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Hybrid computer simulation of the sampled-data modelfor compression ignition enginesC. R. BURROWS a , P. W. VAN EETVELT a & G. P. WINDETT aa Applied Sciences Laboratory, University of SussexPublished online: 03 Apr 2007.

To cite this article: C. R. BURROWS , P. W. VAN EETVELT & G. P. WINDETT (1971) Hybrid computer simulation ofthe sampled-data model for compression ignition engines , International Journal of Control, 14:4, 737-746, DOI:10.1080/00207177108932084

To link to this article: http://dx.doi.org/10.1080/00207177108932084

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INT. J. CONTROL, 1971, VOL. 14, No.4, 737-745

Hybrid computer simulation of the sampled-data modelfor compression ignition engin.est

O. R. BURROWS,t P. W. VAN lDETVELTt and G. P. WINDETTApplied Sciences Laboratory, University of Sussex

[Received 30 October 1970]

The process of fuel injection in a compression ignition engine may be regarded as adiscrete process with the sampling rate varying due to engine speed fluctuations.

The variation in sampling rate theoretically prevents the use of sample-datatheory to study the model unless engine speed fluctuationaare small.

This paper examines the effect of this fundamental limitation on the applicationof sample-data theory.

Notationk = gain

n = number of cylinders

N, w = engine speed

T = sampling time

T = finite time delay

x' = fuel rack displacement

A, B, 0, D, A, B, C", D )

R, Tl, T2, SWl, SW2.. = logic signals, "Q Q' .

(X, (X , (X '{J, {J

1. Introduction

The application of linear continuous control theory to the governing ofcompression ignition engines was discussed :in the classical paper by Welbournet al. (1959). Although Welbourn (1963) recognized that compression ignition­engines should be modelled using sampled-data theory, this approach apparentlyreceived no attention until Bowns (1971) recently applied the theory to athree-cylinder diesel engine.

Hazell and Flower (1971) have reported a theoretical study of four, six andeight-cylinder engines. Using the results published by Welbourn etal. (1959) theyshowed that a sampled-data modelled to better correlation between theoreticaland experimental results.

Sampled-data theory is only applicable when the sampling rate, for a givensampler, is constant whereas in an engine the sampling rate is speed dependent.(We are not considering multi-rate sampling as can occur when there is morethan one sampler in a loop.) Thus, befor~ sampled data theory can be applied

t Communicated by Professor J. C. West.t Member LU.I.E.C.

25

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?as C. R. Burrows et al.

to a diesel engine, it is necessary to examine the effect of this limitation andthis can readily be achieved using a hybrid computer. Moreover, it thenbecomes feasible to study the effects of the commonly occuring non-linearities.

This paper describes thc application of a hybrid computer to the discretemodel of compression ignition engines and compares the results with earlierwork (Welbourn et al. 1959, Hazell and Flower 1971).

2. Discrete model for an engine

The action of fuel injection in a diesel engine is essentially a pulse-widthmodulation process whereby the fuel injected is a function of the control roddisplacement. However, for simplicity, it is assumed that fuel injection can bereprcsented by an ideal sampling process (Bowns 1970, Hazell and Flower197I), as shown in fig. 1.

Fig. 1

O¢5 i r-ed !>pnd + <, _~owf--.,.-'-'--~

'-------J

Output spud

Simplified representation of fuel injection (single cylinder).

The pulses of fuel are burnt and the resulting torque output can be repre­sented by a constant torque which is capable of providing the same energy tothe load. Thus the relationship between the sampled fuel rack position andthe output torque can be represented by a zero order hold (z.o.h.) (Ragazinniand Franklin 1955) and a linear gain factor.

In practice, fuel injection commences at some point before top dead Centre(t.d.c.) and this results in a finite time delay between fuel injection and torqueoutput. The relationship between the fuel rack displacement and the outputtorque can thus be represented as shown in fig. 2.

Fig. 2

FUcz~ ~ac k '---1 Detcy H eo.n. H Gain ~torqUIlposition

(a)

Iii t [•180· 360· Cronk anglll

(b)

t cenotes sampling instant

Relationship between fuel rack position and output torque.

If we now consider a multi-cylinder engine it is obvious that the outputtorque as a function of fuel rack position can be determined by superimposingthe effect of each cylinder. However, for a four-stroke engine, once there aremore than four cylinders there is a torque overlap into the next samplingperiod as shown in fig. 3 and this invalidates the use of a simple zero-orderhold model since the torque level changes between samples. This led Hazelland Flower (1971) to introduce a new term namely, a 'partial zero order hold'

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Simulation of the sampled-data model for compression. ignition enqines 739

(p.z.o.h.) to describe the staircase effect between samples and they derivedthe appropriate pulsed transfer function.

The effect of the number of cylinders on the torque-crank angle relationshipis examined in more detail later.

Fig. 3

16

I I,1

3I I

2

c.2

1:

Feom er-evtcus ~_-L+-_-L,f-_..L,f-_..L:l---_..L~_..L;\----_..Lt-

.!

Cylindl!T" number-

ever-teehot<:hC'd

b S S i

~IgIL.._-=,.--- -----,== ="""""'=

Relationship between output torque and crank angle for a six-eylinder engine.

Figure 3 also illustrates the delay between the sampling instant and thecorresponding torque output. Although the delay is caused by a fixed angleof advance, the time delay represented by this angle is dependent upon thespeed of rotation of the crank shaft, i.e. assume an angle of advance of (jO andthat the engine is operating at a nominal speed of N rev/min then the finitedelay 7' is given by:

(J7' = 6N8. (1)

The sampling rate is also a function of the speed and for a four-stroke engineis given by:

(2)

where n is the number of cylinders.

3. Hybrid computer model3.1. General considerations

It has already been noted that torque overlap into the next samplingnstant can only occur when the number of cylinders is greater than four.However, in considering how to model a four-stroke diesel engine we will, forcompleteness, consider four, six, eight, twelve and sixteen-cylinder engines.

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740 c. R. Burrows et al.

(It will become obvious later that the model proposed for a four-cylindercngine can readily bc adapted when there are less than four cylinders.)

Following the procedure outlined in deriving fig. 3, the effect of the numberof cylinders on the period during which torque overlap occurs can be readilydetermined and is summarized in table 1. (Note each torque pulse lasts for1800 of crank rotation.)

Table I. Effect of the number of cylinders on torque overlap

No. ofCrank rotation

Duration of Duration ofcylinders

between samplestorque pulse torque overlapdegrees

4 180 '1' 06 120 ~T iT8 90 2'1' '1'

12 60 :l1' 2'1'16 45 4'1' 31'

Table 1 can be used to derive a system of samplers and zero-order holds tosimulate a compression ignition engine with any number of cylinders, e.g.consider a six-cylinder engine. The existence of torque overlap requires twosample/hold combinations arranged as follows:

Sampler 1 (fig. 4 (a)) operates at intervals of 2T and the instantaneous valueof X r is held for the interval of 2T. However, since the torque pulse is onlyrequired for an interval of !'l' (table 1), this is followed by a switch 1 (fig. 4 (a))arranged to close when sampler 1 is switched and remain dosed for ~T.

Sampler 2 and switch 2 have a similar function, but sampler 2 is arrangedto close at an interval T after sampler I.

The outputs from the hold circuits are summed and multiplied by anappropriate gain factor.

The above description has ignored the time delay between fuel injectionand output torque. This can easily be included by arranging switches 1 and 2to close T sec after samplers 1 and 2 respectively. (The switches must stillremain closed for ~T.)

Sincc the time delay increases the time during which X r must be held, itcan increase the number of sample/hold combinations required. The numberof sample/hold units can be determined from:

. hold timeNumber of sample/hold units = It".

samp e line

(The right-hand side must be set to the next highest integer value.)This enables the circuit given in fig. 4 (b) to be modified to model any

number of cylinders up to n = 16. The other requirement is to arrange thecorrect timing sequences for the samplers and switches and this can easily beachieved using a hybrid computer.

3.2. Specific example-six-cylinder engine

Details of the computer programme can best be described by considering aspecific example; for this purpuse we proceed to describe the arrangement forsimulating a 360 b.h.p. six-cylinder engine operating at a nominal speed of

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Simulation of the sampled-data model for compression ignition engines 741

500 rev/min as this allows a comparison with earlier published results (Welbournet al. 1959, Hazell and Flower 1971).

The computer diagram is shown in fig. 5 and the details of the logic circuitrequired to operate the samplers and switches are shown in fig. 6. Comparingfigs. 4 (a) and 5 it is seen that the sample/hold combination can be replacedby a track/store unit.

Fig. 4

F'u~1 rackpos! non

e.o.n.

52~ ;,o.h. fs.:;----'(0) n - 6

Fuel rack

position

".o.h fs:,---..

L-__..J~-

.a.c.n. ~3

.e.o.b. ~4

~5(T 'F 0)(b) I," n' "16

KEY S - scmcter-SW Switch

Schematic arrangement for obtaining output torque-fuel rack position.(a) n = 6; (b) l';;;n';;; 16 (T;OO).

WeI bourn et al. (1959) represented the engine by a first-order lag plus afinite delay, which was taken to be the time between successive firing strokesof the engine. This assumption has been examined by Hazell and Flower (1971).In this paper the engine is represented by a combination of sample and holdunits and a first-order lag (amplifier 70, fig. 5). The gain relating the samplefuel rack position and output torque is included in amplifier 70.

For the engine being considered, the sampling rate, from eqn. (2) is:

27TT = 3[(507T/3) +~w]' (3)

where dw, measured in rad/sec, is the change in output speed from the nominalvalue.

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742 C. R. Burrows et al.

Fig. 5

I

t DClMotl2S logic signals

Computer diagram.

Fig. 6

Co)

..'OPi.c

GPR G¢nll:l"'o' purpow nroiSllfr

rc 'nitiol conditionOP Operou

CLR Cleor

S Set

A RcutA,A,B,B Bit

C.C.O.O

+01

"" r-. A r-. /"'- r-R

V V V V V-0 ,

'" I I t I I{>' t t t t I0

c I I

B I

A I

","n n n n n nTlh nT2 Il

sw, I

SW2

( b).

Details of logic circuit. (a.) Components; (b) timing diagram.

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Simulat·ion of the sumpled-data model for c01npression ignition engines 743

The finite delay using eqn. (1) and taking (J = 300 (Hazell and Flower 1971)is:

From the earlier disoussion:

7' = iT.

Hold time = (~T+T).

(4)

(5)

Eqnations (3), (4) and (5) provido the conditions which must be fulfilled bythe logio oircuit to control the track store units and switohes SWl, SW2(fig. 5) and this is aohieved by generating a triangular waveform of period tTusing amplifiers 00, 01, 05, fig. 6. Figure 6 (b) shows that Tl, T2, SWl, SW2operate in the desired mode. (Note that the track store units only sample thefuel rack position at intervals separated by 2T, and Tl and T2 are out ofphase by period T.)

The period of the triangular waveform is varied by feeding back a signalproportional to the change in output speed. This is fed into amplifier 00,fig. 6 (a). Variation of the period of the triangular wave alters the samplingrate and the finite time delay as required by eqns. (3) and (4).

4. Results

To allow comparison with earlier work (Welbourn ct al. ID59, Hazell andFlower 1971), the computer model was subjeoted to sinusoidal variation ofthe rack position and the resulting changes in speed were noted. The experi­ments were performed using a digital transfer funotion analyser.

The action of the sampling prooess is clearly seen from fig. 7.

Fig. 7

Output of sampler unit.

The triangular waveform (signal R, fig. 6 (a)) was monitored on a C.R.a.and variations in the periodio time were observed.

Figure 8 shows an inverse locus of engine speed ehange to sinusoidal move­ment of the fuel rack.

The computer results obtained by ignoring the time delay show goodagreement with results obtained by Hazell and Flower (1971). Inolusion ofthe time delay leads to significantly improved correlation with measuredresults. (It must be stressed that in the absence of engine charaoteristics theangle of advance used was an assumed typical value.)

Table 2 shows that variation of sampling rate and delay time with enginespeed does not lead to any signifioant change in the results. This is furthersupported by the agreement of the curve in fig. 8 (with no timo delay) withearlier published results (Hazell and Flower 1971).

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744 C. R. Burrows et al.

Fig. 8

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Simulation of the sampled-data model for compression ignition engines 745

The larger input variation resulted in 2, maximum speed variation of 50°1<>.It is significant that even this large change in output speed did not result inany appreciable difference between the fixed and variable sampling rate models.

Welbourn et al. (1959) only reported experimental results up to a frequencyof 0·7 Hz and the discrete model gives good correlation up to this value.However, it is clear from fig. 8 that at higher frequencies the model is notsatisfactory, although it still provides closer agreement with experimentalresults than the model used by Welbourn et al. A more exact model could bedeveloped by accounting for the non-linearities occurring due to the fuel rodfunction and the pump characteristics, etc., and this is the main advantageof a hybrid simulation compared with analytical techniques.

The method adopted for including the finite delay has the limitation thatonly angles of advance corresponding to fractions of the sampling rate can beconsidered. However, this does not represent a severe praeticallimitation to themethod.

5. Conclusions

The effect of variations in sampling rate and time delay due to variationsin engine speed have been examined and it is shown that constant-rate samplingprovides an adequate model; thus sampled-data theory can be applied.

The use of a hybrid computer provides the capability of studying the effectsof non-linearities in the components and the load, and this area is receivingattention by the authors.

It may be possible to obtain a better model of a compression ignition engineby treating it as an example of pulse-width modulation, and the authors areinvestigating this possibility. The use of 2, sampled-data. model, including theeffect of the finite delay between fuel injection and torque output, is seen toyield better correlation between experimental and theoretical results comparedwith earlier published work.

REFERENCES

BOWNS, D. E., 1970, 'The dynamic transfer eharacteristics of reciprocating engines'(Bath University of Technology Report).

HAZELL, P. A., and FLOWER, J. 0., 1971, Int. J. Control 2, 121, 146.RAGAZINNI, J. R., and FRANKLIN, G. F., 1958, Sampled-data Control Systems (New York:

McGraw-Hill), Chapter 3.'VELBOURN, D. B., 1934, Essentials of Control Theoru (London: Edward Arnold), p. 94.WELBOURN, D. B., ROBERTS, D. K., and FULLER, R. A., 1959, Proc, Instn meek. Engrs,

173, 575.

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