Multirate feedforward control with mode decomposition for intersample performance inmultivariable motion systems
Masahiro Maea,∗, Wataru Ohnishia , Hiroshi Fujimotoa
a The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan

Abstract
Multirate feedforward control enables perfect tracking control for the desired state trajectory at every sample as the same numberof the model order. The aim of this paper is the comparison of perfect tracking control approaches for intersample performance inmulti-modal motion systems. The multirate feedforward control has a trade-off between the number of states for perfect tracking
control and the reference sampling frequency. To balance the trade-off, the states for the perfect tracking control can be selected by
the mode decomposition. Intersample performance of each approach in a multi-modal motion system is compared in both frequency
domain and time domain.
Keywords: Multirate feedforward control; Perfect tracking control; Zero-order-hold; Mode decomposition; Intersample
performance

1. Introduction

frequency. To balance the trade-off, the multirate feedforward
control approaches based on modal form with additive decomFeedforward control based on exact model inversion enposition [12, 13] and multiplicative decomposition [14] are preables perfect tracking control [1] for the model of the consented. Both approaches select the states for perfect tracking
trolled system. The quality of the feedforward controller dicontrol and balance the trade-off to improve intersample perrectly results in tracking performance in high-precision mechaformance. Note that the concept of perfect tracking control istronic systems such as wafer scanners [2], wire bonders [3], anddefined in the sampled-data controlled systems as “the output
ball-screw-driven stages [4]. In industrial applications, the sysperfectly tracks the reference with zero tracking error at every
tem is controlled in discrete time but the tracking performancesampling point” [1]. The concept of perfect tracking control
should be improved in continuous time.
is in discrete time, and it is distinguished from the concept of
The exact model inversion has a challenge when the model
perfect control in continuous time [15].
has nonminimum-phase zeros such as intrinsic and discretizaThere are other finite sample preview feedforward control
tion zeros [5]. Several approximated inverse approaches inapproaches [16, 17, 18] based on minimizing the two-norm ofclude zero-phase-error tracking control (ZPETC) [1], nonminimum- the tracking error with optimization or least square calculation.
phase zeros ignore (NPZI) [6], and zero-magnitude-error trackThese approaches can deal with constraints using the reduning controller (ZMETC) method [7], and comparisons are prodancy of the preview samples. However, these concepts are notvided in [8, 9]. These approaches cannot provide exact onthe same as the exact model inversion approaches such as the
sample tracking due to approximation. The single-rate stablesingle-rate stable inversion and the multirate feedforward coninversion approach [9] generates the noncausal bounded feedtrol which provide exact on-sample tracking for the model, andforward input for the model with nonminimum-phase zeros andthe number of preview samples is normally larger than that ofprovides perfect output tracking for every sample. However, it
the exact model inversion approaches.
cannot compensate for the zeros around −1 of the discrete-time
Although several approaches are available to design the permodel that cause the oscillating feedforward input and deterifect tracking controller, the choice of the feedforward controller
orate intersample performance when the relative degree of thecan be arbitrarily and there is no comparison in terms of intercontinuous-time model is 2 or more [5].
sample performance for perfect tracking controllers. The aim
To improve intersample performance, the multirate feedof this paper is the analysis of pre-existing perfect tracking conforward control [10, 11] is presented. The multirate feedfortrollers in both frequency domain and time domain and provides
ward control provides perfect n states tracking for every n samthe guideline to design the feedforward controller to improveple and prevents intersample oscillation. There is a trade-off
intersample performance. The main contributions of this paperin the multirate feedforward control between the number ofare as follows.
states for perfect tracking control and the reference sampling
Contribution 1. Perfect tracking control approaches are described focusing on improving intersample performance in multi∗ Corresponding author
modal motion systems.
Email address: mmae@ieee.org (Masahiro Mae)
Preprint submitted to Control Engineering Practice

r(t)

S

r[k]

u[k]
F

H

u(t)

y(t)
G

z

The discrete-time system Gd = ( Ad , Bd , Cd , O) = SGc H isgiven by

e(t)

− +

x[k + 1] = Ad x[k] + Bd u[k],y[k] = Cd x[k].

Figure 1: Block diagram of tracking control. The continuous-time system G iscontrolled by the discrete-time controller F with sampler S and zero-order-hold
H. The objective is to minimize the continuous-time error e(t).

(6)
(7)

The control objective considered in this paper is to minimize the continuous-time error e(t) that includes both on-sample
and intersample performance for the continuous-time referencer(t) that is assumed to be known in advance.

Contribution 2. Intersample performance of each approach isevaluated in both frequency domain and time domain.

2.2. Single-rate feedforward control based on discrete-time model
inversion
The one-sample forward shifted system G̃d of Gd from u[k]to y[k + 1] is given by

The theory is described in a general multi-input multi-output(MIMO) system and the verification is conducted in a singleinput single-output (SISO) system.
The outline is as follows. In Section 2, the tracking problem for intersample performance is formulated. In Section 3,
the desired state trajectory generation method is presented for
SISO and MIMO systems. In Section 4, the multirate feedforward controller design methods with additive and multiplicative decomposition are presented. In Section 5, the intersample
performance is validated in a multi-modal motion system. In
Section 6, conclusions are presented.

x[k + 1] = Ad x[k] + Bd u[k],(8)y[k + 1] = Cd Ad x[k] + Cd Bd u[k].

(9)For the system H = ( A, B, C, D) with nonsingular D, the inverse system H−1 is generally defined as"#
A − B D−1 C B D−1
−1
H =
.
(10)
− D−1 C
D−1
By inverting G̃d , the input u generated by the single-rate feedforward controller is given by

2. Problem formulation
In this section, the problem to improve continuous-time tracking performance is formulated. The perfect tracking control
methods based on the single-rate and multirate feedforward aredescribed. From the trade-off of these two approaches, the requirements of the optimal perfect tracking controller design arepresented.
2.1. Intersample performance in sampled-data control

u[k] = G̃−1
d r[k + 1],(11)

where the single-rate feedforward controller G̃−1d is given by#"
Ad − Bd (Cd Bd )−1 Cd Ad Bd (Cd Bd )−1
.
(12)
G̃−1
=
d
−(Cd Bd )−1 Cd Ad(Cd Bd )−1
When G̃−1d has unstable poles, it can be decomposed as
"
# "
#"
# " #
x s [k + 1]
A
O x s [k]B= s+ s r[k + 1],
xu [k + 1]
O Au xu [k]Buhi " x s [k]#u[k] = C s Cu
+ Dr[k + 1],
xu [k](13)

The considered tracking control configuration is shown in
Figure 1, with input u ∈ Rnu , output y ∈ Rny , reference r ∈ Rny ,(14)
and error e ∈ Rny . In this paper, the system is assumed towhere |λ( A s )| ≤ 1 and |λ(Au )| > 1. The bounded feedforward
be square as nu = ny = m. The m-input m-output nth orderscontinuous-time linear time-invariant system Gc = ( Ac , Bc , Cc , O) input u is given by
is given byu[k] = C s x s [k] + Cu xu [k] + Dr[k + 1]
(15)
ẋ(t) = Ac x(t) + Bc u(t),(1)y(t) = Cc x(t),(2)

where x s follows from solving
x s [k + 1] = A s x s [k] + B s r[k + 1], x s [−∞] = 0
forward in time and xu follows from solving

where Ac ∈ Rn×n , Bc ∈ Rn×m , and Cc ∈ Rm×n . The discretetime system Hd of the continuous-time system Hc discretizedby sampler S and zero-order-hold H in sampling time δ is generally defined as
#
"
Ac Bc
s
,
(3)
Hc =
Cc Dc
"
# " Aδ
#
Ac δ
Ad Bd
e c
A−1
− I)Bczc (eHd = SHc H =
=
,
Cd Dd
Cc
Dc
(4)
x[k] = x(kδ).

(16)

xu [k + 1] = Au xu [k] + Bu r[k + 1], xu [∞] = 0(17)backward in time [9]. This stable inversion approach is basedon that the unstable poles |λ( Au )| > 1 forward in time are stable
poles |λ( Au )−1 | < 1 backward in time. The generated feedforward input u provides perfect output tracking for every sample.
Note that although the feedforward input generated by thesingle-rate stable inversion approach is bounded, the oscillating
poles around λ = −1 cannot be compensated. The oscillating feedforward input can deteriorate intersample performance.
The single-rate feedforward controller has unstable or oscillating poles when the relative degree of the continuous-time modelis 2 or more as Euler-Frobenius polynomials [5].

(5)2sample, the sampling time of the desired state trajectory is Nδ,and the higher N is, the lower the reference sampling frequency
1/Nδ is.

2.3. Multirate feedforward control for full-state tracking
To compensate for oscillating poles of the feedforward controller due to discretization, multirate feedforward control [10]based on perfect state tracking is presented.
z
The n samples lifted system Hd of Hd = ( Ad , Bd , Cd , Dd )is generally defined as
"
#
Ad Bd
zn
−1
Hd = Ln Hd Ln =C d Dd


n
n−1
A
Ad Bd
An−2
· · · Ad Bd Bd 

d Bd

 C d
Dd
O
···
···
O d.. 
..
.
 Cd Ad
Cd Bd
Dd. 
= 
..
..
.. 
..
..
..

.
.
.
.
.
. 


 C An−2 C An−3 B C An−4 B . . .

D
O
d d
d
d d
d
d
 d d

n−1
n−2
n−3
Cd Ad
Cd Ad Bd Cd Ad Bd · · · Cd Bd Dd
(18)
h
iT
u[in ] = Ln u[k] = u[nin ] · · · u[nin + (n − 1)]hiT= u1 [nin ] · · · um [nin ] · · · um [nin + (n − 1)] ∈ R(m×n) ,
(19)
h
iT
y[in ] = Ln y[k] = y[nin ] · · · y[nin + (n − 1)]hiT= y1 [nin ] · · · ym [nin ] · · · ym [nin + (n − 1)] ∈ R(m×n) ,(20)

2.4. Problem description
From these discussions, the optimal perfect tracking controller should be designed by considering the following requirements.
Requirement 1. Oscillating poles of the feedforward controller
due to discretization is compensated by state tracking.
Requirement 2. States for perfect tracking control are selected
to make reference sampling frequency enough high.
The state tracking can be provided by multirate feedforward
control and the states can be selected based on the mode decomposition. In this paper, two kinds of multirate feedforward
controllers with mode selection in additive decomposition [12,13] and multiplicative decomposition [14] are described and intersample performance is compared with pre-existing perfect
tracking control approaches.
In this paper, square systems that have the same numberof inputs and outputs are assumed. For applications in industrial mechatronic systems, it is usual to design a static decoupling controller by coordinate transformation and make the controlled system square and statically decoupled in the rigid-bodymode. The over-actuated non-square systems are not assumed
in this paper because the exact and causal inversion can beachieved by the dynamic squaring-down approach [21].
Note that the controller design is addressed in both continuous and discrete time to deal with the sampled-data system.
Basically, the desired state trajectory generation is conductedin continuous time and the multirate feedforward controller design is conducted in discrete time.

where u[in ] and y[in ] are column vectors, and Ln is n samples
lifting operator [19].
The N (≤ n) samples lifted system of Gd is given by
"
#
Ad Bd
zN
Gd = LN Gd L−1
=
.
(21)
N
C d DdNote that the number of lifting samples is N = n in SISO systems but it is not the case in MIMO systems, see [20]. The
desired state trajectory of Gd is given by the multirate sampler
for every N sample SN that is defined asx̂[iN ] = SN x̂(t) = x̂(iN Nδ),

3. Desired state trajectory generation
In this section, the desired state trajectory methods are presented for the SISO and MIMO systems. For the SISO system,
the desired state trajectory is generated in controllable canonical form. For the MIMO system, the desired state trajectory is
generated by the state transformation using the singular value
decomposition of the input matrix. The bounded desired state
trajectory is generated by the stable and unstable decomposition
and the non-causal convolution with time axis reversal. This paper substantially extends the preliminary result in [11, 22] withgeneralization and theoretical proof in MIMO systems.

(22)

where x̂(t) is the desired state trajectory in continuous time. By
inverting the state equation of Gd , the input u generated by themultirate feedforward controller is given by


−1
−1
u[k] = L−1
N Bd x̂[iN + 1] − Bd Ad x̂[iN ]
−1
−N
= L−1
Ad ) x̂[iN + 1],
N Bd (I − z(23)where z is shift operator in sampling time δ. The generated
feedforward input u provides perfect state tracking for every N
sample and improves intersample performance.
Note that the desired state trajectory x̂ is given by the reference and its derivatives in continuous time for the model without zeros in controllable canonical form. When the model has
zeros, the desired state trajectory generation method that is described in the next section is used. Although the multirate feedforward controller provides perfect state tracking for every N

3.1. Desired state trajectory generation for SISO system
The single-input single-output continuous-time linear timeinvariant nth order system is given by
Gc (s) =3

B(s)bm sm + · · · + b1 s + b0
= n,A(s) s + an−1 sn−1 + · · · + a1 s + a0(24)

where n > m and b0 , 0. Gc in controllable canonical formsGc,cc f = (Ac,cc f , bc,cc f , cc,cc f , 0) is given byẋcc f (t) = Ac,cc f xcc f (t) + bc,cc f u(t),−1
The state transformation matrix Tcc
f from the system in cons

trollable canonical form Gcc f to the system Gc = ( Ac , bc , cc , 0)with the states x is given by(25)

y(t) = cc,cc f xcc f (t),

(26)

where

"

Ac,cc f
cc,cc fh−1
Tcc
f = Bc 0
# 
bc,cc f=  00 −a0

b0

1
..
.
···
···

0

..

.
0
···
bm

1
−an−1
0


0 

.. 
. 
0  .

1 

0

(27)

···


 a1i  ...
n−1
Ac Bc an−1
1

···
.
..
..

an−1
.
..

.


1 



 ,


O

(39)

where

The filter for the state trajectory generation is given byhiβ(t) = L−1 B(s)−1 ,(28)

Gc,cc f = T (Gc , Tcc f ),(40)

xcc f (t) = Tcc f x(t).

(41)

3.2. Desired state trajectory generation for MIMO system
The m-input m-output nth order continuous-time linear times
invariant system Gc = ( Ac , Bc , Cc , O) is given by

where L[·] is the unilateral Laplace transform. The desired state
trajectory in the controllable canonical form is given by
Z tx̂cc f (t) =β(t − τ)rn (τ)dτ,(29)ẋ(t) = Ac x(t) + Bc u(t),(42)y(t) = Cc x(t),

(43)

0

where Ac ∈ Rn×n , Bc ∈ Rn×m , and Cc ∈ Rm×n . Singular Value
Decomposition (SVD) of Bc is given by

where
h
iT
x̂cc f (t) = x̂cc f,0 (t) · · · x̂cc f,n−1 (t) ,
"
#T
dn−1
rn (t) = 1 · · ·
r(t).
dtn−1

(30)

Bc = UΣV H ,(31)

where U ∈ Rn×n and V ∈ Rm×m are unitary matrices so thatU−1 = UH and V −1 = V H . The elements of Σ ∈ Rn×m are given
by
" #
∆
Σ=
,
(45)
O

When B(s)−1 has unstable poles, it can be decomposed as−1
B(s)−1 = B−1
s (s) + Bu (s),(32)

where all poles p s ∈ C of B−1
s (s) are Re(p s ) ≤ 0 and all polespu ∈ C of B−1
u (s) are Re(pu ) > 0. The filters of stable andunstable parts for the state trajectory generation are given byhiβ s (t) = L−1 B−1(33)s (s) ,hi
βu (t) = L−1 B−1(34)u (−s) .

∆ = diag(σi )(i = 1 · · · m ∈ N),(46)

where σi (i = 1 · · · m ∈ N) are the singular values of B.
s
The system G svd = (A svd , B svd , C svd , O) = T (Gc , UH ) withH
the states x svd = U x is given by#  Om×m ∆V H  " Wu (s) x svd (s)(47)=  O(n−m)×m  , Wl (s) O(n−m)×m 
u(s)


C svd
Om×m
y(s)

The stable and unstable parts of the desired state trajectory are
given byZ tx̂ s (t) =β s (t − τ)rn (τ)dτ,
(35)
Z−∞
∞
x̂u (t) =βu (t − τ)rn (τ)dτ,
(36)

 
where u(s) = L [u(t)], x(s) = L [x(t)], y(s) = L y(t) , Wu (s) ∈Rm×n , Wl (s) ∈ R(n−m)×n , and the Rosenbrock system matrix
Π svd (s) is given by"#  Wu (s)∆V H 

A svd − sI B svdΠ svd (s) ==  Wl (s) O(n−m)×m  .
C svd
O


C svd
Om×m
(48)

t

and the bounded desired state trajectory in controllable canonical form x̂cc f is given byx̂cc f (t) = x̂ s (t) + x̂u (t).

(44)

(37)

The following theorem shows that Wl (s) contains the property
for the invariant zeros of the system.

The state transformation of the system H = (A, B, C, D)
with the state transformation matrix T is generally defined as"#
T AT −1 TB
T (H, T) =
.
(38)
CT −1
D

Theorem 1 (Identity of invariant zeros). Invariant zeros of G svd
are the values when Wl (s) is not full row rank.
4

Proof. The invariant zero is defined as the value when the Rosen- and the bounded desired state trajectory x̂ is given bybrock system matrix is not full rank. It is assumed that rank(∆V H ) =x̂(t) = x̂ s (t) + x̂u (t).
(60)
m and the upper m rows of Π svd (s) are full row rank. It is assumed that rank(Cc ) = m and rank(U) = n. From Sylvester’s
Note that the unitary matrix U that is used as the state transrank inequality, for X ∈ Rl×n and X ∈ Rn×k it generally follows
formation matrix is not unique in singular value decomposition
but the desired state trajectory is generated uniquely for the derank(X) + rank(Y) − n ≤ rank(XY).
(49)
sired state-space representation. The calculation of the singular value decomposition can be numerically ill-conditioned andSylvester’s rank inequality is applied to X as Cc and Y as U. It
the state-space representation should be properly formulated so
follows that rank(C svd ) = rank(Cc U) = m and the lower m rowsthat the state transformation matrix U is not numerically illof Π svd (s) are full row rank. Therefore, the values when Wl (s)conditioned.
is not full rank are the same as the values when the Rosenbrocksystem matrix is not full rank.
From (47),"

4. Multirate feedforward control with mode decomposition
Wl (s)
C svd#"
x svd (s) =O(n−m)×my(s)In this section, the multirate feedforward control with mode
decomposition is presented. First, the model of the multi-modal
motion system is defined. Second, the multirate feedforward
control with additive decomposition is described. Third, themultirate feedforward control with multiplicative decomposition is described. The intersample performance of these two
approaches is verified in the next section.

#
,

(50)

and it follows that"x svd (s) =#−1 "Wl (s)C svd

O(n−m)×my(s)#.

Let the state transformation be
"
#−1 "
#
Wl (s)
O(n−m)×mx(s) = Ux svd (s) = U
.
C svd
y(s)

(51)

4.1. Definition of multi-modal motion system
The m-input m-output continuous-time multi-modal motion
system [23] is defined as

(52)

nm
X
ckm bkm=Gc,mod,km (s),Gc (s) =s2 + 2ζkm ωkm s + ω2km km =1
km =1
nm
X

Let the inverse Laplace transform be β(t) = L−1 [B−1 (s)], where"B−1 (s) = U

Wl (s)
C svd

#−1
.

The desired state trajectory is given by"#
Z tO(n−m)×mx̂(t) =β(t − τ)
dτ.
r(τ)
0

(53)

where ω is the resonance angle frequency, ζ is the damping coefficient, and nm is the number of modes. The vectors b ∈ R1×m
and c ∈ Rm×1 are associated with the inputs, the outputs, and thesmode shapes. Gc in modal form Gc,mod = (Ac,mod , Bc,mod , Cc,mod , O)
is given by(54)

ẋmod (t) = Ac,mod xmod (t) + Bc,mod u(t),When B(s)−1 has unstable poles, it can be decomposed as
B(s)

−1

−1
= B−1
s (s) + Bu (s),(61)y(t) = Cc,mod xmod (t),

(55)

(62)
(63)

where

where all poles p s ∈ C of B−1
s (s) are Re(p s ) ≤ 0 and all polespu ∈ C of B−1
(s)
are
Re(p
)
> 0. The filter matrices of stableuu"

and unstable parts for the state trajectory generation are givenbyhi
β s (t) = L−1 B−1(56)s (s) ,hi
−1−1βu (t) = L Bu (−s) .
(57)

Ac,modCc,modBc,modO# Ac,mod,1= 
O
Cc,mod,1hxmod (t) = xmod,1 (t) · · ·The stable and unstable parts of the desired state trajectory are
given by"#Z tO(n−m)×mx̂ s (t) =β s (t − τ)
dτ,
(58)
r(τ)
−∞
"
#
Z ∞
O
x̂u (t) =βu (t − τ) (n−m)×m dτ,
(59)
r(τ)
t

O
..

.

···

iT
xmod,nm (t) ,sAc,mod,nm
Cc,mod,nm

Bc,mod,1
..
.





 ,
Bc,mod,nm 

O
(64)
(65)

and the subsystem Gc,mod,km = (Ac,mod,km , Bc,mod,km , Cc,mod,km , O)
is given by#  0
"
1
O 


Ac,mod,km Bc,mod,km2=  −ωkm −2ζkm ωkm bkm  . (66)Cc,mod,km
O


ckm
O
O
h
iT
xmod,km (t) = xmod,km ,0 (t) xmod,km ,1 (t) .
(67)
5

Note that the number of lifting samples is N = ν in SISO systems but it is not the case in MIMO systems, see [20]. By inverting the state equation of Gd,µ , the input u generated by themultirate feedforward controller with additive decomposition is
given by
−1
−1u[k] = L−1
N Bd,µ x̂µ [iN + 1] − Bd,µ Ad,µ x̂µ [iN ]Although there are other representations such as the numerical decomposed model, the multi-modal model is reasonable for the intersample performance improvement by the state
tracking control, and mode selection should be relied on thephysical meaning of the states. The state tracking is only guaranteed on-sample and the intersample performance improvement is achieved because of the physical relationships between
each state. The gray box modeling approach with white box
structure and black box parameter tuning such as in [24] ispreferable to obtain the high-order multi-modal model in which
each mode has a physical interpretation of the mode shapes.

−1
−N
= L−1
Ad,µ ) x̂µ [iN + 1],
N Bd,µ (I − z(68)where x̂µ [iN ] = SN Tν Tµ x̂mod (t). The generated feedforward
input u provides perfect state tracking for every N sample forthe states corresponding to the selected modes µ.
Note that although perfect state tracking for selected states
does not guarantee perfect output tracking, it can provide betterintersample performance because the desired state trajectory is
generated by the model with full states, and the reference sampling frequency for selected ν states becomes higher to 1/Nδ
where N ≤ ν ≤ n.

(69)

4.3. Multirate feedforward control with multiplicative decomposition

4.2. Multirate feedforward control with additive decomposition
The overview of multirate feedforward control with additive
decomposition [12, 13] is shown in Figure 2. The indices µ of
the selected modes are defined asµ = {km |km ∈ 1, . . . , nm },and the order ν of the selected modes is defined as
ν = 2 × number{µ}.

The overview of multirate feedforward control with multiplicative decomposition [14] is shown in Figure 3. The onesample forward shifted systemzG̃d,mod = ( Ãd,mod , B̃d,mod , C̃d,mod , D̃d,mod )of the discrete-time system in modal formz
Gd,mod = ( Ad,mod , Bd,mod , Cd,mod , O) = SGc,mod H
from u[k] to y[k + 1] is given by# "#"Ãd,mod B̃d,mod
Ad,modBd,mod=. (82)
Cd,mod Ad,mod Cd,mod Bd,modC̃d,mod D̃d,mod

The permutation matrix for the selected modes is defined as" #ETµ = µ ,(70)
E×where Eµ and E× consist of standard basis vectors of selected
and unselected modes, and the standard basis vectors of themode km is defined ashi
Ekm = O2×2(km −1) I2 O2×2(nm −km ) .
(71)
The model reduction matrix extracting upper ν states is definedashiTν = Iν Oν×(n−ν) .
(72)

When ν states corresponding to the modes µ are selected,"#IνOν×(n−ν) −1
Π=S
S
(83)
O(n−ν)×ν O(n−ν)

The system of the selected modes Gc,µ is given byẋµ (t) = Ac,µ xµ (t) + Bc,µ u(t),(73)y(t) = Cc,µ xµ (t),(74)

xµ (t) = Tν Tµ xmod (t),(75)

Ac,µ = Tν Tµ Ac,mod Tµ−1 TνT ,(76)

Bc,µ = Tν Tµ Bc,mod ,(77)

Cc,µ = Cc,mod Tµ−1 TνT .

(78)

h
i
is defined with full rank S = V V× , where V ∈ Rn×ν andV× ∈ Rn×(n−ν) are a column space of an invariant subspace of
A = Ãd,mod and A× = Ãd,mod − B̃d,mod D̃−1d,mod C̃d,mod that correspond to the poles of Gmr and the zeros of G sr . Then thestate-space realizations are given by"#Ãd,mod Π B̃d,mod D̃−1zd,mod
Gmr f =,
(84)
C̃d,modI"#Ãd,modB̃d,modz
G sr f =.
(85)
C̃d,mod (I − Π) D̃d,mod

where

The discrete-time system of Gc,µ is given by#"
Ad,µ Bd,µz,
Gd,µ = SGc,µ H =
Cd,µ
O

(81)

Let the permutation matrix Tµ be such that AmrT (Gmr f , Tµ ) =  O
Cmr Amrz T (G sr f , Tµ ) =  O

O

(79)

z 


and the N (≤ ν) samples lifted system of Gd,µ is given by"#
Ad,µ Bd,µzN−1
Gd,µ = LN Gd,µ LN =.
(80)
Cd,µ Dd,µ
6

O
A sr
Cmrr
O
A sr
C sr


Bmr 
O  ,I
B srr 
B sr  .

D sr

(86)

(87)

x̂µ (t)x̂mod (t)Tµx̂µ [iN + 1]

SN

Tν

z

u[iN ]

−N
B −1
Aµ )
µ (I − z

N

u[k]

L−1
N

Figure 2: Block diagram of multirate feedforward control with additive decomposition.

x̂mod (t)x̂mr (t)

Tµ

SN

Tms

Tν

r sr [iN ] rsr [k]
u[k]
−1
−1
−1
−N
Gsr
LN
B mr (I − z Amr )x̂mr [iN + 1]z

N

Figure 3: Block diagram of multirate feedforward control with multiplicative decomposition.

Gmr with states xmr and G sr with states x sr are given by"#
Amr Bmr
z
Gmr =
,
Cmr
I
"
#
A sr B srz
G sr =
.
C sr
D

where
"
G−1
sr =

(88)

(91)

∗
y[k] = Cmr
xmr [k] + D∗mr C∗sr x sr [k].

z

(99)

In this section, the intersample performance of the perfect
tracking control approaches is validated in a multi-modal motion system. The intersample performance is evaluated in both
frequency and time domains. The nominal and robust performance is verified in the simulation using the model without andwith modeling error. The experimental validation is conductedwith feedback controller.

(94)5.1. Conditions
The validation is conducted in a single-input single-output
multi-modal motion system in Figure 4. The frequency response of the controlled system is shown in Figure 5. For the(95)

where x̂mr [iN ] = SN Tν Tms Tµ x̂mod (t). Then, the input u generated by the multirate feedforward controller with multiplicative
decomposition is given byu[k] = G−1
sr r sr [k],(97)

5. Application to multi-modal motion system

Note that the number of lifting samples is N = ν in SISO systems but it is not the case in MIMO systems, see [20]. By inverting the state equation of Gmr , the reference for the single-rate
inversion r sr is given by


−1
−1
r sr [k] = L−1
N Bd,mr x̂mr [iN + 1] − Bd,mr Ad,mr x̂mr [iN ]
−1
−N
= L−1
Ad,mr ) x̂mr [iN + 1],
N Bd,mr (I − z.

It shows that the approach provides perfect output tracking forevery N sample with D∗mr = O because the multirate inversion
provides perfect state tracking of xmr for every N sample. If thesystem is decomposed as D∗mr , O, there is no perfect output
tracking because perfect state tracking of x sr is not guaranteed.
Therefore, V and V× should be selected such that D∗mr = O.

(92)

G̃ms = ( Ãms , B̃ms , C̃ms , d) = T (G̃d,mod , Tms Tµ ) = Gmr G sr isgiven by#  Amr Bmr C sr Bmr d "Ãms B̃ms
A sr
B sr  .
=  O
(93)


C̃ms
D
Cmr
C sr
D
The N (≤ ν) samples lifted system of Gmr is given by"#
Amr BmrzN−1
Gmr = LN Gmr LN =
.
Cmr
O

#

where D∗mr D = O and the output is given by

where X ∈ Rν×(n−ν) is the solution of the Sylvester equation
Amr X − X A sr = Bmr C sr .

B sr D−1
sr
D−1
sr

Note that the one-sample backward shifted system of G̃ms
is given by"#
Ams BmsGms = T (Gd,mod , Tms Tµ ) =
Cms
O


"
#  Amr Bmr C sr Bmr D 


Ãms
B̃ms

A sr
B sr 
==  O−1
 ∗C̃ms ÃmsO
Cmr D∗mr C∗sr D∗mr D#"#"
A sr B sr
Amr Bmr
,
(98)
=
∗
Cmr
D∗mr
C∗sr
D

(89)

The product of the system H1 = (A1 , B1 , C1 , D1 ) and H2 =( A2 , B2 , C2 , D2 ) is generally defined as A1 B1 C2 B1 D2 
A2
B2  .
(90)
H1 H2 =  OC1 D1 C2 D1 D2The state transformation matrix Tms is given by
"
#−1
Iν
X
Tms =
,
Oν×(n−ν) I(n−ν)

A sr − B sr D−1
sr C sr
−D−1
C
sr
sr

(96)
7

For the controller design, a low-order continuous-time model
Gc is given by
Gc (s) = G1 (s) + G2 (s)2.44
1.1= 2 + 2ss + 2 × 0.024 × (2π × 30)s + (2π × 30)2
3.54
s2 + 2 × 0.02 × (2π × 25)s + (2π × 25)2
= 2 × 2ss + 2 × 0.024 × (2π × 30)s + (2π × 30)2N1 (s) N2 (s)
=
×
.
(101)
D1 (s) D2 (s)uyThe controller is designed by the low-order continuous-timemodel Gc and the continuous-time performance is verified by
the simulation in the low-order continuous-time model Gc without modeling error and the high-order continuous-time modelĜ with modeling error. The verification is conducted in the frequency domain and the time domain. The sampling time of thecontroller is δ = 10 ms. The intersample performance is evaluated in the sampling time δ/20 = 0.5 ms.
The compared 8 approaches are shown in Table 1. The difference in the approach between using additive decomposition
and using multiplicative decomposition is the provided perfect
tracking to states or to outputs. If the dynamics of the system are clearly modeled and the states of the state-space representation have physical meaning, the approach using additive
decomposition is useful because it is clear to which state theperfect tracking is provided. If the dynamics and the physicalmeaning of the system are not clear and the controller design
only focuses on the output, the approach using multiplicative
decomposition is useful because it provides the perfect tracking
to on-sample output but not to state.

∠G [deg]|G| [dB]

Figure 4: High-precision positioning stage with input current u [A] generating
force with linear motor and output displacement y [m] measured by linear encoder with 1 nm resolution.

0
−40
−80
−120
−160
−200
0
−90
−180
−270
−360
100

101

102

103

f [Hz]
Figure 5: Bode diagram of the controlled system: frequency response data
( ), a high-order continuous-time model Ĝ ( ), a low-order continuoustime model Gc ( ), and a low-order discrete-time model for controller design
Gd ( ). Vertical lines ( ) and ( ) show Nyquist frequency 1/2δ and sampling frequency 1/δ, respectively.

5.2. Frequency domain verification
In frequency domain verification, the intersample performance is verified by the performance frequency gain |Er | [25,26, 27] in the simulation that is the steady state continuous-timetracking error normalized by the step sine wave reference andis defined asverification of the modeling error, the high-order continuoustime model Ĝ is given by|Er (jω)| =2.44
Ĝ(s) = 2s

RMS(ejω (t)),
RMS(rjω (t))(102)

where rjω (t) can only contain a single frequency at each frequency and the intersample performance is evaluated in thesampling time δ/20 = 0.5 ms.
The performance frequency gain of the continuous-time tracking error in Gc without modeling error is shown in Figure 6. It
shows that Case 5 = Case 7 and Case 6 = Case 8 in the performance frequency gain. It is because the discretization only
affects the dynamics of zeros, and the choice of the pole does
not affect the performance. Case 4 makes a large error for thewhole frequency range because it cannot compensate for rigid
body dynamics. The approaches like Case 1, Case 6, and Case
8 that cannot compensate for oscillating poles of the feedforward controller due to discretization make large errors around
Nyquist frequency. In low frequency, |Er | is smaller in order ofCase 1 < Case 5 = Case 7 < Case 2 < Case 3. From these analyses, Case 2 with multirate feedforward control for full-state

1.1+ 2
s + 2 × 0.024 × (2π × 30)s + (2π × 30)2
−2.44
+ 2
s + 2 × 0.038 × (2π × 89)s + (2π × 89)2
−1.1
+ 2
s + 2 × 0.07 × (2π × 297)s + (2π × 297)2= G1 (s) + G2 (s) + G3 (s) + G4 (s).
(100)

8

Table 1: Comparison of 8 approaches. T r is the reference sampling time. Gc,mr and Gc,sr are the continuous-time model for the multirate and single-rate inversion.
Intersample performance is evaluated by the Root Mean Square of the tracking error eRMS = RMS(e(t)) with the sampling time δ/20 = 0.5 ms. The evaluations are
conducted by the simulation of Gc without feedback control, Ĝ without feedback control, and Ĝ with feedback control, and by the experiment.

Line

ApproachSingle-rate
Multirate
Additive
Additive
Multiplicative
Multiplicative
Multiplicative
Multiplicative

Tr
δ
4δ
2δ
2δ
2δ
2δ
2δ
2δ

Gc,mr
Gc
G1
G2
N1 /D1N2 /D1N1 /D2N2 /D2Gc,srGc
N2 /D2
N1 /D2N2 /D1N1 /D1

Gc w/o FB
9862 nm
5174 nm
8199 nm>1 mm9309 nm
8559 nm
9309 nm
8559 nmĜ w/o FB
16 811 nm
11 646 nm
12 176 nm>1 mm
11 280 nm
14 547 nm
11 280 nm
14 547 nm

80

80

40

40
|Er | [dB]

|Er | [dB]

Case
1
2
3
4
5
6
7
8

0
−40
−80
−120
100

Ĝ with FB
17 642 nm
12 406 nm
13 397 nm
12 510 nm-

Experiment
27 152 nm
15 027 nm
17 210 nm
22 579 nm
-

101

102

0
−40
−80

101

102

−120
100

103

f [Hz]

103

f [Hz]

Figure 6: Performance frequency gain of the continuous-time tracking error in
Gc . Vertical lines ( ) and ( ) show Nyquist frequency 1/2δ and sampling
frequency 1/δ, respectively. Each line corresponds to 8 approaches from Case
1 to Case 8 in Table 1, respectively.

Figure 7: Performance frequency gain of the continuous-time tracking error inĜ. Vertical lines ( ) and ( ) show Nyquist frequency 1/2δ and sampling
frequency 1/δ, respectively. Each line corresponds to 8 approaches from Case
1 to Case 8 in Table 1, respectively.

tracking provides the best performance in steady state, and Case2, Case 3, Case 5, and Case 7 are preferable approaches.
The performance frequency gain of the continuous-time tracking error in Ĝ with modeling error is shown in Figure 7. Thereis the same trend around Nyquist frequency compared to thesimulation in Gc without modeling error. In low frequency, |Er |is around the same performance in all approaches except Case
4. It is because the controller cannot compensate for the lowfrequency compliance of the unmodeled high-frequency dynamics.

The time series error e(t) in Ĝ with modeling error is shownin Figure 10. The result shows that Case 5 and Case 7 achieve
the best performance because the control input contains relatively low-frequency components compared to Case 2, and it
does not excite the resonances of the unmodeled dynamics. The
performance of Case 1, Case 6, and Case 8 is getting worse in Ĝ
because the oscillating feedforward input due to discretization
excites the unmodeled high-frequency dynamics. From theseanalyses, multirate feedforward control with mode decomposition can provide better intersample performance in transient response than that of multirate feedforward control for full-state
tracking.

5.3. Time domain verification
In the time domain verification, the intersample performanceis verified by the continuous-time tracking error in the simulation for the continuous-time reference trajectory shown in Figure 8. The intersample performance is evaluated by Root Mean
Square error eRMS = RMS(e(t)) in Table 1 with the samplingtime δ/20 = 0.5 ms.
The time series error e(t) in Gc without modeling error isshown in Figure 9. The result shows that Case 2 achieves thebest performance because there is no modeling error between
the controller and the controlled system and perfect tracking
for all states provides smooth intersample behavior.

5.4. Experimental validation
From the result of the simulation verification, the experimental validation is conducted in Case 1, Case 2, Case 3, andCase 5 as shown in Table 1 with the same reference that isshown in Figure 8. The overview of the experimental validation is shown in Figure 11. A feedback controller K is used forstabilization and compensation for low-frequency modeling errors in the experimental validation. For the feedback controller,
the PD controller is designed as 5 Hz bandwidth and 6 dB modulus margin. In the two-degree-of-freedom control structure,
9

0

0.1

0.2
t [s]

0.3

0.4

e [mm]

0.1

e [mm]

0−5

e [mm]
×104

5

0.1
0
−0.1

Figure 8: Continuous-time 4th order polynomial trajectory reference r(t) and its
derivatives. ( ), ( ), and ( ) show sampling points every δ, 2δ, and 4δ.

the feedforward controller handles tracking performance, andthe feedback controller handles modeling error and disturbancerejection, independently. As shown in Figure 11, the feedback
controller compensates for the modeling error and disturbancein low frequency. The simulation with the feedback controller
and the quantization of the linear encoder is also conducted forthe validation of the experimental results.
The time series error e(t) in the experiment is shown in Figure 12. The result has a similar trend to that in the simulation ofĜ with feedback control. Note that due to experimental conditions such as model mismatches, the exact on-sample tracking
is not provided. The intersample performance is evaluated by
Root Mean Square error eRMS = RMS(e(t)) in Table 1 with thesampling time δ/20 = 0.5 ms. It shows that Case 2 with multirate feedforward control for full-state tracking provides the best
performance in simulation and experiment with feedback control. The amplitude spectrum of error e(t) in the experiment isshown in Figure 13. The result shows that Case 1 and Case 5
which contain single-rate filters have a large error in over sampling frequency because the frequency components of the feedforward input over sampling frequency excite the unmodeled
high-frequency dynamics.

0
−0.1
0.1
0
−0.1
0.1

e [mm]

0−5

e [mm]
×102

5

0.1
0
−0.1

0
−0.1

e [mm]

0
−5

0.1
0
−0.1

e [mm]

5

0.1
0
−0.1

e [mm]

0.1
0
−0.1

˙˙˙˙
r [m/s4 ] ˙˙˙r [m/s3 ] r̈ [m/s2 ]ṙ [m/s]

r [m]

×10−2
1
0.5
0

0.1
0
−0.1

0

0.1

0.2
t [s]

0.3

0.4

Figure 9: Error e(t) in the simulation of Gc with sampling time δ/20 = 0.5 ms.
( ), ( ), and ( ) show sampling point every δ, 2δ, and 4δ. Each figure from
top to bottom corresponds to 8 approaches from Case 1 to Case 8 in Table 1,
respectively.

system show that state tracking should be used to compensate
for the oscillating poles of the feedforward controller due to discretization. In summary, the feedforward controller to improve
intersample performance should be designed with the following
conditions.
• State tracking approach with lifted samples at least thesame number as the relative degree can compensate foroscillating poles of the feedforward controller due to discretization when the relative degree of the continuoustime model is 2 or more.
• For steady-state performance without modeling error, multirate feedforward control for full-state tracking provides
the best performance.

6. Conclusion
In this paper, perfect tracking control approaches are described focusing on intersample performance in multi-modal
motion systems. The model of the multi-modal motion system is decomposed into combinations of the states that can beselected for the perfect tracking control. The simulation verification and experimental validation in a multi-modal motion• Multirate feedforward control with mode decomposition
can provide the best intersample performance in transient
response depending on the frequency components of thereference and the unmodeled dynamics of the controlled
system.
10

e [mm]e [mm]

0.1
0
−0.1

0.1
0
−0.1

|e| [m]
0

0.1

0.2
t [s]

0.3

0.4

|e| [m]e [mm]

0.1

e [mm]0.1
0−0.1|e| [m]

0
−0.1

0.1
0
−0.1

0
−0.1

0

0.1

0.2
t [s]

0.3

0.4

Figure 12: Error e(t) in the experiment with sampling time δ/20 = 0.5 ms.
The solid black line shows that in the simulation of Ĝ with feedback control.
Each figure from top to bottom corresponds to Case 1, 2, 3, and 5 in Table 1,
respectively.

0.1

0
−0.1

0.1

e [mm]e [mm]

0.1
0
−0.1

0
−0.1

e [mm]e [mm]

0.1
0
−0.1

0.1

e [mm]e [mm]e [mm]0.1
0−0.1

e [mm]

0.1
0−0.1

uf f [k]+u[k]uf b [k] +

u(t)

H

|e| [m]Figure 10: Error e(t) in the simulation of Ĝ with sampling time δ/20 = 0.5 ms.
( ), ( ), and ( ) show sampling point every δ, 2δ, and 4δ. Each figure from
top to bottom corresponds to 8 approaches from Case 1 to Case 8 in Table 1,
respectively.

y(t)

10−9
10−3
10−6
10−9
10−3
10−6
10−9
10−3
10−6
10−9

102

103

104

Figure 13: Amplitude spectrum of error e(t) in the experiment with samplingtime δ/20 = 0.5 ms. The solid black line shows that in the simulation of Ĝ
with feedback control. Vertical lines ( ) and ( ) show Nyquist frequency
1/2δ and sampling frequency 1/δ, respectively. Each figure from top to bottom
corresponds to Case 1, 2, 3, and 5 in Table 1, respectively.

e[k]ŷ[k] + − y[k]10−6f [Hz]

G

K
Gd

10−3

S

Figure 11: Block diagram of the experimental validation. G, Gd , K, S andH denote a controlled system, a low-order discrete-time model for controller
design, a feedback controller, sampler and zero-order-hold, respectively.

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