Stability of Feedback Control of Boost in Variable

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DOI: 10.1243/PIME_PROC_1989_203_023_02

1989 203: 163Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy R J Backhouse, P C Franklin and D E Winterbone

Stability of Feedback Control of Boost in Variable Geometry Turbocharged Automotive Diesels

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163

Stability of feedback control of boost in variable

geometry turbocharged automotive dieselsR J Backhouse, MA, MSc, PhDNapier Turbochargers Limited, Lincoln

P C Franklin, MAHolset Engineering Company Limited, Huddersfield

D E Winterbone, BSc, PhD, DSc, CEng, FIMechEDepartment of Mechanical Engineering, University of Manchester Institute of Science and Technology, Manchester

This paper is concerned with the control of turbocharged automotive diesel enginesjtted with variable geometry turbines, in particular,for the case where a feedb ack loop is used to control the boost pressure. Interaction b etween the loops governing speed and controlling

boost will affect the stability of the overall system.The effect of this interaction is considered fo r the case of automotive diesel engines in the 200-250 kW class at peak torque speed, bothwith and without engine speed governing, and fo r the cases of idle and maximum speed governing. T he e fe ct s of vehicle load, turbo-charger rotor inertia and actuator response are also considered.Th e analysis uses the characteristic equ ation of the control system to determine lines of system instability in the gain plane.

C

dB

Ff

I E

It,,1jk

m

n W

NEpb

It

12

R

TF

Xf

S

I;

ANE

ddt

NOTATION

a constantdecibeltransfer function of speed sensortransfer function of engine speed to rack position

polar moment of inertia of engine (including fly-wheel and clutch) (kg mZ)total polar moment of inertia (kg mZ)polar moment of inertia of road wheel (kg mZ)square root of minus onetransfer function of rack motion to speed [mm/

mass of vehicle (kg)design gross vehicle weight (kg)number of road wheelsengine rotational speed (r/min)boost pressure (absolute) (bar)gearbox gear ra tiogear ratio of driving axlerolling radius of driving wheels (m)Laplace variable s - )transfer functionload torque (kg m)fuel rack position (mm)error in engine speed (r/min)angular frequency (rad/s)

C(r/min)/mmI

(r/min)I

first derivative with respect to time s-l)

The subscript ref indicates a control reference value.

1 INTRODUCTION

Turbocharging is a well-established means of improvingthe specific power (kilowatts per kilogram) and cost of

The MS was received on 16 M a y 1988 and wa s accepted fo r publication on4 April 1989.

diesel engines for trucks and buses, but has the dis-advantage of poor low-speed torque and poor transientresponse compared to an equivalent naturally aspiratedengine. Use of variable geometry (VG) turbines allowsboost to be improved at low speed without over-speeding the turbocharger at rated engine speeds.Smoke-limited torque is increased at low speeds andbrake specific fuel consumption can be improved overmuch of the engine operating range (1, 2). In addition, ifthe turbine inlet area (Ati) is restricted at part load toraise the minimum turbocharger speed, then the enginetransient response can be improved. Figure 1 shows anidealized Ati schedule for steady state operation; whileopen-loop scheduling in this manner is beneficial,closed-loop (feedback) control of boost pressure givesthe added advantage of automatic compensation forvariations in ambient pressure (e.g. due to altitude) andturbocharger wear, and the control of turbine areaduring transient performance is also improved.

An engine having feedback control of boost pressure

constitutes a multi-variable control problem with twoinputs and two outputs (Fig. 2); coupling or interactionbetween the two feedback loops (one controlling engine

Rated

\aximum turbineinlet area ( A r i )

Engine speed

Fig. 1 Idealized steady state schedule of turbine inlet area

A01988 Q IMechE 1989 0954-4046/89 2.00 + OS Proc Instn Mech Engrs Vo1203

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164 R J BACKHOUSE,P C FRANKLIN AND D E WINTERBONE

Diesel engine

Rack Torque

speed position I s p e d

LoadI torque I12 I

Reference Turbine Boostg22

-pressurerea

boost

Fig. 2 Transfer function diagram of a turbocharged diesel engine with VGturbine and separate feedback control of engine speed and boost

pressure

speed and the other boost pressure) may cause insta-bility, even where each of the loops Considered separ-ately is stable. It is quite feasible to produce amulti-variable controller to control engine speed andboost simultaneously, making due compensation forinteraction; one of the authors has designed amicroprocessor-based controller of this type and used itto control an automotive diesel with a variablegeometry turbine (3). Such an approach is likely toincrease the first cost of the engine and requires integra-tion of the control of turbine area with that of the

engine speed governor; the latter has until now been theresponsibility of the fuel injection equipment (FIE)manufacturer. In the short term, at least, there isbelieved to be a market for variable geometry turbo-chargers using simple control technology such that theycan be fitted to the engine without modification to theFIE. This approach also allows easy fitting to enginesalready in service.

One simple control strategy is to adjust turbine inletarea to try to maintain the inlet manifold pressure con-stant. This leads, under steady state conditions, to thevariation of turbine inlet area of the form shown in Fig.3. This system gives good full-load and good transientperformance and does not require sensing of the engine

ReferenceP , line

I

Engine speed

Fig. 3 Turbine inlet area schedule when actuator is operatedby boost pressure

Part A : Journal of Power Engineering

speed. The force required to change the turbine inletarea may be obtained by means of a pneumatic actu-ator, operating from the boost pressure, and hence thesystem can be contained in the turbocharger itself. Thedisadvantage of this schedule is that the part-load efi-ciency of the engine is impaired by the high back pres-sure, although in some applications this is not a majorproblem.

The application of such a control system is con-sidered for the case of diesel engines in the power rangefrom 200 to 250 kW. The speed governor may be of

either the two-speed or all-speed type. The former typeprovides speed governing at low-speed idle, to avoidstall, and at maximum speed, to prevent damage to thevalve gear and vehicle transmission, with no governingof speed at intermediate speeds. In the case of all-speedgoverning, the governing at intermediate speeds issimilar to that at maximum speed (that is giving a verysteep rise in engine fuelling with a fall in engine speed),with the demanded speed being determined by theaccelerator position. The latter type of governing ismore common in off-road applications, such as mecha-nical digging and bulldozing.

A brief resume of some control theory is given beforecovering the hardware and data used and the assump-tions made. The paper then proceeds to consider theeffect of feedback loop interaction under different engineoperating conditions. The effects of vehicle mass, turbo-charger rotor inertia and improved actuator responseon stability are also considered.

2 THEORY

2.1 Introduction to feedback controlThe basic concepts of feedback control can be obtainedfrom many standard textbooks, e.g. Raven (4). Thissection will relate some of the control parameters to theresponse of engine speed NE) o fuel pump rack posi-tion (xJ. If the fuel delivery to the cylinders (cubic milli-metres per stroke) is assumed to be proportional to rackposition, engine torque proportional to fuel deliveryand load torque proportional to engine speed, then the

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STABILITY OF FEEDBACK CONTROL OF BOO5

Xf- g

;T IN TURBOCHARGED AUTOMOTIVE DIESELS 165

NE .-

(a)

II - - - - - -

(b)Fig. 4 a) Open-loop block diagram

(b) Closed-loop block diagram

equation describing the variation of engine speed is

This first-order differential equation may be rearrangedto give the transfer function of the engine, viz. theresponse of engine speed to a change in rack position :

where D s the differential operator.The derivative in equation (2 )may be replaced by jo,

in the case of sinusoidally ranging parameters (or by s,

the complex Laplace operator, for a more general solu-tion in terms of e ).The transfer function may be represented by the

block diagram shown in Fig. 4a, and it can be evaluatedexperimentally by varying the rack position sinusoidallyat different frequencies and measuring the amplitudeand phase of the resulting speed perturbation relative tothat of the rack motion. The steady state gain can beobtained by making a step change in rack position andmeasuring the resulting change in speed, allowing suffi-cient time for the speed to reach steady state.

Often it is convenient to express the gain at a givenfrequency relative to the steady state gain. Normallygain is expressed in decibels :

gain a t frequency o

steady state gainelative gain (dB) = 20 x log,,

An additional parameter used in control system assess-ment is the break frequency, which is that frequency atwhich the relative gain has fallen to dB; this may beloosely termed the bandwidth of the response.

When feedback is applied, that is when the speed issensed and corrective action made to the rack positionin an attempt to restore the engine speed to its desired(or reference) value, the block diagram becomes that ofFig. 4b, where the section enclosed in dashed lines isnormally performed by the fuel injection equipment

(FIE). The box represents the response (transferfunction) of the speed sensor to changes in enginespeed; box k represents the movement of the rack inresponse to errors in sensed speed. The reference speedmay be constant (that is idle or maximum speed) for

@ IMechE 1989

max-min governing or, in the case of all-speed govern-ing, a movable value that varies with the position of thedriver's accelerator pedal.

The overall response of engine speed to changes inreference speed (that is the closed loop transfer function)is found from

NE = gk Ere( - f N E ) (3)

NE

Thus

(4)k-

NErsf + gk fClosed-loop stability is determined by the denominatorof the transfer function, which is called the closed-loopcharacteristic polynomial. In simple terms, instabilityoccurs when this becomes zero :

1 + gkf= 0 (54

gkf= -1 (5b)

or

The left-hand side of equation (5b) is the system loopresponse; this can be expressed in terms of gain andphase at a given frequency. Figure 5 shows the gain(decibels) and phase (degrees) plotted separately againstthe log of frequency (called a Bode diagram) for a first-order response with additional time delay. The timedelays may result from sampling in digital systems or,for example, from the delay between fuel injection andthe resulting torque appearing at the crankshaft. Thefirst-order response is typical of an engine in which theload is a function of speed.

The loop gain is dimensionless: instability occurswhen at the same frequency the loop gain is 1 (0 dB)and the phase lag round the loop is 180 degrees. In

' f b \ log\ 20 dB/decade

Extra phase lag due to added time delay\irst-order system

\\ -\ \ / z

Increasing \time delay \

\

Fig. 5 Bode diagrams of a first-order system with time delay

Proc Instn Mech Engrs Vol 203



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166 R J BACKHOUSE P C FRANKLIN AND D E WINTERBONE

general, the two effects occur at different frequencies :the gain crossover frequency and the phase crossover fre-quency respectively. The phase margin is the angle bywhich the phase can be increased before reaching minus180 degrees at the gain crossover frequency. The gainmargin is the amount by which the gain can beincreased at the phase crossover frequency before thegain reaches unity.

Oldenburger (5) suggests that 30 degrees phasemargin together with 8 dB gain margin gives satisfac-tory stability for such systems. Garvey 6) observednoticeable roughness in diesel engine running when thephase margin in the speed-governing loop fell muchbelow 30 degrees. In automotive diesel systems thecrossover frequencies are generally in the range 1-5 Hz,although as electronic control becomes more widelyapplied and faster actuators come into use the upperlimit may increase to 10 Hz.

2.2 Multi-variable systemThe basic theories for multi-variable systems are devel-oped by Rosenbrock (7), MacFarlane 8) and others.The system in question has two outputs (engine speedand boost pressure) and two inputs (rack position andturbine inlet area) (Fig. 2). Feedback control of speedtakes place in the governor. Using the symbols of Fig. 2,but with g l l and g12 now incorporating the effects ofthe load, the closed-loop transfer functions of the twoengine outputs can be obtained:

where

c = ( I + f l k l g l l ) ~ f 2 k 2 9 2 2 ) - f 1 f 2 k k l k 2 9 1 2 9 2 1The denominator C is common to each of the closed-loop transfer functions so that any instability affectsboth feedback loops. Instability occurs when C, thecharacteristic polynomial of the system, is zero; that is

(1 +fl k l g11X1 + f 2 k2 gz2) - f 1 f 2 k l k2 912 Q21 = 0( 6 4

or

f l k l g l l +f2k2922 +flfiklk2 911922 --12921) = - 1

(6b)

If each of the individual responses is known, the aboveequation can be used to determine which combinationsof feedback gains (k, and k2 will produce instability.The results may be plotted in the gain plane, where thetwo feedback gain values form the two axes.

Stability margins are harder to define for multi-variable systems than for single-loop ones. It will beassumed, in this case, that a safe margin exists when

Part A: Journal of Power Engineering

both feedback gains can be increased by a factor ofthree without crossing the instability lines determinedfrom the characteristic equation. This corresponds towhat may loosely be called a gain margin of about 10dB.

More general approaches to the analysis of multi-variable systems under feedback control are available7, ), and these are particularly powerful with complex

systems. While it is usually safe to assume that inter-action reduces stability margins (as Rosenbrock'sinverse Nyquist array method does) this is not necessar-ily the case. In this simple case, where speed and boostfeedback characteristics are already known, use of theCharacteristic equation allows the actual stability limitsto be explored under different system assumptions.

3 EXPERIMENTAL

Frequency response measurements were made using

Solartron frequency response analysers on two directinjection turbocharged diesel engines (of 11 litre and 10litre displacement) each fitted with a Holset VH2C tur-bocharger. Some of these results are presented in refer-ence (9). The turbocharger has a prototype variablegeometry mechanism and pneumatic (boost pressure)actuation of the change in the turbine area. The fre-quency responses of the boost control loop were mea-sured on a rig and on the engine itself.

The responses of several mechanical governorstypical of the type fi tted to these engines were examinedand a response equivalent to the best of these wasassumed for this investigation (see Fig. 6) .

FrequencyHz

-~

0.1 0.2 0.4 1 2 4 10

12 litre turbocharged diesel60 load1500 rlmin (engine)

8

O A

FrequencyHZ

_ _ _ _

0.1 0.2 0.4 1 2 4 1001

2010

3 780 Governor typeo Type AA TypeB

A\

overnor response assumed A- 40 t in present analysis- 60

Fig. 6 Assumed governor response compared with responsemeasured for similar FIE pumps

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STABILITY OF FEEDBACK CONTROL O F BOOST IN TURBOCHARGEDAUTOMOTIVE DIESELS 167

0.8

c 0.4-

'8

B-a 0.1

3 0.2

2

a

v

0.05

Typical range of steady state feedback gains mea-

Speedgovernor 1&14 per cent increase in(at maximum maximum fuel per 1 per centspeed) change in engine speed

sured :-

-

-

-

Boost actuator 2.5-60 per cent A,i (as a percentageof fully open area) per 1 per centchange in boost absolute pres-sure

The preferred value of the boost actuator steady stategain, for good transient and steady state performance, is5 per cent Ati per 1 per cent change in boost absolutemeasure P,. This implies a narrow pressure rangewithin which the turbine area actuator may operate freeof the end stops (that is the turbine tends to be fullyclosed or fully open over much of the engine operatingregion) (Fig. 3).

4 RESULTS AND DISCUSSION

4.1 Without speed governor action

When a two-speed max-min governor is used speedfeedback due to governor action occurs only atmaximum speed and low-speed idle. However, someminor intrinsic feedback does occur at intermediatespeeds, because the fuel injection characteristics are gen-erally chosen to give a small increase in fuelling with adecrease in speed in order to achieve the desired torquecurve.

The steady state gains of the interaction terms

become greatest at maximum load and near peaktorque speed; this is likely to be the condition whereinteraction will have the greatest influence on stability.

Typical steady state gain values of the transfer func-tions are given in Table 1 for the 11 litre engine at thisoperating point. Estimated variations for other enginesof this class are given in brackets.

The terms in equation (2) then have the followingsteady state magnitudes, assuming that gain of theboost feedback loop has the desired value of 5 per centAti per cent change in absolute boost pressure:

11 922 O.qO.24 - 0.6)g1 2 0.042(0.026 - 0.08)

rated speed . absolute boost

maximum fuel * Ati

g12921/911922

11 rising to 15% above 1 Hz (dimensionless)

Stable

Without

I /

- Withinteraction

~~

5 10 20 40

Boost feedback gain

Lines indicating instability

Linesof safe stability margin_ -

Region of desired feedb ack gains@ Actuator responses normalTurbocharger inertia standard

Fig. 7 Stability region in the gain plane: peak torque condi-tion without speed governing

At steady state, the product of the interaction termsg12 and g21 s small compared to the product of thedirect terms g l l and g2,; at higher frequencies of inter-est (2-10 Hz) the interaction becomes more significantand may affect the stability of the system.

By substituting the measured frequency responses

into the characteristic equation the values of the feed-back gains ( k , and k,) required to cause instability canbe determined. The lines of instability are plotted forthis operating point in Fig. 7, together with lines sug-gesting a safe gain margin.

In the absence of speed governor action there isclearly sufficient margin in the speed feedback loop;Fig. 7 also indicates that stability in the boost feedbackloop is sufficient and that , if anything, interactionimproves the stability slightly.

4.2 With speed governor action

The stability of the system with speed governing needsto be considered not only at the peak torque conditionbut also at idle and a t maximum engine speed.

4.2.1 Idle speed control

At engine idle speed the boost pressure is well below thelevel at which actuator motion occurs, so that k2 is

Table 1 Typical steady state gain values at peak torque condition in the absence of governoraction

Term DescriDtion

f lkg1fzkgZ2gg Boost response to fuelling

Response of FIE speed to engine speedChange in fuelling with speedEngine speed response to fuellingActuator pressure response to boostActuator rod response to boostBoost response to a ctuator rodSpeed response to a ctuator rod

Steady state value

Unity0.4 max. fuel/% rated speed 0.04.6)0.8 rated speed/% max. fuel 0.61.0)Unity5 turbine area/% change in boost (absolute)(2.5-10)0.5 change in boost (abs.)/%& 0.4-0.6)0.03 rated speed/% A,i (0.02-0.04)1.4 boost ch ange (abs.)/% max. fuel(1.3-2.0)

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168 R J BACKHOUSE,P C FRANKLIN A N D D E WINTERBONE

Table 2 Steady state gains at maximum engine speedTerm Steady state gain Dimension

gl] 0.8 (0.6-1.0) rated spee d/l max . fuelg 1 2 -0.005 (0.03S O.05) rated speed/ turbine areagzl 1.0 (0.8-1.2) boost change (abs.)/ max . fuelg 9 0.35 (0.3-0 .45) abs boost pressure/ turbine area

zero; there is therefore no interaction and stability isdetermined solely by the stability of the speed feedbackloop.

4.2.2 Maxim um speed control (governo r runou t)

Typical steady state gain values for the transfer func-tions under maximum speed governing (point 3 in Fig.3) are given in Table 2 with estimated ranges inbrackets.

In Fig. 8 the lines of instability are shown in the gainplane under maximum speed governing, for the casewhere the load inertia is zero; that is the rotationalinertia is that of the engine and flywheel alone. Stabilityin the speed feedback loop is marginal, but this is neces-sary for fast response. Again, interaction appears toimprove the stability maigins.(a) Eflect of engine load inertia. Since the break fre-quency in the speed response loop is generally wellbelow the range determining control stability (that isbelow the gain crossover frequency), any increase inload inertia reduces the gain more significantly than itincreases phase lag and so improves the stabilitymargin. Instabilities that appear on the test-bed, there-

fore, will not necessarily appear when the engine isoperating normally in a vehicle.The engine is only under load, other than momen-

tarily, when a gear is engaged; the total effective inertiaincreases with vehicle load and inversely with gear ratio.The minimum effective inertia under load is estimatedto be 1.8 times that of the engine and flywheel alone (seethe Appendix). Also shown in Fig. 8 are the lines

instability with 1.8 times engine inertia

Withinteraction.- 40

3 Instability with engine inertian _ _ ~ ~_ _ _ _ _ _ _ - - - - -

**LLllI .I interaction

v

5 10 20 40

Boost feedback gain

Actuator responses normalTurbocharger inertia standard

Fig. 8 Stability region in the gain plane: maximum speed,showing effect of load inertia


With

5 10 i0 40Boost feedback gain

Turbine area actuator response improved10 Hz band width with digital contro l)

Load inertia total) s 1 .8 times engine inertia

Fig. 9 Stability in the gain plane: maximum speed govern-ing, showing effect of turbocharger inertia

obtained after altering the speed responses to simulatean increase in rotational inertia of this magnitude: thegain margin is now ample.

It is assumed that the transmission is torsionallyrigid, and this is valid provided that the natural fre-quency of torsional oscillations is not close to the cross-over frequencies. At frequencies close to thetransmission natural frequency, which is expected to bebetween 5 and 10 Hz, the effective load inertia seen bythe engine may be substantially reduced; this in turnwill reduce the stability margin and may cause hunting.The usable bandwidth of the actuators controllingengine fuelling and turbine area may well be limited bythese considerations.(b) Improved boost actuator response. Figure 9 showsthe lines of instability when the response of the boost

sensor and turbine area actuator are improved to thatof a digitally controlled actuator of 10 Hz bandwidthwith a total time delay of 20 ms. In the 1-5 Hz range, inwhich stability is determined, such a feedback loop hasnegligible attenuation of gain but moderate phase lag.While improving response time this can be expected toreduce stability margins in the boost feedback loop;compare Fig. 9 with Fig. 8. (Lines showing stabilitylimits with interaction indicate the worst case represent-ed by the ranges of values given in Table 2.)(c) Reduced turbocharger moment of inertia. The breakfrequency associated with the turbocharger response isusually well below 1 Hz, so that reducing the turbo-charger moment of inertia increases the boost gain in

the 1-5 Hz range without significantly reducing thephase lag. This reduces the stability margin; neverthe-less the margin appears to be sufficient even when theturbocharger moment of inertia is reduced to 40 percent of its original value (Fig. 9).

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STABILITY OF FEEDBACK CONTROL OF BOOST IN TURBOCHARGED AUTOMOTIVEDIESELS 169

Lines of instability

Without Withinteraction interaction

5 10 20 40

Boost feedback gain

Turbine area actuator response improvedLoad inertia (total) is 1.8 times engine inertiaLines B-B indicate 40 reduction in turbocharger inertia

Fig. 10 Stability region in the gain plane: peak torque condi-tion with speed governing

4.2.3 Mid speed range

With all-speed governing, the stability at higher enginespeeds (point 2 in Fig. 3) is assumed to be similar tothat at maximum speed.

For the region near peak torque speed (point 1 inFig. 3) the limits of stability are shown in Fig. 10. The

load inertia is 1.8 times that of the engine. When theturbocharger inertia is reduced the stability margin inthe boost loop is again reduced; improvement in theboost actuator response further reduces the boost loopstability. Interaction actually increases the gain marginin the boost loop and in the case of reduced turbo-charger inertia (lines labelled A-A) will help to maintainthe stability margin in the desired feedback gain oper-ating range. If improved boost feedback response iscombined with reduced turbocharger inertia (lines B-B)the stability margin becomes inadequate.

5 CONCLUSIONS

In an automotive turbocharged diesel of 2W250 kW,having simple feedback control of boost pressure, asdescribed in this paper, interactions between the feed-back loop controlling boost pressure and that control-ling speed are small and often improve the stabilitymargin. Stable feedback control of boost pressure cantherefore be applied without the need for multi-variableprecompensation.

Stability margin in the loop controlling boost is leastat the peak torque condition and instability is likely toresult at this operating point if a reduced turbochargerinertia (for example, by use of ceramic components) iscombined with fast actuation and digital control of theturbine inlet area.

Provided that the natural frequency of torsionalvibrations in the transmission is not close to the fre-quency determining control stability, the minimum loadinertia seen by an engine when it is operating underload in a vehicle will be significantly greater than that of

IMechE 1989

the engine and flywheel alone. This implies that controlinstabilities seen on the test-bed may not appear whenthe engine is operating in a vehicle.

ACKNOWLEDGEMENTS

The authors are grateful to Holset EngineeringCompany Limited for permission to publish. Part of thework was carried out under SERC sponsorship. Theyare also grateful to Professor F. J. Wallace for per-mission to use the facilities of the Wolfson Engine TestUnit at Bath University, and for the assistance of theWolfson Unit staff.

REFERENCES

1 Wallace, F. J. Way, R. J. B. and Baghery, A. Variable geometryturbocharging-the realistic way forward. SAE paper 810336, 1981.

2 Flaxington, D. and Szczupak, D. T. Variable area radial inflow tur-bines. IMechE Conference on Turbocharging and turbochargers,1982 (Mechanical Engineering Publications, London).

3 Backhouse, R. J. The dynamic behaviour and feedback control of aturbocharged automotive diesel engine with variable geometryturbine. P hD thesis, 1986, University of Manchester.

4 Raven, F. H. Automatic control engineering, 1961 (McGraw-Hill,New York).

5 Oldenburger, R. C. Frequency response data, standards and designcriteria. ASME paper 53-All, 1953.

6 Gamey, D. C. A digital control algorithm for diesel engine govern-ing. SAE paper 850174,1985.

7 Rosenbrock, H. H. Computer aided control system design, 1974(Academic Press, London).

8 MacFarlane, A. G. J. (Ed.) Complex uariable methods for linearmultiuariable feedbak systems, 1980 (Taylor and Francis, London).

9 Backhouse, R. J. and Winterbone, D. E. Dynamic behaviour of aturbocharged diesel engine. SAE paper 860453,1986.

APPENDIX

Estimation of minimum total effective load inertia

Assuming torsional rigidity of the transmission, totaleffective inertia seen by the engine is given by

(7)

This assumes that the inertias of gearbox output shaft,propeller shaft, differential, half-shafts and driving hubsare negligible compared with the aggregate inertia ofthe road wheels.

From equation (7) it is clear that minimum effectiveinertia occurs when r l r2 is maximum (that is in bottomgear).

If it is assumed in addition that the gearing is chosensuch that bottom gear is sufficient to enable the engineto take the vehicle, fully laden, up the steepest gradientencountered, and that the gear ratio, r l r 2 , required toclimb this hill is proportional to the design gross vehicleweight (GVW), that is

rI r2 M

then the third term in equation (7) can be expressed as

The third term of equation (7) is clearly proportional tovehicle mass (as a proportion of GVW), and theminimum total effective inertia occurs in bottom gearwith the vehicle unladen.

Taking, as the worst case, an articulated vehicleProc Instn Mech Engrs Vo1203



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170 R J BACKHOUSE, P C FRANKLIN AND D E WINTERBONE

tractor without its trailer, and assuming the followingamroximate values:

then

I = 3 kg mI = 20 kg mR = 0.55 m

r l r2 = 5 5m = 9 x lo3 kg

n, = 6


l t o t f f 3 + 2.2 + 0.9 kg m2 = 6.2 kg mz= 2.1 1,

Allowing for errors in the assumptions, a value of 1.8

times engine inertia is assumed a safe estimate of theminimum total effective inertia seen by the engine underload in service.

@ IMechE 1989


Documents

Stability of Feedback Control of Boost in Variable