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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)

Boston Marriott Copley Place

July 6-8, 2016. Boston, MA, USA

978-1-4673-8682-1/$31.00 ©2016 AACC 839

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

Xejω=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) =

Cej(ΩTsmod 2π),Ω∈[2kπ

Ts,(2k+1)π

Ts]

Cej(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

2cosωod[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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