Multirate Feedforward Control based on
Modal Form with Mode Selection Applied toMulti-Modal High-Precision Positioning Stage
Masahiro Mae, Wataru Ohnishi and Hiroshi Fujimoto
The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan
Phone: +81-4-7136-3881
Email: mmae@ieee.org, ohnishi@ieee.org, fujimoto@k.u-tokyo.ac.jp

Abstract—Multirate feedforward control provides perfect
tracking control for the desired state trajectory in the model.
The aim of this paper is the analysis of multirate feedforward
control based on modal form with mode selection. The proposed
approach is applied for a mechanical system that has several
modes and a multirate feedforward controller is designed depending on the mode selection. The approach is successfully applied to
an 8th order motion system. The proposed approach improves theintersample behavior of tracking error compared with multirate
feedforward control based on controllable canonical form.
Index Terms—multirate feedforward control, modal form,
perfect tracking control, intersample behavior

I. I NTRODUCTION
Inverse model feedforward control plays an important rolein tracking control problems in high-precision mechatronicsystems, such as wafer and LCD scanners [1], [2], atomic
force microscopes [3], and hard disc drives [4]. If the model
has unstable zeros, the feedforward controller has unstable
poles because it is designed as the inverse of the model.
The feedforward controller with unstable poles generates anunbounded control input and it is not applicable for the real
motion systems. When the model does not have unstable
intrinsic zeros in continuous-time, the discretized model has
unstable discretization zeros when the relative degree of themodel is two or more such as many rigid systems in mechatronic systems [5]. The inverse model feedforward controller
designed for the model with unstable discretization zeros has
unstable poles and unbounded or oscillating control inputs are
generated.
To overcome the unstable zero problems of the model, several approximate inversion approaches are developed, such as
zero phase error tracking control (ZPETC) [6], zero magnitude
error tracking control (ZMETC) [7], and nonminimum-phasezeros ignore (NPZ-Ignore) [8]. However, these approaches areusing approximate inversion of the model and perfect tracking
control defined in [6] cannot be achieved, theoretically.
Multirate feedforward control [9] is proposed to solve
unstable discretization zero problems and to achieve perfect
tracking control. The multirate feedforward control provides
an exact state tracking at every n samples for nth order model.

The discrete-time stable inversion approach [10], [11] is also
proposed, however, this approach treats the unstable intrinsic
and discretization zeros in the same way, and preactuatedcontrol inputs are generated for not only the unstable intrinsic
zeros but also unstable discretization zeros. On the other
hand, the multirate feedforward control approach designed
the independent stable inversion of unstable intrinsic anddiscretization zeros [12]. Therefore, the advantage is that themultirate feedforward controller does not generate preactuatedfeedforward control inputs for unstable discretization zeros.
A higher-order accurate model improves the tracking performance of inverse model feedforward controllers. However,
the multirate feedforward control has the trade-off between
the number of states which achieve perfect tracking and thesampling intervals when the perfect tracking is achieved. For
the nth-order model with the sampling period of control input
Tu , the exact state tracking is achieved every nTu and it is thesame case with a minimum-time dead-beat control [13]. This
problem is also known that the higher-order model is numerically ill-conditioned in designing inverse model feedforward
controllers.
Multirate feedforward control based on the modal form [14]is proposed to improve these problems because of higher-ordermodels. Previous research shows that multirate feedforward
control based on the modal form achieves the exact state
tracking for selected modes in shorter intervals and improves
continuous-time tracking error. The condition number is also
improved compared with conventional multirate feedforward
control. In this approach, there is a degree of freedom toselect which modes for state tracking. The aim of this paper
is to clarify the relationship between mode selection andcontinuous-time tracking error in multirate feedforward control
based on modal form.
The outline of this paper is as follows. In Section II,
the control objective is formulated. Desired state trajectory
generation method for multirate feedforward control is presented in Section III. Multirate feedforward control based onmodal form is presented in Section IV. The advantages of
the approach are demonstrated in Section V. Conclusions arepresented in Section VI.

yd (t)Syd [k]

u[k]
F

H

u(t)

Gc

y(t)

e(t)

IV. M ULTIRATE FEEDFORWARD CONTROL BASED ON
MODAL FORM

− +Fig. 1. Block diagram of tracking control. The continuous-time system Gc
is controlled by the discrete-time controller F with sampler S and holder H.
The objective is to minimize the continuous-time error e(t).

II. P ROBLEM FORMULATIONIn this section, the control problem is formulated. The
overview of the control problem is shown in Fig. 1, withdesired output trajectory yd (t) ∈ R, control input u(t) ∈ R,
output y(t) ∈ R, digital controller F , sampler S, and zeroorder hold H. The continuous-time linear-time-invariant (LTI)
nth-order system Gc is given by
Gc (s) =bm sm + · · · + b1 s + b0
B(s)
= n
,
A(s)
s + an−1 sn−1 + · · · + a1 s + a0

(1)

where m, n ∈ N, n > m, and b0 6= 0. It is assumed that Gc
is minimal realization, stable, and with no unstable intrinsic
zero.
The control objective of this paper is minimizing thecontinuous-time error e. The desired output trajectory yd isassumed to be known a priori. The preactuated control input
which is used in negative time is not used in this control
problem.

In this section, the design procedure of the multirate feedforward control based on modal form is presented.
A. Transformation to modal form
The system Gc in (1) is decomposed into second-ordermodes as follows:
Gc (s) =

nX
mod
l=1

b1,l s + b0,l,s2 + a1,l s + a0,l(8)

where nmod ∈ N is the number of modes and l ∈ N is theindex of the modes. It is assumed that the system Gc is amechanical system and consists of a second-order rigid modeand second-order vibration modes. From these assumptions, n
is even and nmod = n/2.
The state space realization of (8) in which each mode issecond-order controllable canonical form is given byẋmod (t) = Ac,mod xmod (t) + bc,mod u(t),(9)y(t) = cc,mod xmod (t),

(10)

where
T
xmod,nmod (t) ,xmod (t) = xmod,1 (t) xmod,2 (t) · · ·III. S TATE TRAJECTORY GENERATION BASED ON
CONTROLLABLE CANONICAL FORM

(11)

In this section, the state trajectory generation method basedon controllable canonical form is presented. Multirate feedforward control achieves perfect tracking control for the desired
state trajectory xd at every reference sampling period. The
desired state trajectory xd is generated from the desired output
trajectory yd with the system of controllable canonical form.
The controllable canonical form of (1) is given byẋccf (t) = Ac,ccf xccf (t) + bc,ccf u(t),(2)y(t) = cc,ccf xccf (t),0  0 ..
bc,ccf=  .
0
 0
 −a0
1

T
xn−1,ccf (t) , (4)

1
0
..
.

0
1
..
.

···
···
..
.

0
0
..
.

0
−a1

0
−a2

b1
b0

b2
b0

···
···
···

1
−an−1
0


0
0 

.. 
. 
.
0 

b0 
0

(5)

(l = 1, . . . , nmod ),(12)Ac,mod,1

O




,
.

Ac,mod,nmod bc,mod,nmod 
cc,mod,2 · · · cc,mod,nmod
0

  0
1
0
bc,mod,l

−a
−a
b
0,l
1,l
0,l .
=
b1,l
0
1
0
b0,l
..

O
cc,mod,1Ac,mod,lcc,mod,l

bc,mod,1bc,mod,2..
.

Ac,mod,2




=




bc,mod
0

(13)

(14)

The transformation matrix from controllable canonical form
is given by a1b0

T = bc,mod Ac,mod bc,mod · · ·hIt is assumed that the desired output trajectory yd (t) is n−1
times differentiable, and
h
iT
d
dn−1
yd (t) = 1 dt
yd (t).
(6)· · · dtn−1
The desired state trajectory xd (t) is given by
1−1
xd,ccf (t) = L
yd (s) ,B(s)Ac,modcc,modxccf (t) = x0,ccf (t) x1,ccf (t) · · ·Ac,ccf
cc,ccf



(3)

where




T
xmod,l (t) = x0,mod,l (t) x1,mod,l (t)i  ..
 .
(nmod −1)
Ac,modbc,mod  an−1

b0
1
b0

···
.
..

an−1
b0

.

1 
b0

..






1
b0

.

(15)

O

The desired state trajectory in modal form is given by
xd,mod (t) = T xd,ccf (t).

(16)

B. Mode selection(7)

where L−1 (·) is the inverse uni-lateral Laplace transform [15].

The state space realization (13) is decoupled between themodes. Therefore, the feedforward control input can be generated for selected modes.

The mode selection matrix Mµ is defined as
Mµ = diag{Ml } (l = 1, . . . , nmod ),(I(2,2) (selected mode)
Ml =,O(2,2) (not selected mode)

(17)
(18)

where µ is the indicies of selected modes. The rows of M
whose all elements are zeros are eliminated, andν = number{µ}.

(19)where ν ∈ N is the number of selected modes µ and M ∈R2ν×n (e.g. µ = (1, 2) then ν = 2).
The state equation with selected modes in (9) is given byẋµ (t) = Ac,µ xµ (t) + bc,µ u(t),(20)

xµ (t) = Mµ xmod (t),(21)

Ac,µ = Mµ Ac,mod ,(22)bc,µ = Mµ bc,mod .

(23)

where

The desired state trajectory with selected modes is also givenby
xd,µ (t) = Mµ xd,mod (t).

(24)

C. Feedforward control input generation
The feedforward control input is generated with selectedmodes. The discretized system of (20) with a sampling periodof control input Tu is given byxµ [k + 1] = As,µ xµ [k] + bs,µ u[k],(25)

tracking in every Tr = 2νTu for selected modes. Note that thisapproach does not guarantee the perfect state tracking for othermodes, but, they also reasonably tracking. It is because thedesired state trajectory is generated considering other modesin (16).
Compared to the multirate feedforward control based oncontrollable canonical form that archives the perfect state
tracking for all modes in every Tr = nTu , the proposed
approach achieves the perfect state tracking for selected modes
in a shorter period. The shorter tracking period improves
the intersample behavior and leads to reduce continuous-timetracking error.
It is also the advantage of the proposed approach that theorder of the matrix Bµ in (32) is reduced because of mode
selection. The matrix Bµ is the same as the controllability
matrix and is numerically ill-conditioned in higher-order systems. The previous research shows that the proposed approach
outperforms the conventional multirate feedforward for themodel with simple mode reduction [14].
In the proposed approach, the problem of the unstable intrinsic and discretization zeros is treated in the state trajectory
generation and multirate feedforward control, separately [12].
Therefore, when the system does not have intrinsic zeros, theadvantage of the proposed approach is that it does not use
preactuation because of the unstable discretization zeros.
V. S IMULATIONIn this section, the proposed approach is applied to a motion
system. The results demonstrate the advantage of the proposed
approach.
A. Setup

whereAs,µ = eAc,µ Tu , bs,µ =Z Tu

eAc,µ τ bc,µ dτ.

(26)

0

The state equation of the lifted system with input multiplicity 2ν is given byxµ [i + 1] = Aµ xµ [i] + Bµ u[i],(27)where a sampling period of reference Tr = 2νTu andxµ [i] = xµ (iTr ),(28)

Aµ = A2ν, s,µBµ = A2ν−1s,µ bs,µA2ν−2s,µ bs,µ· · · u2ν

(29)
···


u[i] = u1 u2= u(kTu ) u((k + 1)Tu ) · · ·bs,µ ,(30)

u((k + 2ν − 1)Tu ) .
(31)

From (27), the feedforward control input uf f [i] is generated
with 2ν sample previewed desired state trajectory xd,µ [i + 1]
as follows:
uf f [i] = Bµ−1 (I − z −1 Aµ )xd,µ [i + 1],(32)

where z = es2νTu .
The overview of the proposed approach is shown in Fig. 2.
The feedforward control input uf f realizes the perfect state

The motion system of a high-precision positioning stage isillustrated in Fig. 3(a). The continuous-time transfer function
from the input current u [A] generating the force with thelinear motor to the output displacement y [m] measured by
the linear encoder is given by
4.5759 × 106 (s2 + 8.132s + 2.518 × 104 )s(s + 2.101)(s2 + 10.89s + 3.665 × 104 )(s2 + 84.73s + 8.497 × 105 )× 2(s + 45.4s + 3.139 × 105 )(s2 + 262.2s + 3.507 × 106 )(33)

Gc (s) =and is minimal realization, stable, and with no unstable intrinsic zero. The Bode diagram of the motion system Gc is shownin Fig. 3(b).
Gc is decomposed to four second-order modes as follows:
Gc (s) = G1c (s) + G2c (s) + G3c (s) + G4c (s)−0.00027976(s − 8674)−0.00012742(s − 9494)+ 2
s(s + 2.101)(s + 10.89s + 3.665 × 104 )
0.00032577(s − 7779)
8.1406 × 10−5 (s − 1.312 × 104 )+ 2+.
(s + 45.4s + 3.139 × 105 )(s2 + 262.2s + 3.507 × 106 )=(34)
The Bode diagrams of four second-order modes from G1c toG4c are shown in Fig. 3(c).

xd,ccf (t + Tr ) xd,mod (t + Tr ) xd,µ (t + Tr )yd (t + Tr )

L−1

h

i

1
y (s)
B(s) dMµTxd,µ [i + 1]

S
(Tr )

uf f [i]Bµ−1 (I − z −1 Aµ )uf f [k]

u(t)

H

L−1

y(t)

Gc (s)(Tu )Fig. 2. Block diagram of state trajectory generation and multirate feedforward control based on modal form. z, S, H, and L denote esTr , sampler, holder,
and lifting operator [16], respectively.

10-6
10
5

u

0
0

0.002

0.004

0.006

0.008

0.01

y
Fig. 4. Desired output trajectory yd consists of 15th-order polynomials.

10-6(a) High-precision positioning stage with input current u[A] generating force with linear motor and output displacement y [m] measured by linear encoder.

10-8

0
-100
-200
100
0
-90
-180
-270
-360
100

101

102

10-10

103

2

4

6

8

6

8

(a) RMS(e)

101

102

10-6

103

(b) Bode diagram of 8th-order motion system: Gc (s) (

).

0

10-8

-100
-200
100
0
-90
-180
-270
-360
100

101

102

10-10

103

2

4

(b) MAX(|e|)

101

102

103

(c) Bode diagram of four second-order modes: G1c (s) (G2c (s) (), G3c (s) (), and G4c (s) ().

),

Fig. 3. High-precision positioning stage used in simulations.

Fig. 5. Preview samples versus root mean square errors RMS(e) andmaximum absolute errors MAX(|e|) of simulations depending on mode
selection µ. ◦ shows the errors of multirate feedforward control based onmodal form, e.g., ◦: µ = (3), ◦: µ = (3, 4), ◦: µ = (2, 3, 4), and ◦:µ = (1, 2, 3, 4). × shows the errors of conventional ZPETC.

10-6
2

10

0

5

-2

0

10-9

1
0
-1

0

0.005

0.01

0

0.005

(a) Input u

0.01

0

0.005

(b) Output y(c) Error e

Fig. 6. Simulation results of input u, output y, and error e using conventional ZPETC (

10-6
2

10

0

5

-2

0

0.01

).

10-9

1
0
-1

0

0.005

0.01

0

(a) Input u

0.005

0.01

0

(b) Output y

0.005(c) Error e

Fig. 7. Simulation results of input u, output y, and error e using multirate feedforward control with mode selection µ = (3) (

10-6
2

10

0

5

-2

0

0.01

).

10-9

1
0
-1

0

0.005

0.01

0

0.005

(a) Input u

0.01

0

0.005

(b) Output y(c) Error e

Fig. 8. Simulation results of input u, output y, and error e using multirate feedforward control with mode selection µ = (3, 4) (

10-6
2

10

0

5

-2

0

0.01

).

10-9

1
0
-1

0

0.005

0.01

0

0.005

(a) Input u

0.01

0

0.005

(b) Output y(c) Error e

Fig. 9. Simulation results of input u, output y, and error e using multirate feedforward control with mode selection µ = (2, 3, 4) (

10-6
2

10

0

5

-2

0.01

).

10-9

1
0

0
-1
0

0.005

(a) Input u

0.01

0

0.005

(b) Output y

0.01

0

0.005

0.01

(c) Error e

Fig. 10. Simulation results of input u, output y, and error e using multirate feedforward control with mode selection µ = (1, 2, 3, 4) ().

TABLE I
ROOT MEAN SQUARE ERRORS RMS(e) AND MAXIMUM ABSOLUTEERRORS MAX(|e|) OF SIMULATIONS DEPENDING ON MODE SELECTION µ.
µ

RMS(e) [m]MAX(|e|) [m]

ZPETC
(1)
(2)
(3)
(4)
(1, 2)(1, 3)(1, 4)(2, 3)(2, 4)(3, 4)(1, 2, 3)(1, 2, 4)(1, 3, 4)(2, 3, 4)(1, 2, 3, 4)

1.16 × 10−08
3.90 × 10−10
3.80 × 10−10
3.12 × 10−10
1.03 × 10−09
2.89 × 10−10
2.79 × 10−10
1.90 × 10−10
2.78 × 10−10
1.89 × 10−10
1.79 × 10−10
2.47 × 10−10
2.05 × 10−10
2.02 × 10−10
2.02 × 10−102.64 × 10−10

2.06 × 10−08
7.96 × 10−10
7.80 × 10−10
6.36 × 10−10
2.11 × 10−09
5.81 × 10−10
5.73 × 10−10
4.79 × 10−10
5.71 × 10−10
4.72 × 10−10
4.60 × 10−10
6.07 × 10−10
5.64 × 10−10
5.51 × 10−10
5.50 × 10−10
7.73 × 10−10

VI. C ONCLUSION
Multirate feedforward control based on modal form withmode selection is developed. The proposed approach improves
the intersample behavior of tracking error compared withmultirate feedforward control based on controllable canonicalform. The advantages of the approach are validated in thesimulations of the 8th-order motion system. Ongoing research
focuses on the relationship between the reference trajectory
and the mode selection in multirate feedforward control basedon modal form.
R EFERENCES

B. Conditions
The reference of the desired output trajectory yd is given
by a 15th-order polynomial which change from 0 to 10 µm in
0 s to 10 ms as shown in Fig. 4. The sampling period of thecontrol input is set to Tu = 400 µs.
In this simulation, the indicies of selected modes µ is
defined as follows:µ =(1), (2), (3), (4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4),(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2, 3, 4).

The results demonstrate the advantage of the proposed
approach in the objective of minimizing the continuous-timeerror.

(35)

Note that the case with µ = (1, 2, 3, 4) selected all modes andit is the same to the conventional multirate feedforward control
based on controllable canonical form. The proposed approachis compared with the ZPETC [6] as the conventional method.
The system is controlled in open loop.
C. Results
The root mean square errors RMS(e) and the maximum
absolute errors MAX(|e|) of simulations in all cases with thenumbers of preview samples are illustrated in Fig. 5. The root
mean square errors RMS(e) and the maximum absolute errors
MAX(|e|) are shown in TABLE I. From these results, allcases of the proposed approach outperform the conventional
ZPETC, and the mode selection µ = (3, 4) is the best tracking
performance for the desired output trajectory.
The mode selections µ = (3), (3, 4), (2, 3, 4), (1, 2, 3, 4) thatare the best control performance in each preview samples
are compared with the conventional ZPETC. The simulation
results are shown from Fig. 6 to Fig. 10. From these results, the proposed approaches outperform the conventional
ZPETC. It is also confirmed that the proposed approacheswith mode selections, µ = (3), (3, 4), (2, 3, 4), do not achieve
perfect on-sample tracking to the desired output trajectorycompared with the conventional multirate feedforward control,µ = (1, 2, 3, 4), theoretically. However, the approaches with
µ = (3, 4), (2, 3, 4) achieve better tracking performance in theobjective of the continuous-time error reduction.

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