Multirate Forward-model Disturbance Observer for Feedback Regulation beyond Nyquist Frequency

Abstract

A fundamental challenge in digital and sampled-data control arises when the controlled plant is subjected to fast disturbances with, however, a slow or limited sensor measurement. Such intrinsic difficulties are commonly encountered in many novel applications such as advanced and additive manufacturing, human-machine interaction, etc. This paper introduces a discrete-time regulation scheme for exact sampled-data rejection of disturbances beyond Nyquist frequency. By using a model-based multirate predictor and
a forward-model disturbance observer, we show that the intersample
disturbances can be fully attenuated despite the limitations in sampling and sensing. In addition, sharing the main properties of all-stabilizing control, the proposed control scheme offers several advantages in stability assurance and lucid design intuitions.

Full text

Multirate Forward-model Disturbance Observer for Feedback Regulation
beyond Nyquist Frequency
Xu Chen†and Hui Xiao
Abstract— A fundamental challenge in digital and sampled-
data control arises when the controlled plant is subjected
to fast disturbances with, however, a slow or limited sen-
sor measurement. Such intrinsic difficulties are commonly
encountered in many novel applications such as advanced
and additive manufacturing, human-machine interaction, etc.
This paper introduces a discrete-time regulation scheme for
exact sampled-data rejection of disturbances beyond Nyquist
frequency. By using a model-based multirate predictor and
a forward-model disturbance observer, we show that the in-
tersample disturbances can be fully attenuated despite the
limitations in sampling and sensing. In addition, sharing the
main properties of all-stabilizing control, the proposed control
scheme offers several advantages in stability assurance and
lucid design intuitions.
Index Terms— disturbance beyond Nyquist frequency
I. INTRODUCTION
Let a continuous-time plant be subjected to a disturbance
dc(t)and controlled under a discrete-time controller. A fun-
damental challenge arises in feedback control if the sampling
is not fast enough to capture the major frequency components
of dc(t)—or more specifically, when significant disturbances
occur beyond Nyquist frequency. Such a scenario, however,
is becoming increasingly important in modern control sys-
tems, due to, on the one hand the continuous pursuit of
higher performance and robustness using slow or limited
sensing mechanisms (e.g., vision servo [1], chemical process
[2], human-machine interaction, etc); and on the other hand,
the significant interest in novel applications such as in-
situ sensing in advanced additive manufacturing [3], [4]
and virtual and mixed reality [5]. In these applications and
the like, significant disturbances beyond Nyquist frequency
are unattended under conventional servo design. Such large
intersample/hidden disturbances are extremely dangerous, as
they cause unobserved performance loss in the actual system,
increase system fatigue, and can even lead to hardware
failures.
Particularly for advanced manufacturing such as the laser
based additive manufacturing process [3], there are sig-
nificant challenges and opportunities in high-speed, high-
precision sensing and metrology. For instance, [6]–[8] used
an optoelectronic sensor and an infrared camera, respectively,
for the sensing and control of metal powder delivery and
molten pool profile in laser cladding. The sampling for the
†: corresponding to: 191 Auditorium Road U-3139, Storrs, CT, USA,
06269-3139, Tel.:860-486-3688.
Xu Chen (email: xchen@engr.uconn.edu) and Hui Xiao (email:
hui.xiao@uconn.edu) are with the Department of Mechanical Engi-
neering, University of Connecticut.
first sensor is performed at 10 Hz. Although using a high-
speed camera at 800 frames per second, the control of the
closed-loop in the second task is performed at 30 Hz, as it
takes time for the raw image data to be processed and for the
signature characteristics in the molten pool to be estimated.
From general principles of feedback design, the closed-loop
bandwidth is typically 10% to 20% of Nyquist frequency.
Overcoming the limitations from the slow/limited sampling
is thus key for unlocking the full potentials of the highest
performance and robustness in the next-generation additive
manufacturing.
In this paper, we provide a control algorithm to reject
disturbances at arbitrary frequencies beyond the conventional
Nyquist limitation. Let the plant output be sampled at Tssec.
Two principle directions can be pursued for control beyond
Nyquist frequency. Based on fundamental principles in loop
shaping and feedback control [9], the first direction is to
design a continuous-time controller, CCT , that achieves high-
gain control beyond the original Nyquist frequency; and then
digitize CCT at a sampling time greater than Ts. In this regard,
[10] and the references therein discussed beyond-Nyquist
servo control via conventional loop shaping techniques. The
second approach seeks state-space solutions at a sampling
time smaller than Ts[11]. The main contribution of this paper
is the introduction of a beyond-Nyquist exact disturbance-
rejection scheme, which enables full rejection of structured
disturbances at both the sampling and any uniformly spaced
inter-sample instances. Such exact rejection is one of the
very first in regulation control beyond Nyquist frequency.
In addition, the structure of the proposed multirate forward-
model disturbance observer (MR-FMDOB) shares many
advantages of internal model control (IMC) [12] and all-
stabilizing Youla-Kucera parameterization [13], [14], which
offer multiple theoretical and practical benefits in stability
assurance and clear design intuitions.
Notations: Rdenotes the set of real numbers. x[n]
and xc(t)denote, respectively, a discrete sequence
and a continuous-time signal. X(ejω) = Fd{x[n]},
∑∞
n=−∞x[n]e−jωndenotes the discrete-time Fourier transform
(DTFT) of x[n].Xc(jΩ) = F{xc(t)},R∞
−∞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. Hdenotes a
zero order hold (ZOH). When the sampling time is Ts, the
transfer function of a ZOH is H(s) = (1−e−sTs)/s.
II. PRELIMINARIES AND PROBLEM FORMUL ATION
We consider the control system in Fig. 1, where the solid
lines represent continuous-time signal flows, and the dashed
2016 American Control Conference (ACC)
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lines are for discrete-time signals. The main elements here
include the continuous-time plant Pc(s), the sampler that
samples the continuous output at Tssec, the discrete-time
controller C(z), and the signal holder H.
dc
+

//Huc//Pc(s)yc0
+//◦yc◦
Ts
−
yd//
C(z)
ud◦
e
oo
Fig. 1: Block diagram of a sampled-data control system
Assumption 1: Pc(s) = P0(s)e−sτwhere τ≥0; P0(s)and
C(z)are both LTI, proper, and rational.
Fundamentally, the overall closed loop is a sampled-data
control system. As a hybrid of continuous- and discrete-time
systems, its key characteristics include [15]–[17]:
Lemma 1: If Xc(jΩ)exists, the sampling process convert-
ing xc(t)to x[n] = xc(nTs)satisfies
Xejω=1
Ts
∞
∑
k=−∞
Xc(j(ω
Ts
−2π
Ts
k)).(1)
In other words, after discretization, the DTFT X(ejω)con-
tains not only Xc(jω
Ts)(at the corresponding continuous-
time frequency ω/Ts), but also aliased components whose
frequencies extend to infinity.
Lemma 2: If x [n]//H//xc(t)then
Xc(jΩ) = H(jΩ)X(ejΩTs)(2)
where H(jΩ) = (1−e−jΩTs)/(jΩ).
We consider the regulation problem when dccontains
significant energy components beyond Nyquist frequency.
Before the formal definition of the problem, several obser-
vations are made for completeness of analysis:
Digital-control equivalence: After sampling in Fig. 1, the
effect of dcon ydis equivalent to that of a discrete-time
disturbance added right before yd. Let dd[n] = dc(nTs), i.e.
(by using Lemma 1)
Dd(ejω) = 1
Ts
∞
∑
k=−∞
Dc(j(Ω−2π
Ts
k))Ω=ω
Ts
.(3)
Then the relationship between ydand the sampled distur-
bance falls into the discrete-time control problem in Fig. 2,
where Pd(z)is the ZOH equivalent of Pc(s)(sampled at Ts),
and the DTFT of ydis
Yd(ejω) = Sd(ejω)Dd(ejω) = Dd(ejω)
1+Pd(ejω)C(ejω)(4)
where Sd(z),1/(1+Pd(z)C(z)) is the closed-loop sensitiv-
ity function.
Digital control beyond Nyquist frequency: Consider a
real-coefficient discrete-time transfer function C(z). Take an
arbitrary ω=ΩTs,C(ejω)is an image of the frequency
response below Nyquist frequency:1
1Note that in computing the frequency response via computer programs
such as MATLAB, only the responses below Nyquist frequency are plotted.
dd[n]+

//Pd(z)+//◦
−
yd//
C(z)
ud◦
e
Fig. 2: An equivalent of Fig. 1if ydis the signal of concern
Fact 1: Let C(ejω)be the frequency response of a real-
coefficient discrete-time transfer function, then ∀Ωin Hz
C(ejΩTs) = 


Cej(ΩTsmod 2π),Ω∈[2kπ
Ts,(2k+1)π
Ts]
Cej(2π−(ΩTsmod 2π)),Ω∈[(2k+1)π
Ts,(2k+2)π
Ts](5)
where k =0,±1, ... ; and mod denotes the modulo operation.
The result can be readily proved with the facts that C(ejω)
is periodic and conjugate symmetric—C(e−jω) = C(ejω).
In particular, let Ω=π/Tsbe the line of symmetry. Then
C(ejΩTs)in the region [π/Ts,2π/Ts]—i.e. between Nyquist
and sampling frequencies—is conjugate symmetric with that
in [0,π/Ts]. In the Bode Plot, the two response lines have
symmetric (w.r.t. Ω=π/Ts) magnitude response, and oppo-
site phase values. Therefore, 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).
Based on the concept of loop shaping, a conjecture may thus
be formed that increasing the magnitude of C(z)at proper
frequencies below π/Tscan attenuate disturbances beyond
Nyquist frequency.
Fundamental limitation in sampled-data control:
With Ud(ejω) = −C(ejω)Yd(ejω),Yc(jΩ) = Dc(jΩ) +
Pc(jΩ)H(jΩ)Ud(ejω)ω=ΩTsin Fig. 1, and the derived (4),
the continuous-time output satisfies
Yc(jΩ) = Dc(jΩ)−Pc(jΩ)H(jΩ)×
C(ejΩTs)Sd(ejΩTs)Dd(ejΩTs)(6)
or equivalently, after using (3),
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=−∞
k6=0
Dc(j(Ω−2π
Ts
k)).(7)
Note that the magnitude of the two gains on the right
of (7) are conflicting in structure. Reducing one leads to
amplification of the other. Therefore, despite that increas-
ing |C(ejω)|leads to reduced |Y(ejω)|in (4) under pure
discrete-time control, it is fundamentally unattainable to
fully reject the continuous-time disturbance in sampled-data
control. In fact, one can recognize that the second gain,
|Pc(jΩ)H(jΩ)Sd(ejΩTs)C(ejΩTs)|, increases with high-gain
control. Conventional discrete-time loop shaping and the
conjecture after Fact 1 are thus not applicable to fully address
sampled-data control beyond Nyquist frequency.
840

Suppose an additional fictitious faster sensor, at a sampling
time of T0
s=Ts/L(Lis a positive integer), is available
between ycand the existing Tssampler in Fig. 1. Over-
coming the described challenges in sampled-data systems,
we provide a control solution such that the T0
s-sampled
ycasymptotically converges to zero in the presence of a
continuous-time disturbance beyond Nyquist frequency.
III. MULTI RATE MODEL BAS ED DISTURBANCE
REJECTION BEYOND NYQUIST FREQ UEN CY
From the preceding section, sub-Nyquist servo design is
fundamentally limited to reject disturbances above Nyquist
frequency. Fig. 3shows the proposed servo scheme to
overcome such design limitations. Here, the closed loop con-
sists of continuous-time signals and two groups of discrete
signals, with their different sampling rates indicated by the
dashed (slower) and dotted (faster) signal flows.
dc(t)
+

+//◦˜uL[n]//HL
uc(t)//Pc(s)+//◦yc(t)◦
Ts
yd[n]
−
//
//Pd,L(z)//↓L−
ˆyd[n]//◦+
oo
−
OO
QL(z)
cL[n]MP
dL[n]
ood[n]
oo
IL(z)
uL[n]↑L
ue[n]
ooC(z)
u[n]
oo◦
oo
Fig. 3: The proposed multirate disturbance rejection scheme
The upsampling and interpolation process
u[n]//↑Lue[n]//Interpolator IL(z)//uL[n](8)
generates a fast signal uL[n]sampled at Ts/L(L∈Z+),
as illustrated in Fig. 4(with a ZOH interpolator). The
2 4 68 10 12 14 16
0
0.5
1
1.5{u[n]}
{ue[n]}\{u[n]}
{uL[n]}\{u[n]}
Fig. 4: The upsampling and interpolation process
upsampled signal then passes through a Ts/L-based ZOH
HLand enters the plant.
The beyond-Nyquist disturbance rejection is centered
around a multirate forward-model disturbance observer
(MR-FMDOB) that consists of two fast-sampling transfer
functions QL(z)and Pd,L(z), a downsampling operator, and
a multirate signal processing module MP . In the subsequent
derivations, we show that although ydonly contains informa-
tion sampled at Ts, the inter-sample information in dccan be
fully reconstructed with the multirate algorithm block MP
in Fig. 3, if dc(t)satisfies a disturbance model; and in that
case, cL[n]—the output of QL(z)—can fully remove the effect
of the beyond-Nyquist sampled disturbance at a fast sampling
period of Ts/L.
A. Forward-model Disturbance Observer
Ignore the MP block first. If the sampling time in Fig.
3was actually Ts/Land the downsampling block ↓Lis
removed, then the top part of the block diagram is equivalent
to the structure in Fig. 5. Here, Pd,L(z)is the ZOH equivalent
of the continuous-time plant Pc(s), with a fast sampling
time Ts/L;dL[n]and yL[n]are the Ts/L–sampled disturbance
dL[n] = dc(nTs/L)and plant output, respectively.
dL[n]
+
uL[n]+//◦˜uL[n]//Pd,L(z)+//◦
+

yL[n]//
//Pd,L(z)ˆyL[n]−//◦
−
OO
QL(z)
cL[n]oo
Fig. 5: The proposed forward-model disturbance observer
The disturbance compensation scheme is a forward-model
disturbance observer (FMDOB) that is structurally branched
from internal model control (IMC) [12], a special case of
all-stabilizing Youla-Kucera parameterization [18]. Straight-
forward block-diagram analysis gives that the system in Fig.
5has guaranteed stability if QL(z)and Pd,L(z)are stable. The
inputs and the output satisfy, in Z domain,
YL(z) = Pd,L(z)UL(z)+(1−Pd,L(z)QL(z))DL(z)(9)
where the relationship between uL[n]and the output remains
intact compared to the case without FMDOB. Therefore,
the design of FMDOB is advantageously decoupled from
that of C(z)in regular single-rate servo. With this decou-
pled servo design principle, we can now use the affine
Q-parameterization 1 −Pd,L(z)QL(z)to design QL(z)for
disturbance rejection.
Remark 1: When combined with the baseline feedback
design, Pd,L(z)—if unstable—is stabilized by C(z).
Observe the structure of 1 −Pd,L(z)QL(z). To achieve an
exact rejection of the disturbance at a particular frequency
ωoin (9), it must be that
1−Pd,L(ejωo)QL(ejωo) = 0.(10)
In other words, QL(ejωo) = Pd,L(ejωo)−1so that QL(z)inverts
the dynamics of Pd,L(z)at ωo. Recall that QL(z)must be
stable. Certainly, unless for special minimum-phase plants
with a relative degree of zero, it is not feasible to always
assign an exact full inversion Pd,L(z)−1to Q(z)due to
841

instability and non-properness. The following proposition
achieves a point-wise stable inversion at ω=ωoand highly
regulated loop gain at other frequencies.
Proposition 1: Let π/Ts<Ωo<Lπ/Ts(in rad/s) and ωo=
ΩoTs/L be the frequency of a major disturbance component
beyond the baseline Nyquist frequency π/Tsin Fig. 3. Let ϕ=
phase(Pd,L(ejωo)) be the phase response of the Ts/L–sampled
discrete-time plant at ωo; and assume that Pd,L(ejωo)6=0. Let
QL(z) = Qo(z)(b0+b1z−1)(11)
b0=cosϕ−sinϕcot ωo
|Pd,L(ejωo)|,b1=1
|Pd,L(ejωo)|
sinϕ
sinωo
(12)
Qo(z) = 1−k2
2
(1+z−1)(1−z−1)
1−cosωo(1+k2)z−1+k2z−2(13)
k2=1−tan(Bw,r/2)
1+tan(Bw,r/2)(14)
where Bw,r(in rad) is the 3-dB disturbance-rejection band-
width of Qo(z)centered around ωo. Then 1−Pd,L(z)QL(z)
in (9) equals 0at ωoand has almost unity gain at other
frequencies if Bw,ris small—in other words, the feedback
system in Fig. 5fully rejects all disturbances at ωo, without
major disturbance amplification at other frequencies.
Proof: Qo(z)in (13) is a lattice-based band-pass filter
whose bandwidth Bw,ris related to k2by (14) [19], [20].
At the center frequency ωo, using the fact cosωo= (ejωo+
e−jωo)/2, one can verify that Qo(ejωo) = 1∀k2. Hence
QL(ejωo) = (b0+b1e−jωo).(15)
Equation (10), the condition of exact disturbance rejection,
is then equivalent to
|bo+b1e−jωo|=1
|Pd,L(ejωo)|,(16)
phase(bo+b1e−jωo) = −ϕ(17)
Solving the equation set gives (12), which proves the asserted
disturbance rejection at ωo.
For the last part of the proposition, with the bandpass
nature of Qo(z),|Qo(ejω)|is small outside its passband,
yielding 1 −Pd,L(ejω)QL(ejω)≈1. The approximation sign
here can be made arbitrarily close to equality, by reducing
the bandwidth of Qo(ejω).
Recall (9). Given a target frequency ωo, the achieved
frequency-domain properties of QL(z)yields
YL(ejω)(=Pd,L(ejω)UL(ejω),ω=ωo
≈Pd,L(ejω)UL(ejω) + DL(ejω),ω6=ωo
(18)
The first equality in (18) provides the desired disturbance-
rejection performance. Although the same mathematical rela-
tionship is achieved if one had assigned Qo(z) = 1 in (11), the
latter design is highly sensitive to the practically inevitable
noises in dL[n]. By designing the band-pass Qo(z)in (13),
robustness is added to the algorithm, such that the second
equality in (18) holds, to avoid amplification of DL(ejω)if
ω6=ωo.
B. Multirate Model Based Prediction
In this subsection, design of the multirate prediction block
MP in Fig. 3is provided, to establish the equivalence of
the MR-FMDOB to the fast-rate FMDOB in Fig. 5.
Block diagram analysis gives that the input to QL(z)in Fig.
5is YL(z)−ˆ
YL(z) = Pd,L(z)˜
UL(z) + DL(z)−Pd,L(z)˜
UL(z) =
DL(z)—or equivalently, dL[n]in time domain; while the input
to MP is the slow Ts–sampled d[n]. As a fundamental
limitation in signal reconstruction, if the sampling in d[n] =
dc(nTs)did not contain aliasing, perfect reconstruction of
the Ts/L–sampled dL[n]for a general disturbance signal can
only be achieved if MP contains an upsampler and an
interpolator in the form of an ideal low-pass filter (with DC
gain Land cutoff frequency π/L), which is acausal and not
interpretable using a transfer function [17].
The next result shows that the above fundamental limi-
tation can be overcame if dL[n]satisfies an internal signal
model. For brevity, we present the case with L=2. The
design principle applies analogously to the general case.
Theorem 1: Let L =2, d[n] = dc(nTs), and d2[n],
dL=2[n] = dc(nTs/2). If ∃A(z−1) = 1+a1z−1+a2z−2+
... amz−m(am6=0), m ≥2, such that A(z−1)d2[n] = 0at
steady state,2then d2[n]can be perfectly reconstructed from
the slowly sampled d[n]in the form of
d2[2n] = d[n](19)
d2[2n+1] = W(z−1)d[n](20)
=w0d[n] + w1d[n−1] + · · · +wnwd[n−nw](21)
where nw≥m−1. The minimum-order solution of W(z−1)
is obtained at n∗
w=m−1, in which case the coefficients of
W(z−1)come from the unique solution of the matrix equation
˜
Me1e3... e(2m−1)
|{z }
,M∈R(2m−1)×(2m−1)
×










f1
.
.
.
f(m−1)
w0
.
.
.
wm−1










+












a1
a2
.
.
.
am
0
.
.
.
0












=












0
.
.
.
.
.
.
0












(22)
where ejis the elemental column vector whose entries are
all zero except for the j-th entry, which equals 1; and
˜
M,












1
a1
...
.
.
....1
am
...a1
....
.
.
am











(2m−1)×(m−1)
(23)
2Here, z−1is a one-step delay operator such that z−1u[n] = u[n−1].
842

where the unmarked off-diagonal entries in the top-right and
bottom-left corners are all 0.
Proof: By definition, d2[2n] = dc(2nTs/2) = dc(nTs) =
d[n]. Hence (19) holds. To establish (20) and (21), construct
F(z−1)A(z−1) + z−1W(z−2) = 1 (24)
with
W(z−2) = w0+w1z−2+w2z−4+· · · +wnwz−2nw(25)
F(z−1) = 1+f1z−1+f2z−2+· · · +fnfz−nf(26)
where wnw6=0, fnf6=0, and W(z−2)is obtained by replacing
each z−1in W(z−1)with z−2. As A(z−1)d2[n] = 0 at steady
state, it must be that F(z−1)A(z−1)d2[n]→0, which gives,
after substituting in (24),
(1−z−1W(z−2))d2[n]→0.(27)
For n=2 ˜n+1, this implies that at steady state,
d2[2 ˜n+1] = z−1W(z−2)d2[2 ˜n+1]
=W(z−2)d2[2 ˜n] = w0d2[2 ˜n] + w1d2[2(˜n−1)]+
w2d2[2(˜n−2)] + · · · +wnwd2[2(˜n−nw)].
But by definition d2[2(˜n−i)] = d[˜n−i]. Hence, with a change
of notations, the result simplifies to the asserted (20)-(21).
Consider solving (24), which is a special constrained
Diophantine equation. Matching the coefficients of z−i’s
(i=1,2,...,m+nf), one can obtain a linear m+nf–equation
set with the nf+nw+1 parameters of F(z−1)and W(z−2)as
the unknowns. A solution thus exists if nf+nw+1≥m+nf.
Additionally, matching the order of the highest-order term
on the left side of (24) gives m+nf=1+2nw.Hence the
minimum-order solution is achieved with
n∗
w=m−1,n∗
f=2(m−1)−m+1=m−1.(28)
Under (28), the coefficients of z−l’s l∈ {1,2,...,(2m−1)}
in A(z−1)F(z−1) + z−1W(z−2)are given by
wp+
i+j=2p+1
∑
i,j=0,1,...
aifj: for z−2p−1,p=0,1,...,m−1
i+j=l
∑
i,j=0,1,...
aifj: for z−l,l6=2p+1∀p∈ {0,1,...,m−1}
where f0=1 and a0=1. All the above coefficients must be
zero for (24) to hold. Confining so yields, after some algebra,
the matrix equation (22). The column vectors of Min (22)
are all linearly independent. Hence Mis non-singular and a
unique solution always exists.
Transient response: It is recognized that W(z−1)in (21)
is a finite-impulse-response (FIR) filter. The transient of the
signal-reconstruction process in (21) equals nwdiscrete time
steps, which is usually very fast compared to the transient
of the feedback servo control.
Corollary 1: ∀φ∈Rand ωo∈(π,2π), let the sampling
time of d2[n] = cos(ωon+φ)be Ts/2. Upsampling by L =2is
sufficient to fully reconstruct d2[n]using the slow Ts–sampled
d[n]. The reconstruction formulas are: d2[2n] = d[n]and
d2[2n+1] = 2cosωo−1
2cosωod[n]−1
2cosωo
d[n−1]
(29)
Proof: From the table of Z transforms,
d2[n] = (1−z−1cosωo)cosφ−z−1sin ωosin φ
1−2cosωoz−1+z−2δ[n](30)
and hence A(z−1)d2[n]→0 where
A(z−1) = 1+a1z−1+a2z−2=1−2cosωoz−1+z−2.
Therefore m=2. Apply the minimum-order solution in The-
orem 1. Only one intersample point needs to be reconstructed
in (20) (nw=1). (22) simplifies to


1 1 0
a10 0
a20 1 


f1
w0
w1
+

a1
a2
0
=

0
0
0
(31)
where a1=−2cosωoand a2=1. The solution of (31) is
f1=−a2/a1,w0=−a1+a2/a1,w1=a2
2/a1. Substituting the
result in d2[2n+1] = wod[n] + w1d[n−1]gives the explicit
form of (29).
IV. NUMERICAL VER IFIC ATI ON
Consider a plant Pc(s) = 3.74488 ×109/(s2+565.5s+
319775.2)with an input delay of 10 microseconds. Let
the sampling time be limited at Ts=1/2640 second, and
the baseline discrete-time controller be a PID controller
C(z) = [kp+ki/(z−1) + kd(z−1)/z]/1665 with kp=1/8,
ki=1/20, and kd=3/5.
Vibrations are applied to the plant, above Nyquist fre-
quency. To see the limitation (and danger) of sub-Nyquist
designs, the narrow-band disturbance observer [21] is applied
on top of the PID controller, to enable infinite-gain control at
selective frequencies. Such design provides perfect compen-
sation of sinusoidal signals below Nyquist frequency, and is
termed 1x compensation. Fig. 6presents the corresponding
plant output. When the disturbance occurs at 2376 Hz
(i.e. 1.8ΩN), although the sub-Nyquist servo enhancement
enforces the Ts-sampled output to converge to zero in Fig.
6a, the actual output is significantly amplified. The 3σ(σ
is the standard deviation) value of the Ts/2-sampled output
increased from 15.7115 to 20.9562, yielding more than 130%
of error amplification.
In Table I, a series of disturbances between ΩNand 2ΩN
are applied to the plant. Based on the 3σvalue of the
continuous-time output, the largest servo degradation under
1x compensation occurred when the disturbance frequency
is closest to the Nyquist frequency, where the tracking errors
get amplified by 165.12% compared to the baseline value.
Fig. 7shows the efficiency of the proposed algorithm. As
the disturbance lies in (π/Ts,2π/Ts),L=2 was adopted in
the upsampler of MR-FMDOB. The outputs were signifi-
cantly improved compared to the baseline results. At steady
state, the Ts/2-sampled output is zero. The overall (including
the transient response) 3σvalue of the output was reduced by
843

0 0.02 0.04 0.06 0.08 0.1
−10
−5
0
5
10
Time (sec)
Normalized output
Sampled output (sampling time: Ts)
1x compensation off: 3σ = 17.2544
1x compensation on: 3σ = 0.61329
(a) output sampled at Ts
0.08 0.085 0.09 0.095 0.1
−15
−10
−5
0
5
10
15
Time (sec)
Normalized output
Sampled output (sampling time: Ts/2)
1x compensation off: 3σ = 15.7115
1x compensation on: 3σ = 20.9562
(b) output sampled at Ts/2
Fig. 6: Plant output for the case with disturbance at 1.8ΩN
TABLE I: Servo degradation under different disturbances
disturbance frequency
3σvalue of yc(t)
amplification1x compensation
off on
1.3ΩN14.7606 24.3727 165.12%
1.7ΩN15.2522 21.0052 137.72%
1.9ΩN17.3914 20.9352 120.38%
91.32%—in contrast to the 165.12% amplification in Table I.
Analogous verifications were performed on all cases in the
table, where the proposed algorithm achieved a consistent
performance improvement of over 91% 3σreduction.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (sec)
-8
-6
-4
-2
0
2
4
6
8
Normalized output
Sampled output (sampling time: Ts/2)
MR-FMDOB off: 3< = 14.7717
MR-FMDOB on: 3< = 1.2815
Fig. 7: Performance of MR-FMDOB (disturbance at 1.3ΩN)
V. CONCLUSION
In this paper, the problem of sampled-data regulation
control against structured disturbances beyond Nyquist fre-
quency is addressed. Based on models of the plant and the
disturbance, a multirate forward-model disturbance observer
is proposed to enable full rejection of beyond-Nyquist dis-
turbances. Such capabilities open a pathway to solve various
challenging problems where feedback control is subjected to
slow or limited sensor measurement.
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