Control Technologies for In-Vehicle Electric Actuators

Technology
Explanation

Control Technologies for In-Vehicle Electric Actuators
MATSUMOTO Daisuke, ENOMOTO Daisuke, KENJO Kazunori, FUKUSHI Natsuru

Abstract
KYB develops and manufactures various in-vehicle
actuators including electric power steering (EPS). The
control specifications of the actuator differ depending on
the function, but it is a common issue to achieve a systemthat operates stably and satisfies the required functions
and performance. In addition, when controlling the actuator, the control system may become unstable dependingon the design of the control compensator, so it is neces-1

Introduction

Compared with electric power steering (EPS) for passenger vehicles, EPS for all-terrain and utility task vehicles, particularly for off-road vehicles, is required todeliver more highly responsive steering assist and kickback control performance. To satisfy these performancerequirements, it is a precondition that the EPS hardware(including the motor described later and its driving circuit,gear assembly, etc.) has been properly designed. In addition, it is necessary to maximize the hardware capabilities
by using software (i.e., with a proper control methodology). Specifically, the open-loop gain of the control
system (a gain of the loop transfer function) must be set toa level as large as possible within the frequency ranges inwhich the steering assist and kickback control performance are required. However, care must be exercised insetting the open-loop gain because just using a high openloop gain would reduce the gain and phase margins (hereinafter "stability margin"), resulting in an unstable control
system in some cases. As an approach to ensure a stability
margin while using a high open-loop gain (hereinafter"stabilization"), a phase compensator Note 1) has been used.
For stabilization using this approach, the phase compensator is designed with a focus on the gain and phase
information of the frequency characteristics of the looptransfer function of the control system (called a "nonparametric model"). It is relatively easy to implement aphase compensator, but designers need to adjust design
parameters through trial and error while carefully observing the frequency characteristics. This means that suchdesigners need to have a certain level of experience andsary to design the compensator after grasping the characteristics of the actuator to be controlled.
In this report, we explain the compensator design
method for the steering assist control of EPS for all terrain
vehicles and utility task vehicles and for steering angle
control with an eye toward future application in automatic
steering, as well as the implementation method of thisdesign method. This report also describes the future
outlook.

knack. As described later, the compensator is assumed tobe redesigned in phases during the development stage.
For higher efficiency in development, it is desirable toautomate the compensator development processes, including from design to implementation, to some extent.
However, design tool automation can hardly be achievedby conventional approaches.
Note 1) The compensator in a control system is a computing unit
intended to generate control input to impart desirable
characteristics to the system. A phase compensator isdesigned with a focus on the gain and phase of the looptransfer function of the control system.

On the other hand, robust control theory, typified by the
H∞ control theory, involves approaches that use the transfer function and state equation of the controlled object(called a "parametric model") to design the compensator
through inverse operation from the desirable characteristics (target tracking and disturbance response) of thecontrol system. In these approaches, the controlled object,
the characteristics of the control system, and disturbances
are represented by parametric models and the related
various equations are solved to determine the compensator. This means that design tools can relatively easily be
automated, delivering the benefit that compensators canbe designed more efficiently. These design approaches
may be effective to catch up with the rapid development
specific to the all-terrain and utility task vehicles market.
In general, compensators designed based on robust
control theory tend to be more complicated (with higher
degree) than phase compensators. However, in-vehicle
microprocessors mounted with floating point units (FPUs)
have been generally introduced in recent years to allow―8―

KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

implementation of compensators of relatively high degree.
This report explains how to design steering assist andsteering angle control compensators for EPS based on theparametrization of stabilizing compensators 3), which isone of the basic theories of robust control.
2

EPS Systems

2.1 Components of EPS System
Fig. 1 shows the components of the EPS for all-terrain
and utility task vehicles.
The electromechanical brushless motor shown in the
Figure (hereinafter "the power pack") is a brushless motor
integrated with a controller (hereinafter "the motor") as
the name implies. This component performs computations
for all controls including steering assist control.
The torque angle sensor (TAS) is another component
that detects the torque and angle of the steering wheel
operated by the driver. The steering assist control uses
torque values detected by the TAS as described later. The
steering angle control uses angle values detected by the
TAS to set the origin of the steering angle that is calculated from the motor rotation angle.
The gear assembly has a mechanism by which thetorque generated by the motor is amplified by a worm
reducer and this rotary motion is converted into a linear
motion by a rack and pinion device. This component
transfers the steering torque produced by the driver andthe steering assist torque generated by the motor to thevehicle's wheels as a cornering force.

TAS (Torque Angle Sensor)

communication function to communicate with the vehicle
and other sub systems via a CAN network, and ③ Control
functions including motor vector control, steering assist
control, and steering angle control.
With a focus on the steering assist and steering angle
controls, the next chapter describes how to design compensators for these controls.

Controller

Driving
circuit

Gear
assy

Motor

TAS

Torque signal
Angle signal

Rotation angle
sensorCurrent sensor
Motor rotation angle, current

Fig. 2 Block diagram of EPS system

3

Overview of Compensator Design

3.1 Components of Control System
Fig. 3 shows an overview of the control system (thecontrolled object + compensator) with a focus on thesteering assist control or steering angle control.
The control input to this control system is from motor
current commands. The control output from the system issteering angle or torque values from the TAS (the steering
angle is calculated from the motor rotation angle). The
load torque is attributable to the driver's steering andkickback transferred to the motor via the gear assembly
and the friction torque of the gear assembly.
In the Figure, the area enclosed by the red broken line
represents the controlled object, which is a mixed systemof hardware and software. The former includes the motor,
the driving circuit, the controller, and the gear assembly,
and the latter is for motor vector control and steering
angle computation.
Target value
signal(CAN)

Vehicle speed and othersignals (CAN)

Motorcurrent
commandCompensator
Control
inputElectromechanicalbrushless motor (Power pack)

Controlled
objectLoad torque
TorqueVectorcontrol

Driving
circuit

Motor

Current
sensorRotation
angle
sensor

Gear
assy

TAS

Angle

Current
Control
outputGear assy

Steering
angle SteeringEPS components for all-terrain and utility task
vehicles

2.2 Block Diagram of EPS System
Fig. 2 is a block diagram of the EPS system.
The controller has three major functions; ① A monitoring function to detect any abnormality of the CPU, driving
circuit and sensors, and perform fail-safe processing, ② A

Motor rotation angle

angle
computation

Fig. 3Fig. 1

Steering
kickback

Vehicle speed andother signals (CAN)

Overview of control system

3.2 Compensator Design Policy
Within the controlled object shown in Fig. 3 in the previous section, the hardware including the driving circuit,motor, and gear assembly behaves electrically or mechanically in real time. With these physical phenomena, thehardware constitutes a continuous time system. On theother hand, the software is a discrete time system Note 2)
because vector control and steering angle computation are―9―Control Technologies for In-Vehicle Electric Actuators

performed by digital computers. Therefore, the actual
controlled object is a mixed system of continuous anddiscrete time systems. The control input to and output
from the controlled object are updated or sampled at eachof their own control frequencies. That is, the controlled
object ranging from the control input to the control output
can be represented by a discrete time model Note 3). The
compensator can also be represented by a discrete timemodel because its computation is performed by digitalcomputers.
Note 2) A system whose behavior is defined with discrete time(hereinafter "sampling time").
Note 3) A discrete time model is a difference equation or transfer
function representation of the behavior of the controlled
object under the control of a digital computer at a sampling time or the behavior of a discrete time system.

Conventionally, the set of the controlled object andcompensator has been considered as a continuous timesystem. In this approach, the compensator is designed
according to continuous time and then discretized using,for example, a bilinear expression before implementation.
This approach has the problem that design stability
cannot be ensured in actual systems. One major factor isthat a compensator that has been discretized with a bilinear expression, for instance, cannot necessarily become astabilizing compensator for the discrete time system (i.e.,the stability margin for the discrete time system cannot
be guaranteed) (Challenge ①). Another major factor
relates to modeling error of the controlled object (discrepancy between the model and the actual system). The
system cannot be stable if the designed compensator was
not a stabilizing compensator for the actual system in thefirst place (Challenge ②). To address Challenge ①, it isuseful to represent the controlled object as a discrete timemodel and design the compensator according to the discrete time system. For Challenge ②, it is effective to use
the frequency response method or the method of identification 4) using maximum-length sequence signals todetermine a model for designing the compensator (hereinafter a "compensator design model"). Then, we havedecided to introduce the approach of representing thecontrolled object in a discrete time model and designing
the compensator in a discrete time system. We also identify the system to be controlled upon completion of the
hardware fabrication and design the compensator using
the identified compensator design model. We use δ (delta)
operator 5) as a discrete time operator representing thecontrolled object and compensator model. The δ operator
can be expressed as δ = (z - 1)/Tc (where z is the z operator
and Tc is control frequency). A major advantage of using
the δ operator is that relatively high accuracy can beobtained with compensator computation with a low
number of bits. For more information, see References 4), 5).
3.3 Block Diagram of Control System
The controlled object included in the control system
shown in Fig. 3 can be approximately represented by ablock diagram as shown in Fig. 4:

Zero-orderholdFig. 4

SamplerB lock diagram of controlled object (continuous timesystem)P(s) in Fig. 4 is the continuous time model (transfer
function) of the controlled object. k is an integer not lessthan 0 (zero) used to indicate the sampling time [ts(k) =kTc (k = 0, 1, ...)]. u(k) is the output from the compensator,
namely, control input (motor current command). u(t) isu(k) held every "time" ts(k). y(k) is the control output thatis y(t) sampled every "time" ts(k). Control output y(k) will
be a TAS torque signal for steering assist control or asteering angle for steering angle control. d(t) is the disturbance attributable to the load torque shown in Fig. 3 andthe rotation of the steering wheel operated by the driver.
d0 is a constant. Gd(s) is the transfer function representing
a disturbance generator. Note that u(t) and u(k) are signalsthat are different from y(t) and y(k) signals, but for convenience these are not differentiated by symbols.
The block diagram of Fig. 4 may be represented by adiscrete time system as shown in Fig. 5. P(δ) and Gd (δ)
are transfer functions (discrete time models) obtained byzero-order holding and discretization P(s) and Gd (s) witha sampler, respectively. Note that P(s) and P(δ) are transfer functions that are different from Gd (s) and Gd (δ)
transfer functions, but for convenience these are also notdifferentiated by symbols.

Fig. 5

Block diagram of controlled object (discrete timesystem)Thus, the control system in Fig. 3 can be represented
using the transfer function model P(δ) of the controlled
object and the transfer function model C(δ) of the compensator as shown in Fig. 6. In Fig. 6, r(k) is the target
steering angle signal. r0 is a constant. Gr(δ) is the transfer
function representing the target value signal generator.
This report describes how to design a compensator C(δ)
for the control system in Fig. 6. Note that C(δ) is assumed
to be a two-degree-of-freedom controller as follows:
C(δ) = [Cr(δ) Cy(δ)](1-1)u(k) = Cr(δ)r(k)－Cy(δ)y(k)(1-2)― 10 ―KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

Fig. 6Block diagram of control system

3.4 Compensator Design Flow
Fig. 7 shows the compensator design flow as a V-shaped
process diagram. This section describes what to do in eachof the design processes (the flow to the left in Fig. 7).

Setting design specifications and modeling
controlled object and disturbance
Checking behavior on actual vehicle

Requirement
specification

Reflecting system
identification results andredesigning compensator

Systemevaluation

s

es

pr
oc
on

Unit
evaluation

ati

oc
pr

Agreement
verification
Model
verification

ific

n

Detailed
design

Ve
r

sig

De

es

s

Basic
designActual vehicle
evaluation

Automatic code
generation

Fig. 7

V-shaped process of compensator design

3.4.1 Requirement Specification
The following describes what to do in the Requirement
Specification stage.
(1) Setting design specifications
For steering assist control, an assist map where motor
current commands are mapped against TAS torque signals
is used to set the assist torque. A higher assist torquemeans a larger open-loop gain. A major requirement forthe design specifications is that the required assist torquecan be generated stably (i.e., without issuing harmful
vibration). In other words, the control system will remain
stable even with the maximum assumed open-loop gain.
For steering angle control, the steering angle will follow
the target steering angle signal r(k) in Fig. 6. Major design
specifications include, for example, delay time, rise time,
setting time, overshoot, and steady-state deviation to r(k)that may change stepwise or in ramp rate.
(2) Modeling controlled object and disturbances
Controlled object models can roughly be divided into
two types. One is a high-precision simulation model 6) thatcan estimate in relative detail the behavior of the actual
system (hereinafter "the simulation model"). The other isa simplified model for designing compensators (the aforementioned compensator design model). A compensator
design model can be derived from a differential equationthat has linearized and simplified the controlled object. A

compensator design model can also be derived throughidentification of a system using the simulation model or
actual machine. The identification of systems using an
actual machine will be described later. The simulation
model can be created and implemented mainly by using
tools such as MATLAB®/Simulink® and SimulationX®.
The disturbances that may be applied to the controlled
object include; ① Rotation of the steering wheel operated
by the driver (for steering assist control), ② Load torquedue to kickback, and ③ Friction torque of the gear assembly. For steering assist control, these disturbances can be
represented by a transfer function model with an assumption of steering frequency as described later. For steering
angle control in turn, the disturbances should be represented by a step or ramp function model based on theassumption that, with the driver's hands off the steering
wheel, the steering angle can follow the steering angle
command without steady-state deviation under disturbances ② and ③.
3.4.2 Basic Design
In the basic design stage, a compensator is designed
using the compensator design model and the assumed
disturbance model. The compensator can be expressed bya transfer function model C(δ) in this stage. The simulation model and the designed compensator are used tocarry out simulation and verify the adequateness of thecompensator during this stage. The compensator design
will be described in detail in Chapter 4.
3.4.3 Detailed Design
In the detailed design stage, the compensator designedin the basic design stage is developed into an implementation model to be contained in the controller as software.
This stage should verify that the compensator C(δ) derived
in the basic design stage is equivalent to the implementation model (i.e., identical inputs will produce the same
output).
3.4.4 Automatic Code Generation
Embedded Coder® is used to automatically generate
codes from the implementation model.
4Compensator Design

This chapter describes the details of the basic design ofcompensators for steering assist control and steering angle
control. The compensator design for steering assist control
is different from that for steering angle control in control
output, design specifications, and disturbances. The twocompensators can be applied with the same design
approach.
4.1 Basic Design of Steering Assist Control Compensator
4.1.1 Deriving a Compensator Design Model
An EPS model is shown in Fig. 8:
The EPS shown in Fig. 8 is a single-pinion type. Thismodel can also be applied to double-pinion or column― 11 ―

Control Technologies for In-Vehicle Electric Actuators

Torsion bar

the equations below:
δx(k)＝APδ x(k)＋BPδu(k)(3-1)y(k)＝CPδ x(k)(3-2)
where

Motor

Worm reducer

APδ＝（APz－I2）/Tc , I2: 2 × 2 unit matrix

BPδ＝BPz /Tc, CPδ＝CPz＝CP

APz＝e A T , BPz＝∫T0 e A T dτ BP
P cSpring load

Fig. 8

Rack and pinion

EPS model

EPS. The symbols used in the figure indicate the following:θh : Rotation angle of steering wheel [rad]θp : Rotation angle of pinion [rad]im : Motor current [A], τm: Motor torque [N·m]Kt : Motor torque constant [N·m/A]
KTB : Spring constant of torsion bar [N·m/rad]Rw : Reduction ratio of worm reducer
Ipe : Equivalent inertia moment of pinion shaft [kg·m2]
Cpe : Equivalent viscous resistance coefficient of pinion
shaft [N·m/(rad/s)]
Rp : Specific stroke of rack and pinion [m/rad]Kr : Spring constant of rack load [N·m/rad]The TAS torque signal can be expressed by the equation
below:
τs＝KTB(θh－θp)(2-1)
Assuming that im is equal to u (the actual current completely follows the motor current command) and that all
the parts except the torsion bar are rigid bodies, the rotary
motion of the pinion can be expressed by the differential
equation below:
Ipeθ̈p＋Cpe θ̇ p＝RwKtim＋KTB (θh－θp)－KrRp2θp(2-2)
where
im＝u
From Eqs. (2-1) and (2-2), the following state andoutput equations can be obtained:
x＝A
̇P x＋BPu＋EPθh(2-3)
y＝CPx＋FPθh(2-4)
whereT: State variable, y = -τs: Control outputx＝［θp θ̇p］
u: Control input (motor current command)
⎤

⎡

｜
｜
0
1
2
｜，K ＝K ＋K R
AP＝｜
pe
TB
r p
｜
｜－K /I
/I
－C
pe
pe
pe
pe ⎦
⎣

In the initial development stage where the detailed
design of hardware has not been completed, Eqs. (3-1) to(3-3) should be used as compensator design models todesign compensators. However, these models may need tobe revised when specific hardware specifications are madeavailable or when the hardware fabrication has been completed. With consideration given to these cases, the degreeof the compensator design models (degree of the denominator polynomial dp (δ)) is assumed to be 'n' in the following discussion. Then, the transfer function P(δ) isexpressed by the equation below:n（δ）
n δ n－1＋…＋nP1δ＋nP 0
P（δ）＝ P
＝ n Pn－1 n－1
δ ＋dPn－1δ ＋…＋dP1δ＋dP 0
d（δ）
P
(3-4)
4.1.2 Setting Disturbance Models
This section discusses disturbance models of degree lthat can be expressed by the equations below:
d＝Gd(s)d0(4-1)
n（s）
n sl－1＋…nd 1s＋nd 0
G（s）＝ d＝ l dl－1 l－1ds ＋ddl－1s ＋…dd 1s＋dd 0
d（s）
(4-2)d

The state and output equations of Eq. (4-1) can be
expressed by the equations below:x＝A
̇
d x＋Bd d0(4-3)
d＝Cd x(4-4)
where
⎡
⎤
1
0
…
0 ｜
｜ 0
｜
｜
｜ 0
0
1
…
0 ｜
｜
｜
｜
Ad＝｜
｜
｜
｜ 0
｜
0
0
…
1
｜
｜
｜－d
｜
－d
－d
－d
d0
d1
d2
dl－1⎦
⎣
⎡
｜
｜
｜
｜
｜
｜
｜
⎣

…

⎤

0 ｜
｜，F ＝－KP
TB
KTB /Ipe｜
⎦

n（δ）
P
d（δ）
P

n δ＋nP 0＝ 2 P1
(3-3)
δ ＋dP1δ＋dP 0

…

EP＝

－1
P（δ）＝CPδ（δI2－APδ）
BPδ＝

…

⎤

｜
0
｜，C ＝
BP＝
0］
P ［KTBRw Kt /Ipe ｜⎦P cAPz, BPz and CPz are a system/control matrix where Eq.
(2-3) or (2-4) is discretized by the z operator each. Thesecan be determined using the discretization method "zoh"using the MATLAB® function "c2dm". Eqs. (3-1) and(3-2) are called step invariant models 5) of Eqs. (2-3) and(2-4). When Eqs. (3-1) and (3-2) are selected for the compensator design model, the transfer function P(δ) in Figs.
5 and 6 can be expressed by the equation below:

…
⎤

0｜

In Eqs. (2-3) and (2-4), if θh is 0, the state and output
equations for the discrete time model can be expressed by― 12 ―

Bd＝

｜
｜
｜
｜
｜
｜
⎦

…

⎡
｜
｜
｜
⎣
⎡
｜
｜
｜
⎣

c

0
1

，Cd＝［nd 0 nd 1 nd 2 … ndl－1］

KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

The state and output equations of the discrete time modelof disturbances can be expressed by the equations below:
δx(k)＝Adδ x(k)＋Bdδ d0(4-5)d(k)＝Cdδ x(k)(4-6)
where
Adδ＝
（Adz－Il）/Tc , Il: l × l unit matrix

δ planeStable region

Imaginary
axis

Real axis

Bdδ＝Bdz /Tc , Cdδ＝Cdz

Adz＝e A T , Bdz＝∫T0 e A T dτ Bd
d c

c

d c

Adz, Bdz and Cdz are a system/control matrix where Eq.
(4-3) or (4-4) is discretized by the z operator each. Thesecan be determined using the discretization method "zoh"or "matched" (matched pole-zero model 5)) using the
MATLAB® function "c2dm". Note that the matched polezero model has the same poles as those of the step invariant model.
From Eqs. (4-5) and (4-6), the transfer function Gd(δ) of
the discrete time model of disturbances can be expressedby the equations below:
n（δ）
d
d（δ）
d
ndl－1δ l－1＋…nd1δ＋nd0＝ 1
(4-7)
δ ＋ddl－1δ l－1＋…dd1δ＋dd0

Fig. 9Stable region of δ operator

4.1.4 Deriving a Compensator
For steering assist control, assume that the compensatorCr(δ) in Eqs. (1-1) and (1-2) is Cr(δ) = 0. In this case, thecontrol system can be represented by Fig. 10:

－1
G（δ）
＝Cdδ
（δIl－Adδ）
BPδ＝d

Note that the numerator polynomial of the disturbancemodel nd(δ) will not be used for compensator design.
4.1.3 Coprime Factorization of Compensator
Design Model
To get ready for designing the compensator based on
the parametrization of the stabilizing compensator, thecompensator design model of Eq. (3-4) is expressed by
the coprime factorization below. If the controlled object isa system with multiple inputs/outputs, the model needs tobe expressed in right and left coprime factorizations. This
controlled object is a system with single input/output,
which means that the right coprime factorization is identical to the left coprime factorization. Therefore, this report
simply uses the term "coprime factorization".
P(δ)＝NP(δ)/DP(δ)(5-1)n（δ） nPn－1δ n－1＋…＋nP1δ＋nP0
(5-2)
N（δ）
＝ P
＝ n
P
δ ＋fn－1δ n－1＋…＋f1δ＋f0
（δ）
f
D（δ）
＝
P

d（δ）
δ n＋dPn－1δ n－1＋…＋dP1δ＋dP 0
P＝δ n＋fn－1δ n－1＋…＋f1δ＋f0（δ）
f(5-3)f(δ) is a stable polynomial and any given parameter thatcan be set by the designer. When f(δ) is said to be stable,
this means that its root exists at the center of the complex
plane -1/Tc within the circle of radius 1/Tc (see Fig. 9).
How to set f(δ) will be described later.

Fig. 10Block diagram of steering assist control system

The compensator Cy(δ) based on the parametrization of
the stabilizing compensator can be expressed by the equations below:
Cy(δ)＝NC(δ)/DC(δ)(6-1)
NC(δ)＝XP(δ)＋R(δ)DP(δ)(6-2)
DC(δ)＝YP(δ)－R(δ)NP(δ)(6-3)
X（δ）
＝
P

n（δ）
n δ n－1＋…＋nX 1δ＋nX 0
X＝ n－1Xn－1
ɡ（δ） δ ＋ɡn－2δ n－2＋…＋ɡ1δ＋ɡ0

(6-4)
Y（δ）
＝
P

n（δ）
n δ n－1＋…＋nY 1δ＋nY 0
Y＝ n－1Yn－1
ɡ（δ） δ ＋ɡn－2δ n－2＋…＋ɡ1δ＋ɡ0

(6-5)
g(δ) is a stable polynomial that is any given parameter tobe set by the designer. R(δ) is a stable-proper transfer
function (degree of numerator polynomial is equal to orless than degree of denominator polynomial) that is a freeparameter to be selected by the designer. How to set theseparameters will be described later.
Xp(δ) and Yp(δ) are solutions to Bezout equations.
XP(δ)NP(δ)＋YP(δ)DP(δ)＝１(6-6)
Firstly, Xp(δ) and Yp(δ) should be derived. Multiplying
both sides of Eq. (6-6) by h(δ) = f(δ)g(δ) yields an identical equation below:nX(δ)nP(δ)＋nY(δ)dp(δ)＝h(δ)(6-7)
whereh(δ)＝δ2n－1＋h2n－2δ2n－2＋…＋h1δ＋h0
When np(δ) and dp (δ) do not share a common divisor,nx(δ) and ny(δ) that satisfy Eq. (6-7) will be uniquely
decided. Their coefficient can be determined using theequation below 5):
ΘT＝Ψ TE－1(6-8)― 13 ―

Control Technologies for In-Vehicle Electric Actuators

where

where

ΘT＝［q0 q1 … q2n－2 nR 0 nR 1 … nRl－1］

Θ T＝［nY0 … nYn－1 nX0 … nXn－1］
0
0

dd 1

⎤

0
0

0｜
｜
0｜
…

dd 0

…
…

…

…

…
…

0｜
｜
0｜

dd 2 … ddl－1 1

…

w0

w2

：l ×（2n＋l－1）matrix… w2n－2 …

｜
｜
｜
｜
⎦

⎤

｜
｜
｜
｜
⎦

…

The coefficient of nR (δ) can be determined using theequation below:
ΘT＝Ψ TE－1(7-6)

0

0
0
…

The following identical equation should be set so thatthe numerator polynomial of Eq. (7-3) includes dd (δ):dR(δ)f(δ)nY(δ)－ɡ(δ)nP(δ)nR(δ)＝dd(δ)q(δ)(7-4)
Modifying Eq. (7-4) yields the equation below:
dd(δ)q(δ)＋w(δ)nR(δ)＝γ(δ)(7-5)wherew(δ)＝ɡ(δ)nP(δ)＝δ2n－2＋w2n－3δ2n－3＋…＋w1δ＋w0
γ(δ)＝dR(δ)f(δ)nY(δ)＝δ2n＋l－2＋γ2n＋l－3δ2n＋l－3＋…＋γ1δ＋γ0q(δ)＝q2n－2δ2n－2＋q2n－3δ2n－3＋…＋q1δ＋q0

0

…

（δ）
f
n（δ）
－ɡ
（δ）
n（δ）
n（δ）
d（δ）
Y
P
R
＝ R
(7-3)
（δ）
f
（δ）
ɡ
d（δ）
R

w1 … w2n－20…
w0 … w2n－3 w2n－2 …
…

n（δ）
n（δ） n（δ）
YP－ R（δ） d（δ）
ɡ
（δ）
f
R

E2＝

w0
0

…

(7-2)
where dR (δ) is a stable polynomial. How to set the polynomial will be described later.
Developing Dc(δ) yields the equation below:

⎡
｜
｜
｜
｜
｜
｜
｜
⎣

…

n（δ）
n δ l－1＋nRl－2δ l－2＋…＋nR1δ＋nR0
R＝ Rl－1l－1
δ ＋dRl－2δ l－2＋…＋dR1δ＋dR0
d（δ）
R

D（δ）
＝Y（δ）
－R
（δ）
N（δ）
＝
C
P
P

0

ddl－1

0
1

：（2n－1）×（2n＋l－1）matrix

Next is the process to derive R(δ). In Fig. 10, the control
output y(k) in response to the disturbance d(k) can be
expressed by the equation below:y(k)＝S(δ)d(k)＝DC(δ)DP(δ)Gd(δ)d0(7-1)
Gd(δ) may have unstable poles or a pole that can delay
the attenuation of d(k) regardless of its stability. In orderto minimize fluctuations of y(k) due to the effect of suchpoles as quickly as possible, the zero of Dc(δ) (the root of
the numerator polynomial) must include the poles ofGd(δ). Therefore, it is necessary to select R(δ) so that thenumerator polynomial of Dc(δ) includes the denominator
polynomial dd(δ) of Gd(δ). Since the degree of the disturbance model of Eq. (4-7) is l, the degree of R(δ) is set tol-1. R(δ) is expressed by the equation below:
R（δ）＝

0

1

…

nP2 … nPn－1 0

dd 1 … ddl－1
dd 0 … ddl－2
…

E1＝

d
0

…

⎡
｜ d0
｜
｜
｜
｜
｜
｜
⎣

…

…

…

nP1

｜
｜
｜
｜
｜
｜
｜
｜
｜
｜
｜
｜
⎦

…

…

…

nP0

…

…

…

0

⎡
⎤
｜ E1｜
｜
E＝｜
｜E ｜
⎣ 2⎦

0⎤
｜
｜
0｜

…

…
…

dP1 dP2 … dPn－1 1
0 … dP0
nP1 … nPn－1
0
0 … 0
0
nP0 … nPn－2 nPn－1 0 … 0
0

…

…
0

…

0
n
0

…

E＝

…

…

｜
｜
｜
｜
｜
｜ P0
｜
｜
｜
｜
｜
｜
⎣

0
1

Ψ T＝［γ0 γ1 … γ2n＋l－3 1］…

Ψ T＝［h0 h1 … h2n－2 1］⎡
1｜dP0 dP1 … dPn－1｜
dP0 … dPn－2 dPn－1｜ 00

4.1.5 Example of Basic Design of Compensator(1) Determining the compensator design model
When Ap, Bp, Cp, Ep, and Fp in Eqs. (2-3) and (2-4) are
given their physical parameter values and then discretized,
the transfer function of Eq. (3-3) will be:
P（δ）＝

n（δ）
n δ＋nP 0P＝ 2 P1δ＋dd（δ）
PP 1δ＋dP 0

7.807×10－3（δ＋1.980）＝ 2
(8-1)－2
δ ＋7.964×10 δ＋2.163×10－2
The δ operator and the coefficients of the numerator anddenominator of Eq. (8-1) were made dimensionless with
the inverse of the control period 1/Tc (hereinafter "thecontrol frequency"). In other words, δ in the equation is δ
= Tc δ. All figures appearing in the following examples
indicate dimensionless transfer functions.
The poles and zeros of Eq. (8-1) are all within the stableregion, implying a stable controlled object. Note that thezeros in Eq. (8-1) have been added through discretization.
Next, the coprime factorization of Eq. (8-1) P(δ) =Np(δ) / Dp(δ) is set as follows:n（δ） 7.807×10－3（δ＋1.980）
(8-2)
N（δ）
＝ P
＝
P
2
（δ＋0.2583）
（δ）
f
D（δ）
＝
P

d（δ）
δ2＋7.964×10－2δ＋2.163×10－2
P＝2（δ＋0.2583）（δ）
f

(8-3)
1/f(δ) in Eqs. (8-2) and (8-3) is the matched pole-zeromodel of the following stable transfer function 1/f(s) thathas been made dimensionless with the control frequency
1/Tc:
1/f(s)＝1/(s2＋2ζf ωf s＋ωf2)(8-4)
where 0 < ωf, 0 < ζf
If 1/f(s) is stable, its matched pole-zero model 1/f(δ)isalso stable. To allow the control system to deliver highresponsivity, it is desirable to set ωf to a value as high as
possible. ζ f, which is the attenuation factor of the poles of
the control system, should be set to 1 or more. For convenience, these are set as follows:― 14 ―KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

angularfrequency
ζf＝1 ，ωf＝2 Kpe /Ipe (naturaldecided by the equivalent stiffness and equivalent inertia
moment of the pinion shaft multiplied by 2).
The solutions to Bezout equations Xp(δ) and Yp(δ) canbe expressed in the equations below:n（δ） 8.179δ＋0.2314
X（δ）
＝ X
＝
(8-5)P（δ）
ɡ
δ＋0.2583
(8-6)n（δ） δ＋0.6314
Y（δ）
＝ Y
＝
P
（δ） δ＋0.2583
ɡ

nR1＝10.135
nR0＝1.888
nR1＝3.747
nR0＝0.8549

C（δ）
＝
y

n（s）
n（s）
d
G（s）
＝ d
＝ 2
d
2
（s）
＋2ζ
d
s
(9-1)dd ωd s＋ωd
where ζd = 1. The following settings will be used according to the steering assist torque level:ωd＝0.5ωd＝0.9ωd＝1.5

Kpe /Ipe : Large steering assist torque
Kpe /Ipe : Medium steering assist torque
Kpe /Ipe : Small steering assist torque

C（δ）
＝
y

: Large steering assist torque(10-2)

11.926（δ＋0.205）（δ＋0.1938）（δ＋0.07162）（δ＋0.7172）（δ＋0.2008）2: Small steering assist torque(10-3)

According to the equations above, the poles of Cy (δ)
include the pole dd(δ) of the disturbance model in Eq.
(9-2) for all the torque cases. Fig. 11 shows the poles andzeros of the compensator of Eqs. (10-1) to (10-3) plottedon a complex plane. The figure indicates that all the poles
and zeros are within the stable region. Note that both thevertical and horizontal axes have been made dimensionless with the control frequency 1/Tc.

Imaginary axis × Tc

n（δ）
n（δ）
n（δ）
d
d
＝ 2
＝ d
δ ＋2ζd ωd δ＋ωd2 （δ＋ωd）2d（δ）
d
(9-2)
where ζd = 1. ωd is as follows:ωd = 0.07198: Large steering assist torqueωd = 0.1258: Medium steering assist torqueωd = 0.2008: Small steering assist torquewhere
nR1＝15.640
nR0＝2.429

18.314（δ＋0.1513）（δ ＋0.2392δ＋2.031×10－2）（δ＋0.8172）（δ＋0.1258）2

C（δ）
＝
y

G（δ）
＝
d

n（δ） nR1δ＋nR0(9-3)
R（δ）＝ R＝
δ＋0.2583
d（δ）
R(10-1)

2: Medium steering assist torque

In these settings, the step invariant or matched pole-zeromodel of the transfer function in Eq. (9-1) that has been
made dimensionless with the control frequency 1/Tc canbe expressed by the equation below:(3) Deriving free parameter R(δ)
It is assumed that the pole dR (δ) of the free parameter isthe same as g(δ) of Eqs. (8-5) and (8-6). Substituting Eqs.
(8-2), (8-6), and (9-2) as well as dR (δ) = g(δ) = δ + 0.2583
in Eq. (7-6) yields the following free parameter R(δ):

23.819（δ＋0.121）（δ2＋0.2204δ＋2.358×10－2）（δ＋0.8819）（δ＋0.07198）2: Large steering assist torque

KTB /Ipe is used.

(2) Setting disturbance models
The disturbance model for the continuous time system
is set as follows. The numerator polynomial nd(s) of the
disturbance model is hereinafter omitted because it will
not be used for compensator design.

: Small steering assist torque(4) Frequency characteristics of control system
From the above, substituting Eqs. (8-2), (8-3), (8-5),(8-6), and (9-3) in Eqs. (6-1) to (6-3) yields Cy(δ) as
follows:

1/g(δ) in Eqs. (8-5) and (8-6) is the matched pole-zeromodel of the following stable transfer function 1/g(s) thathas been made dimensionless with the control frequency
1/Tc:
1/ɡ(s)＝1/(s＋ωɡ)(8-7)
where ωɡ＝2

: Medium steering assist torqueStable region (inside the circle)
Zero of controller
Pole of controller

Real axis × Tc

Fig. 11

Poles and zeros of compensator (basic design)

Next, the frequency characteristics of Cy(δ), Cy(δ)P(δ)(loop transfer function), and S(δ) are shown in Fig. 12.
For plotting of the Bode diagrams in Fig. 12, Cy(δ), Cy(δ)P(δ), and S(δ) have been converted into the form of the z
operator and applied with the MATLAB® function"dbode". The horizontal axis of these diagrams has been
made dimensionless with the control frequency 1/Tc.
According to the diagrams, the gain of the compensator orloop transfer function depends on the steering assist
torque level.
This design approach is not intended to design compensators with a focus on the loop transfer function or stability margin. Still, it should be noted that eventually the― 15 ―

Gain [dB]Control Technologies for In-Vehicle Electric Actuators

Large steering assist torque
Medium steering assist torque
Small steering assist torquePhase [deg]Dimensionless frequency f × Tc

Large steering assist torque
Medium steering assist torque
Small steering assist torqueDimensionless frequency f × Tc(a) Frequency characteristics of compensator Cy(δ)

Gain [dB]Gain margin of
10 dB or more
Large steering assist torque
Medium steering assist torque
Small steering assist torquePhase [deg]Dimensionless frequency f × Tc

Large steering assist torque
Medium steering assist torque
Small steering assist torquePhase margin of
40 deg or more

Dimensionless frequency f × Tc

Gain [dB](b) Frequency characteristics of loop transfer function Cy(δ)P(δ)
Large steering assist torque
Medium steering assist torque
Small steering assist torquePhase [deg]Dimensionless frequency f × Tc
Large steering assist torque
Medium steering assist torque
Small steering assist torquedcl(δ)＝｛ f(δ)｝2 ɡ(δ)dR(δ)＝(δ＋0.2583)6(10-4)
That is, f(δ), g(δ) and dR (δ) that were set during design
become the poles of the closed-loop transfer function.
This implies that this design approach is to design a compensator by setting the poles of the closed-loop transfer
function (or poles of the control system).
4.2 Basic Design of Steering Angle Control
Compensator
4.2.1 Deriving a Compensator Design Model
Fig. 13 shows a model of EPS including the steering
wheel. For steering angle control, the steering angle determined from the motor rotation angle must follow thetarget steering angle signal while suppressing vibration
caused by the turn of the steering wheel. Therefore, thecontrolled object is the steering system including therotary motion of the equivalent inertia moment of thesteering wheel shaft. The figure uses symbols to indicate
the following meaning. The other symbols not on the list
below have the same meaning as those used in Fig. 8.
θm : Motor rotation angle [rad] (θm = RPθP)
Equivalent inertia moment of steering wheel
Ih : 
shaft [kg·m2]
Ch : Equivalent viscous resistance coefficient ofsteering wheel shaft [N·m/(rad/s)]
τh : Steering wheel input torque [N·m] (assume τh = 0
with the driver's hands off the wheel)
The control output, namely, the steering angle signal θ̂ p
can be determined using the equation below:θ̂ p＝θm /Rw(11-1)
Assuming again that im is equal to u (the actual current
completely follows the motor current command) and that
all the parts except the torsion bar are rigid bodies, therotary motion of the pinion can be expressed by the differential equation below:
Ipeθ̈p＋Cpe θ̇ p＝RwKtim＋KTB (θh－θp)－KrRp2θp(11-2)
where
im＝u

Steering wheel

Dimensionless frequency f × Tc(c) Frequency characteristics of transfer function S(δ)
Fig. 12

Frequency characteristics of steering assist control
systemgain margin is not less than 10 dB and the phase margin
not less than 40 degrees for all cases.
S(δ), which is called the sensitivity function, is lesslikely to be affected by the parameter variations of thecontrolled object or disturbances with its smaller gain.
The smaller gain means a smaller control output to thesteering wheel rotation, in other words, a larger steering
assist torque. The poles of S(δ) are the poles dcl(δ) of theclosed-loop transfer function of the control system in Fig.
10 and can be expressed by the equation below:― 16 ―

Torsion bar

Motor

Worm reducer

Spring load

Fig. 13

Rack and pinion

Model of EPS including steering wheel

KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

The rotary motion of the wheel can be expressed by theequation below:Ihθ̈h＋Chθ̇ h＝－KTB (θh－θp)(11-3)
From Eqs. (11-1) to (11-3), the following state and output
equations can be obtained:
x＝A
̇P x＋BPu(11-4)
y＝CP x(11-5)
wherex＝［θp θh θ̇ p θ̇ h］T: State variable

ble-proper transfer function (hereinafter a "pre-compensator"). In this case, the control system can be represented
by Fig. 14:y＝θˆp＝θm /Rw＝θp: Control output
⎡
｜
｜
｜
｜
｜
｜
｜
⎣
⎡
｜
｜
｜
｜
｜
｜
｜
⎣

Fig. 14
⎤
｜
｜
｜
｜
｜
｜
｜
h⎦

0
0
1
0
0
0
0
1
AP＝
0
－Kpe /Ipe KTB /Ipe －Cpe /Ipe
KTB /Ih －KTB /Ih
0－Ch /I⎤｜
0｜｜0｜, CP＝
BP＝［1 0 0 0］
Rw Kt /Ipe ｜｜｜0⎦

In Eqs. (11-4) and (11-5), the state and output equations
for the discrete time model can be expressed by the equations below:
δx(k)＝APδ x(k)＋BPδu(k)(11-6)y(k)＝CPδ x(k)(11-7)
where
APδ＝
（APz－I4）/Tc , I4 : 4 × 4 unit matrix
BPδ＝BPz /Tc , CPδ＝CPz＝CP

In the control system shown in Fig. 14, the transfer
function Gry(δ) ranging from the target steering angle
signal r(k) to the control output y (hereinafter "target
tracking characteristics") can be expressed by the equation below:
Gry(δ)＝FC(δ)NP(δ)(13-1)
Then, Gry (δ) is set as follows:
n（δ）
G（δ）
＝ M
ry
d（δ）
M
nMm－1δm－1＋nMm－2 δm－2＋…＋nM 1δ＋nM0＝
δm＋dMm－1δm－1＋…＋dM 1δ＋dM 0
(13-2)where dM (δ) is a stable polynomial.
Next, the deviation er(k) of the output yr(k) of Gry(δ)
from r(k) can be expressed by the equation below:
e（k）＝r（k）－y（k）＝r（k）－G（δ）
（k）
r
r
r
ry

APz＝e A T , BPz＝∫T0 e A T dτ BP
P c

c

d（δ）
－n（δ）
M
M
（k）
r
d（δ）
M
(13-3)d（δ）－n（δ）
M
＝ M
G（δ）
r0
r
d（δ）
M

P c

When Eqs. (11-6) and (11-7) are selected as the compensator design model, the transfer function P(δ) in Figs. 5
and 6 can be expressed by the equation below:
n（δ）
P
d（δ）
P
nP 3δ3＋nP2δ2nP1δ＋nP0
(11-8)＝ 4
δ ＋dP3δ3＋dP2δ2＋dP1δ＋dP0
－1
P（δ）
＝CPδ
（δI4－APδ）
BPδ＝

The setting of disturbance models and coprime factorization of the compensator design model are omitted here
as these are as described in sections 4.1.2 and 4.1.3. The
following describes how to derive a compensator.
4.2.2 Setting a Target Value Signal Model
This section discusses the target value signal model ofdegree l represented by the equation below:n（δ）
n δ l－1＋…nd1δ＋nd0
(12)
G（δ）
＝ r
＝ l dl－1 l－1rδ ＋ddl－1δ ＋…dd1δ＋dd0d（δ）
rNote that the numerator polynomial of the target value
signal model nr(δ) will not be used for compensator
design.
4.2.3 Deriving a Compensator
For steering angle control, assume that the compensatorCr(δ) of Eqs. (1-1) and (1-2) is Cr(δ) = Fc(δ) / Dc(δ) basedon the parametrization 3) of the stabilizing compensator forthe two-degree-of-freedom control system. Fc(δ) is a sta-

Block diagram of steering angle control system

｝
（k）
r ＝
＝
｛ 1－G（δ）
rydM (δ) 's degree m may be any number, but needs to be set
from the following perspectives:①If the compensator design model has an unstablezero, in other words, if np(δ) or dr(δ) has an unstableroot, nM (δ) must include the root.
②In order for the deviation er(k) to converge to zero(0) asymptotically, {dM (δ) - nM (δ)} in Eq. (13-3) musthave the unstable pole of Gr(δ) as its root.
Then, assuming that np(δ) has an unstable root, an identical equation shown below is set so that {dM (δ) - nM (δ)} in
Eq. (13-3) has the pole dr(δ) of Eq. (12) as a divisor:dM(δ)－nP(δ)βM(δ)＝dr(δ)αM(δ)(13-4)
wherenM(δ)＝nP(δ)βM(δ)(13-5)
αM (δ) and βM (δ) are polynomials of degree n-1 and l-1,
respectively. Degree of dM (δ) is m = n + l - 1. Modifying
Eq. (13-4) gives the equation below:nP(δ)βM(δ)＋dr(δ)αM(δ)＝dM(δ)(13-6)
The coefficient of βM (δ) can be determined using theequation below:
ΘT＝Ψ TE－1(13-7)― 17 ―

Control Technologies for In-Vehicle Electric Actuators

where
Θ T＝
［αM0 … αM n－1 βM 0 … βM l－1］
Ψ T＝
［dM0 dM1 … dMn＋l－2 1］
⎡
⎤
｜ E1｜
｜
E＝｜
｜E ｜
⎣ 2⎦

dr1

⎤

0
0

0｜
｜
0｜
…

dr0

…
…

…

…

drl－1

0
1
…

0

1

…

0

…

dr1 … drl－1
dr0 … drl－2
…

E1＝

d
0

…

⎡
｜ r0
｜
｜
｜
｜
｜
｜
⎣

…
…

0｜
｜
0｜

dr2 … drl－1 1：n ×（n＋l）matrix

…

nP0

nP2

… nPn－1 …

（n＋l）matrix
：l ×

⎤

｜
｜
｜
｜
⎦

…

0

0
0
…

0

…

…

nP1 … nPn－10…
nP0 … nPn－2 nPn－1 …

…

E2＝

n
0

…

⎡
｜ P0
｜
｜
｜
｜
｜
｜
⎣

｜
｜
｜
｜
⎦

0

How to set dM(δ) will be described later.
Substituting Eq. (13-2) in Eq. (13-1) and modifying it
gives the equation below:
（δ）
f
n（δ）
M
(13-8)
F（δ）
＝
C
d（δ）
n（δ）
P
M

factor), which is decided by the equivalent inertia momentof the steering wheel shaft and the stiffness of the torsion
bar, appear as an antiresonance point in the zeros of P(δ).
Causing either compensator Cr(δ) or Cy(δ) to have thesezeros as divisors in its poles could prevent the antiresonance point from appearing in the zeros of the transfer
characteristics ranging from the target steering angle
signal r(k) to the control output y(k). In this case, however,
the antiresonance point inevitably appears as a pole of thetransfer characteristics from r(k) to the wheel rotation
angle to be a resonance point. So, if the target steering
angle signal abruptly changes, vibration may occur as thesteering wheel is turned. Then, the compensator should
be so designed that the poles of the compensator cannot
have the zeros of the compensator design model as theirdivisors. This can be achieved by determining the compensator Fc(δ) using Eq. (13-9) and making settings sothat the denominator polynomial f(δ) in Eqs. (5-2) and(5-3), the denominator polynomial g(δ) in Eqs. (6-4) and(6-5), and the denominator polynomial dR (δ) in Eq. (7-2)
cannot have the zeros of the compensator design model as
their divisors.

Substituting the relation of Eq. (13-5) in Eq. (13-8) yields
the equation below:

Poles and zeros corresponding tooriginal continuous time system

Imaginary axis × Tc

（δ）
f
β（δ）
M
(13-9)
F（δ）
＝
C
d（δ）
M

Stable region (inside the circle)
Zero of controlled object model
Pole of controlled object model

Eq. (13-9) can be used to determine a pre-compensator
Fc(δ).
A description of how to derive the compensator Cy(δ) isomitted here as it is the same as for steering assist control.
The next section provides an example of basic design ofthe compensator.
4.2.4 Example of Basic Design of Compensator(1) Determining a compensator design model
When Ap, Bp, Cp, Ep, and Fp in Eqs. (11-4) and (11-5) are
input with their physical parameter values, discretized andmade dimensionless with the control frequency 1/Tc, thetransfer function of Eq. (11-8) is as follows:

Next is to set the coprime factorization P(δ) = Np(δ) /Dp(δ) in Eq. (14-1) as follows:

n（δ）
P（δ）
＝ P
(14-1)
d（δ）
P

n（δ）
N（δ）
＝ P
(14-2)P（δ）
f

Zero added throughdiscretization

Real axis × Tc

Fig. 15

Poles and zeros of compensator design model

d（δ）
D（δ）
＝ P
(14-3)
P
（δ）
f

where
nP(δ)＝3.406×10－5(δ＋1.980)
×(δ2＋8.741×10－3δ＋5.718×10－5)
dP(δ)＝δ(δ＋1.233×10－2)
×(δ2＋7.599×10－2δ＋2.685×10－2)
Fig. 15 shows the poles and zeros of P(δ) in Eq. (14-1)
plotted on a complex plane. With a focus on the numerator of Eq. (14-1), there are three zeros. Of these, a real root
is a zero that has been added through discretization. The
remaining complex roots correspond to the original continuous time system. These complex roots are what makethe resonance point (generally with a low attenuation

1/f(δ) in Eqs. (14-2) and (14-3) are the matched pole-zeromodel of the following stable transfer function 1/f(s) thathas been made dimensionless with the control frequency
1/Tc:
1/f(s)＝1/｛ f1(s)f2(s)｝(14-4)
where― 18 ―

f1(s)＝s2＋2ζf 1ωf 1s＋ωf 12

f2(s)＝s2＋2ζf 2ωf 1s＋ωf 22
0＜ωf 1，0＜ωf 2，0＜ζf 1，0＜ζf2

KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

＝2.3
，ω
where ζf1 ζ=f 1＝ζ
ζf2 =f 22.3
and
ωff11＝ω= ωff22＝0.7= 0.7 Kpe /Ipe .
Reducing ζf1 and ζf2 or raising ωf1 and ωf2 will tend to cause
the controller to have unstable poles. So, these parameters
are set to smaller values than for steering assist control.
The solutions to Bezout equations Xp(δ) and Yp(δ) canbe expressed by the equations below:
n（δ）
X（δ）
＝ X
(14-5)
P
（δ）
ɡ
n（δ）
Y（δ）
＝ Y
(14-6)
P
（δ）
ɡ
where
nX(δ)＝1910.8(δ＋0.4936)
×(δ2＋2.550×10－2δ＋1.781×10－4)
nY(δ)＝(δ＋0.5236)(δ＋0.5953)
×(δ＋1.284×10－3)
ɡ(δ)＝(δ＋2.364×10－2)(δ＋9.930×10－2)
×(δ＋0.3669)
1/g(δ) is the matched pole-zero model of the following
stable transfer function 1/g(s) that has been made dimensionless with the control frequency 1/Tc:
1/ɡ(s)＝1/｛ ɡ1(s)ɡ2(s)｝(14-7)
whereɡ1(s)＝s＋ωɡ1
ɡ2(s)＝s2＋2ζɡ2ωɡ1s＋ωɡ22
where ζɡ2＝2.3 ，ωɡ1＝ωɡ2＝0.7

KTB /Ipe .

(2) Setting target value signal models
To ensure that the control output tracking to the target
steering angle signal, which changes in ramp rate, converges to zero (0) asymptotically, a target value signal
model is set as follows:n（δ） 1
(15)
G（δ）
＝ r
＝ 2
r
d（δ）
δ
r
(3) Setting disturbance models
To ensure that the control output tracking disturbances,which change in ramp rate, converges to zero (0) asymptotically, a disturbance model for the continuous timesystem is set as follows:n（s） 1
(16-1)
G（s）
＝ d
＝ 2
d
d（s）
s
d
A step invariant model or matched pole-zero model of thetransfer function of Eq. (16-1) that has been made into adimensionless transfer function with the control frequency
1/Tc can be expressed by the equation below:n（δ） 1
(16-2)
G
（δ）
＝ d
＝ 2
d
d（δ）
δ
d
(4) Deriving free parameter R(δ)
It is assumed that the pole dR (δ) of the free parameter isdR (δ) = δ + 9.930 × 10-2, which is the same as one of thedivisors of g(δ) in Eqs. (14-5) and (14-6). Substituting
Eqs. (14-2), (14-6) and (15-2) as well as dR (δ) in Eq. (7-6)
gives the following free parameter R(δ):n（δ） 6590δ－9.002
R（δ）
＝ R
＝
(16-3)d（δ）
δ＋9.930×10－2
R(5) Deriving compensator Fc(δ)
The target tracking characteristics are set as follows:n（δ） n（δ）
β（δ）
M(16-4)
G（δ）
＝ M
＝ P
ry
（δ）
d（δ）
d
M
M
where
nM (δ)＝1.601×10－3(δ＋8.925×10－3)
×(δ＋1.980)(δ2＋8.741δ＋5718)
dM (δ)＝(δ＋4.383×10－2)51/dM (δ) in Eq. (16-4) is the matched pole-zero model ofthe following stable transfer function 1/ dM(s) that has
been made dimensionless with the control frequency 1/Tc:
1/dM (s)＝1/｛dM1(s)dM2(s)dM3(s)｝(16-5)
where
dM1(s)＝s＋ωM 1
dM2(s)＝s2＋2ζM 2ωM 1s＋ωM 22
dM3(s)＝s2＋2ζM 3ωM 3s＋ωM 32
0＜ωM 1，0＜ωM 2，0＜ωM 3，0＜ζM 2，0＜ζM 3
ωM1
where ζM2ζ=M2ζ＝ζ
1＝1
and，ω
= ωM2
= ωM3＝0.3×= 0.3 x Kpe /Ipe .
M1＝ω
M2＝ω
M3 =M3βM (δ) in Eq. (16-4) has been determined by substituting
np(δ) of Eq. (14-1), dr(δ) of Eq. (15), and dM (δ) of Eq.
(17-2) in Eq. (13-7).
When f(δ) of Eqs. (14-2) and (14-3) and βM (δ) and dM (δ)
of Eq. (16-4) are substituted in Eq. (13-9), the compensator Fc(δ) can be expressed by the equation below:n（δ）
(16-6)
F（δ）
＝ F
C
d（δ）
F
where
nF(δ)＝46.993(δ＋8.925×10－3)
×(δ＋2.364×10－2)2(δ＋0.3669)2
dF(δ)＝(δ＋4.383×10－2)5(6) Frequency characteristics of control system
From the above, substituting Eqs. (14-2), (14-3), (14-5),(14-6), and (16-3) in Eqs. (6-1) to (6-3) yields Cy(δ) as
follows:
Cy(δ)＝NC(δ)/DC(δ)(17-1)
where
n （δ）
N（δ）
＝ NC
(17-2)
C
d（δ）
C
n （δ）
D（δ）
＝ DC
(17-3)Cd（δ）
CnNC(δ)＝8500.8(δ＋1.381×10－2)
×(δ2＋0.2211×10－2δ＋1.974×10－4)
×(δ2＋0.2360δ＋6.286×10－2)2nDC(δ)＝δ (δ＋1.107)
×(δ2＋0.1766δ＋0.1137)
dC(δ)＝(δ＋0.2364×10－2)2
×(δ＋0.9930×10－2)(δ＋0.3369)2
Next, the frequency characteristics of Cy(δ), Cy(δ)P(δ)(loop transfer function), S(δ), and Gyr(δ) are shown in Fig.
16.

― 19 ―

Control Technologies for In-Vehicle Electric Actuators

ode1 (Euler) and the fixed step size to one (1) allowed
computation of a dimensionless discrete time systemusing the δ operator. Major parameters that were set forthe simulation are shown in Table 1.
The results of the simulation are shown in Fig. 17.
According to Fig. 17, the control output y(k) follows
asymptotically to the target steering angle r(k) thatchanges in ramp rate until the sampling count k is 500.
When a ramp disturbance d(k) is applied at k = 1000, thecontrol output y(k) almost does not change and continues
following the target steering angle r(k). No vibration canbe found with the steering wheel angle θh (k) as well.

Gain [dB]Gain margin of
10 dB or more

Phase [deg]Dimensionless frequency f × Tc

Table 1

Phase margin of

Dimensionless40
frequency
f × Tcdeg or more

Parameter
Target steering angle signal
source r0
Disturbance signal source d0
Disturbance generator
SolverFixed step size(a) Frequency characteristics of Cy(δ), P(δ), and Cy(δ)P(δ)
Bandwidth of Gry(δ)

Gain [dB]60dB/decade

Major parameter settings for simulation
Settings/Description
1/500 (0 at k = 500)
1/500 (applied at k = 1000)
Gd(δ) = P(δ)/(δ)2Fixed step ode1 (Euler)
1 (dimensionless time)

Reponses

Phase [deg]Dimensionless frequency f × Tc

Dimensionless frequency f × Tc(b) Frequency characteristics of transfer functions S(δ) andGry (δ)Fig. 16

Frequency characteristics of steering angle control
systemAs a result of the design, the blue solid line Cy(δ)P(δ) in
diagram (a) shows a gain margin of 10 dB or more and aphase margin of 40 degrees or more. It is generally said
that the desirable gain margin for a servo system is 10 dB
or more and the desirable phase margin is 40 degrees ormore. In the dimensionless frequency range below thefrequency (bandwidth) at which Gry (δ) indicated by thegreen solid line shows a gain of approx. -3 dB in diagram(b), S(δ) indicated by the blue solid line shows low-frequency cutoff characteristics of 60 dB/decade. The control
system is thus expected to deliver robustness against disturbances or fluctuations of the controlled object in the
low-frequency range.
(7) Verifying the basic design
The steering angle control system in Fig. 14 wasmodeled on MATLAB®/Simulink® and subjected to simulation by setting P(δ) of Eq. (14-1), Fc(δ) of Eq. (16-6),
Nc(δ) and Dc(δ) of Eqs. (14-2) and (14-3), and Gr(δ) of Eq.
(15), as well as parameters of the disturbance generator
Gd(δ). Transfer Fcn of the Continuous library was used asa transfer function model. Setting the solver to a fixed step

Sampling count k

Fig. 17

Results of simulation of steering angle control
system4.3 Detailed Design and Implementation ofCompensators
4.3.1 Detailed Design of Compensators
This section describes the detailed design of the compensators. Fig. 18 shows a block diagram of a compensator implementation model. In Fig. 18, the six transfer
functions represented by the aforementioned Transfer Fcn
are collectively called a "function model" and the transfer
function of such a function model implemented in an
appropriate way is called an "implementation model" (theterm "implement" here refers to expressing a transfer
function using state and output equations). For the steering assist control compensator, the pre-compensator in the
Figure is Fc(δ) = 0. In R(δ), its numerator polynomial
parameters depend on the magnitude of the steering assist(assist gain), although a detailed description is omitted
here. The area enclosed by the blue broken line in Fig. 18
indicates a block with state variable estimation and feed-― 20 ―KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

Note 4) An instrument that uses a controlled object model toestimate its state variable and disturbances
Pre-compensator

Stabilizing compensator Cy(δ)

Eq.(13-9)State variable estimation and feedback

Output limiter

Eq.(6-4)

Eq.(7-2)
Eq.(6-5)

Variable according to assist gain(for steering assist control)

Note 5) An acronym of MathWorks® Automotive Advisory
Board. A guideline that specifies protocols for MathWorks products including description rules.

Disturbance estimation and feedback

Block diagram of compensator implementation
model

The following introduces an example of implementation of the pre-compensator Fc(δ). The transfer function of
Eq. (13-9) can be expressed as follows:
FC(δ)＝nF(δ)/dF(δ)＋DF(18-1)
wherenF(δ)＝nFn＋l－2δ n＋l－2＋…nF1δ＋nF0
dF(δ)＝δ n＋l－1＋dFn＋l－2δ n＋l－2＋…dF1δ＋dF0
DF: constant representing a feedthrough term of FC(δ)
A possible realization method for lower computation
error may be balanced realization (calculation using the
MATLAB® function "balreal" for instance). To reduce thecomputation complexity, we decided to select controllable
canonical form (this calculation can also be achieved with
the MATLAB® function "canon").
The controllable canonical form of Eq. (18-1) can be
expressed by the equation below:
δxF(k)＝AFxF(k)＋BFr(k)(18-2)
yF(k)＝CFxF(k)＋DFr(k)(18-3)
where
T
x（k）
＝
［xF1
（k） xF2（k） … xFn＋l－1
（k）
］
F

0
1
…

0
0
0
－dF0 －dF1 －dF2
⎤

…
…

⎤
｜
｜
｜
｜
｜
｜
｜
｜
｜
Fn＋l－2⎦

0
0

…

1
0
…

AF＝

0
0
…

⎡
｜
｜
｜
｜
｜
｜
｜
｜
｜
⎣
⎡
｜
｜
｜
｜
｜
｜
｜
⎣

－d

1

0｜
｜
｜
｜
｜
｜
｜
⎦

…

BF＝

0
1

⎫
｜
｜
｜
（k）＝xFn＋l－2（k－1）
xFn＋l－2｜＋Tc xFn＋l－1
（k－1）
⎬
｜
xFn＋l－1（k）＝xFn＋l－1（k－1）｜⎡n＋l－1
⎤
｜
｜｜
｜
－Tc｜
｛ dFi－1 xFi（k－1）｝＋r（k－1）
｜Σ
｜｜
⎣ i＝1
⎦
⎭
(18-4)
A compensator implementation model is created on
MATLAB®/Simulink® in such a manner that computation
is carried out in the order of Eq. (18-4) and Eq. (18-3). For
higher model readability, the model creation should be
conducted according to the modeling guideline in which
the description rule was established based on MAAB Note
5) 7)
.
xF（k）
＝xF（k－1）＋Tc xF（k－1）
1
1
2

Eq.(5-2)

Eq.(5-3)

Fig. 18

Eq. (18-2) can be developed into the equation below:…

back functions and that enclosed by the red broken line isa block with disturbance estimation and feedback functions. This implies that the parametrization of a stabilizing
compensator can be basically applied with the same
theory as for compensators using the state variable anddisturbance estimation observer Note 4).

，CF＝［nF0 nF1 … nFn＋l－2］

Compensator computation by microprocessors uses the
single precision floating-point format for memory saving.
Therefore, control parameters (variables having a fixedvalue) in Eqs. (18-3) and (18-4) and state variables (whose
value changes with time) are all defined with single precision floating-point variables in the implementation modelstage.
4.3.2 Verifying the Compensator Implementation
Model
The function and implementation models consisting of
the six transfer functions in Fig. 18 are each subject to a
back-to-back test to verify that these models are equivalent to each other.
4.3.3 Implementing Compensators
As described in section 3.3, Embedded Coder® is used
to automatically generate a C code from the implementation model. The generated C code is checked for conformance to the MISRA-C® Note 6) rules using a static analysis
tool. Any nonconforming items are remedied 7). Verification of the implementation results will be described in thenext chapter.
Note 6) Coding standards intended to ensure safe, portable, andreliable software (in C-language)

5

Design Verification

This chapter provides an example of the results of verification of the compensator design for steering assist
control according to the verification process.
5.1 Compensator Unit Evaluation
Software integrated with a designed compensator was
implemented in the power pack. The power pack was thenapplied with sinusoidal signals as motor current commands. From input and output signals of the compensator,
the frequency characteristics of the compensator Cy(δ)
were measured. The results are shown in Fig. 19.
― 21 ―

Control Technologies for In-Vehicle Electric Actuators

Gain [dB]

The measurement results closely match the functionalmodel, verifying that the compensator has been implemented as designed. Note that the results shown in Fig. 19
are for another design different from the basic design
described in 4.1.5 Example of Basic Design of Compensator.

Measurements
Functional model

Phase [deg]Dimensionless frequency f × Tc
Measurements
Functional model

Dimensionless frequency f × Tc

Fig. 19

Example of results of unit evaluation ofcompensator Cy(δ)

5.2 System Evaluation and Redesigning of Compensators
The system shown in Fig. 8 was applied with sinusoidal
waves as motor current commands to measure the frequency characteristics of the controlled object. The frequency characteristics of the controlled object are shownin Fig. 20. The figure plots the frequency characteristics ofthe compensator design model of Eq. (8-1) (hereinafterthe "basic design model"), of the measurement results,
and of a model identified from the measurement results.
Gain [dB]

Basic design model
Measurements
Identified model

(19-1)
wherenP(δ)＝5.317×10－3(δ＋0.1864)
×(δ2－0.2701δ＋1.600)
2dP(δ)＝(δ ＋0.5513δ＋8.201×10－2)
×(δ2＋9.755×10－2δ＋2.135×10－2)
All the gains of the basic design model, measurements,
and identified model decrease after around a dimensionless frequency of 0.02 at nearly -40 dB/decade, showing
relatively good agreement with each other. However, thephase of the measurements is much behind the basic
design model after around a dimensionless frequency of
0.03. Possible causes include a response delay of theactual current from the motor current command, thecontrol computation time of the software, frictional torqueand small rattling of the machinery, and deflection of theelements that were assumed to be rigid bodies, whichwere not taken into account in Eq. (8-1).
Since the gain and phase of the measurements are ingood agreement with those of the identified model, thecontrolled object can be approximately expressed in atransfer function of degree 4 as shown in Eq. (19-1).
In fact, the frequency characteristics of the actual controlled object show a larger phase delay than that of thebasic design model. This means that a compensator
designed based on the basic design model may fail to keep
the control system stable. Then, we have redesigned thecompensator using the identified model of Eq. (19-1). The
redesigned compensator can be expressed by the equation
below:nNC（δ）
nDC（δ）
(19-2)where① For large steering assist torque:nNC(δ)＝22.847(δ＋8.494×10－2)
×(δ2＋0.5513δ＋8.201×10－2)
×(δ2＋0.1579δ＋1.889×10－2)
nDC(δ)＝(δ＋0.1864)(δ＋3.666×10－2)2
×(δ2＋1.295δ＋0.7986)

Basic design model
Measurements
Identified model

Dimensionless frequency f × Tc

Fig. 20n（δ）
n δ3＋n δ2＋n δ＋nP 0
P＝ 4 P3 3 P2 2 P1
δ ＋dP 3δ ＋dP 2δ ＋dP 1δ＋dP 0
d（δ）
P

C（δ）
＝
y

Dimensionless frequency f × Tc

Phase [deg]P（δ）＝

Frequency characteristics of controlled object

The identified model was obtained by calculating thefrequency response vector from the measurement datausing the MATLAB® function "fft" and then determinedwith the MATLAB® function "invfreqz". Transfer function parameters determined with "invfreqz" are written in
the z operator form. The obtained parameters were thus
converted into the δ operator form. The identified modelin Fig. 20 can be expressed by the equation below:― 22 ―② For middle steering assist torque:nNC(δ)＝18.246(δ＋0.1022)
×(δ2＋0.5513δ＋8.201×10－2)
×(δ2＋0.1646δ＋1.701×10－2)
nDC(δ)＝(δ＋0.1864)(δ＋6.502×10－2)2
×(δ2＋1.262δ＋0.7211)③ For small steering assist torque:nNC(δ)＝12.111(δ＋0.1336)
×(δ2＋0.1745δ＋1.250×10－2)

KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

Gain [dB]×(δ2＋0.5513δ＋8.201×10－2)
nDC(δ)＝(δ＋0.1864)(δ＋0.1127)2
×(δ2＋1.200δ＋0.6005)
1/f(δ) in Eqs. (5-2) and (5-3) is the matched pole-zeromodel of the following stable transfer function 1/f(s) thathas been made dimensionless with the control frequency
1/Tc:
1/f(s)＝1/｛f1(s)f2(s)｝(19-3)
where
f1(s)＝s2＋2ζf 1ωf 1s＋ωf 12
f2(s)＝s2＋2ζf 2ωf 1s＋ωf 22
0＜ωf 1，0＜ωf 2，0＜ζf 1，0＜ζf 2

Dimensionless frequency f × Tc

Phase [deg]Large steering assist torque
Medium steering assist torque
Small steering assist torqueDimensionless frequency f × Tc(a) Frequency characteristics of compensator Cy(δ)

Gain [dB]，ωwhere ζf1ζ=f 1＝ζ
ζf2 =f 2＝1
1 andωf f11＝ω
= ωff22＝2= 2 Kpe /Ipe . 1/g(δ) in
Eqs. (6-4) and (6-5) is the matched pole-zero model of thefollowing stable transfer function 1/g(s) that has been
made dimensionless with the control frequency 1/Tc:
1/ɡ(s)＝1/｛ɡ1(s)ɡ2(s)｝(19-4)
whereɡ1(s)＝s＋ωɡ1，ɡ2(s)＝s2＋2ζɡ2ωɡ1s＋ωɡ22

Gain margin of
5 dB or more
Large steering assist torque
Medium steering assist torque
Small steering assist torqueDimensionless frequency f × Tc

KTB /Ipe .
Phase [deg]where ζɡ2＝1 ，ωɡ 1＝ωɡ2＝2

Large steering assist torque
Medium steering assist torque
Small steering assist torque

Next, the parameter for the pole 1/dd(δ) of the disturbance model of Eq. (9-1) was set as follows:ωd＝0.9Kpe /Ipe : Medium steering assist torqueωd＝1.5

Kpe /Ipe : Small steering assist torquePhase margin of

Dimensionless frequency
30 degf ×orTcmore(b) Frequency characteristics of loop transfer function Cy(δ)P(δ)

In all these cases, ζd = 1.
Also, it is assumed that the pole of the free parameter R(δ)
in Eq. (9-3) is the matched pole-zero model of 1/g1(s)stated above.
The frequency characteristics of the redesigned Cy(δ),Cy(δ)P(δ) (loop transfer function), and S(δ) are shown in
Fig. 21. As a result, the gain margin is 5 dB or more andthe phase margin is 30 degrees or more in all cases. It has
been verified through system evaluation and actual vehicle
evaluation that a stable control system has been obtainedwith the redesigned compensator, although a detailed
description is omitted here.
5.3 Actual Vehicle Evaluation
In in-house actual vehicle evaluation, a vehicle of an
existing model is used to mainly test stationary steering(steering of a stopped vehicle). The test driver should start
to turn the steering wheel and then turn it back to checkthe effect of the steering assist. The driver should also
check that they feel no vibration via the wheel. The tuning
parameters include the parameters of Eqs. (19-3) and(19-4), the parameters of the pole of the disturbance modelof Eq. (9-1), and the parameters of the poles of Eq. (9-3).
In in-house actual vehicle evaluation, several compensator candidates of different designs should be prepared bymodifying the tuning parameters stated above toward
actual vehicle evaluation at customer sites.

Gain [dB]

Kpe /Ipe : Large steering assist torque

Large steering assist torque
Medium steering assist torque
Small steering assist torqueDimensionless frequency f × Tc

Phase [deg]ωd＝0.5Large steering assist torque
Medium steering assist torque
Small steering assist torque

Large steering assist torque
Medium steering assist torque
Small steering assist torqueDimensionless frequency f × Tc(c) Frequency characteristics of transfer function S(δ)
Fig. 21

Frequency characteristics of redesigned steering
assist control system

In actual vehicle evaluation at customer sites, a driving
test is carried out on a dedicated off-road test course, in
addition to the stationary steering evaluation. During thedriving test, particularly the driver's feeling of steering at
the start of cornering and of kickback during driving
straight ahead or cornering should be checked.

― 23 ―

Control Technologies for In-Vehicle Electric Actuators

6

Future Outlook

The approach explained in this report is just a compensator design method applicable to existing EPS that is alittle bit different from the traditional approach. The
importance of electronically-controlled in-vehicle equipment will rise as in-vehicle equipment automation andautonomous driving is further accelerated. As an in-vehicle equipment manufacturer, KYB needs to develop
various value-added products. In the course of the development, not only electrical and machine element engineers but also control technology engineers should play
two major roles of:①crystallizing and screening product ideas during the
product planning stage to create new products, and;② developing a robust design to minimize uncertainty
throughout the product lifecycle.
For role ①, just think about sensors, for instance. What
are the minimum sensors that can deliver the functions
and control performance required by the product (for cost
reduction)? What physical amounts should be detected
and controlled to satisfy the requirements (for higher
added-value)? An effective means to solve these possible
challenges is to identify controllability and observabilityof the system from the control technology perspective andto utilize the observer theory. For role ②, a parametricmodel involving uncertainty may be identified throughsimulation using a model with product quality (production variations) and secular changes in quality taken into
account or through their testing on an actual machine.
Such a parametric model can be effectively used to designcompensators, ensuring robust stability of the control
system.
Based on the above, we would like to strengthen our
product design and proposal capabilities.
7

Concluding Remarks

Focusing on EPS for all-terrain and utility task vehicles,
this report explains KYB's control technologies for invehicle electric actuators by introducing some numericexamples. The basis of the technologies is a design
approach based on parametrization of stabilizing compensators, which has been developed into a discrete timesystem in the δ operator form. The various equations used
for compensator calculation were converted into software

programs with m-file Note 7). This allows automatic calculation of parameters for the function and implementation
models of compensators just by setting the tuning parameters described in Section 5.3.
Note 7) A text file that describes programs to be executed on
MATLAB®

The control technologies explained in this report can beapplied not only to electric actuators but also to electric
pumps, hydraulic actuators, and other various electronically-controlled components.
Although compensators only make up a small proportion of in-vehicle software-controlled components, they
are an essential technology directly affecting security,
safety, and comfort. We would like to continue contributing to the improvement of quality and added-value of
KYB products with our system analysis and evaluation
techniques, control technologies, and software technologies.
Finally, we would like to take this opportunity to sincerely thank all those concerned who have extended guidance and cooperation to us in implementing and evaluatingthese control technologies.
・MATLAB®, Simulink®, Embedded Coder®, and MathWorks®
are registered trademarks of The MathWorks, Inc.
・SimulationX® is a registered trademark of ESI ITI GmbH.
・MISRA-C® is a registered trademark of The MISRA Consortium Limited.

References
1) MIYAKE, TOMITA: Development of EPS for Non-Passenger
Vehicles, KYB Technical Review No.60, pp 60-63, (2020).
2) 
INAMITSU, HORI: Development of Active Suspension by
KYB, KYB Technical Review No.63, pp 9-14, (2021).
3) MAEDA, SUGIE: System Control Theory for Advanced Control,
Asakura Publishing, (1990).
4) ADACHI: System Identification Theory for Users, The Societyof Instrument and Control Engineers, (1993).
5) 
KANAI, HORI: An Introduction to Digital Control System-Application of Delta Operator-, Maki Shoten, (1992).
6) KITAMURA: Development of Simulation Technology for EPS
System, KYB Technical Review No.59, pp 17-24, (2019).
7) KOBAYASHI: Application of MBD to Development of ECU
Prototype for EPS, KYB Technical Review No.52, pp 49-53,(2016).

― 24 ―KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

Author 
MATSUMOTO Daisuke

ENOMOTO Daisuke

Joined the company in 1995.
Development Sect. No.2, Electronics
Technology Dept., Engineering
Headquarters, Automotive
Components Operations
Special Sect. Manager, BasicTechnology R&D Center,
Engineering Div.
Engaged in analysis of in-vehicle
actuators and development of theircontrols

Joined the company in 2018.
Development Sect. No.2, Electronics
Technology Dept., Engineering
Headquarters, Automotive
Components Operations
Engaged in analysis of in-vehicle
actuators and development of theircontrols

KENJYO Kazunori

FUKUSHI Natsuru

Joined the company in 2015.
Chief Researcher, Electronics
Technology Sect., Basic Technology
R&D Center, Engineering Div.
Engaged in research anddevelopment of control software forin-vehicle actuators

Joined the company in 2017.
Chief Researcher, Electronics
Technology Sect., Basic Technology
R&D Center, Engineering Div.
Engaged in research anddevelopment of control software forin-vehicle actuators― 25 ―