MIMO multirate feedforward controller design
with selection of input multiplicities
and intersample behavior analysis
Masahiro Mae∗, Wataru Ohnishi, Hiroshi Fujimoto
The University of Tokyo, Japan

Abstract
Inversion-based feedforward control is a basic method of tracking controls. The aim of this paper is to design MIMO multirate
feedforward controller that improves continuous-time tracking performance in MIMO LTI systems considering not only on-sample
but also intersample behavior. Several types of MIMO multirate feedforward controllers are designed and evaluated in terms ofthe 2-norm of the control inputs. The approach is compared with a conventional MIMO single-rate feedforward controller insimulations. The approach improves the intersample behavior through the optimal selection of input multiplicities with MIMO
multirate system inversion.
Keywords: Feedforward control, Inverse system, Multirate, Multi-input/multi-output systems, Digital control, Position control

1. Introduction
Inversion-based feedforward controllers play an importantrole in the tracking control of many high-precision mechatronicsystems, such as wafer and LCD scanners, and industrial robots[1]. For the demands of high-performance, high-speed, andflexible tasks, many high-precision mechatronic systems havemultiple degree-of-freedoms and are multi-input multi-output(MIMO) systems.
Many high-precision mechatronic systems are usually controlled by single-input single-output (SISO) controllers under
the assumption that they are mechanically decoupled, and coupling problems between each axis can be ignored. Several highprecision mechatronic systems with severe coupling problem
between each axis, such as a 6-degree-of-freedom high-precision
positioning stage, are controlled with MIMO controllers, such
as SISO controllers with a continuous-time pre-compensator[2], feedforward input shaping approach [3], and feedforward
H∞ approach [4]. However, these continuous-time controllers
are usually discretized by Tustin transform for digital implementation. Therefore, the effect of discretization by the zeroorder hold is not strictly considered and perfect tracking control
cannot be achieved for a discrete-time nominal system.
In high-precision positioning systems with multiple actuators and sensors, such as a 6-degree-of-freedom high-precision
positioning stage, it is common that the number of actuatorsand sensors are imaginarily converted by coordinate transformation to the same number of degrees of freedom of motion[2]. In this framework, this paper mainly focuses on MIMO
linear-time-invariant (LTI) systems with an equal number of∗ Corresponding author

Email address: mmae@ieee.org (Masahiro Mae)
Preprint submitted to Mechatronics

inputs and outputs. For the tracking control of MIMO LTI
systems, MIMO feedforward controllers are needed to achieve
good tracking performance by considering the coupling problems and redundancy of MIMO LTI systems.
The continuous-time inversion-based approaches such as [5,6, 7] can be used for continuous-time systems. However, practical tracking controllers are often implemented by digital systems for large flexibility and low cost [8]. Therefore, the tracking control is conducted with digital control and has some limitations attributed to discretization. The main problem of theinversion-based feedforward controllers is the unstable discretizedzeros, which are out of the unit circle on the z plane, of thecontrolled system discretized by a sampler and a holder. The
inversion-based feedforward controllers are designed by the inverse of controlled systems and they have unstable poles due tothe unstable zeros of the controlled systems.
To overcome the discretized unstable zero problems, several
approximated inverse approaches are presented in the singlerate feedforward control, such as nonminimum-phase zeros ignore (NPZI) [9], zero-phase-error tracking control (ZPETC)[10] and zero-magnitude-error tracking controller (ZMETC) method[11]. However, these methods cannot achieve the exact tracking
on sampling points because of the approximation.
An exact inverse approach, discrete-time stable inversion[12, 13], is presented, but this method cannot cope with thediscretized zeros around z = −1 that become oscillating polesof the inversion-based feedforward controllers [14, 15]. It isnoted that these single-rate feedforward control approaches canbe extended to MIMO LTI systems [16, 17]. FIR filter tuningwith a gradient approximation-based algorithm is presented fordecoupling control of MIMO systems with a discrete-time controller [18]. However, this approach uses an optimization in theyd (t)Syd [k]

u[k]
F

H

u(t)

Gc

y(t)

e(t)

2.2. Discretization and sampling periods
The discrete-time system Gd discretized by the zero-orderhold with Gc and the generalized sampling period δ is given by− +x[k + 1] = Ad x[k] + Bd u[k],Figure 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).

(3)

y[k] = Cd x[k],(4)where k ∈ Z. Ad , Bd , and Cd are given byZ δAc δAd = e , Bd =e Ac τ Bc dτ,algorithm and it is not suitable when many kinds of references
are used.
Based on these approaches in the single-rate feedforward
control, a multirate control approach is presented [19]. Compared with the single-rate system, the multirate system has all
zeros at z = 0. Therefore, the multirate feedforward controller
has all poles at z = 0, and exact on-sample tracking can beachieved, and intersample behavior is also improved.
Previous researches show that the multirate feedforward control approach can be extended from the SISO LTI systems to the
MIMO LTI systems [20]. MIMO multirate feedforward controller is effective to reject cross-couplings effects comparedwith the basic pre-compensator approach [21]. The MIMO
multirate feedforward controller can be designed for several
kinds because of the redundancy of MIMO LTI systems andmultirate sampling periods. In this paper, a procedure of designing an optimal MIMO multirate feedforward controller ispresented.
The outline of the paper is as follows. In Section 2, theproblem of tracking with digital control is formulated. In Section 3, the conventional MIMO single-rate feedforward control
approach and its limitations are presented before introducing
the proposed method. In Section 4, the proposed MIMO multirate feedforward control approach is presented. In Section 5,
the advantages of the approach are demonstrated by application
to a MIMO motion system in the simulation. In Section 6, theconclusion of this paper is presented.

Cd = Cc .

(5)

0

In the discrete-time system, three sampling periods exist:
T r , T y , and T u , which represent the sampling periods of a reference r(t), an output y(t), and a control input u(t), respectively.
Three sampling periods T r , T y , and T u are the same in the
single-rate system and are different in the multirate system.
2.3. Perfect tracking control and intersample behavior
In the problem of tracking control, the discrete-time controller F should be designed as Gd F = I, where Gd = SGc H,at every sampling point and achieves perfect tracking control.
The perfect tracking control is defined as follows [10]:Definition 1. The perfect tracking control is defined as a methodwith which the plant output perfectly tracks the desired trajectory with zero tracking error at every sampling point.
It is important that the perfect tracking control only guarantees the tracking error on the discrete-time sampling points,
but not in the continuous-time. In the problem of tracking control, the objective is to minimize the continuous-time error e(t).
Therefore, not only on-sample tracking error but also intersample tracking error should be considered in the design of thediscrete-time controller F.
In this paper, two types of discrete-time controllers are mentioned, the first is a single-rate feedforward controller and thesecond is a multirate feedforward controller.

2. Problem formulation

In this section, the control problem is formulated. The overview 3. Single-rate feedforward control for multi-input multi-output
of tracking control is shown in Figure 1.
system
2.1. Definition of multi-input multi-output system
The state equation and the output equation of an m-input moutput nth-order continuous-time linear time-invariant system
Gc are given byẋ(t) = Ac x(t) + Bc u(t),y(t) = Cc x(t),hBc = bc1 · · ·The single-rate system G s discretized by the zero-order holdwith Gc and the sampling period δ = T u is given byx[k + 1] = A s x[k] + B s u[k],y[k] = C s x[k].

(1)

bcm ,

h

Cc = cc1

···

ccm

iT

(7)

From the state space representation of the single-rate system
G s , control inputs u f f [k] of the single-rate feedforward controller F sr for the reference of the desired output trajectory r[k] =yd [k + 1] are given by

(2)
i

(6)

,

where the state variables are x(t) ∈ Rn×1 , inputs are u(t) ∈Rm×1 , outputs are y(t) ∈ Rm×1 , and the matrices are Ac ∈ Rn×n ,Bc ∈ Rn×m , and Cc ∈ Rm×n . This paper mainly focuses on the
MIMO LTI systems that have the same number of inputs andoutputs. This is a natural assumption for mechatronic systems
to achieve both the state controllability and the hardware cost
reduction.

u f f [k] = F sr yd [k + 1],
where F sr is given by"
A s − B s (C s B s )−1 C s A s
F sr =−(C s B s )−1 C s A s
2

B s (C s B s )−1(C s B s )−1

(8)

#
.

(9)

There is exact tracking of the desired output trajectory yd at every sample in the systems with the single feedforward control.
However, the single-rate feedforward controller has a problem. It is known that a single-rate system discretized by thezero-order hold has discretized zeros depending on the relative order of the continuous-time system [22]. The discretizedzeros appear around z = −1 on the real axis on the z plane.
The single-rate feedforward controller is designed as the inverseof the single-rate system, and the zeros of the single-rate system become the poles of the single-rate feedforward controller.
When the pole of the system is around z = −1 of the z plane, thesystem becomes oscillated or diverged. Therefore, the singlerate feedforward controller has the problem that the generated
control inputs may be oscillated or diverged. If the single-rate
feedforward controller F sr has unstable poles, a stable inversion approach or an approximated inverse approach is used, seedetails in [12, 17].
On the other hand, the multirate feedforward controller isdesigned so that all poles are at z = 0 and the generated control
inputs are not oscillated or diverged. In this paper, MIMO multirate feedforward controller is proposed to make the continuoustime error smaller than that of a MIMO single-rate feedforward
controller.

where l ∈ N is the index of the inputs. The plant order n is equal
to the sum of the input multiplicities σl asmX

σl = n.

(12)l=1

In MIMO LTI systems, n vectors are selected from the generalized controllability indices, and the full row rank matrix B
can be designed for almost all discretized sampling periods1 .
Therefore, several types of multirate systems are designed according to the selection of input multiplicities.
From the selection of input multiplicities, T ul , which is the
sampling period of lth input ul , is defined by
T ul =

N
Tu.
σl

(13)

It is noted that the sampling period T u is the smallest value of
T ul .
A sampling period T f is defined as the frame period whichis the largest value between T r , T y and T u . In this paper, theframe period T f of the multirate system is defined by
T f = T r = NT y = NT u .

(14)

The multirate system G discretized by the zero-order hold
4. Multirate feedforward control for multi-input multi-output with G and the sampling period δ = T is given byculsystemx[i + 1] = Ax[i] + Bu[i],(15)
In this section, the design method of the MIMO multirate
y[i] = Cx[i],(16)feedforward controller is proposed for the tracking control of
MIMO LTI systems. The multirate feedforward control has an
where i ∈ Z. A, B, x[i], and u[i] are given byadvantage of intersample behavior compared with the singlerate feedforward control [13].
A = e Ac T f ,(17)hi
B = B1 · · · Bl · · · Bm ,(18)
4.1. Design of input matrix from generalized controllability indices
C = Cc ,(19)
The generalized controllability indices are defined as folx[i] = x(iT f ),(20)lows [20]:hiTu[i] = u1 [i] · · · um [i]Definition 2. The generalized controllability indices of Ac ∈
h
iT
n×n
n×m
R and Bc = [bc1 , · · · , bcm ] ∈ R
are defined as follows:= u11 [i] · · · u1σ1 [i] u21 [i] · · · umσm [i] , (21){bc1 , · · · , bcm , Ac bc1 , · · · , Ac bcm , · · · , An−1
c bcm }.

and Bl , A sl and b sl are defined ash
Bl = Aσsll −1 b sl Aσsll −2 b sl · · · A sl b sl
Z Tul
Ac T ule Ac τ bcl dτ.
A sl = e, b sl =If (Ac , Bc ) is a controllable pair, n linearly independent vectors
be selected from the he generalized controllability indices.
The generalized controllability indices are the sets of the input
multiplicities σl .
The input multiplicities σl is defined as follows [20]:N = max(σl ),

(11)

(23)

The input matrix B in a multirate system is designed by thegeneralized controllability indices depending on the input multiplicities σl . It becomes a nonsingular square matrix becauseof the definition of the generalized controllability indices. The
state and input of the multirate system is shown in Figure 2.

Setting φ as a set of n vectors selected from the generalized
controllability indices, σl and N are defined by
(10)

(22)

0

Definition 3. Input multiplicities σl are defined as the numberof the input which comes from the same input in the same frame
period T f .

σl = number{k| Ak−1
c bcl ∈ φ},ib sl ,

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 and η j havethe same real parts, and the discretizing sampling period T satisfies ηi = η j +j 2kπ
T (k = ±1, ±2, . . .); furthermore, it is limited to only several cases [23].

3x[i + 1]

From the state equation of a multirate system (15), the partin which the control input u affects to the state x is given by

x[i]
Tul

Bu[i] = x[i + 1] − Ax[i].

ulσl [i]ul1 [i]u11 [i]iTf

In the multirate feedforward control, there is exact tracking of
the state x[i] and x[i + 1]. The difference of the state v[i] isdefined as

u1σ1 [i]Tu1u12 [i](i + σ11 )Tftv[i] = x[i + 1] − Ax[i],)Tf (i + 1)Tf(i + σ1σ−11u[i] = B−1 v[i].

∥u[i]∥22 = vT [i](B−1 )T B−1 v[i],4.2. Controller design and control input generation(30)

and ∥u[i]∥22 becomes a quadratic form of v[i].

From the state equation of the multirate system (15), control
inputs u f f [i] of the multirate feedforward controller Fmr for thereference of the desired state trajectory r[i] = xd [i+1] are givenbyFor the normalization of the difference of the state, v[i] isdefined as the unit sphere:∥v[i]∥22 = v21 + · · · + v2n = 1.

(31)

According to the relationship between the range of a quadratic
form with the unit sphere and eigenvalues [26], the range of∥u[i]∥22 is given by(24)

where Fmr and z is given byz = e sT f .

(29)

The square of the 2-norm of the control input ∥u[i]∥22 = u21 +· · · + u2n is given by

Figure 2: MIMO multirate input control.

Fmr = B−1 (I − z−1 A)
#
"
O
I
,
=
−B−1 A B−1(28)

and the control input u[i] is represented as(i + σ1l )Tf (i + σlσ−1)Tflu f f [i] = Fmr xd [i + 1],(27)

λn ≤ ∥u[i]∥22 ≤ λ1

(λn ≤ λ(n−1) ≤ · · · ≤ λ1 ),(32)

where λi is the eigenvalue of (B ) B . λci , which is the eigenvalue of BBT , is the reciprocal of λi as
1(33)
λci = ,
λi
−1 T

(25)
(26)

There is exact tracking of the desired state trajectory xd at every N samples in the nominal system with the multirate feedforward control. It is noted that all poles of the multirate feedforward controller Fmr are z = 0 because the state matrix of Fmr
is O, and smooth control input is generated compared with thesingle-rate feedforward controller F sr . For the details of the desired state trajectory generation, see [24, 25]. A block diagramof the control system is shown in Figure 3. L is a discrete-time
lifting operator [8]. L−1 outputs the elements of the Nth dimensional vector u f f [i], which are inputs at every period T f , in theorder from 1 to σl by T ul .

−1

and the range of ∥u[i]∥22 given by11≤ ∥u[i]∥22 ≤(λc1 ≤ λc2 ≤ · · · ≤ λcn ).
(34)
λcn
λc1
σci , which is the singular value of the input matrix B is thesquare root of λci asp
σci (B) = λci (BBT ),(35)
and the range of 2-norm of the control input ∥u[i]∥2 is given by11≤ ∥u[i]∥2 ≤(σc1 ≤ σc2 ≤ · · · ≤ σcn ).
(36)
σcn
σc1
If 2-norm of the control input ∥u[i]∥2 is too large, it is not suitable for the limitation of the mechatronic systems. Making theupper bound of the 2-norm of control input ∥u[i]∥2 smaller isequal making the smallest singular value σc1 (B) larger. From
this consideration, the input multiplicity is selected so that thesmallest singular value σc1 (B) becomes the largest. Therefore,
the optimal design of the MIMO multirate feedforward controller to make the maximum value of 2-norm of control inputs
smaller is proposed. The MIMO multirate feedforward controller cannot specify the band of the continuous-time error because it only guarantees the exact tracking of the desired state
trajectory xd at every frame period T f in the nominal system,
but the intersample behavior becomes smoothly connected between the discrete sampling points in continuous time with thecontrol inputs of the optimally designed controller. The analysis of the bound of the continuous-time error is an open issue.

4.3. Optimal selection of input multiplicities
Several types of multirate feedforward controllers can be
designed depending on the multirate system G with the selection of input multiplicities σl . There is exact tracking of thedesired state trajectory xd at every N samples in the systems
with all kinds of multirate feedforward controllers [21]. However, the control inputs and intersample behavior are different depending on the multirate system G. For the applicationof high-precision positioning control in mechatronic systems,
continuous-time tracking error is preferred to be small, and also
control input u should be smaller because of the limitation of
mechatronic systems. An approach of designing the optimal
MIMO multirate feedforward controller is proposed to make 2norm of control inputs smaller in the rest of this section.
4

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

H

u(t)

(Tu )

y(t)

Gc (s)

Figure 3: Block diagram of MIMO multirate feedforward control. S, H, and L denote sampler, holder, and lifting operator [8], respectively. z and z s denote e sT r
and e sT u , respectively.

4.4. Example: intersample behavior of multirate feedforward
in different sets of input multiplicities

5.1. System modeling
The approach is validated on the two-inertia system motor bench shown in Figure 6(a). The two-inertia system motor
bench has two motors on the left and right side and two motors are connected by the flexible shaft as shown in Figure 6(b).
The two-inertia system motor bench is used for theoretical andapplicability validation. The two-inertia system motor bench
has 20 bit/rev optical encoder for both sides which is enoughhigh resolution for high-precision mechatronic systems. In thispaper, the two-inertia system motor bench is modeled as a twoinput two-output 4th-order system. The block diagram of thesystem is shown in Figure 7. The two inputs u are the left andright side torque, τl and τr , and the outputs y are the left andright side angle, θl and θr , respectively. The Bode diagram ofa frequency response function measurement of the system isshown in Figure 8. The measurement is obtained through theidentification experiment with a multisine input [27] from 1 Hz
to 1249 Hz, and a sampling frequency is 2.5 kHz. From the frequency response function measurement, the parameters of the
two-inertia system motor bench are given as shown in Table 2
and the identified continuous-time system Gc is given by thestate space model with the state equation (39) and the output
equation (40).

The optimal design of the MIMO multirate feedforward controller is validated with the example of a numerical simulation.
The continuous-time system Gc is defined as the transfer
function matrix (37). The reference of the desired output trajectory yd is given by 7th-order polynomials which change from 0
to 1 in 0 s to 400 µs for each output. The sampling period of thecontrol input is set to T u = 400 µs. For the design of MIMO
multirate feedforward controller, seven types of sets of input
multiplicities are selected as(σ1 , σ2 ) = (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0). (38)The examples of multirate inputs are shown in Figure 5.
The smallest singular value σc1 (B) and the simulation results are shown in Table 1. From the procedure of designing the
optimal MIMO multirate feedforward controller, the set of input multiplicities in which the smallest singular value σc1 (B) is
the largest is the optimal set of input multiplicities for the controlled system. The advantage of this procedure is that the optimal MIMO multirate feedforward controller is designed without numerical simulations. When the order of the system ishigh or the number of inputs and outputs is large, the numberof the set of input multiplicities becomes enormous. Therefore,
testing all sets with several references in numerical simulations
spends a large amount of time, and the proposed design procedure is effective.
The validity of the procedure can be confirmed from theroot mean square and the maximum absolute value of control
inputs u and tracking errors e in Table 1. The trend is that
control inputs u and tracking errors e become small when thesmallest singular value σc1 (B) is large. From Table 1, the optimal MIMO multirate feedforward controller is designed withthe set of input multiplicities (σ1 , σ2 ) = (4, 2) which makes thesmallest singular value σc1 (B) largest, and the root mean squareof tracking errors e are the smallest in all sets.
In summary, the proposed procedure is validated, and the
optimal MIMO multirate feedforward controller can be designedwith the set of input multiplicities which makes the smallest singular value σc1 (B) largest, without spending time on numericalsimulations.

0   θl (t)   0  1 
0   θ̇l (t)   Jl
 + 

01  θr (t)  0

Dr 
K
0
− Jr − Jr θ̇r (t)
r


"
# "
#  θl (t) θl (t)
1 0 0 0  θ̇l (t) =θr (t)
0 0 1 0 θr (t)
θ̇r (t)  θl (t)   0
d  θ̇l (t)  − KJl
= 
dt θr (t)  0θ̇r (t)  JK

1− DJll
0
0

0
K
Jl


0 
#
"
0  τl (t)0  τr (t)
1

(39)

Jr

(40)

5.2. Conditions
The conventional MIMO single-rate feedforward controller
F sr and the proposed MIMO multirate feedforward controller
Fmr are compared in the tracking control of the continuous-timesystem Gc . With the proposed procedure, the optimal MIMO
multirate feedforward controller is designed for Gc with theset of input multiplicities (σ1 , σ2 ) = (2, 2) which makes thesmallest singular value σc1 (B) largest. The poles and zeros of
the feedforward controllers F sr and Fmr are shown in Figure 9.
From Figure 9(a), the conventional MIMO single-rate feedforward controller F sr has one pole around z = −1 which leads toan oscillation, and from Figure 9(b), the proposed MIMO multirate feedforward controller Fmr has all poles on z = 0. The
reference of the desired output trajectory yd is given by 7thorder polynomials which change from 0 to 100 µrad in 0.8 ms
to 2 ms for each output. The sampling period of the control input is set to T u = 400 µs. From these conditions, the reference
signal is steep enough compared with T u .

5. Verification in multi-input multi-output positioning system
In this section, the tracking performance considering the intersample behavior of the optimal MIMO multirate feedforward
controller is verified compared with that of a MIMO single-rate
feedforward controller.
51
Gc (s) = 6
s + 8895s5 + 3.979 × 107 s4 + 2.428 × 109 s3 + 9.099 × 1012 s2 + 4.382 × 1013 s + 24"#
4.702 × 1010 s2 + 2.294 × 1011 s + 5.477 × 1015
1.387 × 108 s2 + 1.233 × 1012 s + 5.477 × 1015
15
4
7 3
10 2
12
15
5.477 × 10
1220s + 1.085 × 10 s + 4.835 × 10 s + 1.462 × 10 s + 5.477 × 10

(37)

Table 1: σc1 (B), the smallest singular value of B, and root mean square and maximum absolute value 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 |)(0, 6)
1.86 × 10−160.00 × 1000
0.00 × 1000
9.81 × 1004 2.27 × 1005
4.48 × 10−01 9.94 × 10−01
7.34 × 1000
1.53 × 1001
(1, 5)
6.77 × 10−13
5.54 × 10−12 7.83 × 10−12 1.71 × 1005 4.27 × 1005
4.01 × 10−01 9.88 × 10−01
9.30 × 1000
2.47 × 1001
(2, 4)
8.09 × 10−07
2.24 × 10−10 5.01 × 10−10 3.95 × 1005 8.37 × 1005
3.28 × 10−01 9.70 × 10−01
1.40 × 1001
3.67 × 1001
(3, 3)
4.70 × 10−07
4.70 × 1003
1.48 × 1004
6.72 × 1005 1.73 × 1006
2.26 × 10−01 8.50 × 10−01
4.01 × 1001
1.28 × 1002
(4, 2)
1.54 × 10−04
2.06 × 1003
4.78 × 1003
8.12 × 1002 1.29 × 1003
2.77 × 10−01 9.20 × 10−01 3.09 × 10−01 8.99 × 10−01(5, 1)
5.70 × 10−07
5.84 × 1005
1.51 × 1006
2.27 × 1003 3.21 × 1003
3.44 × 1001
1.01 × 1002
7.66 × 10−01
1.70 × 1000
(6, 0)
1.01 × 10−06
3.44 × 1005
7.45 × 1005
0.00 × 1000 0.00 × 1000
1.05 × 1001
2.58 × 1001
4.11 × 10−01 9.96 × 10−01

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

(σ1 , σ2 )(3, 3)

Tu1 = Tu12

3

Tu2 = Tu

u2

1

2

3

Tu1 = Tu

u1

4Tu = Tf

0

0
0

1

2

3

0

4

1

2

3

4
10-3

10-3

(a) y1d (t)(4, 2)(b) y2d (t)(6, 0)
Table 2: Parameters of two-inertia system motor bench.

8.40 × 10−4 kgm2
4.00 × 10−3 Nms/rad
95.5 Nm/rad

Jr
Dr

1

Tu2 = 2Tu u2

2

3

1

4
2

6Tu = Tf

Figure 4: Desired output trajectory yd (t) = [y1d (t) y2d (t)]T : they are 7th-orderpolynomials, respectively.

Jl
Dl
K

3Tu = Tf
u1

8.20 × 10−4 kgm2
4.00 × 10−3 Nms/rad

Tu1 = Tuu1

not used

u2

1

2

3

4

5

6

Figure 5: Examples of multirate inputs. Two inputs u1 and u2 are generated
according to the input multiplicities (σ1 , σ2 ). A control input with 0 input
multiplicity is not in use.

ventional MIMO single-rate feedforward controller in smooth
control inputs and continuous-time tracking errors.

5.3. Results: intersample behavior of multirate feedforward
and single-rate feedforward
The simulation results are shown in Figure 10. Figure 10(a)and Figure 10(b) show that the control inputs of the conventional MIMO single-rate feedforward controller F sr oscillate
and the proposed MIMO multirate feedforward controller Fmr
generates the smooth control inputs. Figure 10(c) and Figure 10(d) show that the outputs of the single-rate feedforward
controller are oscillated because of the oscillated control inputs, and the outputs of the multirate feedforward controller
are settled after 2 s. Figure 10(e) and Figure 10(f) show thatthe continuous-time tracking error of the multirate feedforward
controller is smaller than that of the single-rate feedforward
controller, thus the effectiveness proposed method is verified.
MIMO multirate feedforward controller is used in the twodegree-of-freedom robust control with feedback controllers which
reduce modeling error and disturbances. The role of the feedforward controller is the nominal tracking performance in thetwo-degree-of-freedom control scheme, and the simulation validations accurately verify it. In summary, the proposed optimal
MIMO multirate feedforward controller outperforms the con6

6. Conclusion
The procedure of the optimal MIMO multirate feedforward
controller design is proposed. The optimal MIMO multirate
feedforward controller makes the upper bound of the 2-normof control input ∥u[i]∥2 smaller, and as a result, the continuoustime tracking errors become smaller. The numerical simulationis conducted for the 6th-order system, and the proposed procedure of the selection of input multiplicities is validated.
The continuous-time tracking errors of the proposed MIMO
multirate feedforward controller Fmr are compared with the conventional MIMO single-rate feedforward controller F sr withthe two-inertia system motor bench. Depending on the polesof each controller, the conventional single-rate controller generates oscillated control inputs and the proposed multirate controller generates smooth control inputs. As a result, continuoustime tracking errors of the multirate controller are better thanthat of the single-rate controller in the MIMO LTI system.

50
20
0
-20
-40
-60

0
-50
100
180
90
0
-90
-180
0
10

(a) Photograph of two-inertia system motor bench.

τl

θl
Jl
Dl

θr
K

Left Side

τr

+

101

180
90
0
-90
-180
0
10

103

180
90
0
-90
-180
0
10

10

1

10

2

10

3

(b) From τr to θ̇l .

10

102

1

103

2

3

10

10

-50
100

101

102

103

180
90
0
-90
-180
0
10

10

1

10

2

10

(c) From τl to θ̇r .

3(d) From τr to θ̇r .

Figure 8: Bode diagram of two-inertia system motor bench. The black line is afrequency response function measurement of the system and the magenta lineis the identified continuous-time system.

∆θ

−
+

1

102

θr

1
Jr s2 +Dr s

−

3

10

101

0

100

+

τl

2

10

100

50

Figure 6: Details of two-inertia system motor bench. In this paper, the twoinertia system motor bench is modeled as a two-input two-output system. The
two inputs are left side torque τl and right side torque τr , respectively. The two
outputs are left side angle θl and right side angle θr , respectively.

K

1

103

20
0
-20
-40
-60

Right Side

τs

10

102

(a) From τl to θ̇l .

JrDr(b) Model of two-inertia system.

τr +

101

θl

Jl s2 +Dl s

Figure 7: Block diagram of two-inertia system.

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

Ongoing research focuses on MIMO LTI systems whichhave a different number of inputs and outputs, and the combination of single-rate and multirate controllers.

-1
-1

-0.5

0

(a) F sr

0.5

1

-1

-0.5

0

0.5

1

(b) FmrFigure 9: Poles and zeros of multirate feedforward controller Fmr and singlerate feedforward controller F sr with unit circle on z plane.

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