Periodicity of the Idle Speed of a diesel engine

Copyright @ IFAC Periodic Control Systems. Cemobbio-Como. Italy. 200 I

PERIODICITY OF THE IDLE SPEED OF A DIESEL ENGINE N. Kositza, Ch. Fleck, A. Schlo6er and H. Rake

Institute of A utomatic Control Aachen University of Technology, 52056 Aachen, Germany Phone: ++49-241-807481 , Fax: ++49-241-8888296 E-Mail: N.Kositza@irt .rwth-aachen .de

Abstract: This paper presents studies of speed fluctuations of a diesel engine at idle speed. Especially fluctuations because of cylinder imbalance were taken into consideration. Therefore a model of the engine was developed. Simulated and measured speed were analysed by means of the fourier and wavelet transform. The aim was to point out possibilities for the design of a periodic controller. Copyright o20011FAC

Keywords: Automotive, Diesel Engines, Idle Speed Control, Modelling, Signal Analysis, Spectrum, Time-Frequency Representation

1. INTRODUCTION

friction etc., the engine stops. This phenomenon is called lean misfire limit (LML) . Furthermore differences between single fuel injection pumps lead to periodic disturbances. Especially these are regarded in this study. They can be reduced by cylinder balancing.

The importance of high-speed direct-injection diesel engines has increased in the last few years. This is due to the low fuel consumption concomitant with excellent dynamic performance in vehicle movement. But further reduction of fuel consumption is still desired and needed. Vehicles driving in the city consume 30 % of their fuel while idling (Thornhill et al. 2000) . Reducing the idle speed by 100 rpm for a vehicle with a fuel consumption of 10 1/100 km can save 4,25 % of fuel (Hrovat and Sun 1997) .

As customers want engines that operate reliably with low vibrations and are able to overcome load steps (e.g. by switching on the air conditioning system) the idle speed is set to a fairly high value. A reduction of these speed fluctuations would permit a reduction of idle speed and to fulfill both requests of the customer, namely low vibrations and low consumption. But before examining the idle speed control itself, it's important to understand the causes and to detect the periodic vibrations.

There are several reasons which limit the reduction of idle speed: instability, friction and differences between single fuel injection pumps. This must be explained with regard to the injection system. The one of the diesel engine considered here consists of a unit pump system with a control rack. The injection map shows at lower speed a shape which leads to instability of the engine and can be noticed in speed fluctuations (Kamata et al. 1986). This phenomenon is of periodic behaviour and is called hunting. On the other hand if at low speed the energy supplied by the fuel injected is not sufficient to overcome the losses due to

In this study different methods based on the speed measurement to permit the possibility of recognising defective pumps or analysing their influence on the speed are compared. Many methods exist like pattern recognition or correlation analysis , etc. The focus of this study has been the Fourier and the Wavelet Transform. Therefore the speed fluctuations of a diesel engine in the lower speed range have been examined based on a mathematical model of the engine and additional data ob-

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In order to investigate the speed oscillations the crankshaft angle was chosen as independent variable of the engine model (Happe 2000) . The resulting structure of the model for one cylinder is shown in figure 1.

tained from a test bench. Especially the influence of cylinder imbalance was taken into consideration. Classical methods like Fourier transform are compared with newer ones like Wavelet transform. The use of an engine model is able to illustrate the operational behaviour of the engine and to give further information on the measured speed fluctuations. One aim was to point out possibilities for the design of a periodic controller which entails a further reduction of the average idling speed and takes comfort and consumption into consideration.

Assuming an ideal combustion process the working process of the diesel engine is described by the thermodynamic characteristics of the basic engine cycles. The differential equation of the piston and crankshaft motion leads to the resulting rotational speed. Figure 2 shows for comparison the measured and the simulated speed of one engine cycle of a 6 cylinder engine. The simulated signal is in sufficient accordance with the measured one.

2. DIESEL ENGINE MODEL In the literature many models of diesel engines can be found (Schmidt 1995, Kao and Moskwa 1995, Hild et al. 1999, Isermann et al. 1998, Zweiri et al. 1999). They differ in their independent variable (time or crankshaft angle), their accuracy, the parts of the engine regarded, the injection system etc. As already mentioned the injection system of the diesel engine considered here consists of a unit pump system with a control rack. Four or six single pumps driven by a camshaft produce the pressure in the fuel pipes, necessary for injection. A rack controls through its position the quantity of injected fuel. The resultant characteristic of the injection system has large influence on the fluctuations of the idle speed, especially on hunting.

3. FREQUENCY ANALYSIS OF SIMULATED SIGNALS To simulate a defective pump a reduced amount of fuel was given to one cylinder of the model. Figure 3 shows the corresponding simulated speed of the engine model in idling operation. Speed [rpm] 808,-~--~--~--~--~--~~--~--~~

!

806 804 802 800 798 796

794

792

1 1 1 1

790

788

-I

__________________________

1 1 1 J

786

__ __ __ __ __ __ __ 14.05 14. 1 14.15 14.2 14.25 14.3 14.35 14.4 14.45 14.5 Time [5]

L-~

14

~

~~

~

~~

~

~

Fig. 3. Simulated idling speed with one cylinder receiving less fuel than others (see arrows)

Fig. 1. Engine model

3500 3000 2200 CD

2000

"0

1800

a. E <

~

1600

1400

2500 2000 1500 1000

1200

500 1000

800

o

4

20

40

60

80

100

120

140

160

180

200

6 Frequency [Hz]

8

10

12

TIme[s]

Fig. 4. Frequency spectrum of the simulated speed Fig. 2. Comparison of measurement and simulation for an engine cycle

Due to the cyclic combustion the engine speed is a periodic signal with a frequency spectrum

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that ideally includes only the speed and the combustion frequency. The defective pump however produces a disturbance frequency in the speed signal. This disturbance frequency (f ~ 6.67 Hz) depends on the number of cylinders (here: 6) and the average speed (here: 800 rpm) and can be clearly seen in figure 4.

to balance the speed. The peak at order 0.15 exists no longer and has also not been remarked at several other speeds and measurements. MOr---~----~----~--~----~--~

4. FREQUENCY ANALYSIS OF MEASURED SIGNALS 200

4.1 Fourier Transform The rotary-speed measurement is supplied at nonconstant time intervals and the data acquisition system normally takes the values irregularly. For this reason the Discrete Fourier Transform (DFT) couldn't be applied directly to the measured data to calculate the frequency spectrum. Without changing the measurement system several approaches for a better interpretation of the measured signal have been realised. The speed data can be interpolated e.g. with cubic splines and then equidistantly sampled (Nahrath et al. 1999) . Another possibility we used was to approximate the Fourier-coeffiecients by Trapezoidal or Simpson's Rules of the integrals. Both ways led to sufficient results.

400 600 800 Number of Increments

1000

1200

10000

8000 CD "0 :::I

iE cC

6000 .

4000 2000 .

0

2

4

8

10

Order

Fig. 5. Measured speed and corresponding order spectrum 550r---~----~----~--~----~--~

The speed measurement system itself offers a third possibility: Though the speed n is measured at non-constant time intervals, the angel intervals are constant. The Fourier Transform

J 00

FTa(w) =

n(a) · exp(-jwa)·da

(1) 4oo~--~----~----~--~----~----J

o

-00

leads to a spectrum where the abscissa includes orders instead of frequencies . For a 4-stroke engine characterized by two alternate cycles the rotational speed always results in a peak at the second order.

400 600 800 Numbers of Increments

1000

1200

1~r-----~----~----~----~-----.

8000 CD

Figure 5 shows the rotational speed, measured with 60 teeth per rotation, of a controlled 4stroke diesel engine and the corresponding order spectrum. All cylinders behave almost the same. Beside the expected peaks at every 2nd order there can be noticed slight peaks at the order 0.15 and 1.5 and minimal ones at 0.5 and 1. Both slight peaks can also be found in the analyzed input signal of the engine, the diesel mass flow .

~

6000 ..

l

4000 .

cC

10

Order

Fig. 6. Measured speed and corresponding order spectrum with one cylinder receiving no fuel In the next sections only this disturbed measurement will be further examined.

Figure 6 shows the controlled rotational speed and the corresponding order spectrum if one cylinder receives no fuel. The expected peak at order 0.5 can be clearly seen, but is not as high as the one at order 1.5. This leads to the assumption that the controller which works with larger sampling intervals creates the peak at order 1.5 by trying

4.2 Short Time Fourier Transform The main disadvantage of the DFT is the impossibility to locate the occurence of specific frequen-

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cies in time. For example , if one pump is defective (less fuel at low speed) , it's always defective at that speed. Therefore it would be useful to know, at what time interval and speed a special frequency occurs. One answer is the Short Time Fourier Transform (STFT)

As already seen in figure 6, the spectrum of the signal has three (lines of) peaks at orders 0.5, 1.5 and 2. The peak line at order 0.5 confirms the existence of a defective pump and consequently the need either for an exchange of the pump or for algorithms especially developed for cylinder balancing.

STFTa(a,w) =

J

The disadvantage of the larger window is the loss of time information. Comparing figure 7 and 8 it can be clearly seen that the local maxima and minima at order 2 along the time axis (Number of Revolutions) got lost. One thing should at last be regarded: The achieved result depends also on the chosen window. Figure 9 shows the order spectrum using a Hanning window

00

n(cp)·w(a - cp)·exp(-jwcp ) · dcp

(2)

-00

which transforms by means of a so-called window w with constant size only one part of the signal

at a time. The size of the window determines the frequency (order) resolution. The result for a rectangular window and a window size of one revolution is shown in figure 7. The order resolution is 1. To recognize differences , the corresponding order resolution must be at least twice the order wanted to be regarded . Therefore the window size should be four revolutions or longer. The result is shown in figure 8.

w(k

+ 1)

= 0.5·

(1 - cos (27r

n:

1) )

k = 0, . . . , n - 1

(3)

Unfortunately most of the frequency accuracy got lost.

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2000

1500

i

t ooo

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,.

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1000

20

...

15

SOO

10

Number of

Revolution. 0.5

Fig. 9. Order spectrum of the measured speed using a Hanning window and a window size of four revolutions

Fig. 7. Order spectrum of the measured speed using a rectangular window and a window size of one revolution

4.3 Continuous Wavelet transform

.... ....

As already stated there 's still a problem combined with the Short Time Fourier Transform. After the uncertainty principle the resolution of frequency decreases with higher time resolution, which corresponds to the selection of a narrower window . That's why the selection of the constant window size is important. The selection should be adapted to the signal on the basis of a-priori-knowledge.

3000

-I j

~

l

2000

,_ 20

,

10

0

1.5

0.5

0

Another interesting possibility is the Continuous Wavelet Transform (CWT) (Misiti et al. 2000 , Wickerhauser 1996)

-

_01

......

J 00

Fig. 8. Order spectrum of the measured speed using a rectangular window and a window size of four revolutions

WTa(a,a)

80

=

1 IW V lal

a-cp n(cp)'I1(--)dcp (4)

-00

a

which transforms the signal with variable-sized windows. The uncertainty principle restricts only the product of time and frequency intervals therefore the wavelet transform provides precise informations about low frequencies by using a long time window and vice versa. Other transforms which use adapted window sizes are the Wigner distribution or the Choi-Williams Distribution (Meltzer 1999).

der 5. In each decomposition step the approximation of the signal (s) is devided in the next approximation (al ... a5) and the detail (d l ... d 5). Included in the figure is the original signal and the detail of step 5.

•

In contrast to the Short Time Fourier Transform the result of the Wavelet Transform doesn't depend on time and frequency, but on time and scale s. Instead of a shifted window Sinus-Function the wavelet transform uses scaled and shifted versions of a small wave - the wavelet I}I. The scales can be converted to a so-called pseudo-frequency fs

- D..

f s-

fe

s

8,

d,

8

2

d2

3

d3

4

d4

5

d5

.. .-.

-

............ ----------". ... ~

8 8 8

1_~

", __ ""_", __ 1"11,,,,,

(5) Fig. 11. Discrete Wavelet Transform of the measured signal using Daubechies wavelets

with the sampling period D. and the cent er frequency fe of the wavelet . Figure 10 shows the wavelet transform of the measured signal using a Morlet wavelet. The scales were transformed to (pseudo-)frequencies. The figure shows with perfect time resolution the known peaks indicating the rotational speed and the defective pump. Furthermore the transitions between the peaks can be clearly seen.

The DWT can therefore be used to denoise or compress a signal. A possibility to use this information for diagnosis would be to calculate the Crest-Factor (Stockmanns 2000) of the signal which is defined as

c=

max(lnil)

Vir E:=ln;

(6)

The Crest-Factor is used to e.g. determine disturbances in gears. 0.1 ..---,---,---,.---...---,----,----.--~-_, 0.09

! 0.08 ", ~ i3 0.07

1000

S do

~" ~

5

0.06 0.05

o.o.t

0.03l..::==:::::==:::::==:::::::::=::::;:====::l o. 02 L 1

Fig. 10. Continuous Wavelet Transform of the measured signal using Morlet wavelets

2

4

5

•

7

10

_ o l d o u .... RevoIutIono

Fig. 12. Crest-Factor for measured raw and decomposed signal

4.4 Discrete Wavelet transform First considerations on this solution shows figure 12, which includes the difference between the Crest factors of the measured signals of the "disturbed" (CD) and the "healthy" (CH) engine. Furthermore the signals were decomposed and the difference between the Crest factors calculated of the approximization of the third step. For calculation the factor was determined in blocks of two revolutions.

The Discrete Wavelet Transform (DWT) corresponds to filtering and is therefore useful for signal analysis and the detection of failures. It can be developed of the CWT by using scales and positions based on powers of two - so-called dyadic scales and positions (Misiti et al. 2000) . The original signal will be decomposed by a low and a high pass filter in approximations and details. The filters belong directly to the wavelets.

Principally the Crest Factor would be a good possibility to detect defective pumps. The decomposition increases the difference in the factor between

Figure 11 shows the 5 step decomposition of the measured signal using Daubechies Wavelets of or-

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Misiti, M., Y. Misiti, G. Oppenheim and J .-M. Poggi (2000). Wavelet Toolbox User's Guide: For Use with MATLAB. 2nd ed. The MathWorks, Inc. Nahrath, Th., B. Bauer and A. Seeliger (1999). Vibration monitoring at unstable speeds. In: COMADEM 99. 12th International Congress on Condition Monitoring and Diagnostic Engineering Management .. University of Sunderland, England. pp. 554-562. Schmidt, C. (1995). Digitale kurbelwinkelsynchrone Modellbildung und Drehschwingungsdampfung eines Dieselmotors mit Last. Fortschritt-Berichte VDI: Verkehrstechnik/ Fahrzeugtechnik. VDI-Verlag. Diisseldorf. Stockmanns, G. (2000). Wavelet-Analyse zur Detektion von Zustandsanderungen. FortschrittBerichte VDI: Biotechnik/ Medizintechnik. VDI-Verlag. Diisseldorf. Thornhill, M., S. Thompson and H. Sindano (2000). A comparison of idle speed control schemes. Control Engineering Practice 8(5), 519-530. Wickerhauser, M. V. (1996) . Adaptive WaveletAnalysis, Theorie und Software. Vieweg. Braunschweig/Wiesbaden. Zweiri, Y.H., J.F. Whidborne and L.D. Seneviratne (1999). Dynamic simulation of a singlecylinder diesel engine including dynamometer modelling and friction. Proceedings of the institution of mechanical engineers - Part D 213(4),391-401.

the healthy and the defective engine. Further work will reinforce this idea.

5. CONCLUSION The Fourier and the Wavelet Transforms are powerful tools to combine diagnostics with an idle speed controller. In this study beside the diesel engine model the possibilities of these signal analysis methods have been presented regarding the problem of cylinder balancing. Especially the Discrete Wavelet Transform, a fast and simple algorithm, leads to interesting results. First steps in developing a diagnosis and/or balancing tool were made. Further work will examine their applicability for diagnosis and develop an algorithm for balancing in combination with a periodic idle speed controller. One aspect will be a possible use in engine production control.

6. REFERENCES Happe, J. (2000). Dynamische Modellierung und Simulation des Ladungswechsels eines NfZDieselmotors zum Entwurf einer Leerlaufdrehzahlregelung. Master's thesis. RWTH Aachen. Hild, 0., A. SchloBer, K. Fieweger, St. Pischinger and H. Rake (1999). Die Regelstrecke eines PKW-Dieselmotors mit Direkteinspritzung im Hinblick auf Ladedruck- und Abgasriickfiihrregelung. Motortechnische ZeitschriJt 60(3), 186-192. Hrovat, D. and Jing Sun (1997). Models and Control Methodologies for IC Engine Idle Speed Control Design. Control Engineering Practice 5(8), 1093-1100. Isermann, R., M. Hafner, J. Schaffnit, M. Schiiler and S. Sinsel (1998). Modellbildung und Simulation des statischen und dynamischen Verhaltens von Dieselmotoren mit Thrbolader. In: GMA-Kongress '98, Mess- und Automatisierungstechnik. pp. 21-37. VDI Berichte 1397. VDI-Verlag. Diisseldorf. Kamata, M., J. Nishihama and H. Sakai (1986). Improvement of idle speed stability of diesel engines by digital control. JSAE Review 7(2), 16-22. Kao, M. and J.J. Moskwa (1995). Thrbocharged Diesel Engine Modeling for Nonlinear Engine Control and State Estimation. Transaction of the AMSE - Journal of Dynamic Systems, Measurement and Control 117(1), 20-30. Meltzer, G. (1999). Stand und Tendenzen der Schwingungsiiberwachung und -diagnostik/ Innovative Diagnosetechnik. In: Schwingungstagung '99, Schwingungsiiberwachung und diagnose von Maschinen und Anlagen. VDI Berichte 1466. Diisseldorf. pp. 1-29.

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