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Spectral distribution and implications of feedback regulation beyond Nyquist frequency

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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.
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Proceedings of 2016 International Symposium on Flexible Automation
ISFA 2016
1-3 August, 2016, Cleveland, Ohio, U.S.A.
SPECTRAL DISTRIBUTION AND IMPLICATIONS OF FEEDBACK REGULATION
BEYOND NYQUIST FREQUENCY
Dan Wang
Dept. of Mechanical Engineering
University of Connecticut
Storrs, CT, 06269, U.S.A.
Masayoshi Tomizuka
Dept. of Mechanical Engineering
University of California
Berkeley, CA, 94720, U.S.A.
Xu Chen
Dept. of Mechanical Engineering
University of Connecticut
Storrs, CT, 06269, U.S.A.
ABSTRACT
A fundamental challenge in digital and sampled-data con-
trol arises when the continuous-time plant is subject to fast dis-
turbances 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.
1 Introduction
Modern manufacturing systems are increasingly subjected
to the challenge of limited sensing in the design of control sys-
tems. For instance in selective laser sintering—one of the very
few additive manufacturing techniques that are capable to di-
rectly fabricate metallic parts—the scanning speed of the laser
beam can be more than 10 m/sec while the beam diameter can be
below 100 μm. To capture the temperature evolution of the sinter-
ing process via infrared thermography, ideally the camera speed
should be over 100,000 frames per second [1]. A cost-effective
approach to implement large-field-of-view infrared cameras at
such a high speed is currently unavailable. Even if the technology
were feasible, it remains a major obstacle to utilize the significant
amount of image data for real-time feedback control. Such an is-
sue is causing major challenges in additive manufacturing [2,3].
Similar scenarios also appear in many other processes such as
high-speed vision servo and discrete manufacturing.
Corresponding author
Theoretically, let a plant Pc(s)be controlled by a digital
controller C(z)under a sampling time of Tssec (namely, the
Nyquist frequency is π/Ts). It is a standard result in digital
control that |C(ejΩoTs)|=(while maintaining closed-loop sta-
bility) asymptotically rejects sampled disturbances at ΩoHz.
The result is valid regardless of whether Ωois below or above
the sampling frequency 1/Ts. This seems to suggest that dig-
ital control below Nyquist frequency is potentially capable to
attenuate disturbances above π/Ts. On the other hand, based
on sampled-data control [4–8], the inherent periodic sampling
partitions the continuous-time frequency into infinite regions of
[2kπ/Ts,2(k+1)π/Ts),k=0,±1,±2,...; and a continuous-time
disturbance yields a fundamental mode plus an infinite amount
of aliases in the partitioned regions. Literature has analyzed the
system characteristics (i) by treating all regions as a set [4–7]
and (ii) in the particular region of [0,2π/Ts)[8, 9]. Specifically
for case (ii), different from pure continuous-time feedback de-
sign, high-gain control is no longer capable of fully rejecting the
fundamental disturbance if it occurs in [0,2π/Ts)[9].
Unfortunately, the case with general beyond-Nyquist distur-
bances demands additional attention; and analysis of this miss-
ing piece falls out of both the aforementioned categories. A
main result of this paper is the uncovering of a spectral anal-
ysis for sampled-data control to reject real-valued disturbances.
We show that the spectral effects of high-gain control on beyond-
Nyquist disturbances greatly differ from that on disturbances in
[0,π/Ts). Specifically, it is shown that sub-Nyquist servo design
is infeasible to reject beyond-Nyquist disturbances; furthermore,
there also exists an upper bound of frequency for rejecting dis-
turbances even below π/Ts. In fact, at frequencies above such an
upper bound, high-gain sampled-data control tends to harm the
978-1-5090-3466-6/16/$31.00 ©2016 IEEE 23
actual servo performance. This implies a direct fundamental lim-
itation for any feedback regulation to control sampled-data sys-
tems, and also provides an elemental tool that assists the analysis
of sampled-data control. Usage and discussions of the tool are
provided, along with suggested solutions to compensate distur-
bances beyond Nyquist frequency.
We remark that the present paper focuses on analysis of the
system response with respect to beyond-Nyquist disturbances
and the different spectral distributions. For the synthesis of
sampled-data control, readers are referred to the literature of lift-
ing techniques and multirate control, etc.. Some discussions of
the latter aspect will be provided in Section 5.
Notations:x[n]and xc(t)denote, respectively, a discrete
sequence and a continuous-time signal. The periodic function
X(ejω)=Fd{x[n]}
n=x[n]ejωndenotes the discrete-time
Fourier transform (DTFT) of x[n]. The non-periodic Xc(jΩ)=
F{xc(t)}=
xc(t)ejΩtdtis the Fourier transform of xc(t).
Here, Ωis the frequency in rad/s; ω(= ΩTs)is the normalized
frequency in rad. F1{·} is the operator for inverse Fourier
transform (IFT). (c)and (c)denote, respectively, the real and
imaginary parts of a complex number cC. In a discrete-time
linear time invariant (LTI) negative feedback loop consisting of
a single-input single-output plant Pd(z)and a controller C(z),
Sd(z)1/(1+Pd(z)C(z)) denotes the sensitivity function (i.e.
the transfer function between the output disturbance and the out-
put of Pd(z)).
2 Preliminaries
We consider the sampled-data control system in Fig. 1,
where the solid lines represent continuous-time signal flows, and
the dashed lines are for discrete-time signals. The main servo el-
ements here include the continuous-time plant Pc(s), the analog-
to-digital converter (ADC) that samples the continuous output at
Tssec, the discrete-time controller C(z), and the signal holder
H. In this paper, His assumed to be a ZOH. Although gener-
alized sample hold function has shown promise in sampled-data
performance, it has fundamental limitations in closed-loop ro-
bustness and sensitivity [10]. In addition, practical systems often
have hardware limitations on changing the signal holders.
dc
+
//Huc//Pc(s)yc0
+//yc
//ADC
yd//
C(z)
ud
e
oo
FIGURE 1: Block diagram of a sampled-data control system
We focus on the case with Pc(s)=P0(s)esτwhere τ0;
P0(s)and C(z)are both LTI, proper, and rational. In addition, the
closed loop is assumed to satisfy the nonpathological sampling
condition:
Assumption 1. Let the minimal state equation of the plant be
˙x=Ax +Buc. Take any two eigenvalues λiand λjof A, with
Re(λi)=Re(λj). It is assumed that Im(λi)Im(λj)=2πp/Ts
for any nonzero integer p.
Under Assumption 1, 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 (see, e.g. [11–13]).
Throughout the analysis, we assume that this stability condition
is satisfied.
For completeness, we review first several key characteristics
of the sampled-data process:
Lemma 1. [14] If Xc(jΩ)exists, the sampling process convert-
ing xc(t)to x[n]=xc(nTs)gives
Xejω=1
Ts
k=
Xc(j(ω
Ts
2π
Ts
k)).(1)
Following conventions, in the summation in (1), we refer to
Xc(jω/Ts)as the fundamental signal component, and the other
parts as the aliased components.
Lemma 2. [15] The process x [n]//H//xc(t)
satisfies Xc(jΩ)=H(jΩ)X(ejΩTs)where H(jΩ)=(1
ejΩTs)/(jΩ)is the Fourier transform of ZOH.
Lemma 3. The combined sampled-data process
u[n]//H//Pc
Ts
//yd[n]
yields
Yd(ejω)
U(ejω)=1
Ts
k=
Pc(j(ω
Ts
2π
Ts
k))H(j(ω
Ts
2π
Ts
k)),(2)
which is the frequency response of the ZOH equivalent of Pc.
If Pc(jω/Ts)is bandlimited and the sampling is not too slow,
then Pd(ejω)Yd(ejω)/U(ejω)Pc(jω/Ts)at low frequencies.
In particular, Pd(1)=Pc(0)[16].
Let dd[n]=dc(nTs), i.e.
Dd(ejω)= 1
Ts
k=
Dc(j(Ω2π
Ts
k))Ω=ω
Ts
.(3)
24
With the above building elements and the notations defined in the
introduction, it can be shown that
Yc(jΩ)=11
Ts
Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)Dc(jΩ)
1
Ts
Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)
k=
k=0
Dc(j(Ω2π
Ts
k)).(4)
If dc(t)=ejΩot, then dd[n]=ejω0n(ω0=Ω0Ts); Dc(jΩ)=
2πδ(ΩΩo); and Dd(ejω)= 2π
Ts
k=δ(ω
TsΩo2kπ/Ts).
Then (4) gives
Yc(jΩ)=2π11
Ts
Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)δ(ΩΩo)
2π
Ts
Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)
k=
k=0
δ(ΩΩo2π
Ts
k),(5)
which corresponds to, at steady state, an infinite amount of out-
put harmonics:
yc(t)=[1Γ0(Ω0)]ejΩ0t
k=
k=0
Γk(Ω0)ej(Ω0+2π
Tsk)t.(6)
Here, we define the characteristic feedback loop gain
Γk(Ωo)1
Ts
Pc(j(Ωo+2π
Ts
k))H(j(Ωo+2π
Ts
k))×
Sd(ej(Ωo+2π
Tsk)Ts)C(ej(Ωo+2π
Tsk)Ts).(7)
As discrete-time frequency responses are periodic with a period
of 2π, it is immediate that
Γk(Ω0)= 1
Ts
Pc(j(Ωo+2π
Ts
k))H(j(Ωo+2π
Ts
k))×
Sd(ejΩoTs)C(ejΩoTs).(8)
Remark 1. (Spectral properties of disturbances beyond
Nyquist frequency). For general signals, (1) shows the creation
of aliased modes at intervals of the sampling frequency 2π/Ts.
In practice, any real disturbance {dc(t)R}actually creates
aliased harmonics below and above Nyquist frequency π/Ts.
Specifically, consider dc(t)=cos(Ωot)=(ejΩot+ejΩot)/2.
The Fourier Transform of dc(t)and the DTFT of the sampled
dd[n]are (recall Lemma 1)
Dc(jΩ)=πδ(ΩΩo)+πδ(Ω+Ωo)(9)
Dd(ejω)= π
Ts
k=δ(ω
Ts
Ωo2π
Ts
k)+δ(ω
Ts
+Ωo+2π
Ts
k),(10)
where δ(ΩΩo)denotes a shifted Dirac delta impulse. If
π/Ts<Ωo<2π/Ts, then k =1yields an aliased harmonic
δ(ω/TsΩ0+2π/Ts)below Nyquist frequency at 2π/TsΩo,
symmetric to the fundamental mode Ωowith respect to (w.r.t.)
the line Ω=π/Ts. Analogously, when sampled at Ts, it can be
verified that ˜
dc(t)=cos[(2π/TsΩo)t]yields the same sampled
spectrum below 2π/Ts, with the only difference that 2π/TsΩo
(below Nyquist frequency) corresponds to the frequency of the
fundamental mode.
3 Main Results
We introduce part of the main results via a graphical exam-
ple first, then derive the full propositions in formal terms.
Assume π/Ts<Ωo<2π/Tsand let 2π/TsΩobe below a
principle sampled-data bandwidth that is smaller than π/Ts. Fig.
2a and 2c demonstrate the difference in the output spectra when
the disturbance is dc(t)=cos(Ωot)(above Nyquist frequency)
and ˜
dc(t)=cos(Ω
ot)=cos[(2π/TsΩo)t](below Nyquist fre-
quency), respectively.
Based on Remark 1, the sampled disturbance spectra (and
hence the sampled outputs) are the same. Using discrete-time
servo enhancement tools [17], one can create a high gain in C(z)
at Ω0or 2π/TsΩ0, which can asymptotically reject the sam-
pled disturbance and yield yd[n]0. However, the weighting of
individual modes in yc(t)are fundamentally different. Overall,
˜
dc(t)is significantly attenuated in Fig. 2d yet dc(t)is actually
amplified in Fig. 2b. In more details:
1. For Ωobeing above Nyquist frequency, the magnitude 1
Γ0(Ωo)is close to 1 and the aliased harmonic (correspond-
ing to Γ1(Ωo))at2π/TsΩobelow Nyquist frequency has
a relative high gain compared to Γ0(Ωo). With sub-Nyquist
high-gain control at Ωo, the magnitude of Γk(Ωo)increases
towards the limiting case Γ
k(Ωo). In particular Γ
1(Ωo)is
close to 1, as illustrated in Fig. 2b.
2. On the other hand, for Ω
o<π/Ts,Γ
0(Ω
o)can be controlled
to be close to 1, yielding a small 1 Γ
0(Ω
o), as illustrated
in Fig. 2d.
The spectral distribution can first appear counter to time-domain
intuitions. With the sub-Nyquist high-gain control in Fig. 2b, the
Ts-sampled disturbances cos(Ωot)and cos((2π/TsΩo)t)can
be perfectly rejected from digital control, which conventionally
may suggest to yield null gains in the spectrum below Nyquist
frequency. Null gains, however, occur in neither the case for dis-
turbances at Ωonor that at 2π/TsΩo. In fact, Fig. 2b contains
significant components at 2π/TsΩo.
25
Ω
Yc
πe±jφ
||||||||
1Γ0(Ωo)1Γ0(Ωo)
Γ1(Ωo)
Γ1(Ωo)Γ1(Ωo)
Γ1(Ωo)
Γ2(Ωo)
Γ2(Ωo)......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(a) dc(t)=cos(Ωot)
Ω
Yc
πe±jφ
||||||||
1Γ
0(Ωo)
1Γ
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
Yc
πe±jφ
||||||||
1Γ0(Ω
o)1Γ0(Ω
o)
Γ1(Ω
o)
Γ1(Ω
o)Γ1(Ω
o)
Γ1(Ω
o)Γ2(Ω
o)
Γ2(Ω
o)
......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(c) ˜
dc(t)=cos(Ω
ot)=cos((2π/TsΩo)t)
Ω
Yc
πe±jφ
||||||||
1Γ
0(Ω
o)1Γ
0(Ω
o)
Γ
1(Ω
o)
Γ
1(Ω
o)
Γ
1(Ω
o)
Γ
1(Ω
o)Γ
2(Ω
o)
Γ
2(Ω
o)
......
π
Ts
2π
Ts
π
Ts
2π
Ts
3π
Ts
4π
Ts
3π
Ts
4π
Ts
(d) ˜
dc(t)=cos(Ω
ot)=cos((2π/TsΩo)t); with enhanced control at Ωo
FIGURE 2: Illustration (gains of the impulses are actually com-
plex) of the spectrum of yc(t)in sampled-data control when
π/Ts<Ωo<2π/Ts. Dashed spikes: δ(Ω+Ω0)and its aliased
harmonics; solid spikes: δ(ΩΩ0)and its aliases.
Intuitions of the above illustrations can be made partially
from the summation form of yc(t)in (4). Perhaps less intuitive
for now, is the fact that Sub-Nyquist ZOH sampled-data con-
trol cannot reject continuous-time disturbances near and above
Nyquist frequency in Fig. 1. Details of the results, along with
the derivations and further implications, will be provided in the
remaining subsections based on three aspects of considerations:
(i) spacial distribution of the aliased modes; (ii) characterization
of the magnitude for each mode; (iii) implications and funda-
mental limitations.
3.1 Loop-shaping beyond Nyquist frequency
From Remark 1, we are reminded that real disturbances al-
ways contain components below and above Nyquist frequency.
For a real-coefficient discrete-time controller, high-gain control
at Ωoalso equivalently gives high servo gain at 2π/TsΩo,if
Ωo(π/Ts,2π/Ts). To be more complete, as discrete frequency
responses are periodic and conjugate symmetric, we have:
Fact 1. Let M(ejω)be the frequency response of a real-
coefficient discrete-time transfer function, then Ωin Hz
M(ejΩTs)=Mej(ΩTsmod 2π),Ω[2kπ
Ts,(2k+1)π
Ts]
Mej(2π(ΩTsmod 2π)),Ω[(2k+1)π
Ts,(2k+2)π
Ts],(11)
where k =0,±1,...; and mod denotes the modulo operation.
(11) also holds if M(ejω)is the DTFT of a real signal.
With the observations, consider a sampled real disturbance
at Ωo(π/Ts,2π/Ts). Within one period (0,2π/Ts), the sampled
disturbance contains two spectral peaks at Ωoand 2π/TsΩo
(Remark 1); and Fact 1 implies that the discrete-time sensitivity
is always simultaneously reduced/increased at the two symmetric
frequencies. A conjecture may thus be formed that performing
enhanced servo design at 2π/TsΩo(below Nyquist frequency)
rejects both modes in sampled-data control. Later analysis will
disprove the conjecture.
3.2 Spectral distribution and implications
We analyze the output structure under a real-valued funda-
mental disturbance first, then derive the spectral distribution at
the beginning of this section, as well as infer its major implica-
tions. For brevity, all proofs are provided in the appendix.
Let dc(t)=cos(Ωot+φ). Recall from Fourier transform,
that Fejφδ(ta)=ej(Ωa+φ). Applying cos(Ω0t+φ)=
(ej(Ω0t+φ))and linearity in (6) gives
yc(t)=[1Γ0(Ω0)]ej(Ωot+φ)
k=,k=0
Γk(Ω0)ej[(Ωo+2π
Tsk)t+φ].(12)
Denote yc(t)=(x(t)). By properties of Fourier trans-
form, F{(x(t))}=[X(jΩ)+X(jΩ)]/2. Using the re-
sult in (12) gives Yc(jΩ)=πejφ(1Γ0(Ωo))δ(ΩΩo)+
πejφ(1Γ0(Ωo))δ(ΩΩo)πejφ
k=,k=0Γk(Ω0)δ(Ω
Ωo2π
Tsk)πejφ
k=,k=0Γk(Ω0)δ(ΩΩo2π
Tsk).
26
By definition, Γk(Ωo)is conjugate symmetric, with
Γk(Ω0)=Γk(Ω0), as all transfer functions on the right side
of (7) are real-coefficient based. Using additionally the fact that
δ(x)=δ(x)xR,weget
Yc(jΩ)=πejφ(1Γ0(Ωo))δ(ΩΩo)
+πejφ(1Γ0(Ωo))δ(Ω+Ωo)
πejφ
k=,k=0
Γk(Ω0)δ(ΩΩo2π
Ts
k)
πejφ
k=,k=0
Γk(Ω0)δ(Ω+Ωo+2π
Ts
k).(13)
To better explain the main theorem, we denote Td(z)
Pd(z)C(z)/[1+Pd(z)C(z)] = Pd(z)C(z)Sd(z)as the discrete-time
complementary sensitivity function, and introduce an analysis
form of (7):
Γk(Ωo)= Pc(j(Ωo+2π
Tsk))H(j(Ωo+2π
Tsk))
TsPd(ejΩoTs)Td(ejΩoTs).(14)
In addition, we make a common assumption about the baseline
servo:
Assumption 2. Td(z)has low-pass dynamics with approxi-
mately unity DC gain.
Theorem 1. Assume that the closed loop in Fig. 1 is stable. Let
dc(t)=cos(Ωot+φ). Then
Yc(jΩ)dΩ=2πℜejφSd(ejΩoTs).(15)
When |C(ejΩoTs)|=and the closed loop is stable, yd[n]con-
verges to zero at steady state, with the magnitude of the fun-
damental modes at Ωoand Ωobeing |1Γ
0(Ωo)|=|1
Γ
0(Ωo)|=|1Pc(jΩo)H(jΩo)/(TsPd(ejΩoTs))|(nonzero in
general); and the gains of other harmonic modes being Γ
k(Ωo)=
Γ
k(Ωo)=lim|C(ejΩoTs)|→Γk(Ωo), where
Γ
k(Ωo)= Pc(j(Ωo+2π
Tsk))H(j(Ωo+2π
Tsk))
l=Pc(j(Ωo+2π
Tsl))H(j(Ωo+2π
Tsl)) (16)
=Pc(j(Ωo+2π
Tsk))H(j(Ωo+2π
Tsk))
TsPd(ejΩoTs).(17)
(15) links the overall continuous-time closed-loop perfor-
mance on the left with the discrete-time design on the right.
Although reduced |Sd(ejΩoTs)|(high-gain control) reduces the
overall effect
Yc(jΩ)dΩ, it is fundamentally unattainable to
simultaneously reject the fundamental and the harmonic distur-
bances. Similar to the feedback limitation on the infeasibility
to simultaneously reject disturbance and sensor noises, the gains
for ej(Ωot+φ)and e j[(Ωo+2πk/Ts)t+φ]in (13) are conflicting in struc-
ture. Letting C(ejΩoTs)=0 in (7) yields Γk(Ωo)=0, namely,
a null gain for each harmonic and a unity gain for the funda-
mental disturbance in (13). On the other hand, as shown next,
counter to conventional servo design, high-gain control in Fig. 1
can conditionally compensate the fundamental disturbance if it
occurs below Nyquist frequency, yet for most cases will amplify
the aliased harmonics in a sampled-data closed-loop system. To
reveal the detailed spectrum of Yc(jΩ), we make several remarks
on the general shape of |Γk(Ωo)|and |Γ
k(Ωo)|:
Remark 2. Overall, high-gain control at Ωoincreases the mag-
nitudes of Td(ejΩoTs)and Γk(Ωo)(= Γ
k(Ωo)Td(ejΩoTs)).AsT
d(z)
is a low-pass filter whose bandwidth—denoted as BT—is com-
monly 10%-20% of Nyquist frequency [15], for the majority of
frequencies it holds that |Γ
k(Ωo)|>|Γk(Ωo)|. Furthermore:
1. If Ωo+2kπ/Ts[0,BT), then the low-pass H(j(Ωo+
2kπ/Ts))/Ts1, and P(j(Ωo+2kπ/Ts))H(j(Ωo+
2kπ/Ts))/TsPd(ejΩoTs), yielding Γk(Ωo)and Γ
k(Ωo)
to both be approximately 1. Thus, |1Γk(Ωo)|and
|1Γ
k(Ωo)|are both small.
2. If Ωo+2kπ/Ts[BT,π/Ts), then |Td(ejΩoTs)|<1 and
|Γ
k(Ωo)|>|Γk(Ωo)|. For the majority of the frequencies,
|Γ
k(Ωo)|≈1 and |1Γ
k(Ωo)|1.
3. If Ωo+2kπ/Ts>π/Ts, the ZOH |H(j(Ω+2kπ/Ts))|re-
duces quickly outside its approximate bandwidth π/Ts.
High-gain control still makes |Γ
k(Ωo)|>|Γk(Ωo)|, but the
overall magnitudes |Γ
k(Ωo)|and Γk(Ωo)are too small such
that 1 Γ
k(Ωo)and 1 Γk(Ωo)both approximate 1.
Fig. 3 shows the magnitude responses1of Γk(Ωo),Γ
k(Ωo),
1Γk(Ωo), and 1 Γ
k(Ωo)for a typical servo design in Sec-
tion 4. The Nyquist frequency is indicated by the vertical line at
1230 Hz. As an analysis tool, Fig. 3 verifies Remark 2, and re-
veals several fundamental characteristics of sampled-data control
above and below Nyquist frequency:
1. Unless at very low frequencies (below the baseline closed-
loop bandwidth) where Γk(Ω)Γ
k(Ω), the aliased harmon-
ics are amplified by high-gain control below Nyquist fre-
quency.
2. Sub-Nyquist sampled-data control in Fig.1cannot reject
continuous-time disturbances near and above Nyquist fre-
quency. Above π/Ts, the effect of feedback control on
1Γ0(Ωo)is greatly limited by Remark 2.3. Let k=0 in the
1It is sufficient to plot the function values for positive frequencies. Other
values can be obtained from the conjugate symmetry of Γk(Ωo).
27
bottom plot of Fig. 3. High-gain control in C(z)only rejects
the fundamental disturbance mode up to the intersection fre-
quency of 1Γ
k(Ω)and 1 Γk(Ω). Around π/Ts, the capa-
bility of feedback control is limited by the aliasing-induced
distortion in (16). In addition, the achievable maximum at-
tenuation—indicated by the distance between |1Γ
0(Ω)|
and |1Γ0(Ω)|—decreases with increasing frequency.
0 1000 2000 3000 4000 5000 6000 7000 8000
−200
−150
−100
−50
0
50
100
Magnitude (dB)
Γk
*(Ω)
Γk(Ω)
0 1000 2000 3000 4000 5000 6000 7000 8000
−150
−100
−50
0
50
Ω + 2πk / Ts (Hz)
Magnitude (dB)
1−Γk
*(Ω)
1−Γk(Ω)
FIGURE 3: Magnitude responses of Γ()
k(Ω)and 1 Γ()
k(Ω)as
a function of Ω+2πk/Ts. The first three vertical lines indicate,
respectively, Nyquist frequency (1230 Hz), sampling frequency,
and 3π/Ts.
Definition 1. The first intersection frequency between |1
Γ
0(Ω)|and |1Γ0(Ω)|above the discrete-time closed-loop
bandwidth is called the principle sampled-data bandwidth.
For brevity and without loss of generality, let π/Ts<Ωo<
2π/Tsand 2π/TsΩobe below the principle sampled-data band-
width. Based on (13) and the magnitude properties of Γk(Ωo),
one can obtain the output spectra in Fig. 2 under dc(t)=
cos(Ωot)and ˜
dc(t)=cos(Ω
ot)=cos[(2π/TsΩo)t]. Although
the sampled disturbances (and hence the sampled outputs) are
the same, the weighting of individual modes are fundamentally
different for the two modes in (0,π/Ts)and (π/Ts,2π/Ts), as has
been explained at the start of Section 3.
If Ωo>2π/Ts, an alias will still occur below Nyquist fre-
quency. Analogous analysis can be made based on the spectral
properties of the loop gains for the individual modes.
For the case where the disturbance is beyond Nyquist fre-
quency in Fig. 2b, with little actual control over 1 Γ0(Ωo),
Γ±k(Ωo)(k=0) is amplified such that the aliasing effect cancels
the fundamental component after sampling. Fig. 2d on the other
hand achieves zero sampled output by actually reducing the mag-
nitude of 1Γ0(Ω
o). From Parseval’s theorem, the large spectral
spikes in Fig. 2b thus will yield significant (hidden) actual per-
formance degradation in the time domain.
Remark 3. Mathematically, the principle sampled-data band-
width occurs at
|1Γ
k(Ω)|=|1Γk(Ω)|=1Γ
k(Ω)Td(ejΩTs).(18)
For systems satisfying Assumption 2, the small gain of Td(ejΩTs)
near Nyquist frequency renders 1Γk(Ω)to be close to 1 in Fig.
3 and hence the location of the principle sampled-data band-
width below Nyquist frequency. In practice, such characteristics
are common in motion control systems (i.e. systems with inertia
type of dynamics) and robust control design with high-frequency
model uncertainties.
4 Numerical Verification
We provide a numerical example on high-speed motion con-
trol under vision feedback. Consider a plant Pc(s)=3.74488 ×
109/(s2+565.5s+319775.2)with an input delay of 10 mi-
croseconds. Let the sampling time be limited at Ts=1/2640 sec-
ond. Let the baseline discrete-time controller be a PID controller
C(z)=kp+ki/(z1)+kd(z1)/zwith gains kp=1/8/1665,
ki=1/20/1665, and kd=3/5/1665. Such a design provides a
3dB frequency of 92 Hz in the discrete-time sensitivity function,
slightly below 10% of the Nyquist frequency.
The narrow-band disturbance observer [18] is applied on top
of the PID controller, to enable high-gain control at selective fre-
quencies. Such design provides perfect compensation of sinu-
soidal signals below Nyquist frequency, and is termed 1x com-
pensation. Continuous-time single-frequency vibrations are ap-
plied to the plant, below and above Nyquist frequency. Fig. 4
presents the limitation of 1x design in beyond-Nyquist distur-
bance rejection, which matches with the prediction in Fig. 2.
In more details, when the disturbance occurs at 2376 Hz (i.e.
1.8ΩN), although the sub-Nyquist servo enhancement achieves
zero steady-state output in the Ts-sampled output in Fig. 4a, the
actual continuous-time output is significantly amplified. The 3σ
(σis the standard deviation) value of the continuous-time output
increased from 15.7076 to 20.8725, yielding a 133% amplifica-
tion. The amplification is also evident in the frequency domain
in Fig. 4c. Due to the performance limit illustrated in Fig. 3,
high-gain sub-Nyquist control barely changes the fundamental
component at 2376 Hz while greatly amplifies the sub-Nyquist
aliased component at 264 Hz.
28
Time (sec)
0 0.02 0.04 0.06 0.08 0.1
Normalized output
-10
-5
0
5
10
Sampled output (sampling time: Ts)
1x compensation off: 3 = 17.2544
1x compensation on: 3 = 0.61329
(a) yc(t)sampled at Ts
0.08 0.085 0.09 0.095 0.1
−20
−10
0
10
20
Time
(
sec
)
Normalized output
1x compensation off: 3σ=15.7076
1x compensation on: 3σ=20.8725
(b) Continuous-time output yc(t)
0 5 10 15 20 25
spectrum amplitude
0
2
4
6
81x compensation on
Frequency (kHz)
0 5 10 15 20 25
spectrum amplitude
0
2
4
6
81x compensation off
(c) Fast Fourier Transform (FFT) of yc(t)sampled at Ts/10
FIGURE 4: Plant output for the case with disturbance at 1.8ΩN
Fig. 5 verifies the case with regular sub-Nyquist distur-
bances. Similar to Fig. 4, the Ts-sampled output is also zero
at steady state. With the fundamental disturbance falling below
Nyquist frequency, sub-Nyquist high-gain control was able to at-
tenuate the disturbance component at 924 Hz. The aliased har-
monics are however all amplified, as predicted by Fig. 2. Corre-
spondingly, the actual continuous-time output contains intersam-
ple ripples, as predicted from Parseval’s Theorem.
0.08 0.085 0.09 0.095 0.1
−10
−5
0
5
10
Time
(
sec
)
Normalized output
1x compensation off: 3σ=15.6223
1x compensation on: 3σ=3.9487
(a) Continuous-time output
0 5 10 15 20 25
spectrum amplitude
0
2
4
6
1x compensation on
Frequency (kHz)
0 5 10 15 20 25
spectrum amplitude
0
2
4
6
1x compensation off
(b) FFT (sampling time Ts/10)
FIGURE 5: Plant output for the case with disturbance at 0.7ΩN
5 Conclusion and Discussions
In this paper, the problem of sampled-data regulation con-
trol against structured disturbances beyond Nyquist frequency is
analyzed. When the output is sampled at Ts, it is shown that reg-
ular sampled-data control at a sampling time of Tsis infeasible
to attenuate disturbances not only beyond, but also near Nyquist
frequency. A spectral analysis is further proposed for revealing
the fundamental difference between sub- and beyond-Nyquist
sampled-data regulation control. The results imply that the re-
jection of beyond-Nyquist vibration disturbances must either rely
on increasing the hardware sampling frequency, or applying cus-
tomized multirate control. For instance, peak/resonant filters can
be configured at a higher sampling rate. By using model-based
prediction, it is further possible to fully reject structured distur-
29
bances beyond Nyquist frequency. One such example is provided
in [19].
6 Appendix: Proofs
To prove Theorem 1, several lemmas are introduced first.
Lemma 4. It holds that
k=
Γk(Ωo)=Td(ejΩoTs)=1Sd(ejΩoTs)(19)
Proof. Using the impulse modulation formula (2) to represent
Pd(ejω)gives:
Td(ejω)=Sd(ejω)C(ejω)1
Ts
k=
Pc(j(ω
Ts
2π
Ts
k))H(j(ω
Ts
2π
Ts
k))
=
k=
Γk(ω
Ts
)=
k=
Γk(ω
Ts
)(20)
which yields the first equality in (19). The second equality fol-
lows from Sd(z)+Td(z)=1z.
Lemma 5. Let the closed loop be stable in Fig. 1. For dc(t)=
ejΩot, the closed-loop steady-state output satisfies
Yc(jΩ)dΩ=2πSd(ejΩoTs)(21)
Proof. Integrating the Fourier transform of (6) and applying
Lemma 4 proves Lemma 5.
Proof of Theorem 1. Analogous to Lemma 5, integrating (13)
and applying (20) give (15). The limiting case of Γk(Ωo)is im-
mediate from Td(ejΩoTs)=1if|C(ejΩoTs)|=, which also yields
Sd(ejΩoTs)=1Td(ejΩoTs)=0 and hence the rejection of the Ts-
sampled disturbance at steady state.
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30
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