and not fully aligned sensor pair in real time, and in particular, the rejection of narrow-band disturbances whose frequencies are much higher than the Nyquist frequencies of the sensors.
Conference Paper
Full-text available
A fundamental challenge in digital and sampled-data control arises when the continuous-time plant is subject to fast disturbances that possess significant frequency components beyond Nyquist frequency. Such intrinsic difficulties are more and more encountered in modern manufacturing applications, where the measurement speed of the sensor is physically limited compared to the plant dynamics. The paper analyzes the spectral properties of the closed-loop signals under such scenarios, and uncovers several fundamental limitations in the process.
Article
Full-text available
A fundamental challenge in digital control arises when the controlled plant is subjected to a fast disturbance dynamics but is only equipped with a relatively slow sensor. Such intrinsic difficulties are, however, commonly encountered in many novel applications such as laser- and electron-beam-based additive manufacturing, human–machine interaction, etc. This paper provides a discrete-time regulation scheme for exact sampled-data rejection of disturbances beyond Nyquist frequency. By introducing a model-based multirate predictor and a forward-model disturbance observer, we show that the inter-sample disturbances can be fully attenuated despite the limitations in sampling and sensing. The proposed control scheme offers several advantages in stability assurance and lucid design intuitions. Verification of the algorithm is conducted on a motion control platform that shares the general characteristics in several advanced manufacturing systems.
Book
The hard disk drive is one of the finest examples of the precision control of mechatronics, with tolerances less than one micrometer achieved while operating at high speed. Increasing demand for higher data density as well as disturbance-prone operating environments continue to test designers’ mettle. Explore the challenges presented by modern hard disk drives and learn how to overcome them with Hard Disk Drive: Mechatronics and Control. Beginning with an overview of hard disk drive history, components, operating principles, and industry trends, the authors thoroughly examine the design and manufacturing challenges. They start with the head positioning servomechanism followed by the design of the actuator servo controller, the critical aspects of spindle motor control, and finally, the servo track writer, a critical technology in hard disk drive manufacturing. By comparing various design approaches for both single- and dual-stage servomechanisms, the book shows the relative pros and cons of each approach. Numerous examples and figures clarify and illustrate the discussion. Exploring practical issues such as models for plants, noise reduction, disturbances, and common problems with spindle motors, Hard Disk Drive: Mechatronics and Control avoids heavy theory in favor of providing hands-on insight into real issues facing designers every day.
Article
By A. A. Karatsuba and S. M. Voronin: 396 pp., DM. 198.-/US$112.00, ISBN 3 11 013170 6 (Walter de Gruyter, 1992).
Article
This paper develops an automated vascular access system (A-VAS) with novel vision-based vein and needle detection methods and real-time pressure feedback for murine drug delivery. Mouse tail vein injection is a routine but critical step for preclinical imaging applications. Due to the small vein diameter and external disturbances such as tail hair, pigmentation, and scales, identifying vein location is difficult and manual injections usually result in poor repeatability. To improve the injection accuracy, consistency, safety, and processing time, A-VAS was developed to overcome difficulties in vein detection noise rejection, robustness in needle tracking, and visual servoing integration with the mechatronics system.
Article
Nonlinear vibration control systems (both passive and active) always involve parameter design and performance optimization tasks. A systematic and novel frequency-domain method is established to this aim in this study based on a newly developed concept - nonlinear characteristic output spectrum (nCOS). The nCOS function can be any system output function or multiobjective performance function to be optimized. It is shown for the first time that the nCOS function can be expressed into an explicit and analytical polynomial function of any model parameters which define underlying linear dynamics of the system. A simple least square algorithm is provided for the determination of this nonlinear parametric relationship. This novel nCOS function can obviously facilitate parameter analysis and design of nonlinear vibration control systems and provide a useful tool for a simple linear control design, while simultaneously considering inherent nonlinear dynamics of a system. A case study in vehicle suspension control demonstrates these new results.
Article
The output regulation problem for sampled-data feedback systems is considered in a general setup. Necessary and sufficient conditions are derived based on the state-space approach in the lifted domain. The results are applied to synthesis problems for servo systems based on H-infinity control. The validity is demonstrated by a design example of repetitive control.
Book
*** http://link.springer.com/content/pdf/10.1007%2F978-3-319-12391-2.pdf A monograph is recently published by Springer in the series “Understanding Complex Systems” with the Founding Editor: S. Kelso. The monograph is mainly a summary of Dr Jing ( and Prof Lang) 's research focusing on Nonlinear analysis and design in the frequency domain in the past about 10 years. The book addresses fundamental theory and methods related to the analysis and design of nonlinear systems in the frequency domain and presents most of the recent important advances both in theory and applications about the Volterra series approach. The monograph is featured by: * A state-of-the-art summary of most important and recent advances in the area of frequency domain methods for nonlinear analysis developed in the past 20 years * A systematic frequency domain method for nonlinear analysis and design based on Volterra series expansion, which is of both theoretical and application significance to all those researchers related to nonlinear systems * A very novel insight into nonlinear dynamics in the frequency domain, which is different from all the other existing commonly-used methods such as harmonic balance and describing functions * Solid analysis and design results which demonstrate how to employ nonlinearity for a better system performance * A very engineering point of view, which can facilitate nonlinear analysis and design in practice This book targets those readers who are working in the areas related to nonlinear analysis and design, nonlinear signal processing, nonlinear system identification, nonlinear vibration control, and so on. It particularly serves as a good reference for those who are studying frequency domain methods for nonlinear systems. More details can be referred to the following links: http://www.springer.com/978-3-319-12390-5 http://link.springer.com/book/10.1007%2F978-3-319-12391-2