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Modeling of a n Internal CombustionEngine for Control Analysis

Jeffrey A. Cook and Barry K. Powell

ABSTRACT: The process mode l for an in-

ternal combustion engine with spark ignition

is inherently nonlinear. F or purposes of con-

trol analysis, it is desirable to have a l inear

model. This paper presents a discussion of

recent activity in nonthermodynamic mod-

eling of automotive internal combustion en-

gines and shows how adequa te l inea r mode ls

can be developed for control analysis.

Introduction

The past several years have witnessed a

good deal of engine model building activity

in the automotive industry in a category re-

ferred to as “control-oriented” or “control

des ign” mode ls . These mode ls a re genera lly

low-frequency engine representations with

uniform pulse, homogeneous charge, and

lumped parameter approximations of engine-

breathing and rotational dynamics. This pa-

per contains a brief review of the mode l ing

literature and presents a fundam ental nonlin-

ear model of a spark ignition, internal com-

bustion engine. A linear control-oriented

model is derived from the nonlinear process,

techniques for experimental verification are

examined, and a practical l inear engine ex-

ample incorporating m ultirate sampling is i l-

lustrated.

Literature Review

Power plant characterizations for four-,

six-, and eight-cylinder engines were devel-

oped in the early 1970sby Hazell and Flower

[1]-[3]. These models and related analysis

are significant in that Hazell and Flow er were

among the first to develop discrete models

wi th sampl ing commensura te wi th engine

crank-angle events, to develop approxima-

tions of the fuel-dependent torque by full-

and partial-pulse representations, an d to per-

form comparative parameter design and sta-

bili ty analy ses.

Eight-cylinder nonlinear representations

with air-to-fuel ratio ( M F ) and exhaus t gas

An early version of this paper was presented at

the 1987 American Control Conference, Minne-

apolis. Minnesota, June 10-12, 1987. Jeffrey A.

Cook and Barry K . Powell are members of the

Research Staff at Ford Motor Company, Dear-

born. MI 48121.

rec ircula t ion (EGR ) dynam ics were deve l -

oped by Garofalo [4], Prabhakar et al . [5],

and Powel l [6]. The Prabhakar mode l was

one of the first attempts at a comprehensive

representation in that i t contained spark ad-

vance, throttle and fuel control variables, and

empirically based approximations for ex-

haust emissions.

A l inea r power p lant mode l , comple te wi th

an approximat ion of emiss ion behavior , w as

developed by Cassidy et al . [7]and was used

for l inear quadratic control design. More re-

cently, l inear quadratic control was applied

by Kamei et al . [8] to a 23rd-order l inear

perturbation model. Static and dynamic dy-

namom eter test results were used to estimatemodel parameters using a statist ical identi-

fication method. Nonlinear dynamic models

appropriate to wide speed and load operating

ranges were developed by Powell [9], De-

losh e t a l . [IO], a n d Do b n e r [l l ] . W u e t a l .

[I21 developed a similar nonlinear model

with experimentally based dynamic intake

manifold fuel wall-wetting condensation ef-

fects.

A bibl iography an d more thorough review

of nonthe rmodynamic engine mode ls i s pre-

sented by Powel l and Cook [13]; a recent

comprehensive model review, including in-

put-output models and physically based

models, is presented by Powell [14]. Ot h e r

engine models will be referenced in the fol-lowing discussion of specific engine ele-

ments.

Linear Model Development

The founda t ion for th i s paper i s the fun-

damenta l , nonl inea r engine mode l deve loped

by Powell [9], which is illustrated in Fig. 1.

The model contains representations of th e

throt tle body, engine pumping phenomena ,

induction process dynamics, fuel system,

engine torque generation, and rotating iner-

tia . The linearized version of the engine

model is i l lustrated in Fig. 2. Subsequent

paragraphs will proceed from the general

nonl inear mode l of F ig . 1, briefly reviewing

each subsystem and developing the l inear-

ized relationships i l lustrated.

Throttle Bod4

The basic throttle body model is devel-

oped by assuming one-dimensional, steady,

isentropic compressible flow of an ideal gas.

Air mass flow rate through the throttle body

may be thought of as a nonlinear separable

function of manifold pressure and throttle

angle. The partial derivative of the a i r mass

flow rate equation with respect to throttle

angle 0 defines the l inearized airflow rate

sensitivity KO n the mode l of F ig . 2.

Intake Manifold Dynamics

The dynamic equa t ions for the mani fold

system are developed by emplo ying the prin-

ciples of conservation of mass and thermo-

dynamic ene rgy, and assuming tha t the va -

porized constituents satisfy the equation of

state. Conservation of mom entum is satisfiedby assuming that a uniform pressure exists

in the intake manifold between the throttle

body and the intake valves. This latter as-

sumption, although valid for a low-fre-

quency dynamic representation, precludes

simulation of high-frequency acoustic prop-

agation. Yuen [I51 and Serva t i [I61describe

the dynamic equations that result from using

these principles. A method of developing a

manifold model that accounts for concentra-

t ion va ria t ion due to EGR, fue l , and a i r i s

presented by Moskwa and Hedrick [17].

For certain applications, i t may be as-

sumed that temperature is approximately

constant and that the dominant manifold in-

take mass rate is the airflow rate, lit“. M a k -ing these assumptions provides a first-order

differential equation relating the rate of

change of manifold pressure ( P ) o the flow

rates into (lit,) and out of (M ) he mani fold:

(1 )

where K p is a function of the gas constant,

gas molecular weight, specific heats, mani-

fold volume, and nominal assumed manifold

temperature. Recalling that the mass airflow

rate is, in general, a fu nction of throttle an gle

and manifold pressure P , and noting that the

engine pumping mass flow rate, M, s a

func t ion of mani fold pressure and sp eed, N ,

linearization of Eq . (1 ) yields the following:

P = Kp(lit, - ik’)

A P = Kp(am,/aP - a i k ’ i a p ) A p

+ KpKOA$ - K,,K,vAN (2)

where KN is the pumping feedback defined

by a h f l a N . Defining the inverse of the coef-

0272-1 08/88/0800-002001 000 988 IEEE

iU I t € € Control Syiterns M n q a i i n e

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F U E LCOMMAND

THROTTLE MANIFOLDe NGINE ENGINE4 BODY ma PLENUM PUMP POWER

ImfCHARGE MASS RATE

SPEEDINERTIA -

I

SPARKCOMMAND

'AF- f

r

G6 ' DISTURBANCE

e -2STAP* GP

PKPrps + 1

A 6- O

A NK R TR

b

T R S + 1

ficient of incremental pressure in Eq. ( 2 ) to

be the manifold t ime constant, rP , esults in

the l inearized expression for manifold dy-

namics contained in Fig. 2 .

Fuel Systems

Operation of an engine a t or near a partic-

ular air-to-fuel ratio requires management of

both air and fuel flow into the manifold sys-

tem. Numerous methods exis t for sens ing

airflow in ord er to provide a proportional fuel

command, many of which a re desc r ibed by

Bowler [ 181 a n d M a n g e r [19]. These meth-

ods generally fall into the categories of vol-ume rate measurement, mass rate measure-

ment , or indirect estimation methods based

on the measurement of related variables. T his

latter method of airflow determination is re-

ferred to as speed density. Stated simply, a

AIR BYPASS

COMMAND

speed density air sensing system is based on

the calculation of an estimate, defined here

as A M , of the mass flow rate quantity &f.A n

example of an estimate in terms of pressure

and engine speed i s shown,

A M = c P N (3)

where A M is the air mass flow rate (Ibm/

min) , P the manifold absolute pressure

(psia), N the engine speed (rpm), and c th e

proportionality constant (Ibm-in.*/lbf-rev).

Linea rizing Eq . (3) about the nomina l engine

MANIFOLD FILLING 240° IP LAG ROTATIONAL

DYNAMICS

~~

speed, N o , and pressure, P o , yields

A A M = c ( P o A N + N o A P )

Fo r control to a specific air-to-fuel ratio,

engine fuel flow rate is proportional to

airflow rate. The actual amount of fuel

(4)

th e

th e

in -

jected at any one event is proportional to the

airflow rate divided by engine speed (which

is a ssumed to be proport iona l to the a i r

charge).

Th e other aspect of fuel management that

must be addressed fo r electronically con-

trolled, port fuel-injected en gines is injection

timing. Two injection timing strategies will

be discussed here.

Sequential electronic fuel injection meters

fuel individually to each cylinder during the

appropriate portion of the engine cycle. For

example, fuel might be injected into each

cylinder in turn immediately before the in-take valve opens. Thus, each cylinder re-

ceives a fuel charge delayed the sam e amount

from the t ime of injection.

Another method of injection timing is re-

ferred to as bank-to-bank injection. In one

Augu5 t 1988 21

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implementation of this strategy for a six-cyl-

inder engine, the amount of fuel to be in-

jected IS calculated once per engine revolu-

tion based on the speed density airflow. For

purposes of economy and simplicity, the in-

jectors are slaved in groups of three and fired

alternately at 360-deg crank-angle incre-

ments delayed 240 deg from the fuel meter-

ing calculation.

Engine Pumping

An estimate for the air, fuel, and EG R

mass flow rates out of the manifold and into

the cylinders can be developed by treating

the engine as a pump. Mass flow rate would

thus equal the product of engine speed, en-

gine displacement, and volumetric effi-

ciency. For constant intake manifold tem-

perature and exhaust gas pressure, the

volumetric efficiency may be expressed as a

high-order polynomial in speed and manifold

pressure. Using such a relation in the prod-

uc t to form manifold mass Row rate egress,

M , yields a polynomial function of speed

and pressure. An example of a plot of massflow rate as a function of manifold pressure

and speed is shown in Fig. 3 . These da ta a re

referred to as an induction map. The partial

derivative of this function with respect to

engine speed at a particular engine operating

point defines the pumping feedback gain K, v

in the linearized model.

Induction-Po~3erStroke Delay

As in a reciprocating pump, individual

samples of the mass flow rate for each cyl-

inder are taken into the engine in a contin-

uous speed-dependent sequence. These sam-

ples eventually produce torque via the

combustion process after being delayed by

1200

a= 1000m?

800

a

3 600

LL 400

a

\

l-LL

5ln

ln

= 200

1 IFWMPINGLOOP- WORK (NEGATIVE)I

I

CYLINDER VOLUME-DC

-

-

-

-

-

-

Fig. 4 . Cylinder pressure versus volume diagram.

the combustion to power stroke lag included

in Fig. 2 as a transport delay.Successive 180-deg increments of crank-

shaft rotation delineate the basic pheno mena

of the four-stroke cycle, spark ignition en-

gine . These fundamenta l events a re the

ingestion of a combustible aidfuel mixture

into the cylinder through the open intake

valve as the piston traverses from top-dead-

cente r (TDC) to bot tom-dead-cente r (BDC)

of the intake stroke, compression of the mix-

ture as the piston returns to TDC, ignition

and rapid expansion during the subsequent

power s t roke dr iv ing the pis ton do wnward in

the cylinder and imparting torque to the

crankshaft , and, finally, elimination of the

products of combustion from the cylinder

O L

0

72.0

12

I I I 1 I I I

2 4 6 8 IO 12 14

M A N I F O L D P R E S S U R E - P S I

Ex a m p l e of an internal combustion engine inductionig . 3.

map.

through the open exhaust port as the piston

re turns to TDC during the exhaus t s t roke .Thes e events are i l lustrated in the cylinder

pressure versus volume diagram of F ig . 4.

It is clear that a delay exists between the

ingestion of the aidfuel mixture and the

torque production related to this mixture.

That is, the torque developed by the engine

at any particular t ime is a function of the

flow and pressure characteristics extant dur-

ing the previous induction event. Hence, the

minimum induc t ion-to-power (IP) stroke lag

is 18 0 deg of cranksh aft rotation. It is to be

expected that this IP lag has significant con-

trol implications. In particular, at low engine

speeds, where the IP lag is the longest, the

delay in torque can have an adverse effect

on engine stabili ty.

Engine Brake and Combustion Torque

Torque is generated from the combustion

process, which is dependent on the ignition

of a cylinder charge of air, fuel, and residual

gas, as well as other variables and parame-

ters that influence combustion efficiency

(such as the cyl inder head geom etry , for ex-

ample ) .

Defining the engine torque in terms of

measurable or physically meaningful inde-

pendent variables yields a quasistatic relation

upon which dynamic elements reflecting

friction effects and breathing delays may be

superimposed. The structure of the torque

equation provides a foundation for experi-mental determination of appropriate numer-

ical values. An estimate for characterization

of the engine torque is obtainable by em-

ploying analytical curve-fit ting techniques to

dynamometer-obtained experimental data as,

22 It€€Con t r o l S y s t r m s M o q o i i n e

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for example , in [20] . For the l inea r mode l ,

the following functional dependence is as-

sumed for the engine brake or output torque:

( 5 )

where M,, is the mass charge delayed by the

IP lag (Ibm). F,, the fue l de layed by the IP

lag (Ibm), and 6 the ignition timing (degreesbefore TDC).

As previously developed, mass charge is

a func t ion of mani fold pressure and engine

speed . Because pressure is the dominant vari-

able. engine output torque can be considered

to be an implicit function of delayed pres-

sure, P,,. The linearized relationship is then

A T , = G,, A P d + Gf A Fd

+ G6A6 + FNAN ( 6 )

where G,> s the influence of delayed pressure

on torque, Gf the influence of delayed fuel

on torque, G6 he spark advance influence on

torque, and FN he engine friction defined as

the partial derivative of engine torque with

respect to speed. The first three terms of Eq.

(6) define what is usually referred to as the

combustion torque, T,..

For a four-cylinder engine , the fundamen-

tal engine events are offset 180 crank-angle

degrees per cylinder, such that each cylinder

produces power over one-quarter of the two

engine revolutions per cycle. Assuming that

a uniform torque pulse is produced over the

entire I80 deg of the power s t roke based on

the sample value taken at the end of the pre-

vious intake stroke, the combined output of

the four cylinders would describe a nonin-

termpted pulse torque function extending

over the entire cycle.

For the six-cylinder engine, the basic cycle

events are offset 120 deg per cylinder so that

the power stroke of any particular cylinder

begins 60 deg be fore the pow er s t roke of the

previous cylinder in the rotation ends. If the

terminal 60 de g of the power s t roke a re ne -

glected, a 120-deg sample-rate-and-zero-or-

der hold , beginning a t TDC of the power

stroke, provides a continuous, nonoverlap-

ping torque output for the combined cylin-

ders with any individual torque component

comprised of a 120-d eg torque pulse, the

magnitude of which is based on the value

sampled 60 deg be fore BDC of the in take

stroke. Note that the IP delay extends from

60 deg before BDC of the intake stroke until

TDC of the power stroke, an interval of 2 4 0

de g or two sample pe riods a t the 120-degrate. These events are i l lustrated in Fig . 5.

Power-Train Rotational Dynamics

The rotational motion of the engine crank-

shaft is given in terms of the engine polar

.1 Si\hIPLE INTERVAL = 120 DEGREES

CYL 1

I NTAKE COMPRESSI ON POWER EXHAUST

I NTAKE COMPRESSI ON PO WER

Fig. 5. Six-cylinder engine sampling times.

moment o f ine r t ia , angula r accele ra tion, an d

the difference between the net torque gen-

erated by the engine and the load torque of

the shaft . Crankshaft acceleration is given by

Newton’s second law:

J e N = ( 3 0 / a ) T ,- ( 3 0 / a ) T L ( 7 )

where J , is the engine in ertia (ft-lbf-sec’/rad),

T, the engine output torque (Ibf-ft), and TL

the engine external torque load (Ibf-ft).

For a vehic le employing an automat ic

transmission, the external torque load con-sists of the load applied by the torque con-

verter plus external torque disturbances,

which may arise as a result of auxiliary loads

imposed on the engine (engagement of the

air conditioner compressor, for example).

The torque from the converte r i s genera l ly

specified as the square of the ratio of engine

speed to a converter input capacity factor,

K ,. Linearizing Eq. (7) and substituting the

expressions for T, an d TL provides a differ-

ential equation describing the incremental

engine acceleration, N :

= (e)G,,AP + G f A F d

+ G6A6 - A Td) (8 )

If a constant KR is defined to inco rporate the

polar moment of inertia term and the inverse

of the coefficient of A N is defined as the

rotating inertia t im e constant 7 R , hen the in-

clusion of a disturbance torque, A T d , esults

in the l inear transfer-function relationship for

engine acceleration il lustrated in Fig. 2.

Model Param eter Estimation

and Validation

Model parameter estimation and valida-

tion experiments might include static and dy-

namic tests compatible with allowable dy-

namomete r or vehicle measurements. First ,

static experimental data may be used to cal-

ibrate the throttle body, estim ate engine pos-

it ive crankcase ventilation and oth er leakage,

and generate an engine pumping induction

map. This informat ion combined wi th the

prede termined engine torque da ta may be

used to develop an algebraic expression for

brake torque as a function of A IF , mass flow

rate, speed, and spark advance. Subsequent

to static calibration, dynamic tests should be

performed using a throttle kicker (at various

throttle-angle levels) and spark advance step-

type inputs. A representative engine system

response to a small throttle-angle step input

is shown in Fig . 6 for a four-cylinder, car-buretted engine along with modeling results

showing relatively good correlation. Adjust-

ment in pressure-sensitive induction map pa-

rameters would reduce the steady-state pres-

sure error, and modification of the simulated

transmission damping would reduce the tran-

sient error exhibited in the i l lustration.

A method for pe rforming a number of s im-

ple tests to obtain model parameter values is

delineated by Coats and Fruechte [21]. The

experiments essentially consist of throttle ,

spark, and load inputs that give perturbation

responses of engine outputs, the correspond-

ing measurement of which allows direct es-

timates of model parameter values. The lin-

earized structure of the engine model alsoforms the foundation for the use of identifi-

cation techniques to determine model param -

eters. This approach was employed by M or-

ris et al . [22] and M o m s and Powell 1231.

Generally, in these approaches, the engine

model is grouped into dynamic effects as-

sociated with the intake manifold and the

rotating inertia . Landau’s identification tech-

nique [24] may then be applied to the mul-

tiple-inputisingle-output ubsys tems. A ben-

eficial aspect of the identification approach

is that the signal measurement and control

implementation effects are incorporated into

the mode l pa rameters .

Linear Multiple Sampling RateExample

A block diagram of a l inearized six-cyl-

inder engine model and idle speed controller

is illustrated in Fig . 7. This engine repre-

sentation is the basic l inear engine model of

A u y u b t 1988 i J

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-LM A N F O L D "1A / V a h i c l e

PR ESSU R E" H g ) 15 -.------ *.II \

\ModeI1

U-1 2 0 1 2

T I M E ( S E C )

Fig. 6. Sample transient response for validation.

tern that is one-third the fundamental en gine

rate. The samplers in F ig . 7 are included to

emphasize this sampling rate discrepancy

between the discrete-time subsystems. The

multiple sampling rates provide a nonstan-

dard but tractable stabili ty analysis problem

as described by Powell et al . [ 2 5 ] . Closed-

loop idle speed feedback control is effectedby pure integral control of airflow via a

closed throttle bypass valve to provide

steady-state accuracy and proportional con-

trol of spark t iming to enhance speed of re-

sponse. Typical engine parameters for the

six-cylinder engine are enumerated in the

Table . F igure 8 i l lustrates the open- and

closed-loop response of the system to a unit

torque disturbance. Th e oscillatory nature of

the open-loop response is due to the injection

timing delays in combination with the IP lag

and the manifold fi l l ing dynamics.

Although the model i l lustrated is l inear-

ized about a particular idle speed, it should

be emphasized that the model is actually ac-

curate within a reasonably large neighbor-

hood of operating points by virtue of th e

speed-dependent sampling. Such use of state-

(o r feedback-)variable transformation to rep-

F ig . 2 evolving at the fundamental six-cyl-

inder period, T, equal to 120 crank-angle

degrees. To this model has been added speed

density airflow estimation as described by

Eq . (4) and the bank-to-bank fuel-injection

timing previously described. Note that the

fuel-injection timing period, f, f 360 de g

produces a sampling rate in the fuel subsys-

- (7 = 3 T )

FUEL GAIN

AIR MANIFOLO I I ROTATIONAL

BY PASS FILLING IP LAG

1 GP

PUMPING FEEDBACK Gs

-+

-Z

SPARK GAIN

T

TK N *

AN-

Fig. 7. Linearized six-cylinder engine idle speed control model.

24

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Table

Six-Cylinder Engine M odel Parameters

at 600 rpm

r,,. sec

K,, , (Ibf-h)/(lbm-in.’-sec)

G,,. ft-lbfipsi

T. sec

rR . se cK,. rpni/( t-lbf-sec)

K,,+ Ibm/(rpm-min)

G, . ft-lbfilbm

0.21

0.776

13.37

0.033

3.98

0.08

67.2

36.6

resent a nonlinear system by an equiva lent

linear system is referred to by Kokotovic [26]

and illustrated by Cook and Powel l [ 2 7 ] .

Conclusions

Th c deve lopment of a basic non linear rep-

resentation of an engine dynamic sys tem has

been reviewed. Th e mode l conta ins desc r ip-

tions for the induc t ion process and engine

power system as well as characterization ofthe fuel system. The general description

forms a foundation to which other important

transient characterizations, such as exhaus t

ga s recirculation system dynamics or intake

manifold fuel wall-wetting may be added. In

addition. a l inear model has been developed

for a particular six-cylinder engin e and a t ime

response of the system is presented.

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A RPM .:: 1-2 J

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Fig. 8.

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Sixth IFAC Symp. Identification and Sys-

tem Parameter Estimation, Washington,

DC, June 1982.R . L. Moms and B. K. Powell, “Modem

Control Applications in Idle Speed Con-

trol,” Proc. 1983 Amer. Contr . Con$, vol.

2 , pp. 79-85, June 1983.

I. D. Landau, “Unbiased Recursive Iden-

tification Using Model Reference Adaptive

Techniques,” IEEE Truns. Auto. Contr.,

vol. AC-32, no. 2 , pp . 194-202, Apr. 1986.

B. K. Powell, J. A. Cook, and J . W. Griz-

zle. “Modeling and Analysis of an Inher-

ently Multi-Rate Sampling Fuel Injected

Engine Idle Speed Control Loop,” ASME

J . Dyn. S ) x , Meas., C ontr . , vol. 109, pp.

405-410, Dec. 1987.

P. V. Kokotovic, “Control Theory in the

80’s: Trends in Feedback Design,” Ninth

World Congress of IFAC, Budapest, Hun-

gary, July 1984.[27) J. A. Cook and B. K . Powell. “Discrete

Simplified External Linearization and Ana-

lytical Comparison of IC Engine Fami-

lies,” Proc . 1987A mer . Conrr. Con$, vol.

1, pp . 326-330, June 1987.

[191

[20]

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[26]

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8/6/2019 00007726


Jeffrey A. Cook has been

a Research Engineer in the

Control Systems Depart-

ment, Research Staff,

Ford Motor Company,

since 1976. He received

the B . S . degree in me-

chanical engineering from

The O hio State University

in 197 3 and the master’sdegree in electronics and

computer control from

Wayne State University in1985. His research interests are in the areas of air/

fuel ratio and emissions control for internal com-

bustion engines. He is also an Adjunct Faculty

Member at Lawrence Institute of Technology,

Southfield, Michigan.

Barry K. Powell has been

a Research Engineer in the

Control Systems Depart-

ment, Research Staff,

Ford Motor Company,

since 1976. Prior to that

time, he worked at Ford

Automotive Safety Re-

search and Bendix Cor-

poration Research Labo-

ra tor ie s . His re spons i -

b i l i t i e s h a v e i n c l u d e d

mathematical modelingand control of aerospace and automotive systems.

His current activity is in real-time analysis and

control of automotive power-train systems.

Out of Control

‘‘This year’s mode l comes with a servo amplifier for speed control.”

26I € € € Control Systems Mogoz l n e

00007726

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