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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]e−jωndenotes the discrete-time

Fourier transform (DTFT) of x[n]. The non-periodic Xc(jΩ)=

F{xc(t)}=∞

−∞xc(t)e−jΩtdtis the Fourier transform of xc(t).

Here, Ωis the frequency in rad/s; ω(= ΩTs)is the normalized

frequency in rad. F−1{·} is the operator for inverse Fourier

transform (IFT). ℜ(c)and ℑ(c)denote, respectively, the real and

imaginary parts of a complex number c∈C. 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)e−sτwhere τ≥0;

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−

e−jΩTs)/(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(Ω−2π

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)=1−1

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(Ω−2π

Ts

k)).(4)

If dc(t)=ejΩot, then dd[n]=ejω0n(ω0=Ω0Ts); Dc(jΩ)=

2πδ(Ω−Ωo); and Dd(ejω)= 2π

Ts∑∞

k=−∞δ(ω

Ts−Ωo−2kπ/Ts).

Then (4) gives

Yc(jΩ)=2π1−1

Ts

Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)δ(Ω−Ωo)

−2π

Ts

Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)

∞

∑

k=−∞

k=0

δ(Ω−Ωo−2π

Ts

k),(5)

which corresponds to, at steady state, an inﬁnite amount of out-

put harmonics:

yc(t)=[1−Γ0(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+e−jΩot)/2.

The Fourier Transform of dc(t)and the DTFT of the sampled

dd[n]are (recall Lemma 1)

Dc(jΩ)=πδ(Ω−Ωo)+πδ(Ω+Ωo)(9)

Dd(ejω)= π

Ts

∞

∑

k=−∞δ(ω

Ts

−Ωo−2π

Ts

k)+δ(ω

Ts

+Ωo+2π

Ts

k),(10)

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

π/Ts<Ωo<2π/Ts, then k =−1yields an aliased harmonic

δ(ω/Ts−Ω0+2π/Ts)below Nyquist frequency at 2π/Ts−Ωo,

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/Ts−Ωo)t]yields the same sampled

spectrum below 2π/Ts, with the only difference that 2π/Ts−Ωo

(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/Ts−Ωobe 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/Ts−Ωo)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/Ts−Ω0, 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/Ts−Ωobelow 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/Ts−Ωo)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/Ts−Ωo. In fact, Fig. 2b contains

signiﬁcant components at 2π/Ts−Ωo.

25

Ω

Yc

πe±jφ

||||||||

1−Γ0(Ωo)1−Γ0(−Ωo)

Γ1(−Ωo)

Γ−1(Ωo)Γ1(Ωo)

Γ−1(−Ωo)

Γ2(−Ωo)

Γ−2(Ωo)......

π

Ts

2π

Ts

−π

Ts

−2π

Ts

3π

Ts

4π

Ts

−3π

Ts

−4π

Ts

(a) dc(t)=cos(Ωot)

Ω

Yc

πe±jφ

||||||||

1−Γ∗

0(Ωo)

1−Γ∗

0(−Ωo)

Γ∗

1(−Ωo)

Γ∗

−1(Ωo)

Γ∗

1(Ωo)

Γ∗

−1(−Ωo)

Γ∗

2(−Ωo)

Γ∗

−2(Ωo)......

π

Ts

2π

Ts

−π

Ts

−2π

Ts

3π

Ts

4π

Ts

−3π

Ts

−4π

Ts

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

Ω

Yc

πe±jφ

||||||||

1−Γ0(Ω

o)1−Γ0(−Ω

o)

Γ1(−Ω

o)

Γ−1(Ω

o)Γ1(Ω

o)

Γ−1(Ω

o)Γ2(−Ω

o)

Γ−2(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/Ts−Ωo)t)

Ω

Yc

πe±jφ

||||||||

1−Γ∗

0(Ω

o)1−Γ∗

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/Ts−Ωo)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(Ω−Ω0)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/Ts−Ωo,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/Ts−Ωo

(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/Ts−Ωo(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 Fe−jφδ(t−a)=e−j(Ωa+φ). Applying cos(Ω0t+φ)=

ℜ(ej(Ω0t+φ))and linearity in (6) gives

yc(t)=ℜ[1−Γ0(Ω0)]ej(Ωot+φ)

−

∞

∑

k=−∞,k=0

ℜΓk(Ω0)ej[(Ωo+2π

Tsk)t+φ].(12)

Denote yc(t)=ℜ(x(t)). By properties of Fourier trans-

form, F{ℜ(x(t))}=[X(−jΩ)+X(jΩ)]/2. Using the re-

sult in (12) gives Yc(jΩ)=πejφ(1−Γ0(Ωo))δ(Ω−Ωo)+

πe−jφ(1−Γ0(Ωo))δ(−Ω−Ωo)−πejφ∑∞

k=−∞,k=0Γk(Ω0)δ(Ω−

Ωo−2π

Tsk)−πe−jφ∑∞

k=−∞,k=0Γk(Ω0)δ(−Ω−Ωo−2π

Tsk).

26

By deﬁnition, Γk(Ωo)is conjugate symmetric, with

Γ−k(−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)∀x∈R,weget

Yc(jΩ)=πejφ(1−Γ0(Ωo))δ(Ω−Ωo)

+πe−jφ(1−Γ0(−Ωo))δ(Ω+Ωo)

−πejφ∞

∑

k=−∞,k=0

Γk(Ω0)δ(Ω−Ωo−2π

Ts

k)

−πe−jφ∞

∑

k=−∞,k=0

Γ−k(−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 |1−Γ∗

0(Ωo)|=|1−

Γ∗

0(−Ωo)|=|1−Pc(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))/Ts≈1, and P(j(Ωo+2kπ/Ts))H(j(Ωo+

2kπ/Ts))/Ts≈Pd(ejΩoTs), yielding Γk(Ωo)and Γ∗

k(Ωo)

to both be approximately 1. Thus, |1−Γk(Ωo)|and

|1−Γ∗

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 |1−Γ∗

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),

1−Γk(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

1−Γ0(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 1−Γ∗

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 |1−Γ∗

0(Ω)|

and |1−Γ0(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 |1−Γ0(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/Ts−Ωobe 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/Ts−Ωo)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 1−Γ0(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

|1−Γ∗

k(Ω)|=|1−Γk(Ω)|=1−Γ∗

k(Ω)Td(ejΩTs).(18)

For systems satisfying Assumption 2, the small gain of Td(ejΩTs)

near Nyquist frequency renders 1−Γk(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/(z−1)+kd(z−1)/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)=1−Sd(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=−∞

Γ−k(ω

Ts

)=

∞

∑

k=−∞

Γk(ω

Ts

)(20)

which yields the ﬁrst equality in (19). The second equality fol-

lows from Sd(z)+Td(z)=1∀z.

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)=1−Td(ejΩoTs)=0 and hence the rejection of the Ts-

sampled disturbance at steady state.

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