IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 2, APRIL 2019

863

Preactuated Multirate Feedforward Control for
Independent Stable Inversion of Unstable
Intrinsic and Discretization Zeros
Wataru Ohnishi , Member, IEEE, Thomas Beauduin, Member, IEEE,and Hiroshi Fujimoto, Senior Member, IEEE

Abstract—A plant with unstable zeros is known to be difficult to be controlled, because of the initial undershootof its step response and the unstable poles in its inversion system. There are two reasons why a plant has unstable zeros in the discrete-time domain: 1) noncollocation of
actuators and sensors; and 2) discretization by the zeroorder hold. Addressing problems 1) and 2) simultaneously,approximate and discrete-time stable inversion methods
have been proposed as model-based feedforward control
for unstable zeros. However, the approximated inversion
methods compromise tracking performance, and thediscrete-time stable inversion methods obtain oscillatory
control inputs owing to the direct cancellation of discretization zeros. Therefore, this paper proposes a preactuatedmultirate feedforward control, which is an independent
stable inversion of the two types of zeros: unstable intrinsic zeros from noncollocation and unstable discretization zeros from the zero-order hold. The proposed method
combines both a state trajectory generation based on atime- and imaginary-axis reversal to address problem 1) anda multirate feedforward implementation to address problem2), independently. Moreover, this study verifies the tracking
performance improvement by performing simulations andexperiments in comparison with well-known model-based
feedforward control methods using a high-precision motion stage. The role and importance of multirate feedforward control are also demonstrated by comparing it to acontinuous-time domain approach with preactuation.
Index Terms—Discretization, multirate feedforward, nonminimum phase (NMP) system, preactuation, unstable zero.

I. INTRODUCTIONTRACKING control with zero gain and phase errors
between the desired and output trajectories at every

Manuscript received June 10, 2018; revised October 25, 2018; accepted December 30, 2018. Date of publication January 30, 2019; dateof current version April 16, 2019. Recommended by Technical Editor J.
Yi. This work was supported by the Japan Society for the Promotion of
Science Grants-in-Aid for Scientific Research under Grant 18H05902.
(Corresponding author: Wataru Ohnishi.)W. Ohnishi and H. Fujimoto are with the University of Tokyo, Tokyo
113-8656, Japan (e-mail:, ohnishi@ieee.org; fujimoto@k.u-tokyo.ac.jp).
T. Beauduin was with the University of Tokyo, Tokyo 113-8656, Japan.
He is now with MTT Innovation Inc., Vancouver BC V5Y 1E6, Canada(e-mail:, thomas.beauduin15@ae.k.u-tokyo.ac.jp).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMECH.2019.2896237

Fig. 1. Step response comparison: P 1 is a first-order transfer function without an unstable zero. P 2 , P 3 , and P 4 have one, two, and threeunstable zero(s), respectively.

sampling point can be achieved, in theory, by model inversion(i.e., pole-zero cancellation). However, a plant with unstable(i.e., nonminimum phase, NMP) zeros gives rise to an initial
undershoot when stepped (see Fig. 1) and highly oscillatory or
unstable control trajectories when inverted [1], [2]. Unstable zeros of discrete transfer functions can be classified as: 1) intrinsic
zeros corresponding to the plant dynamics (e.g., noncollocatedplacement of sensor and actuator); or 2) discretization zeros
due to signal sampling [3]. Note that discretization zeros are
unstable when the relative order of the continuous-time plantis greater than two, even without continuous-time unstable zeros [4]. Systems with unstable zeros are, for instance, wafer
stages [5], high-precision positioning stages for flat panel display manufacturing systems [6], [7], hard disk drives [8], boost
converters [9], permanent magnet synchronous motors during
six-step operation [10], backward driving cars [11], and pitch
angle regulation of aircraft [12].
Extensive research has been dedicated to the design of tracking control methods for systems with NMP zeros. The existing proposals can be classified as approximate model-inversion
methods and preactuated model-inversion methods [13]. A
straightforward way to implement an approximate inversion is
to cancel stable poles and zeros while ignoring NMP zeros,i.e., NPZI method [14]. This approach exhibits both magnitude1083-4435 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

864

and phase errors; hence, zero-magnitude-error tracking control(ZMETC) [15] and zero-phase-error tracking control (ZPETC)[16] methods have been proposed and improved upon. Despite
the stable control input, NPZI, ZMETC, and ZPETC may not
yield satisfactory tracking performance owing to the approximations involved (depending on the system and performance
specifications).
Preactuated model inversion can achieve perfect tracking
with infinite preview (i.e., knowledge of future references) andpreactuation (i.e., actuation applied a time interval before theactual output of the system). Continuous-time-domain stable
inversion methods are proposed in [17]–[20]. This study unifies
these proposals as continuous-time preactuated model-inverse(CPMI) methods. However, these approaches do not consider
the effect of the zero-order hold. There lies the key difference
between the CPMI method and the proposed preactuated multirate feedforward control, which explicitly considers the effect of the zero-order hold by constructing a multirate system.
Alternatively, a discrete-time-domain approach is proposed in[21], which compensates both intrinsic and discretization unstable zeros, simultaneously. This method can achieve perfect
tracking at each sampling point, but it can cause high oscillations in the intersampling behavior by direct cancellation of
the discretization zeros [1], [22]. For high-precision motion
systems such as wafer scanners and printing systems, preactuation methods have been proposed and applied, focusing on
iterative learning control and linear periodically time-varying
systems [23], [24].
We propose a multirate feedforward with preactuation [25]and experimentally validate its effectiveness [26]. The proposed method solves problems 1) and 2) independently, regardless of the desired trajectory given that it is n − 1 differentiable (where n denotes the order of the nominal plant).
This paper concludes our findings and provides an in-depth
review and comparison of the effectiveness of the independent inversion with a wide range of established techniques in
the field.
First, the unstable intrinsic zeros in the continuous-time domain are offset through a state trajectory generation using atime-axis reversal. Then, the stable inversion of the discretization zeros is calculated through a multirate feedforward approach [27]. Note that the multirate control is commonly referred to as the set of techniques used to improve feedback control performance [28]–[30]. These methods use multiple sampling times to enhance the performance/cost tradeoff present insingle-rate feedback control. In contrast, this multirate method
is for model-based feedforward control, using multiples of thesampling time to obtain a stable inversion of the discretization zeros. A multirate feedforward method is proposed in [27]and extended to systems with continuous-time stable zeros [7]and adaptive control [31]. Fujimoto et al. [8] proposed a multirate NPZI method, which ignores only intrinsic (continuoustime) unstable zeros, whereas the single-rate NPZI method introduced in Section III-A and [14] ignores both intrinsic anddiscretization zeros simultaneously. This method [8] constructs
a stable inversion of the discretization zeros; however, it is still

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 2, APRIL 2019

affected by the approximation of the unstable intrinsic zeros.
The proposal described in this paper goes a step further, improving upon the model inversion and subsequently tracking
errors of systems with NMP zeros. It is an independent stable
inversion for unstable intrinsic zeros and discretization zeros,
which are inverted by preactuation and multirate feedforward,
respectively.
To demonstrate the effectiveness, this paper thoroughly compares the proposed method with both approximate and preactuated model-inversion methods by simulations and experiments.
This paper shows that it is impossible to compensate for thezero-order hold delay by only previewing the reference of CPMI
method. The considered system has several intrinsic and discretization unstable zeros, for which conventional approximate
methods are unable to achieve perfect tracking. Simulation andexperimental results show that the proposed method effectively
reduces tracking errors. Note that the performance of modelbased methods, including the proposed method, is affected bymodeling errors, which can be mitigated by feedback control.
However, because of the presence of continuous-time unstable zeros in the lower frequency range, it is difficult to design
high-bandwidth feedback controllers. Hence, for a system withunstable zeros, precise system identification inherently plays acrucial role. Albeit, for the range of applications targeted, extensive modeling and identification are at present commonly
employed.
II. NOTATIONS AND DEFINITIONS
The plant in continuous-time domain is defined as Pc (s).
Ps [zs ] denotes the discretized plant of Pc (s) by the zero-orderhold with sampling time Tu , where s denotes a complex variablefor Laplace transform and zs = eT u s .
A continuous-time transfer function of the nominal plant is
Pn (s) =

B(s)
A(s)

(1)

where A(s) is the nth order and B(s) is the mth order, expressedasA(s) =

sn + an −1 sn −1 · · · + a0b0

bm sm + bm −1 sm −1 + · · · + b0
B(s) =.
b0(2)

Note that (1) is irreducible. The state and output equations of(1) are defined asẋ(t) = Ac x(t) + bc u(t), y(t) = cc x(t).

(3)

The discretized plant by a zero-order hold with sampling time
Tu is defined asx[k + 1] = As x[k] + bs u[k], y[k] = cs x[k] Tu
Ac T uAs = e, bs =
eAc τ dτ · bc , cs = cc .
0

(4)
(5)

OHNISHI et al.: PREACTUATED MULTIRATE FEEDFORWARD CONTROL

865

where nu denotes the order of B ust [zs ]. The feedforward controller is then designed as
Cf f [zs ] = P̃n−1 [zs ] =Fig. 2. Approximated plant inverse feedforward control configuration,
where C f f [zs ] = P̃ n−1 [zs ]. P̃ n [zs ] is the approximated nominal plantmodel without unstable zeros. P c (s) and C f b [zs ] denote the continuoustime plant and the discrete-time feedback controller, respectively. H andS denote a holder and a sampler, respectively.

zs−q A[zs ].
B st [zs ]B̃ ust [zs ](9)The difference among the three prominent methods (NPZI,ZPETC, and ZMETC) is the design of B̃ ust [zs ] and q (0 ≤q ∈ Z). Refer to [14] for an overview. A Bode plot of tracking
control performances of single-rate model-inversion methods isshown in Fig. 3, where a plant shown in Fig. 6(b) is used as acase study.
B. CPMI Methods
Several CPMI methods are proposed in [17]–[20]. However,these papers do not consider the effect of the zero-order hold.
The key difference between CPMI methods and the proposed
method is the zero-order hold consideration by multirate feedforward control. A block diagram of the CPMI method is shownin Fig. 4.
1) Step 1. Stable–unstable decomposition: B(s)−1 definedin (2) is decomposed into a stable part F st (s) and an unstablepart F ust (s) as follows:
B(s)−1 =Fig. 3. Example of tracking control performance of single-rate modeln
inversion methods C f f [zs ]P s [zs ]zs pre , where n pre denotes the previewed number of samples for ZPETC. The plant shown in Fig. 6(b)is used for the case study.

In the discrete transfer function, it is defined as
Pn [zs ] = cs (zs I − As )−1 bs .

(6)b0
bm sm + bm −1 sm −1 + · · · + b0

(10)

= F st (s) + F ust (s)f st (t) = L−1 F st (s) , f¯ust (t) = L−1 F ust (−s)

(11)
(12)

where L−1 denotes the inverse unilateral Laplace transform.
Note that F ust (−s) is stable.
2) Step 2. Stable part feedforward control input: Stable part
feedforward control input is calculated by a convolution between
the reference ACPM I r CPM I (t) and f st (t) as tst
f st (t − τ )ACPM I r CPM I (τ )dτ
(13)
uf f (t) =−∞III. SINGLE-RATE MODEL-INVERSION METHODS

where

A. Approximate Model-Inverse Methods
When a nominal plant Pn [zs ] discretized by the zero-orderhold has an unstable zero, the inversion system in the feedforward controller Pn−1 [zs ] becomes unstable. To avoid this problem, several approximate model-inverse feedforward controllers
P̃n−1 [zs ] have been proposed in the literature. A generalized
block diagram is shown in Fig. 2, in which Ty and Tu denotethe sampling and control periods, respectively. In this section,
Ty = Tu because only single-rate (zs = eT u s ) control methods
are considered. Approximations decompose the nominal plantin a stable B st [zs ] and unstable part B ust [zs ], as follows:
B st [zs ]B ust [zs ]B[zs ]
Pn [zs ] ==A[zs ]
A[zs ]
B ust [zs ] = bu n u zsn u + bu (n u −1) zsn u −1 + · · · + bu 0

(7)
(8)

1
a0 a1 · · · an −1b0
Tr CPM I = 1 ρ · · · ρn r(t)ACPM I =

1


(14)

where ρ denotes the Heaviside operator [33]. Equation (13) canbe written as
 t
ust
(t)
=
f st (t − τ )ACPM I r CPM I (τ )dτ
(15)
ff
0

assuming r CPM I (t) = 0 when t < 0.
3) Step 3. Unstable part feedforward control input: Unstablepart feedforward control input is calculated by: 1) a convolution
between the time-axis reversed reference ACPM I r CPM I (−τ̄ )and the stable signal f¯ust (t̄ − τ̄ ); and 2) a time-axis reversal. This procedure results in an infinity time preactuation for866

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 2, APRIL 2019

Fig. 4. CPMI method. The stable inversion of the continuous-time plant is calculated through the dotted boxed blocks. Note that this step does notconsider the zero-order hold of the plant.

Fig. 5. Proposed preactuated multirate feedforward control. S, H, and L denote a sampler, holder, and lifting operator [32], respectively. z and zs
denote es T r and es T u , where T r = nT u , respectively.

A. Controllable Canonical Form Realization

continuous-time unstable zeros compensation asuust
f f (t) =

 t̄
−∞



f¯ust (t̄ − τ̄ )ACPM I r CPM I (−τ̄ )dτ̄ t̄=−t. (16)

4) Step 4. Total feedforward control input: The total feedfor-

ward control input is calculated by a sum of the stable and theunstable part feedforward control input as
ust
uo (t) = ust
f f (t) + uf f (t).

(17)

Then, uf f (t) is sampled into uf f [k] by a zero-order hold. No
consideration of the zero-order hold causes a delay. This will be
discussed in Section V-F and shown in Fig. 11.
IV. PREACTUATED MULTIRATE FEEDFORWARD CONTROL
We propose a multirate feedforward control with preactuation to design a stable inversion feedforward controller forplants with unstable intrinsic and discretization zeros. Thismethod solves the unstable zero inversion problem in threesteps. First, we calculate the controllable canonical form realization for the nominal plant. Second, the stable inversion
for the unstable intrinsic zeros is calculated using a time- andimaginary-axis reversal in a continuous-time domain. Then,stable inversion for unstable discretization zeros is calculatedusing a multirate feedforward proposed in [27]. A block diagram of the preactuated multirate feedforward control is shownin Fig. 5.

We realize the nominal plant shown in (3) as the controllable
canonical form
T
x(t) = x1 (t) x2 (t) · · · xn (t)
⎡
⎤
0
1
0
···
0
⎢
⎥
⎢ 0
0
1
···
0 ⎥
⎢
⎥
⎢
⎥
Ac = ⎢
⎥
..
⎢
⎥
.
⎢
⎥
⎣
⎦
−a0 −a1 −a2 · · · −an −1Tbc = 0 0 · · · b0
cc = 1

b1
b0

···

bm
b0

0 ···

(18)

0 .

The objective to use the controllable canonical form is to
use the differential relationship of the state vector, as shownin (20).
B. State Trajectory xd Generation
First, a desired state trajectory xd for the multirate feedforward is generated. The state trajectory xd is defined asxd (t) = x1d (t) x2d (t) · · ·xn d (t)

T

.

(19)

OHNISHI et al.: PREACTUATED MULTIRATE FEEDFORWARD CONTROL

867

Owing to the controllable canonical form realization [see (18)],
we haveT
xd (t) = x1d (t) ρx1d (t) · · · ρn −1 x1d (t) .
(20)
According to (3), to track the reference position trajectoryr(t), the desired state trajectory xd (t) should satisfy

b1
b0

⎡

···

bm
b0

0

···

x1d (t)⎤⎥⎢⎢ ρx1d (t) ⎥
⎥
⎢
⎥
⎢ 2
ρ x1d (t) ⎥.
0 ⎢
⎥
⎢
⎥
⎢
..
⎥
⎢
.
⎦
⎣

xust
d (t) is calculated as follows: first, a convolution of the time-

reversed reference position trajectory r(−t̄) and the stable signal f¯ust (t̄) is calculated. Next, the time axis is reversed. The
mathematical proof is provided in [34].
4) Step 4. State trajectory generation: The state trajectory
xd (t) is obtained byust
xd (t) = xst
d (t) + xd (t).

ρn −1 x1d (t)(21)
From (2) and (21), x1d (t) is obtained as
x1d (t) =

1
r(t).
B(ρ)

(22)

Therefore, the entire vector xd (t) is obtained as
xd (t) =
where

1
r(t)
B(ρ)


T
r(t) = r1 (t) r2 (t) · · · rn (t)T= 1 ρ · · · ρn −1 r(t).

(23)

bm

sm + bb0
m −1 + · · · + b
m −1 s0= F st (s) + F ust (s)f st (t) = L−1 F st (s) , f¯ust (t) = L−1 F ust (−s) .

(24)

(25)
(26)
(27)

sired state trajectory xst
d (t) for the stable part is forwardly generated as follows:
T
 st
st
xst
xst
d (t) = x1d (t) x2d (t) · · ·n d (t) t=
f st (t − τ )r(τ )dτ.
(28)
−∞

0

assuming r(t) = 0 when t < 0.

C. Feedforward Input Generation From the State
Trajectory
The effect of unstable discretization zeros can be avoidedusing the multirate feedforward control [27]. Fig. 5 shows that
there are three time periods Ty , Tu , and Tr denoting the periods
for y(t), u(t), and r(t), respectively. These periods are set as
Tr = nTu = nTy .
The multirate system of (4) is given as
(32)

where

2) Step 2. Stable part state trajectory generation: The de-Equation (28) can be written as t
st
xd (t) =f st (t − τ )r(τ )dτ(31)

x[i + 1] = Ax[i] + Bu[i], y[i] = cx[i]However, (23) has unstable poles when the plant Pn (s) has
unstable zeros. To prevent the diversion of the state trajectory
xd (t), the stable–unstable decomposition and time-axis reversal
techniques are applied.
1) Step 1. Stable–unstable decomposition: Stable–unstablecomposition is performed as same as the CPMI method shownin Section III-B1
B(s)−1 =

t̄=−t

−∞

r(t) = cc xd (t)= 13) Step 3. Unstable part state trajectory generation: The desired state trajectory xustd (t) for the unstable part is generated
by
T
 ust
ust
xust
xust
d (t) = x1d (t) x2d (t) · · ·n d (t)
 t̄


=
.
(30)
f¯ust (t̄ − τ̄ )r(−τ̄ )dτ̄ (29)
A = Ans , B = Ans −1 bs

Ans −2 bs···

As b sc = cs , x[i] = x(iTr )

bs


(33)

by calculating the state transition from t = iTr = kTu to t =(i + 1)Tr = (k + n)Tu . Here, the input vector u[i] is definedin the lifting form
T
u[i] = u1 [i] u2 [i] · · · un [i]T= u(kTu ) u((k + 1)Tu ) · · · u((k + n − 1)Tu ) .
(34)
According to (32), the feedforward input uo [i] is obtained fromthe previewed state trajectory xd [i + 1] as follows:uo [i] = B −1 (I − z −1 A)xd [i + 1]

(35)

where z = esT r .
V. EXPERIMENTAL VALIDATION
A. Experimental Setup
The experimental setup used to verify the proposed theory is shown in Fig. 6. It is an air-guided single-degree-offreedom flexible stage driven by a set of linear motors. The position of the table and the driven carriage are measured by two
linear encoders with 1-nm precision. By using interior or exterior division, we can measure any vertical imaginary position. In this paper, the height of the measurement point is set as
Lm = 0.300 m by exterior division to have continuous-time unstable zeros. The applied force, i.e., current, is measured through868

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 2, APRIL 2019

Fig. 6. Experimental high-precision positioning stage and its frequency
response function (FRF) data and eighth-order model (L m = 0.300 [m]).
(a) High-precision positioning stage. (b) FRF data and model of (a).

Fig. 7.
(b) P s .

Pole-zero map of the identified model shown in Fig. 6. (a) P c .

an inductive current sensor with a 25-A range and quantized witha 14-bit A/D converter, i.e., 3.05-mA step.
B. System Identification
The dominant system dynamics of the setup are modeled
through frequency-domain identification techniques; see [35].
First, the nonparametric frequency response is measured fromopen-loop experiments using periodic multisine excitations withquasi-logarithmic spacing to cover the wide frequency band ofinterest. Subsequently, an eighth-order parametric transfer function model Pn (s) is estimated iteratively using a Levenberg–
Marquardt method with a maximum likelihood criterion. Some
additional weighting is applied in the middle frequency range(10–400 Hz) to reflect the region in which high model accuracy
is desired for feedforward design.
The measured frequency response and the estimated eighthorder model are shown in Fig. 6(b). The identified continuoustime pole-zero map is shown in Fig. 7(a), in which Pn (s)contains continuous-time unstable zeros at s = +126, +878
and stable zeros at s = −1090, −120, −92.7 ± 1.20j. Additionally, the pole-zero map of the discretized transfer function with Tu = 400 μs is shown in Fig. 7(b), in which Ps [zs ]contains an unstable discretization zero.
C. Conditions
The setup has a current controller as inner loop, which has
a 1-kHz bandwidth, with a 12.5-kHz sampling and a position controller as outer loop with 2.5-kHz sampling. The position feedforward controllers are designed with Tu = 400 μs
sampling time. The position feedback controller is designed

Fig. 8. Feedback control performance. The feedback controller is designed as a PID controller and a second-order shaping filter. The designed gain and phase margins are 14.2 dB (at 10 Hz) and 37.2◦ (at2.8 Hz), respectively. (a) Bode plot of the open-loop system. (b) Nyquist
plot of the open-loop system.

as a proportional–integral–derivative (PID) controller and asecond-order shaping filter. The feedback control performanceis shown in Fig. 8, which shows that it is difficult to achieve
high bandwidth for a plant with continuous-time unstable zeros in the low-frequency range. This indicates that the feedback
controller cannot help the trajectory tracking performance. The
block diagram shown in Fig. 2 is used for NPZI, ZPETC, andZMETC methods. Note that in the configurations of Figs. 2, 4,
and 5, without modeling error or disturbances, the output of the
feedback controller Cfb [zs ] is zero.
The target trajectory is given as a 0.05 second step reference
interpolated by a 15th-order polynomial and is shown in, i.e.,
Fig. 9(a). As for the CMPI and the proposed methods, the control
input is applied from t = −0.0428 s to preactuate the system.
This time length is determined by the current sensor resolution.
This is 5.38 times longer than the time constant of the dominantunstable zero in continuous-time domain. The effect regarding
short time preactuation compared to the time constant of theunstable zeros is discussed in [36].
D. Simulation Results
Simulation results are shown in Fig. 9. Fig. 9(f) demonstrates
that the proposed method can achieve perfect tracking without any undershoot or overshoot. In contrast, Fig. 9(a) and (e)shows that the NPZI, ZMETC, and ZPETC controllers create undershoot and/or overshoot. The feedback only (FB only)
case shows a slow response, which demonstrates that a plantwith a slow continuous unstable zero cannot rely on a feedback
controller only for reference tracking. Fig. 9(g) shows that the
CMPI method creates a similar current reference to the proposed
method. However, due to the lack of considering the zero-orderhold, the current reference is delayed compared to the proposal,
and it results in the tracking error shown in Fig. 9(f). Notethat these simulations contain no modeling error and/or disturbances; hence, the feedback current is zero for NPZI, ZPETC,ZMETC, CMPI, and the proposed method.
E. Experimental Results
The experimental results are shown in Fig. 10. The trend in theexperimental results are in good agreement with the simulations

OHNISHI et al.: PREACTUATED MULTIRATE FEEDFORWARD CONTROL

869

Fig. 9. Simulation with the eighth-order model shown in Fig. 6(b). Note that in (d) and (h), feedback current is zero for NPZI, ZPETC, ZMETC,
CPMI, and the proposal due to no modeling error and disturbance assumption. (a) Position. (b) Tracking error. (c) Feedforward current. (d) Feedback
current. (e) Position. (f) Tracking error. (g) Feedforward current. (h) Feedback current.

Fig. 10. Experiment with the stage shown in Fig. 6. The eighth-order nominal model is used for feedforward controller design. (a) Position. (b)Tracking error. (c) Feedforward current. (d) Feedback current. (e) Position. (f) Tracking error. (g) Feedforward current. (h) Feedback current.

shown in Fig. 9. From Fig. 10(e) and (g), during preactuation,
the output position has almost no motion. After preactuation,
the proposed method has almost no undershoot or overshoot. As
summarized in Table I, the proposed method is experimentally
validated.
Fig. 10(d) and (h) shows that, except the feedback only case,the feedback current references are quite small compared to
the feedforward current references. This is because the nominal
output yo [k], which is calculated by the feedforward current
reference uo [k] and the nominal plant Pn [zs ], almost matches
the actual output y[k] due to the well-identified nominal model.

TABLE IMAXIMUM TRACKING ERROR (IN MICROMETERS) OF FIGS. 9 AND 10*Intersample tracking error.

F. Observations of the Effect of Multirate Feedforward
As described above, the feedforward current reference issimilar between the CPMI method and the proposed method,870

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 2, APRIL 2019

Fig. 11. Error comparison between the proposed method and the shifted CPMI method. Dots are illustrated by every T r = nT u . It shows thatthe proposed method achieves the perfect tracking for T r = nT u , which is not achievable by the CPMI method (simulation). (a) Tracking error.
(b) Tracking error (zoom). (c) Tracking error (zoom2).

as shown in Figs. 9(g) and 10(g). The difference of the two
methods is the zero-order hold consideration by the multirate
feedforward.
Fig. 11 shows the tracking error comparison between the
proposed and shifted CPMI methods. It shows that the zeroorder hold delay cannot be compensated by just shifting the current reference even by a noninteger sample shift. These results
clearly show the importance of multirate feedforward control,
which compensates for the zero-order hold delay. Fig. 11(c)shows that perfect tracking is achieved by the proposed method
for every Tr = nTu .
VI. CONCLUSION
In the discretized domain, there are two types of zeros:
1) the intrinsic zeros, which have counterparts in the continuoustime domain; and 2) the discretization zeros generated bydiscretization by the zero-order hold. The proposed preactuated multirate feedforward control deals with problems 1) and2) independently. On one hand, the unstable intrinsic zeros arecompensated by the preactuation. On the other hand, the unstable discretization zeros are compensated by the multirate
feedforward control with preview. The multirate feedforward
controller generates the feedforward input, which achieves perfect tracking for the designed state trajectory. The simulation
results show that the zero-order hold delay cannot be compensated by just shifting the reference and underline the importanceof multirate feedforward.
This study experimentally validates the proposed method using a high-precision positioning stage with continuous-time unstable zeros. Additionally, this system has a discretization zero.
Owing to a well-identified eighth-order model, the experimental
results follow the simulations. The experimental result obtainedwith the proposed method strongly reduces the tracking error
and achieves almost zero undershoot and overshoot.
REFERENCES[1] K. L. Moore, S. P. Bhattacharyya, and M. Dahleh, “Capabilities and limitations of multirate control schemes,” Automatica, vol. 29, no. 4, pp. 941–
951, 1993.
[2] T. Hagiwara, T. Yuasa, and M. Araki, “Stability of the limiting zeros ofsampled-data systems with zero-and first-order holds,” Int. J. Control,
vol. 58, no. 6, pp. 1325–1346, 1993.

[3] T. Hagiwara, “Analytic study on the intrinsic zeros of sampled-data systems,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 261–263, Feb. 1996.
[4] K. Åström, P. Hagander, and J. Sternby, “Zeros of sampled systems,”
Automatica, vol. 20, no. 1, pp. 31–38, 1984.
[5] H. Butler, “Position control in lithographic equipment,” IEEE Control
Syst. Mag., vol. 31, no. 5, pp. 28–47, Oct. 2011.
[6] A. Hara, K. Saiki, K. Sakata, and H. Fujimoto, “Basic examination on
simultaneous optimization of mechanism and control for high precision
single axis stage and experimental verification,” in Proc. 34th Annu. Conf.
IEEE Ind. Electron. Soc., 2008, pp. 2509–2514.
[7] K. Saiki, A. Hara, K. Sakata, and H. Fujimoto, “A study on high-speed andhigh-precision tracking control of large-scale stage using perfect tracking
control method based on multirate feedforward control,” IEEE Trans. Ind.
Electron., vol. 57, no. 4, pp. 1393–1400, Apr. 2010.
[8] H. Fujimoto, K. Fukushima, and S. Nakagawa, “Vibration suppression
short-span seeking of HDD with multirate feedforward control,” in Proc.
Amer. Control Conf., 2006, pp. 582–587.
[9] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics.
New York, NY, USA: Springer, 2011.
[10] T. Miyajima, H. Fujimoto, and M. Fujitsuna, “A precise model-based
design of voltage phase controller for IPMSM,” IEEE Trans. Power Electron., vol. 28, no. 12, pp. 5655–5664, Dec. 2013.
[11] G. Oriolo, A. D. Luca, and M. Vendittelli, “WMR control via dynamic
feedback linearization: Design, implementation, and experimental validation,” IEEE Trans. Control Syst. Technol., vol. 10, no. 6, pp. 835–852,
Nov. 2002.
[12] R. Stengel, Flight Dynamics. Princeton, NJ, USA: Princeton Univ. Press,
2004.
[13] B. Rigney, L. Y. Pao, and D. Lawrence, “Nonminimum phase dynamic
inversion for settle time applications,” IEEE Trans. Control Syst. Technol.,vol. 17, no. 5, pp. 989–1005, Sep. 2009.
[14] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “Analysis and comparison of three discrete-time feedforward model-inverse control techniques for nonminimum-phase systems,” Mechatronics, vol. 22, no. 5,
pp. 577–587, 2012.
[15] J. Wen and B. Potsaid, “An experimental study of a high performance motion control system,” in Proc. Amer. Control Conf., 2004, vol. 6, pp. 5158–
5163.
[16] M. Tomizuka, “Zero phase error tracking algorithm for digital control,” J.
Dyn. Syst., Meas., Control, vol. 109, pp. 65–68, 1987.
[17] L. Hunt, G. Meyer, and R. Su, “Noncausal inverses for linear systems,” IEEE Trans. Autom. Control, vol. 41, no. 4, pp. 608–611,
Apr. 1996.
[18] S. Devasia, D. Chen, and B. Paden, “Nonlinear inversion-based output
tracking,” IEEE Trans. Autom. Control, vol. 41, no. 7, pp. 930–942,
Jul. 1996.
[19] L. Biagiotti and C. Melchiorri, Trajectory Planning for Automatic Machines and Robots. New York, NY, USA: Springer, 2008.
[20] T. Shiraishi and H. Fujimoto, “A reference trajectory generation forsystem with unstable zeros considering negative-time domain analysis,” in Proc. IEEJ Int. Workshop Sens., Actuation, Motion Control,
2015, pp. 1–6.
[21] L. Marconi, G. Marro, and C. Melchiorri, “A solution technique for almost
perfect tracking of non-minimum-phase, discrete-time linear systems,”
Int. J. Control, vol. 74, no. 5, pp. 496–506, 2001.

OHNISHI et al.: PREACTUATED MULTIRATE FEEDFORWARD CONTROL[22] T. Sogo and M. Joo, “Design of compensators to relocate sampling zerosof digital control systems for DC motors,” SICE J. Control, Meas., Syst.
Integr., vol. 5, no. 5, pp. 283–289, 2012.
[23] J. van Zundert and T. Oomen, “Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems,”
Int. J. Control, doi: 10.1080/00207179.2017.1380315.
[24] L. Blanken, G. Isil, S. Koekebakker, and T. Oomen, “Flexible ILC: Towards a convex approach for non-causal rational basis functions,” IFACPapersOnLine, vol. 50, no. 1, pp. 12107–12112, 2017.
[25] W. Ohnishi and H. Fujimoto, “Multirate feedforward control with state
trajectory generation based on time axis reversal for plant with continuous
time unstable zeros,” in Proc. IEEE Int. Conf. Adv. Intell. Mechatronics,
2016, pp. 689–694.
[26] W. Ohnishi, T. Beauduin, and H. Fujimoto, “Preactuated multirate feedforward for a high-precision stage with continuous time unstable zeros,”
IFAC-PapersOnLine, vol. 50, no. 1, pp. 10907–10912, 2017.
[27] H. Fujimoto, Y. Hori, and A. Kawamura, “Perfect tracking control based on
multirate feedforward control with generalized sampling periods,” IEEE
Trans. Ind. Electron., vol. 48, no. 3, pp. 636–644, Jun. 2001.
[28] S. Bittanti, F. Dell’Orto, A. Di Carlo, and S. M. Savaresi, “Notch filteringand multirate control for radial tracking in high-speed DVD-players,”
IEEE Trans. Consum. Electron., vol. 48, no. 1, pp. 56–62, Feb. 2002.
[29] T. Atsumi and W. Messner, “Compensating for ZOH-induced residual
vibrations in head-positioning control of hard disk drives,” IEEE/ASME
Trans. Mechatronics, vol. 19, no. 1, pp. 258–268, 2014.
[30] J. van Zundert and T. Oomen, “LPTV loop-shaping with application to
non-equidistantly sampled precision mechatronics,” in Proc. 15th Int.
Workshop Adv. Motion Control, 2018, pp. 467–472.
[31] H. Fujimoto and B. Yao, “Multirate adaptive robust control for discretetime non-minimum phase systems and application to linear motors,”
IEEE/ASME Trans. Mechatronics, vol. 10, no. 4, pp. 371–377, Aug. 2005.
[32] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. New
York, NY, USA: Springer, 1995.
[33] G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control System Design.
Upper Saddle River, NJ, USA: Prentice-Hall, 2000.
[34] T. Sogo, “On the equivalence between stable inversion for nonminimum
phase systems and reciprocal transfer functions defined by the two-sided
Laplace transform,” Automatica, vol. 46, no. 1, pp. 122–126, 2010.
[35] R. Pintelon, P. Guillaume, Y. Rolain, J. Schoukens, and H. Van Hamme,“Parametric identification of transfer functions in the frequency domain-asurvey,” IEEE Trans. Autom. Control, vol. 39, no. 11, pp. 2245–2260,
Nov. 1994.
[36] W. Ohnishi, T. Beauduin, and H. Fujimoto, “Finite time optimal preactuation for non-minimum phase systems considering control input andtracking error constraints,” in Proc. IEEE Conf. Control Technol. Appl.,
2018, pp. 1114–1119.

Wataru Ohnishi (S’13–M’18) received the B.E.,
M.S., and Ph.D. degrees from the University ofTokyo, Tokyo, Japan, in 2013, 2015, and 2018,
respectively.
He is currently a Research Associate with the
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, the University of Tokyo. His research interests include high-precision motion control.
Dr. Ohnishi is a member of the Institute ofElectrical Engineers of Japan.

871

Thomas Beauduin (M’16) received the B.S. and
M.S. degrees in mechanical engineering fromthe University of Leuven, Leuven, Belgium, in2012 and 2014, respectively.
In 2014, he joined the Hori-Fujimoto Laboratory, the Department of Advanced Energy, Graduate School of Frontier Sciences, the Universityof Tokyo, Tokyo, Japan, as a Research Assistant. Since 2017, he has been a Research Engineer with MTT Innovation Inc., a startup company from the Department of Computer Science,
University of British Columbia, Vancouver, BC, Canada. His research interests include nanoscale motion control and optics.

Hiroshi Fujimoto (S’99–M’01–SM’12) received
the Ph.D. degree in electrical engineering fromthe University of Tokyo, Tokyo, Japan, in 2001.
In 2001, he joined the Department of Electrical Engineering, Nagaoka University of Technology, Niigata, Japan, as a Research Associate.
From 2002 to 2003, he was a Visiting Scholarwith the School of Mechanical Engineering,Purdue University, West Lafayette, IN, USA. In
2004, he joined the Department of Electrical andComputer Engineering, Yokohama National University, Yokohama, Japan, as a Lecturer and became an Associate Professor in 2005. He has been an Associate Professor with the Universityof Tokyo since 2010. His research interests include control engineering,
motion control, nanoscale servo systems, electric vehicle control, motor
drive, visual servoing, and wireless motors.
Dr. Fujimoto was a recipient of the Best Paper Award from the IEEE
TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 2001 and 2013, the Isao
Takahashi Power Electronics Award in 2010, the Best Author Prize of
SICE in 2010, and The Grand Nagamori Award in 2016. He is a memberof the Society of Instrument and Control Engineers, the Robotics Societyof Japan, and the Society of Automotive Engineers of Japan.