Intersample Behavior Analysis of MIMO
Multirate Feedforward Control dependingon Selection of Input Multiplicities
Masahiro Mae ∗ Wataru Ohnishi ∗ Hiroshi Fujimoto ∗∗The University of Tokyo 5-1-5, Kashiwanoha, Kashiwa, Chiba,277-8561, Japan (e-mail: mmae@ieee.org, ohnishi@ieee.org,
fujimoto@k.u-tokyo.ac.jp)

Abstract: Perfect tracking control method using multirate feedforward control is a very effective
control method in high-precision positioning systems since this method provides zero tracking
error for a nominal plant at every reference sampling point theoretically. When we design
a multirate feedforward controller for a multi-input multi-output system, it is known thatseveral types of controllers can be designed depending on how to select input multiplicities.
Although multirate feedforward controllers provide perfect tracking at every reference sampling
point theoretically, intersample behavior is different. In this paper, we propose the stablestate trajectory generation method and the guideline to design an optimal multi-input multioutput multirate feedforward controller considering the 2-norm of control input and improve
the intersample behavior. The effectiveness of the proposed method is verified in the simulation.
Keywords: Feedforward control, Digital control, Multirate, Multi-input/multi-output systems,
Inverse system
1. INTRODUCTION
Perfect tracking control (PTC) using multirate feedforward control is widely used for controlling precision positioning systems in industrial application fields such as
manufacturing semiconductors and flat panels, and controlling a head of hard disk drives. Multirate feedforward control provides perfect tracking control [Fujimoto
et al. (2001)]. The word “perfect tracking control” isdefined as “the plant output perfectly tracks the desired trajectory with zero tracking error at every sampling point” [Tomizuka (1987)]. Perfect tracking cannot
be achieved by a single-rate control scheme because of
unstable zeros due to discretization of the plant using
zero-order hold [Tomizuka (1987)]. Therefore, several studies are conducted for the approximated plant model inverse approaches in the single-rate control scheme such asnonminimum-phase zeros ignore (NPZI) method [Butterworth et al. (2012)], zero-phase-error tracking controller(ZPETC) method [Tomizuka (1987)] and zero-magnitudeerror tracking controller (ZMETC) method [Wen and Potsaid (2004)]. Multirate feedforward control solves the problem of discretization unstable zeros and provides perfect
tracking control [Fujimoto et al. (2001)].
Recently, many mechatronic systems are used in industrial applications and the precision positioning systems
in these mechatronic systems are becoming multi-input
multi-output (MIMO) systems to improve performance[Butler (2011)]. Because of coupling problems in eachaxis, a sufficient control performance cannot be expected
for MIMO systems when the simple single-input singleoutput (SISO) controllers are designed for each axis, respectively. Therefore, to overcome coupling problems in

MIMO systems, it is necessary to treat controlled systems as MIMO systems and design MIMO feedforward
controllers. When the multirate feedforward controllers
are designed for MIMO systems, several kinds of controllers can be designed depending on the selection of input
multiplicities [Fujimoto (2000)]. All designed multirate
feedforward controllers achieve perfect tracking control,
however, intersample performances are different depending on the controller designs [Mae et al. (2018)]. Perfect
tracking control is defined in the framework of a digital
control system [Fujimoto et al. (2001)]. Therefore, there
is no consideration of the tracking error between sampling
points. For this reason, even if several kinds of multirate
feedforward controllers for MIMO systems achieve perfect
tracking on sample points, the optimal controller can bechosen from the view of the intersample behavior.
In this paper, we propose the state space representation using singular value decomposition which gathers continuous
time invariant zeros into the part of the state trajectory
generation. We solve the problem due to unstable invariant zeros of the controlled system in the state trajectory
generation and improve the intersample behavior. We also
propose the guideline how to choose input multiplicities todesign the optimal MIMO multirate feedforward controller
from viewpoint of the 2-norm of control input. When the
2-norm of control input is too large, it is not suitable forthe mechatronic systems. We aim to make the 2-norm ofcontrol input smaller and the smaller control input also
tend to the smaller tracking error between sampling points.
Conventionally, the intersample behavior is evaluated bysimulation results, but, it costs much more time whenmany kinds of controllers can be designed. We propose

the guideline how to evaluate the 2-norm of control input
and choose an optimal controller before the simulation.
The effectiveness of the proposed method is verified in thesimulation of the two-inertia motor bench system.
2. STATE TRAJECTORY GENERATION FOR
MULTI-INPUT MULTI-OUTPUT SYSTEM

2.1 Definition of multi-input multi-output system
The state equation and the output equation for an m-input
m-output nth order plant are given byẋ(t) = Ax(t) + Bu(t),(1)y(t) = Cx(t),(2)
where the state variables are x(t) ∈ Rn×1 , each input
is u(t) ∈ Rm×1 , each output is y(t) ∈ Rm×1 , and thematrices are A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rm×n .
2.2 State space representation of MIMO system
Singular value decomposition of B matrix Singular value
decomposition of the matrix B is given by
B = U ΣV H ,(3)
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),
where σi (i = 1, 2, . . . , m ∈ N) are the singular values ofthe matrix B.
Multiplying both sides of (3) by U H from the left yields


∆V H
.
(4)
U H B = ΣV H =OState variable conversion using matrix U
The statevariable conversion from (A, B, C, D) to (Ã, B̃, C̃, D̃)using the transformation matrix U H as x̃(s) = U H x(s)is given by(5)

B̃ = U H B = ΣV H =



∆V H

O(n−m)×m,Ã − sI B̃
,
C̃
D̃


W̃u (s)Ã − sI = W̃ (s) =,W̃l (s)(9)(10)

where W̃u (s) ∈ Cm×n , and W̃l (s) ∈ C(n−m)×n .

The proposed method of generating the state trajectoryfrom the output trajectory in the MIMO system will be
described below. Using a proposed state space representation, an intersample problem which is occurred by thecontinuous time unstable invariant zeros is solved in thepart of a state trajectory generation. In this study, theplant is assumed to be an m-input m-output nth orderplant, where m < n and rank(B) = m.

Ã = U H AU ,Π̃(s) =

(6)C̃ = CU ,(7)

D̃ = D = Om×m .

(8)

Zero Separation using system matrix The system matrixΠ̃(s) ∈ R(n+m)×(n+m) of (Ã, B̃, C̃, D̃) and W̃ (s) ∈ Rn×n
are represented as

The system (Ã, B̃, C̃, D̃) is represented by the system
matrix Π̃(s) as follows:
 


On×m
x̃(s)
=
.
(11)
Π̃(s)
u(s)
y(s)
Π̃(s) is substituted each element as follows:






W̃u (s)
∆V H
Om×m
x̃(s)
 W̃l (s) O(n−m)×m =  O(n−m)×m  . (12)
u(s)
y(s)
C̃
Om×m
Since the system matrix Π̃(s) is a square matrix of (n +m) × (n + m), it is full rank if full row rank. The zeros ofthe system are defined as the values s = z for which thepolynomial system matrix Π̃(s) loses rank. When ∆V H
is full rank, each line of the upper m rows is linearly
independent with other lines for ∀s because of the rightside m columns. Therefore, the zeros of the system dependon the lower and left side n × n elements of the system
matrix Π̃(s). Then, the lower and left side n × n elementsof the system matrix are extracted as follows:O(n−m)×mW̃l (s)x̃(s) =
.
(13)
y(s)
C̃
Since multirate feedforward control solves the problems of
unstable invariant zeros due to discretization, the problemsof the continuous time unstable invariant zeros should
be solved in the part of the state trajectory generation.
Therefore, this state space realization is proposed forthat the part of the state trajectory generation has allcontinuous time unstable invariant zero of the controlled
system.
State trajectory generation
The state trajectory x̃(s)from the output trajectory y(s) in the MIMO system isgiven by
−1 
O(n−m)×mW̃l (s)x̃(s) =
.
(14)
y(s)
C̃
From the state variable conversion using the transformation matrix U as x(s) = U x̃(s), the state trajectory x(s)from the output trajectory y(s) in the MIMO system isgiven by
−1 
O(n−m)×mW̃l (s)U x̃(s) = U
y(s)
C̃


−1 
O(n−m)×mW̃l (s)⇔ x(s) = U
.
(15)
y(s)
C̃
By inverse Laplace transform, the desired state trajectory
xd (t) can be obtained from the desired output trajectoryyd (t) in the MIMO system:
" 
−1 
#
O(n−m)×mW̃l (s)−1
xd (t) = L
U.
(16)
yd (s)C̃When the system has discretization unstable invariant
zeros, the desired state trajectories become unstable because the inverse system which appears in the (16) has
unstable poles. In this case, the convolution method withtime axis reversal [Ohnishi et al. (2019)] is used and stable
desired state trajectories with preactuation are generated.
The preactuation means that control input occurs fromnegative time.
3. MULTIRATE FEEDFORWARD CONTROL FOR
MULTI-INPUT MULTI-OUTPUT SYSTEM
Multirate feedforward control provides perfect tracking
control [Fujimoto et al. (2001)]. A digital tracking control
system typically has two samplers for the reference signalr(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 of r(t), y(t), andu(t), respectively. A larger Tr or Ty value is defined as theframe period Tf .
3.1 Definition of multi-input multi-output system
In an m-input p-output nth order MIMO system, the state
equation (17) and the output equation (18) describing thecontinuous-time plant are given byẋ(t) = Ac x(t) + Bc u(t),(17)y(t) = Cc x(t),(18)
T

Bc = [bc1 · · · bcm ] , Cc = [cc1 · · · ccp ] ,where the plant state is x ∈ Rn , the plant input is u ∈ Rm ,and the plant output is y ∈ Rp .
3.2 Design of B matrix from generalized controllability
indices
The generalized controllability indices are defined as follows [Fujimoto (2000)]:
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
The generalized controllability indices are the sets of the
input multiplicities σl . The input multiplicities σl aredefined as follows:
Definition 2. (Input multiplicities). Input multiplicities σlare defined as the number of the input which comes fromthe same input in the same frame period Tf .
Setting ϕ as a set of these n vectors, σl and N are definedbyσl = number{k|Ak−1
bcl ∈ ϕ},
(19)
c
m
X
σl = n,(20)
l=1

the full row rank matrix B can be designed for almost
all discretized sampling periods 1 . Therefore, feedforward
controllers must be designed according to their different
forms.
3.3 Feedforward input generation from state trajectory
From (22), the control inputs uf f [i] required for PTC are
given by (23).
x[i + 1] = Ax[i] + Bu[i],
(22)
−1
−1
uf f [i] = B (I − z A)x[i + 1],(23)
where the matrices A, x[i], u[i], z and Tf are given by
A = eAc Tf ,u[i] = [u1 [i] · · · um [i]](21)

In the MIMO system, n (= plant order) vectors are selected from the generalized controllability indices, andz = esTf ,Tf = N Tu ,TT

= [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 [Chen and Francis(1995)]. L−1 outputs the elements of the nth dimensional
vector uf f [i], which is input at every period Tr , in theorder from 1 to n by Tu = Tr /n.
3.4 Variety of MIMO multirate feedforward controllers
From these discussions, the multirate feedforward controllers for MIMO systems can be designed for several
kinds depending on the selection of input multiplicities andthey have different types of B matrix. It is confirmed thatall designed controllers achieve perfect tracking control[Mae et al. (2018)]. Therefore, it is a problem that which
controller design method is optimal for the controlled
system. In the definition of perfect tracking control, thetracking error at the sampling point is only mentioned
because the multirate feedforward controller is designedin the scheme of digital control. It is also necessary to
reduce the tracking error between sampling points forthe actual mechatronic applications. In this paper, we
discuss the optimal method of selecting input multiplicities
and constructing the B matrix for designing the optimal
MIMO multirate feedforward controller.
4. SELECTION OF INPUT MULTIPLICITIESWhen the multirate feedforward controller is designed with
B chosen from the generalized controllability indices, thecontrol input part of the state equation (22) is represented
as follows:x[i + 1] = Ax[i] + Bu[i]⇔ Bu[i] = x[i + 1] − Ax[i].
(24)The control input u[i] is represented by using the changeof state trajectory v[i] = x[i + 1] − Ax[i] as follows:
Bu[i] = v[i]⇔ u[i] = B −1 v[i].
(25)1

N = max(σl ).

x[i] = x(iTf ),This is possible because the controllability of a continuous-timesystem is not preserved in the discrete system only if the two polesηi and ηj have the same real parts, and the discretizing sampling
period T satisfies ηi = ηj + j 2kπ(k = ±1, ±2, . . .); furthermore, itTis limited to only several cases.

yd (t + Tr )State Trajectory
Generation

xd (t + Tr )

S
(Tr )

xd [i + 1]

uf f [i]B −1 (I − z −1 A)

uf f [k]

+

L−1

u[k]

H

+ uf b [k] (Tu )

u(t)

Plant

y(t)

G(s)

Cf b [zs ]ey [k]

Gn [zs ]yd [k]+

S−y[k] (Ty )Fig. 1. Block diagram of controllers and a plant: a state trajectory generation, a multirate feedforward controller, anda singlerate feedback controller. S, H, and L denote a sampler, holder, and lifting operator [Chen and Francis(1995)], respectively, z and zs denote esTr and esTu , respectively.
To evaluate the size of the control input, ||u[i]||2 = u21 +· · · + u2n is calculated as follows:||u[i]||2 = v T [i](B −1 )T B −1 v[i],(26)2and ||u[i]|| becomes the quadratic form of v[i].
We define ||v[i]||2 as 1 for considering the general desired
state trajectory as follows:||v[i]||2 = v12 + · · · + vn2 = 1.
(27)
2
2
When ||v[i]|| is 1, the range of ||u[i]|| is determined by
the eigenvalue λi ((B −1 )T B −1 ) as follows:
λn ≤ ||u[i]||2 ≤ λ1 (λn ≤ λ(n−1) ≤ · · · ≤ λ1 ).
(28)
The eigenvalue λci (BB T ) has a relationship between the
eigenvalue λi ((B −1 )T B −1 ) as follows:1
λci = .
(29)λiThen, the range of ||u[i]||2 is determined by the eigenvalue
λci (BB T ) as follows:1
1≤ ||u[i]||2 ≤(λc1 ≤ λc2 ≤ · · · ≤ λcn ). (30)λcn
λc1
The singular value σci (B) has a relationship between the
eigenvalue λci (BB T ) as follows:q(31)
σci (B) = λci (BB T ).
Therefore, the range of the 2-norm of the control input||u[i]||2 is determined by the singular value σci (B) as
follows:
1
1≤ ||u[i]||2 ≤(σc1 ≤ σc2 ≤ · · · ≤ σcn ). (32)σcn
σc1
When the 2-norm of control input is too large, it is notsuitable for the mechatronic systems. Therefore, we design
the optimal MIMO multirate feedforward controller fromthe viewpoint of the 2-norm of control input. To makethe upper bound of the 2-norm of control input ||u[i]||2
smaller, it is equal that the smallest singular value σc1 (B)makes larger. From this consideration, we should choose
the input multiplicity so that the smallest singular value
σc1 (B) is the largest.
5. SIMULATIONIn the simulation, we verify the proposed guideline how
to design the optimal MIMO multirate feedforward controller. We compare with the size of σc1 (B), the smallest
singular value of B, and the tracking errors in the simulation.

Fig. 2. Photograph of two inertia motor bench. In this paper, the two inertia motor bench is modeled as a twoinput two-output system. The two inputs are motorside torque τm and load side torque τl , respectively.
The two outputs are motor side angle θm and loadside angle θl , respectively.
5.1 Plant model
In this simulation, we control the two-inertia motor benchsystem, shown in Fig. 2. The system is modeled as the twoinput two-output fourth order plant model. The transfer
function matrix G(s) of the plant is defined as (33). The
T
T
inputs u(t) = [u1 (t) u2 (t)] = [τm (t) τl (t)] of the plantmodel are motor side torque τm and load side torqueTτl , respectively. The outputs y(t) = [y1 (t) y2 (t)] =T[θm (t) θl (t)] of the plant model are motor side angle θmand load side angle θl , respectively. The plant model has
coupling problem between each degree-of-freedom (DOF).
In these MIMO coupled systems, MIMO multirate feedforward controller is needed to achieve perfect tracking
control theoretically.
5.2 Controller design
We design several kinds of MIMO multirate feedforward
controller depending on the selection of input multiplicities. Three sets of input multiplicities (σ1 , σ2 ) =(2, 2), (3, 1), (4, 0) are used to design controllers. We design
the matrix B from (34) to (36).
σc1 (B), the smallest singular value of B, is comparedwith the simulation results in three kinds of controllers.
σc1 (B) is calculated as Table 1. Considering from the sizeof σc1 (B), the sets of input multiplicity (σ1 , σ2 ) = (2, 2) isthe best to design MIMO multirate feedforward controller
for this plant model G(s).
5.3 Conditions
T

The desired output trajectory yd (t) = [y1d (t) y2d (t)] =
 ref
T
ref
is seventh-order polynomials. θm(t) isθm (t) θlref (t)
1
9.7 × 102 (s2 + 1.1 × 105 )1.1 × 108
G(s) = 2 2
1.1 × 108
1.1 × 103 (s2 + 9.6 × 104 )s (s + 2.1 × 105 )(33)(σ1 , σ2 ) = (2, 2) : B = [As bs1 bs1 As bs2 bs2 ](σ1 , σ2 ) = (3, 1) : B = A2s bs1 As bs1 bs1 A2s bs2 + As bs2 + bs2(σ1 , σ2 ) = (4, 0) : B = A3s bs1 A2s bs1 As bs1 bs1

(34)
(35)

from 0 to 100 µrad over a 0 to 0.1 s period. θlref (t) isfrom 0 to 100 µrad over a 0 to 0.1 s period. Tf = 12 ms,
N = max(σ1 , σ2 ) and Tf = N Ty = N Tu = Tr . In thisstudy, the actual plant G(s) and the nominal plant Gn (s)is the same. Therefore, the feedback controller Cf b [zs ] inFig. 1 dose not work in the simulation.
If the experiment is conducted in actual mechatronicsystems, a multirate feedforward controller is used witha feedback controller as a two-degree-of-freedom control
system. A multirate feedforward controller handles the
tracking performance and a feedback controller handles
the rejection of the external disturbance and the modelingerror.
5.4 Simulation results
T

T

The control inputs u(t) = [u1 (t) u2 (t)] = [τm (t) τl (t)]in the simulation are shown in Fig. 3. It is verified that,when the smallest singular value of B is large, the control
Tinputs become small. The output y(t) = [y1 (t) y2 (t)] =T[θm (t) θl (t)] in the simulation are shown in Fig. 4. The
tracking errors become 0 at every sampling period Tf ,demonstrating that perfect tracking control is achieved.
Note that the values of the control input u and theoutput y in (σ1 , σ2 ) = (3, 1) are too large compared with
(σ1 , σ2 ) = (2, 2), (4, 0). Therefore, it can not be shown indetail in this range of plot due to the calculation error.
To evaluate the intersample behavior in the simulation,
The root mean square and maximum absolute value of theTTcontrol inputs u(t) = [u1 (t) u2 (t)] = [τm (t) τl (t)] andTT
the tracking error e(t) = [e1 (t) e2 (t)] = [eθm (t) eθl (t)]are listed in Table 1. Compared with σc1 (B), the smallest
singular value of B, and the intersample behavior in thesimulation results, it is verified that the sets of input
multiplicity (σ1 , σ2 ) = (2, 2) is the best to design MIMO
multirate feedforward controller for this plant model G(s).
The proposed method does not consider the specific casesof the desired trajectories, but the trend of the optimal
controller can be evaluated by the proposed simple calculation method.
6. CONCLUSION
This paper proposes the guideline how to expand multirate
feedforward control to the MIMO systems. Noteworthy,
multirate feedforward control achieves perfect tracking
control at every sampling point. In the actual mechatronic
applications, however, intersample behavior is also a big
issue for tracking control. We solve the problem due tounstable invariant zeros of the controlled system in thestate trajectory generation and improve the intersample
behavior. We also propose the guideline how to choose(36)

input multiplicities to design the optimal MIMO multirate
feedforward controller from viewpoint of the 2-norm ofcontrol input.
The state space representation is calculated using the
singular value decomposition of the B matrix. This representation gathers the continuous time unstable invariant
zeros into a state trajectory generation part. The problemof the discrete time unstable invariant zeros is solved in a
multirate feedforward control part. Therefore, the problem
of continuous and discrete time unstable invariant zeros
can be solved separately.
To design the optimal MIMO multirate feedforward controller, we focus on the 2-norm of control input. When
2-norm of control input is too large, it is not suitable forthe mechatronic systems. We aim to make the 2-norm ofcontrol input smaller and the smaller control input also
tend to the smaller tracking error between sampling points.
It is an easier way compared with doing the simulationin all cases of the sets of input multiplicities. Using thismethod, the optimal MIMO multirate feedforward controller is designed which considers not only on tracking
error at the sampling point but also intersample behavior.
The effectiveness of the proposed method is verified in thesimulation.
The proposed method evaluates the 2-norm of control
input for the generalized desired state trajectory, but did
not think about the relationship between the control input
and the desired output trajectory in specific cases. The
2-norm of control input for the specific desired output
trajectory also can be calculated. An analysis of therelationship between the plant and the desired output
trajectories, the experimental validation and considering
mechatronic constraints are future research.
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Table 1. σc1 (B), the smallest singular value of B, and root mean square and maximum absolutevalue of control inputs u and tracking errors e depending on sets of input multiplicities (σ1 , σ2 ).
(σ1 , σ2 )

σc1 (B)

RMS(u1 )MAX(|u1 |)RMS(u2 )MAX(|u2 |)RMS(e1 )MAX(|e1 |)RMS(e2 )MAX(|e2 |)(2, 2)(3, 1)(4, 0)

3.17 × 10−5
1.84 × 10−14
5.28 × 10−6

5.24 × 10−5
2.22 × 104
1.22 × 10−4

7.76 × 10−5
4.11 × 104
1.98 × 10−4

4.42 × 10−5
1.91 × 104
06.56 × 10−5
2.90 × 104
01.45 × 10−8
1.38 × 102
3.47 × 10−7

3.44 × 10−8
3.54 × 102
7.71 × 10−7

1.45 × 10−8
1.01 × 102
2.18 × 10−7

3.44 × 10−8
2.52 × 102
5.07 × 10−7

10-4

2

10-4

2

1

1

0

0

-1

-1

-2

-2
0

0.02

0.04

0.06

0.08

0.1

0

0.02

(a) τm

0.04

0.06

0.08

0.1

(b) τlFig. 3. Control Input u: Control Input of (σ1 , σ2 ) = (2, 2) and (4, 0) is not so large, but, that of (σ1 , σ2 ) = (3, 1) is toolarge and is not suitable for the mechatronic systems.
10-4

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

10-4

1

0.02

0.04

0.06

0.08

0.1

(a) θm

0
0

0.02

0.04

0.06

0.08

0.1

(b) θlFig. 4. Output y: perfect tracking control is achieved and tracking errors at every sampling points become zero forthe nominal plant, theoretically. However, the intersample behavior is different depending on the sets of the input
multiplicities (σ1 , σ2 ) = (2, 2), (3, 1) and (4, 0).
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