State Trajectory Generation for MIMO Multirate Feedforward
using Singular Value Decomposition and Time Axis Reversal
Masahiro Mae1 , Wataru Ohnishi1 and Hiroshi Fujimoto1
Abstract— Multirate feedforward control provides a perfect
tracking control for a desired state trajectory in ideal theoretical condition. In this study, we propose a state trajectory
generation method from a desired output trajectory for a
multi-input multi-output (MIMO) system using singular value
decomposition and time axis reversal. This method provides
perfect tracking control in MIMO systems for a desired
output trajectory. We apply this method to a MIMO highprecision stage. This method improves the general applicabilityof multirate feedforward control for a MIMO system.

I. I NTRODUCTION
Multirate feedforward control is widely used for controlling precise systems such as high-precision stages andhard disk drives [1]. Multirate feedforward control solves
the problem of unstable zeros due to discretization by using
zero-order hold and provides a perfect tracking control (PTC)[2]. The word “perfect tracking control” is defined as “theplant output perfectly tracks the desired trajectory with zero
tracking error at every sampling point” [3].
High-precision stages are important for manufacturing
semiconductors and liquid crystal panels [4]. Their demandin society is increasing annually [5]. Conventional highprecision stages have one degree of freedom (DOF) andare mechanically constrained except along the long stroke.
However, demand for more precise control in recent years
has spurred development of high-precision stages with sixDOFs that are lifted by air bearings [6]. These systems arecontrolled as a multi-input multi-output (MIMO) system. In
MIMO systems, there is a coupling problem between each
input and output. MIMO systems are typically controlled
by a single-input single-output (SISO) controller after being
decoupled with a precompensator [7]. In this method, perfect
tracking control cannot be achieved because the precompensator does not consider unstable zeros due to the discretization. When a plant has unstable zeros, perfect tracking cannot
be achieved by a single-rate control scheme [3]. Therefore,
several studies are conducted for the approximated plantmodel inverse approaches in the single-rate control schemesuch as nonminimum-phase zeros ignore (NPZI) method [8],zero-phase-error tracking controller (ZPETC) method [3] andzero-magnitude-error tracking controller (ZMETC) method[9]. For these reasons, a MIMO multirate feedforward controller is required for perfect tracking control in a MIMO
system [10]. In multirate feedforward control for a MIMO
system, the controller design is not unique because it depends
1 The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 2778561, Japan, mmae@ieee.org, ohnishi@ieee.org, fujimoto@k.u-tokyo.ac.jp

designing a matrix B from the generalized controllability
indices, and several types of controllers can be designed [11].
In multirate feedforward control, we can obtain a control
input that provides perfect tracking control for a desired state
trajectory [2]. Therefore, the desired state trajectory must be
generated from the desired output trajectory. In conventional
multirate feedforward control in the SISO system, the system
is represented in a controllable canonical form and thedesired state trajectory is generated by convolving the output
equation using a differential relationship between the statevariables [1]. In multirate forward control for a MIMO system, not all state variables can be expressed in a differential
relation in some cases due to the existence of an off-diagonalterm in the matrix C of the output equation. In addition,
when the nominal plant has unstable zeros in continuous-time
domain, the inverse system becomes unstable. Therefore, thecontinuous-time stable inversion [12] is studied. This method
provides perfect tracking at each sampling point, but highoscillations occur in the inter-sampling behavior because thismethod uses the inverse of discretization zeros [13], [14].
For this reason, the independent stable inversion of unstable
intrinsic and discretization zeros by a preactuation and amultirate feedforward [15] is studied. In this method, thestable inversion for the unstable intrinsic zeros is calculatedusing a time axis reversal and imaginary axis reversal ina continuous-time domain. Then, the stable inversion for
unstable discretization zeros is calculated using a multirate
feedforward. To deal with unstable intrinsic and discretization zeros separately, inter-sampling oscillations becomes
smaller.
In this study, we propose a method for generating adesired state trajectory from a desired output trajectory for
multirate forward control in a MIMO system using singularvalue decomposition. In addition, we discuss the convolution
instability due to unstable zeros of the plant in continuoustime domain or several calculation errors. In these cases, we
propose a calculation method that uses time axis reversal[15]. In [15], SISO systems are only considered, but in thispaper, we expand a calculation method that uses time axis
reversal for MIMO systems.
We focus on two-DOFs in a six-DOF high-precision stage
with translation along the x axis and rotation around they axis, which is a dual-input dual-output system with acoupling problem. We use the state trajectory generation
method using singular value decomposition and time axis
reversal. We also design a multirate feedforward controller
for the MIMO system. We verify the effectiveness of the
proposed method through simulation.

II. S TATE TRAJECTORY GENERATION FOR SINGLE - INPUTSINGLE - OUTPUT SYSTEM
The conventional method for generating a desired state
trajectory from a desired output trajectory in the SISO system
will be described below.
The transfer function of the plant is given by
B(s)
A(s)
bm sm + bm−1 sm−1 + · · · + b0=.
sn + an−1 sn−1 + · · · + a0

P (s) =(1) 
x1 (t)0x(t)d  2   0
 . = dt  ..  
xn (t)−a0
y(t)= b0

1
0

···
···
..
.

−a1

···

···

bm

b1

  
0
x1 (t)  x2 (t)  0    ..  +  ..  u(t),  .  .
1
xn (t)−an−1x1 (t)

 x2 (t) 
0 · · · 0  . .
 .. 
xn (t)
0
0



(2)

(3)

Since each state variable has a differential relationship withthe state equation (2), convolution of the output equation (3)in continuous-time
domain gives the desiredT state trajectory
xd (t) = x1d (t) x2d (t) · · · xnd (t) from the desired
output trajectory yd (t) as
x1d (t) = L−1

1
bm sm + · · · + b1 s + b0
yd (s) .

1) Singular value decomposition of B matrix: Singularvalue decomposition of the matrix B is given by
B = U ΣV H ,(7)

where U ∈ Rn×n , Σ ∈ Rn×m , V ∈ Rm×m .
The elements of Σ are given by
 
∆
Σ=
,
O
∆ = diag(σi ) (i = 1 · · · m ∈ N),

The state space representation of this plant in a controllable
canonical form is given by

B. Singular value decomposition for multi-input multi-output
system(4)

where L−1 is an inverse Laplace transform.
III. S TATE TRAJECTORY GENERATION FOR MULTI - INPUT
MULTI - OUTPUT SYSTEM
The proposed method of generating the state trajectoryfrom the output trajectory in the MIMO system will be
described below. In this study, the plant is assumed to be
an m-input m-output nth order plant, where m < n andrank(B) = m. This system does not have continuous-time
unstable zeros.

where σi (i = 1, 2, . . . , m ∈ N) are the singular values ofthe matrix B.
Multiplying both sides of (7) by U H from the left yields
U H B = ΣV H


∆V H
=
.
O

2) Laplace transform for state equation: The Laplacetransform of the state equation (5) is given bysx(s) = Ax(s) + Bu(s).

The state equation and the output equation for an m-input
m-output nth order plant are given byẋ(t) = Ax(t) + Bu(t),(5)y(t) = Cx(t),(6)

where the state variables are x(t) ∈ Rn×1 , each input isu(t) ∈ Rm×1 , each output is y(t) ∈ Rm×1 , and the matrices
are A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rm×n .

(9)

The state variable conversion of (9) using the transformation matrix U H as x̃(s) = U H x(s) is given bys(U H x(s)) = (U H AU )(U H x(s)) + (U H B)u(s)⇔ (sI − U H AU )x̃(s) = (U H B)u(s)


∆V H
u(s).
⇔ W (s)x̃(s) =O
W (s) of (10) is represented as
Wu (s)
W (s) =,Wl (s)(10)(11)

where W (s) = sI−U H AU , Wu (s) ∈ Cm×n , and Wl (s) ∈C(n−m)×n .
From (10) and (11)Wu (s)
∆V H
u(s).
(12)
x̃(s) =Wl (s)O
The extracted lower (n − m) rows of (12) are given byWl (s)x̃(s) = O.

A. Definition of multi-input multi-output system

(8)

(13)

3) Laplace transform for output equation: The Laplacetransform of the output equation (6) is given byy(s) = Cx(s).

(14)

The state variable conversion of (14) using the transformation matrix U H as x̃(s) = U H x(s) is given byy(s) = (CU )(U H x(s))⇔ (CU )x̃(s) = y(s).

(15)

4) State trajectory generation: The concatenate matrix
from (13) and (15) is given by




CU
y(s)
x̃(s) =.
(16)
Wl (s)O
From (16), the state trajectory x̃(s) from the output trajectoryy(s) in the MIMO system is given by

−1 

CU
y(s)
.
(17)
x̃(s) =Wl (s)O
From the state variable conversion of (17) using the transformation matrix U as x(s) = U x̃(s), the state trajectoryx(s) from the output trajectory y(s) in the MIMO system
is given by

−1 

CU
y(s)
(U x̃(s)) = U
Wl (s)
O

−1 

CU
y(s)
⇔ x(s) = U.
(18)
Wl (s)
O
By inverse Laplace transform of (18), the desired state
trajectory xd (t) can be obtained from the desired output
trajectory yd (t) in the MIMO system:
" 
−1 
#
CU
y
(s)
d
xd (t) = L−1 U.
(19)
Wl (s)O
C. Time axis reversal for unstable inverse system
The desired state trajectory in (19) becomes unstable inseveral cases depending on whether the plant has continuoustime unstable zeros or several calculation errors. In this study,
we solve this problem using the state trajectory generation
method with time axis reversal [15]. We generate a stable
desired state trajectory that provides perfect tracking control.
In this study, it is assumed that the differential values of thedesired output trajectory are given up to n−1 differentiations.
1) Stable-unstable decomposition: Fij (s) defined in (20)is decomposed into a stable part Fijst (s) and an unstable part
Fijust (s) as follows:

−1
CU
U
= Fij (s) (i, j ∈ {1, 2, . . . , n})(20)
Wl (s)= Fijst (s) + Fijust (s),


st
fij
(t) = L−1 Fijst (s) ,


ust
f¯ij
(t) = L−1 Fijust (−s) .

(21)
(22)
(23)

Note that Fijust (−s) is stable.
2) Stable part state trajectory generation: The desired
state trajectory xst
d (t) for the stable part is forwardly generated as follows:
 st
T
st
st
xst
d (t) = x1d (t) x2d (t) · · · xnd (t)Z tyd (τ )st=fij (t − τ )
dτ.
(24)
O
−∞
(24) can be written asZ t
yd (τ )stst
xd (t) =fij (t − τ )dτ.
O0assuming yd (t) = 0 when t < 0.

3) Unstable part state trajectory generation: The desired
state trajectory xust
for the unstable part is generated by
d

T
ust
ust
ust
xust
d (t) = x1d (t) x2d (t) · · · xnd (t)Z t̄y (−τ̄ )ust
f¯ij(t̄ − τ̄ ) d
.
(26)
dτ̄
=
O
−∞
t̄=−t

xust
d is calculated as follows. First, a convolution between thetime reversed desired output trajectory yd (−t̄) and the stable
ust
signal f¯ij(t) is calculated. Next, the time axis is reversed.
A mathematical proof is provided in [16].
4) State trajectory generation from stable-unstable parts:
The desired state trajectory xd (t) is obtained byust
xd (t) = xst
d (t) + xd (t).

IV. M ULTIRATE FEEDFORWARD CONTROL FOR
MULTI - INPUT MULTI - OUTPUT SYSTEM
Multirate feedforward control provides PTC [2]. A digitaltracking control system typically has two samplers for thereference signal r(t) and the output y(t), and one holder
on the input u(t), as shown in Fig. 1. Therefore, three time
periods exist: Tr , Ty , and Tu , which represent the periods ofr(t), y(t), and u(t), respectively. A larger Tr or Ty value isdefined as the frame period Tf .
A. Definition of multi-input multi-output system
In an m-input p-output nth order MIMO system, the state
equation (28) and the output equation (29) describing thecontinuous-time plant are given byẋ(t) = Ac x(t) + Bc u(t),y(t) = Cc x(t),
Bc = bc1 · · ·

(28)
(29)




Cc = cc1

bcm ,

···

ccp

T

.

n

where the plant state is x ∈ R , the plant input is u ∈ Rm ,and the plant output is y ∈ Rp .
B. Design of B matrix from generalized controllability indices
The generalized controllability indices are defined as follows [10]:Definition 1 (Generalized Controllability Indices): The
generalized controllability indices of (Ac , Bc ) are defined
below for Ac ∈ Rn×n and Bc = [bc1 , · · · , bcm ] ∈ Rn×m ,respectively. If (Ac , Bc ) is a controllable pair, n linearly
independent vectors including the linear combination canbe selected from{bc1 , · · · , bcm , Ac bc1 , · · · , Ac bcm , · · · , An−1bcm }.
c
Setting ϕ as a set of these n vectors, σl and N are defined
by
m
X

(25)

(27)

σl = number{k|Ak−1
bcl ∈ ϕ},c(30)

σl = n,(31)

N = max(σl ).

(32)

l=1

In the MIMO system, n (= plant order) vectors are
selected from the generalized controllability indices, andthe full row rank matrix B can be designed for almost
all discretized sampling periods1 . Therefore, feedforward
controllers must be designed according to their different
forms.
C. Feedforward input generation from state trajectory
From (33), the control inputs uf f [i] required for PTC are
given by (34).
x[i + 1] = Ax[i] + Bu[i],uf f [i] = B −1 (I − z −1 A)x[i + 1],(33)

Symbol
xm
xg1
xg2
θy
fx
τy
Mx1
Cx1
Kx1
Mx2
Jθy
Cθy
Kθy
Lm
Lg2
Lf x

Meaning
Measured position of the fine stage
Position of the CoG of the planar air bearing and the air gyro
Position of the CoG of the fine stage
Measured attitude angle of the fine stage
Input force of the fine stage in the x direction
Input torque of the fine stage in the θy direction
Mass of the planar air bearing and the air gyro
Viscosity coefficient in the xg1 motion
Spring coefficient in the xg1 motion
Mass of the fine stage
Moment of inertia of the fine stage
Viscosity coefficient of the fine stage in the θy motion
Spring coefficient of the fine stage in the θy motion
Distance between the measurement point of xm and the CoRDistance between the CoR and the CoG of the fine stage
Distance between the CoR of the fine stage and the actuation point

Value
–
–
–
–
–
–
0.077 kg430 N/(m/s)
11000 N/m
5.3 kg0.10 kgm2
1.6 Nm/(rad/s)
1200 Nm/rad−0.0050 m−0.051 m−0.50 m(34)

where the matrices A, x[i], u[i], z and Tf are given by
A = eAc Tf , x[i] = x(iTf ), z = esTf , Tf = N Tu ,Tu[i] = u1 [i] · · · um [i]T= u11 [i] · · · u1σ1 [i] u21 [i] · · · umσm [i] .
A block diagram of the control system is shown in Fig. 1.
L is a discrete-time lifting operator [17]. L−1 outputs theelements of the nth dimensional vector uf f [i], which is input
at every period Tr , in the order from 1 to n by Tu = Tr /n.
V. S IMULATION
A. ModelingIn the simulation, we control the fine stage of the six-DOF
high-precision stage shown in Fig. 2(a). This fine stage is
supported by a six-DOF air bearing gravity canceller. In thisstudy, two-DOFs corresponding to translation x along thex axis and pitching θy around the y axis are controlled, as
shown in Fig. 2(b).
The equations of motion for translation and pitching of
the stage are given by (35) and (36) [18].
(Mx1 + Mx2 )ẍg1 + Cx1 ẋg1 + Kx1 xg1 + Mx2 Lg2 θ̈y = fx

(35)

(Mx2 L2g2 + Jθy )θ̈y + Cθy θ̇y + Kθy θy + Mx2 Lg2 (ẍg1 − gθy )= τy + fx Lf x

(36)

Convert xg1 to observable xm by (37).
xm (s) = xg1 (s) + Lm θy (s)TABLE IM ODEL PARAMETERS .

(37)

The parameters of the stage are shown in Table I.
aij , bik (i ∈ {2, 4}, j ∈ {1, 2, 3, 4}, k ∈ {1, 2}) fromthe expressions (35), (36), and (37), (ẍm , θ̈y ) explained by(xm , ẋm , θy , θ̇y ) are given by (38) and (39).
ẍm = a21 xm + a22 ẋm + a23 θy + a24 θ̇y + b21 fx + b22 τy(38)θ̈y = a41 xm + a42 ẋm + a43 θy + a44 θ̇y + b41 fx + b42 τy(39)The state equation (40) and the output equation (41) for thecontinuous-time plant are given byẋ(t) = Ac x(t) + Bc u(t),(40)y(t) = Cc x(t),(41)

1 This is possible because the controllability of a continuous-time system
is not preserved in the discrete system only if the two poles ηi andηjhave the same real parts, and the discretizing sampling period T satisfies
ηi = ηj + j 2kπ(k = ±1, ±2, . . .); furthermore, it is limited to onlyTseveral cases.

where the vectors x(t), u(t), and y(t), and the matrices Ac ,Bc , and Cc are given by
 
xm
 
 
ẋm 
fxxx(t) =   , u(t) =, y(t) = m ,
θy
τy
θy
θ̇y


0
1
0
0


a21 a22 a23 a24 
 , Cc = 1 0 0 0 ,Ac = 
 0
0 0 1 00
0
1 a41 a42 a43 a4400b21 b22   = bc1 bc2 .
Bc = 
0
0b41 b42
By using the zero-order hold in the state equation (40)and the output equation (41) for the continuous-time plantwith sampling period Tu , the state equation (42) and output
equation (43) for the discrete-time plant are given byx[k + 1] = As x[k] + Bs u[k],y[k] = Cs x[k],

(42)
(43)

where x[k], u[k], y[k], and Bs are given by
xm [k]
ẋm [k] , u[k] = fx [k] , y[k] = xm [k] ,x[k] =  θy [k] τy [k]
θy [k]
θ̇y [k]
Bs = bs1 bs2 .
B. State trajectory generation
In this study, we use a state equation and an output
equation in a balanced realization which is transformed
from the state equation (42) and the output equation (43)for considering the numerical stability of the calculation ina multirate feedforward controller design. We generate adesired state trajectory from (19) using the balanced realized
plant model. If the filter Fij (s) (i, j ∈ {1, 2, . . . , n}) in(20) becomes unstable, we divide it into stable and unstableparts. We generate stable desired state trajectories by adding
a stable part (24) and an unstable part (26) together.

Fig. 1. Block diagram of the controller. S, H, and L denote a sampler, holder, and lifting operator [17], respectively, z and zs denote esTr and esTu ,respectively.

(a) Photograph of the 6-DOF
high-precision stage.

(b) Coupling problem between xm and θy .

Fig. 2.

(d) MIMO multirate sampling control at the same interval. (σ1 , σ2 ) =(2, 2)

Simulation details.

C. Design of multirate feedforward controller for multi-input
multi-output system
We design the matrix B from the generalized controllability indices:{bs1 , bs2 , As bs1 , As bs2 , A2s bs1 , A2s bs2 , A3s bs1 , A3s bs2 }(c) Block diagram of theplant.

(44)

In this study, we design the matrix B as follows:(σ1 , σ2 ) = (2, 2) : B = As bs1 bs1 As bs2 bs2 (45)

20

10

10

5

0

0

-10
-0.02

-0.01

0

0.01

0.02

-5
-0.02

(a) fxwhich is the most basic design because the number of each
input is the same in the frame period Tf . These multirate
inputs are shown in Fig. 2(d).

10

10

8

8

6

6

D. Condition

4

4

The trajectory xref
m is given by a seventh-order polynomialfrom 0 µm to 100 µm over a 0 ms to 20 ms period. The
trajectory θyref is also given by a seventh-order polynomialfrom 0 µrad to 100 µrad over a 0 ms to 20 ms period. Tu =200 µs, N = max(σ1 , σ2 ) = 2 and Tr = 2Ty = 2Tu =Tf . The feedback controller Cf b [zs ] in Fig. 1 is 0 in thesimulation.

2

2

The simulation results are shown in Fig. 3. Magnified
views of the tracking error are shown in Fig. 4. The tracking
errors become 0 at every sampling period Tf , demonstratingthat perfect tracking control is achieved. Magnified views of
the input in −600 µs to 0 µs are shown in Fig. 5. Preactuation
occurs at a negative time, which means that perfect tracking

0.02

-0.01

0

0.01

0.02

0.01

0.02

10

0
-0.02

-0.01

(c) xmFig. 3.

0.01

-5

10

0
-0.02

0

(b) τy-5

E. Simulation results

-0.01

0

(d) θySimulation results of the proposed method (σ1 , σ2 ) = (2, 2).

control is achieved even if the inverse system is unstable.
These results illustrate the effectiveness of the proposed
method.
VI. C ONCLUSION
In this study, we proposed a state trajectory generation
method using singular value decomposition and a multirate

10-10
2

2

1

1

0

0

10-10

-1
-1
-2
-2
8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6
10

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6

-3

10-3

(a) exm (zoom)(b) eθy (zoom)

Fig. 4. Magnified views of the tracking errors in the simulation results forthe proposed method (σ1 , σ2 ) = (2, 2).
1

10-31

5

0

0

-1
-6

10-32

-4

-5
-6

-2
10

-4

-2
10-4

-4

(a) fx (zoom)(b) τy (zoom)

Fig. 5. Magnified views of the inputs in the simulation results for the
proposed method (σ1 , σ2 ) = (2, 2).

feedforward controller designed for a MIMO system. We
can use a multirate feedforward controller more generally
in MIMO systems using this method. We also discussed
the calculation method using time axis reversal when thegenerated state trajectory is unstable due to unstable zeros
in continuous-time domain or several calculation errors. The
effectiveness of the proposed method was verified from asimulation of a dual-input dual-output high-precision stage.
Future research will focus on comparing the effectiveness ofthe proposed method in various cases where the plant has
unstable zeros.
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