Feedforward with Acceleration and Snap
using Sampled-Data Differentiator fora Multi-Modal Motion System
Masahiro Mae ∗ Max van Haren ∗∗ Wataru Ohnishi ∗Tom Oomen ∗∗,∗∗∗ Hiroshi Fujimoto ∗∗Department of Electrical Engineering and Information Systems,
Graduate School of Engineering, The University of Tokyo, Japan.
(e-mail: mmae@ieee.org)∗∗
Department of Mechanical Engineering,
Eindhoven University of Technology, The Netherlands.
∗∗∗
Delft Center for Systems and Control,
Delft University of Technology, The Netherlands.

Abstract: Sampled-data control requires both on-sample and intersample performance in highprecision mechatronic systems. The aim is to design a discrete-time linearly parameterized
feedforward controller to improve both on-sample and intersample performance in a multimodal motion system. The continuous-time performance is considered as state compatibility
by a multirate zero-order-hold differentiator. The developed approach enables the linearlyparameterized feedforward controller design for sampled-data systems with physically intuitive
tuning parameters. The performance improvement is validated by comparing the developed
approach with a conventional approach using a backward differentiator for a multi-modal motion
system.
Keywords: feedforward, discrete-time system, sampled-data control, zero-order-hold, multirate
1. INTRODUCTION

Feedforward control is essential for reference tracking in industrial high-precision mechatronic systems such as semiconductor lithography systems (Steinbuch et al., 2021) andhigh-speed scanners (Ito et al., 2019). The feedforward
controllers are usually implemented in digital hardware,and the parameter of the feedforward controller is first
designed from the model of the controlled system, andsecondly tuned by experimental data. For intuitive tuningof the feedforward controller, it is preferable that thefeedforward controller is represented by the parameterswith physical meaning.
The linearly parameterized feedforward control (Lambrechts et al., 2005) has an advantage because the tuning process is physically intuitive. Model inversion based
feedforward controllers such as zero phase error tracking
control (Tomizuka and Sun, 2021) are widely used to improve tracking performance. However, it is time-consuming
to identify the model of the controlled system and hard totune parameters manually.
The feedforward controller design for higher-order motion
systems has a challenge because of the model complexity
and it results in many non-intuitive parameters in thefeedforward controller. Industrial mechatronic systems aremodeled as the dominant rigid mode at a lower frequency
and several flexible modes at a higher frequency due
to limited mechanical stiffness (Gawronski, 2004). The

feedforward controller can be parameterized physicallyintuitive using the modal characteristics.
The sampled-data characteristics should be considered in
the feedforward controller design because of the limitationof the sampling frequency. The feedforward controllers areusually implemented in discrete-time with sampler andzero-order-hold (Chen and Francis, 1995). Several related
studies using the multirate feedforward control (Fujimoto
et al., 2001) are developed to improve intersample performance by compensating for the oscillation of the Nyquist
frequency.
Although several linearly parameterized feedforward control approaches exist, the on-sample performance is mainly
discussed in conventional approaches and the sampleddata characteristics with sampler and zero-order-hold are
not considered. The conventional linearly parameterized
feedforward control is designed by using the backward differentiator (Lambrechts et al., 2005) and it is not compatible with the states of the continuous-time motion system.
The pre-existing state tracking approaches for both onsample and intersample performance (van Zundert et al.,2020) need the model of the controlled system based on
the system identification and they are not linearly parameterized in tuning parameters.
The main contribution of this paper is the linearly parameterized feedforward control approach considering sampleddata characteristics to improve both on-sample and intersample performance in multi-modal motion systems. The
contributions in this paper are the following:

Contribution 1. The multirate zero-order-hold differentiator is developed to design the discrete-time basis functions
Ψ [k] that satisfy state compatibility for the continuoustime reference r(t).
Contribution 2. The linearly parameterized feedforward
considering sampled-data characteristics is designed witha multirate zero-order-hold differentiator and both onsample and intersample performance improvement is experimentally validated in a multi-modal motion system.
The outline is as follows. In Section 2, the problem thatis considered in this paper is formulated. In Section 3,
the design method of the feedforward controller using
the multirate zero-order-hold differentiator is developed,
constituting Contribution 1. In Section 4, the advantageof the approach is demonstrated in the experiment with amulti-modal motion system, constituting Contribution 2.
In Section 5, conclusions are presented.
2. PROBLEM FORMULATIONIn this section, the problem to improve continuous-timetracking performance is formulated. First, the requirements in this paper are described. Second, the reference
tracking problem is defined in intersample performance.
Third, the low-order feedforward control approach is investigated for reference tracking in a multi-modal motion system. Fourth, the low-order feedforward control approachis implemented in discrete-time. Finally, the problem inthe conventional approach is described.
2.1 Problem description
In this paper, the feedforward controller is designed withrespect to the following requirements:
Requirement 1. The feedforward controller is linearly parameterized with physical parameters for intuitive tuning.
Requirement 2. The sampled-data characteristics withsampler and zero-order-hold are considered in the feedforward controller design.
Requirement 3. The designed feedforward controller canbe applied to multi-modal motion systems.
To improve both on-sample and intersample performance,
the main problem in the linearly parameterized feedforward controller is the discrete-time basis function designthat is compatible with the continuous-time reference r(t)considering sampled-data characteristics.
2.2 Reference tracking for intersample performance
The considered tracking control configuration is shown in
Fig. 1, with reference r(t) ∈ R, control input u(t) ∈ R,
and output y(t) ∈ R. The continuous-time linear timeinvariant system G is controlled by the sampled-data
controller that consists of feedforward controller F (θ),
feedback controller K, sampler S, and zero-order-hold H,
where sampler and zero-order-hold are defined as follows:
Definition 1. (Sampler). The sampler S with samplingtime Ts is defined as
S : r(t) 7→ r[k], r[k] = r(kTs ).
(1)

+
F (θ)
r(t)

S

e(t)

−

uf f [k]

y(t)

+
r[k]

uf b [k]

e[k]

−

K

+

+ u[k]

u(t)

H

y[k]

G

S

Fig. 1. Tracking control diagram. The continuous-timesystem G is controlled by the feedforward controller
F (θ) and the discrete-time feedback controller K withsampler S and zero-order-hold H. The objective is tominimize the continuous-time error e(t). The solid anddotted lines denote the continuous-time and discretetime signals, respectively.
Definition 2. (Zero-order-hold). The zero-order-hold H
with sampling time Ts is defined as
H : u[k] 7→ u(t), u(kTs + τ ) = u[k], τ = [0, Ts ). (2)The control objective in this paper is to minimize thecontinuous-time error e(t). Traditionally, the conventional
discrete-time controller only focuses on the on-sample performance with the discrete-time error e[k]. To improve thecontinuous-time error e(t), both on-sample and intersample performance should be considered.
2.3 Low-order feedforward for multi-modal motion system
The goal of the feedforward controller design is to extend
the rigid mode behavior over a frequency range as highas possible. Note that the controlled system and thecontrollers are assumed to be the continuous-time system
only in this subsection.
Industrial mechatronic systems consist of the dominant
rigid mode at a lower frequency and several flexible
modes at a higher frequency due to limited mechanicalstiffness. The continuous-time single-input single-output
multi-modal motion system is defined as
nf l
X
1
ki
Gc (s) =+,(3)
ms2 i=1 m(s2 + 2ζi ωi s + ωi2 )where m is the total mass of the system, nf l is thenumber of the flexible modes. The resonance frequency,
the damping coefficient, and the mode gain at the ith modeare ωi , ζi , and ki ∈ {−1, 1}, respectively.
To compensate for both rigid mode and flexible modes, thetraditional acceleration feedforward is extended with theadditional snap feedforward (Boerlage et al., 2004), andthe ideal feedforward controller F ∗ (s) is defined as
F ∗ (s) = ms2 + D∗ (s)s4 ,(4)∗where D (s) is the coefficient of the snap feedforward.
The objective of the feedforward F ∗ (s) is to minimize theclosed-loop error given bye(s) = S(s)r(s) − S(s)Gc (s)F ∗ (s)r(s),(5)
where S(s) denotes the sensitivity function and is definedasS(s) = (1 + Gc (s)K(s))−1 .
(6)
∗
−1
∗
It results in F (s) = Gc (s) and D (s) is given by11D∗ (s) = G−1(7)c (s) 4 − m 2 .
ss

F (θ)r(t)ξ2ξ4

Ψa [k]
Ψs [k]

θar(t)uf f [k]

+

S

+

θs

Fig. 2. Linearly parameterized feedforward with acceleration and snap.

r[k]

z



1 − z −1Ts2Ψa [k]z2



1 − z −1Ts4Ψs [k]

Fig. 3. Basis functions of acceleration and snap with abackward differentiator. The solid and dotted lines
denote the continuous-time and discrete-time signals,
respectively.

Assuming the reference trajectory in the mechatronicsystems mainly contains the low-frequency components,
the low-frequency contribution of the snap feedforward isgiven by


1
1
∗
−1
D = lim D (s) = lim Gc (s) 4 − m 2
s→0
s→0
s
s
Pnf l Q−m i=1ki j̸=i ωj2
Qnf l 2=.
(8)i=1 ωi
Finally, the low-order feedforward controller with acceleration and snap is given by
F (s) = ms2 + Ds4 ,(9)where m and D are the tuning parameters in accelerationand snap.

where z = esTs is the shift operator defined as z −n r[k]=nr[k − n]. It is assumed that n is even and the z 2 denotes the phase compensation. When n is odd, the phase
1compensation consists of the half sample shift z 2 and theapproximation of averaging the current and previous value
is considered (Lambrechts et al., 2005).

2.4 Feedforward implementation with basis functions

Finally, the discrete-time feedforward input uf f [k] withbackward differentiator is given by
uf f [k] = Ψbd [k]θ,(14)

The feedforward controller design method is discussed
above in continuous-time. However, the controllers are
implemented to digital hardware in discrete-time. As a result, the continuous-time differentiator s used in the feedforward controller is conventionally replaced by ξ which
consists of the discrete-time differentiator and sampler S.
The linearly parameterized feedforward with accelerationand snap is shown in Fig. 2. The feedforward controller
F (θ) from the continuous-time reference r(t) to design thediscrete-time feedforward input uf f [k] is defined as 
 2 4  θa
F (θ) = ξ ξ
,
(10)
θs
T

where θ = [θa θs ] are the tuning parameters in acceleration and snap.
Finally, the discrete-time feedforward input uf f [k] withacceleration and snap is given by θ
uf f [k] = F (θ)r(t) = Ψ [k]θ = [Ψa [k] Ψs [k]] a , (11)θs
 2 4
where Ψ [k] = [Ψa [k] Ψs [k]] = ξ ξ r(t) are the discretetime basis functions that are compatible with the acceleration and snap of the continuous-time reference r(t),respectively.
2.5 Feedforward with acceleration and snap using
conventional backward differentiator
In the conventional approach (Lambrechts et al., 2005),
the discrete-time basis functions are designed by thecontinuous-time reference r(t) and the backward differentiator that is given bynn
1 − z −1nz 2 S,
(12)
ξbd =
Ts

The basis function design in acceleration and snap withbackward differentiator is shown in Fig. 3 and the basis
functions are given by 2 4
ξbd r(t)Ψbd [k] = ξbd
"
#


4
−1 2
1 − z −1
1−z
2
=
zr[k]
z r[k] ,(13)
TsTs
where r[k] = Sr(t).

Twhere θ = [θa θs ] is the tuning parameter.
Although the on-sample performance is mainly discussed
in the conventional approach using the backward differentiator, the sampled-data characteristics with sampler andzero-order-hold are not considered.
3. LINEARLY PARAMETERIZED FEEDFORWARD
CONTROLLER DESIGN USING MULTIRATE
ZERO-ORDER-HOLD DIFFERENTIATOR
In this section, the linearly parameterized feedforward
controller design method considering sampled-data characteristics is presented. The improvement of both on-sample
and intersample performance is based on the state compatibility in a sampled-data system with zero-order-holdand integrators. The basis functions are designed by themultirate zero-order-hold differentiator. The approach isapplied to the low-order feedforward controller design withacceleration and snap for multi-modal motion systems. It
results in Contribution 1.
3.1 State compatibility of differentiator in sampled-datasystem with zero-order-hold and integrators
The continuous-time differentiator s used in the feedforward controller should be replaced by the sampled-data
differentiator ξ defined as follows:
Definition 3. (Sampled-data differentiator). The nth order sampled-data differentiator ξ n with sampling time Ts
is the conversion from the continuous-time signal r(t) tothe discrete-time signal Ψn [k] that is compatible with thenth order derivative of r(t) and defined asΨn [k] = ξ n r(t).
(15)

In the n samples lifted system, the exact state tracking
can be achieved in every n samples using such as aminimum-time dead-beat control (Goodwin et al., 2000)
and multirate feedforward control (Fujimoto et al., 2001).
In such cases, the states in every n samples are given by
the multirate sampler defined as follows:
Definition 4. (Multirate sampler). The multirate sampler
Sn in every n samples with sampling time Ts is defined as
Sn : r(t) 7→ r[in ], r[in ] = r(knTs ).
(16)
The state-space representation of the continuous-time nthorder integrator in the controllable canonical form is givenby
0 1 0 0 0
.
. .
 n

 
 0 . . . . 0 .. 1
Anc bncs.
= Hnc =
= , (17)cnc 0
 0 0 .. 1 0 s0 0 0 0 1
1 0 ··· 0 0
where Anc ∈ Rn×n , bnc ∈ Rn×1 , and cnc ∈ R1×n . To
improve both on-sample and intersample performance insampled-data systems with zero-order-hold and integrators, the basis functions used in the linearly parameterized
feedforward controller should satisfy the state compatibility defined as follows:
Definition 5. (State compatibility). The discrete-time signal Ψn [k] compatible with the nth order derivative signalof the continuous-time signal r(t) satisfies state compatibility if the output through the system consisted of thecontinuous-time (n − m)th order integrator H(n−m)c andzero-order-hold H is equal to the continuous-time mthorder derivative signal of r(t) in every n samples sampledby multirate sampler Sn and defined asdm
Sn m r(t) = Sn H(n−m)c HΨn [k],(18)dt
where m = 0, 1, . . . , n − 1.
3.2 Multirate zero-order-hold differentiator forintersample performance
To improve the intersample performance in the discretetime system, the states of the reference trajectory are
considered. The multirate zero-order-hold differentiator isdesigned by the inverse of the continuous-time integrator
discretized by sampler and zero-order-hold to satisfy thestate compatibility. In this paper, it is assumed that thecontinuous-time reference r(t) is class C n−1 and differentiable at least n − 1 times.
To satisfy the n states compatibility in every n samples,the lifted signal is considered using the lifted operator
defined as follows:
Definition 6. (Lifting operator). The lifting operator Ln
in every n samples is defined as
Ln : u[k] 7→ u[in ],
(19)
where
T
u[in ] = [u[nin ] u[nin + 1] · · · u[nin + (n − 1)]] ∈ Rn .
(20)
The n sample lifted system is defined as follows:

Definition 7. (Lifted system). Consider a discrete-timezsystem Hd = Cd (zI − Ad )−1 Bd + Dd . The relation
between the input and the output in the n sample lifted
system of Hd is given by
y[in ] = Ln y[k] = (Ln Hd L−1(21)n )(Ln u[k]) = H d u[in ],whereTy[in ] = [y[nin ] y[nin + 1] · · · y[nin + (n − 1)]] ∈ Rn ,(22)
and the lifted system H d is defined as
Ad B dzn−1
H d = Ln Hd Ln =C d Dd


n−1
n
Ad
Ad Bd
An−2
Bd · · · Ad Bd Bd
d
 Cd
Dd
O
··· ··· O 



.. 
..
 Cd Ad

.
C
B
D
.
d d
d


=
.
..
..
.
.
.
.
.
.
.
.

.
.
.
.
.
. 




 Cd An−2 Cd An−3 Bd Cd An−4 Bd . . . Dd O ddd
Cd An−1
Cd An−2
Bd Cd An−3
B d · · · C d B d Dd
d
d
d
(23)
Considering the states in discrete-time, the nth orderintegrator that is discretized by sampler and zero-orderhold is given byAnd bndz
Hnd = SHnc H =
cnd 0
 A T

−1 Anc Tsnc se
Anc (e− I)bnc
=
.
(24)
cnc
0
To design the inverse of the nth order integrator discretizedby sampler and zero-order-hold, the n sample lifted system
is given by
And B nd
zn
−1
,
(25)
H nd = Ln Hnd Ln =C nd D nd
and in state-space representation defined asxn [in + 1] = And xn [in ] + B nd u[in ]y[in ] = C nd xn [in ] + D nd u[in ]whereT
xn [in ] = [x0 [in ] x1 [in ] · · · xn−1 [in ]] ∈ Rn .

(26)
(27)
(28)

Satisfying the state compatibility, the relationship between
the reference and the states is given by
rn [in ] = xn [in ],
(29)
where
T

d
dn−1
rn [in ] = Sn 1
· · · n−1 r(t)dtdtT
= [r0 [in ] r1 [in ] · · · rn−1 [in ]] ∈ Rn .
(30)
From the discussions above, the multirate zero-order-holddifferentiator is given as follows:
Theorem 1. (Multirate zero-order-hold differentiator.)
From (26) and (29), considering the inverse of thecontinuous-time nth order integrator discretized by sampler and zero-order-hold using the multirate feedforward
control (Fujimoto et al., 2001), the nth order multirate
zero-order-hold differentiator that satisfies the state compatibility is given by

r(t)

d2
dt2
d3
dt3

S4

r2 [i2 ]r4 [i4 ]2
B −1
2d (z I2 − A2d )

L−1
2

4
B −1
4d (z I4 − A4d )

L−1
4

u

Ψa [k]yΨs [k]

Fig. 4. Basis functions of acceleration and snap using
multirate zero-order-hold differentiator. The solid line
denotes the continuous-time signal. The dotted, highfrequency dashed and low-frequency dashed lines denote the discrete-time signal sampled by Ts , 2Ts , and
4Ts , respectively.

T
d
dn−1
n
−1 −1 n
ξmr = Ln B nd (z In − And )Sn 1· · · n−1 .
dt
dt
(31)
Proof 1. See Definition 5 and Fujimoto et al. (2001).

□

3.3 Feedforward with acceleration and snap using multirate
zero-order-hold differentiator
To design the feedforward controller with acceleration andsnap, the lifted systems of the double integrator and the
4th integrator discretized by zero-order-hold are given by
A2d B 2dz2
−1
H 2d = L2 SH2c HL2 =
,
(32)
C D
 2d 2d 
A4d B 4dz4
H 4d = L4 SH4c HL−1
,
(33)
4 = C
4d D 4dwhere the continuous-time double integrator H2c and 4th
integrator H4c are represented in controllable canonicalform, respectively.
The basis function design using multirate zero-order-holddifferentiator is shown in Fig. 4 and the basis functions are
given by

 2
4
ξmr
r(t)
Ψmr [k] = ξmr −1 −1 2
T
L2 B 2d (z I2 − A2d )r2 [i2 ]= −1 −1 4.
(34)
L4 B 4d (z I4 − A4d )r4 [i4 ]Finally, the discrete-time feedforward input uf f [k] using
the multirate zero-order-hold differentiator is given by
uf f [k] = Ψmr [k]θ,(35)
T

where θ = [θa θs ] is the tuning parameter.
4. APPLICATION IN MULTI-MODAL MOTIONSYSTEM
In this section, the approach in Section 3 is applied toa multi-modal motion system. The experimental results
demonstrate the performance improvement in both rigid
and flexible modes. It results in Contribution 2.
4.1 Setup
The experimental setup of the two-inertia system is shownin Fig. 5. The system is modeled as the multi-modal
representation and given by

Fig. 5. Experimental setup of a two-inertia system connected via a flexible shaft. The motor on the left side
is used as an input u and the encoder on the rightside is used as an output y, respectively.
|G| [dB]

S2

0
−50
−100
−150
0

∠G [deg]

d
dt

−180
−360
100

101

102
f [Hz]

103

Fig. 6. Bode diagram of experimental setup: frequency
response data ( ), continuous-time model Gc ( ),and discrete-time model Gd ( ). Nyquist frequency
is shown in a black dotted line ( ).
Gc (s) = Grb (s) + Gf l (s)
(36)
1
k
=
+
,
(37)
ms2
m(s2 + 2ζωs + ω 2 )where m = 0.0004, k = −1, ζ = 0.01, and ω = 2π × 54
rad/s.
The frequency response data, the continuous-time model
Gc , and the discrete-time model Gd are shown in Fig. 6.
Note that these models are not directly used for thefeedforward controller design but only used for the physicalanalysis.
4.2 Conditions
The continuous-time reference r(t) is the 4th order polynomial trajectory shown in Fig. 7. The sampling time ofthe discrete-time controller is Ts = 5 ms. The continuoustime output y(t) is obtained by higher sampling frequency
in every 250 µs only for evaluation of the continuous-timeerror e(t).
The approach using the multirate zero-order-hold differentiator ξmr is compared to that using the backward differentiator ξbd . The feedforward controller with accelerationand snap is used in the experiment. The same viscous fricdr(t) istion compensation with the basis function ṙ[k] = S dt
used in each method. The tuning parameters are optimizedby the norm-optimal iterative learning control with several
iterative experiments (Bolder et al., 2014).

5. CONCLUSION

r(t) [rad]20r̈(t) [rad/s2 ]30

5The low-order feedforward control approach considering
the sampled-data characteristics is developed to improve
both on-sample and intersample performance for reference
tracking in multi-modal motion systems. The feedforward
controller is linearly parameterized and the basis functions
are designed using the multirate zero-order-hold differentiator for the state compatibility to the continuous-timereference. Application to the multi-modal motion systemdemonstrates a significant improvement in tracking performance compared to the conventional approach in theexperiment. Ongoing research focuses on learning the tuning parameters from the experimental data, rational basis functions considering the sampled-data characteristics,
and extension to the multi-input multi-output systems.

10
0
×103

0
−5

˙˙˙˙
r (t) [rad/s4 ]×1082

ACKNOWLEDGEMENTS

0
−2
0

0.2

0.4

0.6

0.8

1

t [s]

REFERENCES

Fig. 7. Continuous-time 4 order polynomial trajectoryreference: position r(t), acceleration r̈(t), and snap
˙˙˙˙r (t).
the(t) [rad]

×10−1
4
2
0
−2
−4
0

0.2

0.4

This research is granted from JSPS Overseas Challenge
Program for Young Researchers.

0.6

0.8

1

t [s]
Fig. 8. Tracking error e(t) in experiment with Ψ =[ṙ, Ψa , Ψs ]: using backward differentiator ( ) andmultirate zero-order-hold differentiator ( ). The
feedforward using the multirate zero-order-hold differentiator outperforms that using the backward differentiator in both rigid and flexible modes.

4.3 Experimental validation
The continuous-time error e(t) in the experimental resultis shown in Fig. 8. The tracking performance using themultirate zero-order-hold differentiator is improved in two
points compared to that using the backward differentiator.
First, the acceleration feedforward compensates for rigid
dynamics correctly and the error during acceleration anddeacceleration periods of the reference becomes smaller.
Second, the snap feedforward compensates flexible dynamics correctly and the error because of the mechanical resonance becomes smaller. The results demonstrate that thefeedforward using the multirate zero-order-hold differentiator outperforms that using the backward differentiatorin both rigid and flexible modes.

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