Compensators in Control Systems

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Compensators in Control Systems
Refer to “Control Technologies for In-Vehicle Electric Actuators” (page 8)

MATSUMOTO Daisuke
Development Sect. No.2, Electronics Technology Dept., Engineering Headquarters,
Automotive Components Operations

1

For the control system in Fig. 1, the velocity v of thevehicle can be expressed by the equation below:

What is a Compensator?

v＝1.1 Role of Compensators
The compensator Note 1) in a control system refers to acomponent that generates manipulated variables (controlinput) to attain a desirable response (control output) fromthe system.

As time elapses (s→0), the vehicle velocity v can be
expressed by the equation below:
C（0）P（0）P（0）
v－d1＋C（0）P（0） r 1＋C（0）P（0）
1
(2)
1C（0）
＝v r－d1
1＋1＋1
C（0）P（0）C（0）P（0）v＝Note 1) This document uses the term “compensator”, not theterm “controller” which may sometimes be used.

As an example, a case is considered in which avehicle is applied with a driving force u (control input)
and controlled so that its velocity v (control output) ismaintained at a fixed target velocity vr (setting value)(Fig. 1). The vehicle is affected by disturbances d such as
aerodynamic drag, friction against the road surface, andforces caused by gravity on slopes. It is assumed here that
disturbances d are almost constant.

Fig. 1 Vehicle velocity control

Then, a control system is configured as shown in Fig. 2:

Fig. 2

Vehicle velocity control system (1)whereP(s): Transfer function of controlled object (transferfunction from the force applied to the vehicle
u-d to the velocity v)
C(s): Transfer function of compensators: Laplace operator (or differential operator)

C（s）P（s）
P（s）
(1)
vr－d
1＋C（s）P（s）
1＋C（s）P（s）

When the compensator C(s) is designed so that C(0) is
sufficiently large, 1/C(0) in Eq. (2) is almost zero (0).
The vehicle velocity v is almost v = v r accordingly.
Designing the compensator C(s) in this way will achieve
one of the control objectives, that is, “to keep the velocity
close to the target value”. Behind the control objectives
lie a wide range of needs including cost-effectiveness,safety, and comfort. The role to be played by thecompensator in order to satisfy these needs will bedescribed in Chapter 3.
1.2 Types of Compensators and Example ofControl System Characteristics
Typical compensators include proportional, differential,integral, phase lead, and phase lag compensators. A
detailed description of these compensators is omittedhere.
To achieve one of the control objectives “to keep thevelocity close to the target value” stated in the previous
section, a proportional and integral compensator will besufficient, provided that the transfer function of thecontrolled object can be expressed by the equation
below:
1P（s）＝
(3)Mswhere M indicates the weight [kg] of the vehicle. Eq. (4)represents an ideal vehicle that is applied with a control― 42 ―KAYABA TECHNICAL REVIEW No. 65 OCT. 2022

input u without delay. The transfer function of theproportional and integral compensator can be expressedby the equation below:
KK s＋KIC（s）＝KP＋ I ＝ P
(4)sswhere K P and K I are proportional and integral gains,respectively. When Eqs. (3) and (4) are substituted in Eq.
(1), the vehicle velocity v can be expressed by theequation below:
C（s）P（s）
P（s）
v－d1＋C（s）P（s） r 1＋C（s）P（s）
KPK
1s＋ IsM
M
M
＝v－dK
K r
K
K
(5)
s2＋ P s＋ I
s2＋ P s＋ IMMMM
1s2ζωn s＋ωn2M＝ 2v－ds ＋2ζωn s＋ωn2 r s2＋2ζωn s＋ωn2
v＝

where

ωn＝ KI /M ，ζ＝KP 1 /（KIM）/2
Setting KP and KI to achieve 0 < ωn and 0 < ζ will makethe control system stable. The equation also implies that,as time elapses (s → 0), v = vr will hold.
1.3 How to Configure Compensators
There are two types of compensator configuration:
analog and digital. An analog compensator uses resistors,capacitors, operational amplifiers, and other elements.
The example shown in the previous section assumes
control with an analog compensator. The transfer
function of an analog compensator is represented using
the Laplace operator s (or differential operator). A digitalcompensator is configured using a logic circuit or digitalcomputer. Digital compensators have the advantage thatcomplex compensators can be configured using software,
thereby enabling the user to easily modify parametersettings. Today, many control systems for in-vehicle
actuators use digital compensators because theperformance of microprocessors has been improved. The
transfer function of a digital compensator is represented
using the z and δ operators.

2

a model of the controlled object and the desirable
characteristics of the control system, instead of “firstselecting the type of the compensator”, such as
proportional or integral as introduced in section 1.2,
before designing. In other words, the designer can find acompensator satisfying the desirable characteristics ofthe control system even if the controlled object is quite
complex. Still, a more-or-less accurate model of thecontrolled object (a mathematical model such as transferfunction) will be needed. In some cases, the achievablecharacteristics of the control system may be limited
depending on the characteristics of the controlled object.
For your information, the proportional and integral
compensator stated in section 1.2 can also be derived
from the parametrization of a stabilizing compensator.

3

Two-degree-of-freedom ControlSystem

The control system described in Chapter 1 may haveother control objectives than just “to keep the velocity
close to the target value”, attributable to various demands
including “[1] to reduce the fuel (or electricity for electricvehicles) consumption during acceleration (costeffectiveness)”, “[2] to reduce the shock felt bypassengers during acceleration or deceleration (ridecomfort)”, and “[3] to minimize speed variations during
driving on slopes (safety)”. Demands [1] and [2] mainly
relate to target tracking and [3] relates to disturbance
response. With a focus on Eq. (1) in section 1.1, target
tracking (the first term on the right-hand side) anddisturbance response (the second term on the right-handside) are both decided by the compensator C(s). Thismeans that, in some cases, demands [1] & [2] anddemand [3] can hardly all be satisfied at the same time.
Then, another control system should be considered asshown in Fig. 3:

Fig. 3 Vehicle velocity control system (2)Parametrization of Stabilizing
Compensators

Parametrization of stabilizing compensators wasproposed by Youla and others during the 1970s 1). Thismay be called “Youla parametrization”. Studies on robust
control theory became active toward the 1990s and many
books explaining how to apply Youla parametrization toactual control scenes 2) were published. A stabilizing
compensator design with parametrization should
represent “how the compensator should be” derived from

The velocity v of the vehicle with the control system inFig. 3 can be expressed by the equation below:
F（s）C（s）P（s）
P（s）
(6)
v＝v－d
1＋C（s）P（s） r 1＋C（s）P（s）
With an additional compensator F(s), target tracking(the first term on the right-hand side) and disturbance
response (the second term on the right-hand side) can be
set separately. That is, two-degree-of-freedom design is
now available. In this sense, the control system in Fig. 3
is called a two-degree-of-freedom control system.

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Compensators in Control Systems2) MAEDA, SUGIE: System Control Theory for Advanced Control,

References
1) D.C. Youla, Hamid A. Jabr, J.J. Bongiorno Jr: Modern WienerHopf Design of Optimal Controllers - Part II: The Multivariable
Case, IEEE Trans. Auto Control, AC-21, No. 3, pp. 319-338,(1976).

― 44 ―

Asakura Publishing, (1990).