real time systems: pa rameter identification forspeed sensor wheels

Real Time Systems: Parameter Identification for

Speed Sensor Wheels

Muhammad Nomani Kabir1, Yasser Alginahi1, Munir Ahmed2 and Abdisalam Issa-Salwe2

1Department of Computer Science

2Department of Information System

College of Computer Science and Engineering

Taibah University, P.O. BOX 344, Madinah Munawarah 41411, KSA

Abstract: Modern diesel fuel injection system (com-

mon rail and pump-nozzle-units) requires the reference

of crank angles, engine speed and load to estimate the

two parameters: crank angle at the start of fuel injection

and amount of fuel to inject for a diesel engine. Measure-

ment of engine speed of a combustion engine is carried

out by sensor wheels mounted on the crankshaft. Crank

angles can be estimated from the measured engine speed.

But geometrical tolerances of a sensor wheel result in sys-

tematic errors in the measurement. To get the corrected

speed, we have to determine these errors due to geomet-

ric tolerances. These geometrical tolerances can be esti-

mated by solving a nonlinear system of equations which

is formulated using a suitable physical model of kinetic

energy balance of the crankshaft and the measurement of

engine speed.

Keywords: Real Time Operation; Numerical Methods;

Sensor Wheel; Geometric Tolerances.

1 INTRODUCTION

Generally, measurement of the speed of combustionengines is carried out by sensor wheels held on thecrankshaft. The geometric tolerances of a sensorwheel can cause a systematic deviations that can re-duce the quality of engine speed sensing. Therefore,to obtain the appropriate speed, we have to de-termine the deviations due to geometric tolerances.Here two parameters are very important in combus-tion engine control systems: the crank angle whichis at the start of fuel injection and the amount of fuelto be in- jected. To assess these parameters, a mod-ern diesel fuel injection system (common rail andpump-nozzle-units) requires engine speeds, crankangles, load etc. Engine speeds and crank anglescan be measured by a sensor wheel. In the paper weare going to look at how in real time systems canbe identification the geometric tolerances of sensor

wheel. To get the appropriates speed, we have tode- termine the errors due to geometric tolerances.These geometrical tolerances can be estimated bysolving a nonlinear system of equations which is for-mulated using a suitable physical model of kineticenergy balance of the crankshaft and the measure-ment of engine speed.

2 USING A SENSOR IN REAL TIME

OPERATION

Speed of combustion engines is estimated by sen-sor wheels mounted on the crankshaft. Geometrictolerances of a sensor wheel produce systematic de-viations from the actual speed of the engine. There-fore, the measured speed differs from the actualspeed due to imperfect sensing of engine speed bythe sensor wheel with some geometric tolerances. Toget the actual speed, we have to determine the devi-ations due to geometric tolerances.

FIGURE 1: A 60-2-2 sensor wheel for diesel engineswith 56 teeth and 2 gaps. This figure is reproducedby the courtesy of IAV GmbH, Germany

c© 2012, IJATCSE All rights reserved 43

Muhammad Nomani Kabir et al., International Journal of Advanced Trends in Computer Science andEngineering, 3(3), May - June, 2012, 43-49

Two parameters are very important in combustionengine control systems: the crank angle at the startof fuel injection and the amount of fuel to inject. Toestimate these parameters, a modern diesel fuel in-jection system (common rail and pump-nozzle-units)requires engine speeds, crank angles, load etc. En-gine speeds and crank angles can be measured by asensor wheel.

FIGURE 1 shows a 60-2-2 sensor wheel for dieselengines with 56 teeth with two gaps. The wheelmounted on the crankshaft rotates with it. Thewheel is fitted with teeth sensible by a stationarymagnet which field changes with the movement ofteeth. The change is sensed by the voltage gener-ated in a coil of wire in the magnetic field.

A controller receives each voltage pulse from thesensor and records its time of arrival T0, T1, · · · .With this information, the period ∆Ti = Ti+1 − Ti

can be determined where the index i = 0, 1, · · · indi-cates the tooth number. The speed φ̇e can be mea-sured by

φ̇e =∆φe

∆Ti

.

The teeth are evenly spaced except for missing teethin the gap. The voltage signal is interpolated overtwo missing teeth in each gap. The sensor wheel inFIGURE 1 has 60 teeth (including 4 missing teeth),and therefore ∆φe = 3600/60 = 60. The distance be-tween two successive teeth should be constant. How-ever, geometric tolerances in distances between twosuccessive teeth of sensor wheels lead to systematicdeviations of the actual engine speed. Since geomet-ric tolerances are related to these deviations by somesimple formula, we will investigate the method in [2]to compute the deviations by different variations ofNewton’s method.

3 PHYSICAL MODEL

Identification of geometric tolerances in sensorwheels is based on the balance of the kinetic energyof the crankshaft [1], [5], [9], [10], [11]. The kineticenergy can be modeled by the energy E0 due to theengine speed and the energy E1 due to the load:

Ekin = E0 + E1.

The kinetic energy is related to the actual enginespeed by the equation

Ekin =1

2Θφ̇2,

where Θ is the moment of inertia and φ̇ is the actualengine speed. Combining above equations togetherwe have

1

2Θφ̇2 = E0 + E1.

Due to the sensor wheel deviation, we obtain thedeviated engine speed φe (which is the measuredspeed). The actual engine speed φ̇ can be given by

φ̇e = φ̇(1 − δ), (1)

where δ is the speed correction due to sensor wheeldeviation. Using this relation to the previous equa-tion we get

1

2Θ

φ̇2e

(1 − δ)2= E0 + E1.

This equation can be set up for each tooth i of thesensor wheel:

1

2Θi

φ̇2e,i

(1 − δi)2= E0,i + E1,i, (2)

where the index i = 0, · · · , n− 1 indicates the toothnumber. Notice that the tooth position n corre-sponds again to the tooth position 0. We supposethat the load is constant during measurement of theengine speed.Computing Θ

The moment of inertia Θi depends on the crank an-gle. It can be calculated by the rotating and oscillat-ing masses of the engine. In practice, the stiff partof the moment of inertia cannot be accurately deter-mined. Therefore, an offset Θoffset is applied to theestimated moment of inertia Θ1st,i. Hence,

Θi = Θoffset + Θ1st,i. (3)

Computing E0

The energy E0 due to the engine speed changes withthe friction, load, etc. The speed correction δ0 effectson E0 which must be taken into account. At i = 0,the energy E0 due to the engine speed is given by

E0,0 =1

2Θ0

φ̇2e,0

(1 − δ0)2. (4)

Similarly, at i = n we have

E0,n =1

2Θ0

φ̇2e,n

(1 − δ0)2. (5)

During a crankshaft revolution, the energy due tothe engine speed has approximately a linear varia-tion. Therefore the energy E0,i between E0,0 andE0,n can be computed by the equation

E0,i = E0,0 +E0,n − E0,0

ni, with i = 0, · · · , n − 1. (6)



0 20 40 60 80 100 12080

85

90

95

100

Tooth counter i

Eng

ine

spee

d (r

ad/s

ec)

Measured speedCorrected speed

FIGURE 2: Measured and corrected engine speeds.

1.6188 1.619 1.6192 1.6194 1.6196 1.6198 1.62

x 105

290

300

310

320

330

340

350

Tooth counter i

Eng

ine

spee

d (r

ad/s

ec)



0 20 40 60 80 100 120400

420

440

460

480

500

Tooth counter i

Eng

ine

spee

d (r

ad/s

ec)



Computing E1

The energy E1,i due to the load is stored during thecompression stroke of the engine. It varies linearlywith the load:

E1,i = ploadG1,i. (7)

The values of G1,i are computed by complicated for-mulas which include the rules of the crankshaft kine-matics and polytrope compression [1], [5], [9]. Thegoal of the identification is to estimate the unknownparameters using the known/measured set of data,and determine the actual engine speed φ̇. The knownand unknown quantities/parameters are listed in the



0 50 100 150 200 250 300 350330

335

340

345

350

Tooth counter i

Eng

ine

spee

d (r

ad/s

ec)



300 350 400 450 500 550 600 650330

335

340

345

350

Tooth counter i

Eng

ine

spee

d (r

ad/s

ec)



600 650 700 750 800 850 900 950330

335

340

345

350

Tooth counter i

Eng

ine

spee

d (r

ad/s

ec)



following table. Known/measured quantities Unknown parameters

Θ1st,i, i = 0, · · · , n − 1 Θoffset

pload, G1,i, E1,i, i = 0, · · · , n − 1 δi, i = 0, · · · , n − 1

φ̇e,i, i = 0, · · · , n



Table 1: Solutions by different variants of Newton’s method

4 MATHEMATICAL FORMULATION

From (2), the nonlinear residual is given by

Fi = E0,i + E1,i −1

2(Θoffset + Θ1st,i)

φ̇2e,i

(1 − δi)2, (8)

with i = 0, · · · , n − 1 which has the unknown pa-rameters δ0, · · · , δn−1 and Θoffset. This system con-sisting of n residual components and n + 1 unknown

parameters requires an additional equation for its so-lution. Now we give the (n+1)th necessary equationwhich implies that the mean sensor wheel deviationvanishes:

Fn =n−1∑

i=0

δi. (9)

Hence we have the following nonlinear system ofequations

F (x) = 0, (10)



with

x = [δ0, δ1, · · · δn−1,Θoffset]T , (11)

and

F = [F0, F1, · · · , Fn]T .

This system of nonlinear equations is solved for xand hence the corrected engine speed can be ob-tained using (1).

5 TEST RESULTS

The system (10) was solved by Matlab in [2]. Wesolved the problem by different variants of Newton’smethod [4], [6], [7], [8] and analyzed the convergenceand computational time. Based on this result, wewant to choose an appropriate variant to solve thesystem in the real time operation. Several variantsof Newton’s method were used to solve the nonlinearsystem (10) for δ and Θoffset. All the data of φ̇e,pload, G1 and Θ1st from the sensor wheel with n =60 teeth were given. The Jacobian matrices werecomputed using center differences [3], [8]. The testwas performed by a GCC compiler of version 3.3.1under a Linux operating system, and executed on aPentium 4 processor with 2.40 GHz. We obtainedthe corrected engine speed from δ and φ̇e using (1).The test results are presented in FIGUREs 2–4.

In these figures, measured and corrected enginespeeds (computed from a converged solution) areplotted against the tooth counter (i = 0, · · · , 119).In FIGUREs 2 and 4, the solution δ computed fromthe current revolution (with tooth counters i = 0 to59) were used to correct the measured speeds φ̇e ofthe next revolution (with tooth counters i = 60 to119). FIGURE 4 shows good results in estimatingthe corrected speed during the revolutions. How-ever, In FIGURE 4 we observe a very good resultin estimating the corrected speed during the currentrevolution, while the result deteriorates a little dur-ing the next revolution. The cause of the deterio-ration is the imperfection of computing E1 and theeffect of the crankshaft torsional vibration. The tor-sional vibration increases with higher engine speed.Therefore, FIGURE 3 shows very bad results for therevolutions in that the solution δ computed for FIG-URE 2 (with tooth counters i = 0 to 59) were usedto correct the measured speeds with tooth countersi = 161880 to 162000.

In the real time operation, computation for thecurrent revolution ck may take more time than theperiod δTk+1 of next revolution ck+1. In such a case,computation continues during the next few revolu-tions ck+1, ck+1, · · · ck+p, and the previous solution

xk−1 is set for these revolutions. When computationfor the current revolution ck ends, computation forthe revolution ck+p+1 starts. This process is set tocontinue in the real time operation. As a demonstra-tion, we conducted a numerical test which are shownby FIGUREs 5–7.

From FIGUREs 5–7, we assume that the enginespeed is 350 radian/sec. Therefore, for 1 revolutionit takes (2π/350) ∗ 1000 ≈ 17 ms (millisecond). As-sume that computation takes 60 ms in a real timeoperation, and hence p = 1 + 60/17 ≈ 4, that is,results of computation can be obtained after 5 rev-olutions. For FIGURE 5, we used the data of the1st revolution was used to obtain δ to correct themeasured speeds of the 6th revolution. FIGURE6 shows the measured and corrected speeds wherethe data for the 6th revolution was used to recal-culate δ to correct the measured speeds of the 11threvolution. Note that previous δ was used to cor-rect the measured speeds of 6th to 10th revolutions.For FIGURE 7, the data for the 11th revolution wasused to estimate δ again to correct the speeds of the16th revolution. Previous δ was used to correct thespeeds of 10th to 15th revolutions. This procedurecontinues in the real time operation.

Results from the different variants of Newton’smethod using the measured speeds shown in FIG-URE 4 are listed in TABLE 1. In this table, thefirst four columns provide the initial point, the vari-ant of Newton’s method, the number of iterationsand the 2-norms of residuals. Columns 5, 6 and 7give us whether convergence [3], [4] to a true solu-tion was achieved and the cause of failure and the re-quired time in millisecond. Since a nonlinear systemof equations may have more than one solution, the”true solution” is meant a reasonable and practicallyuseful solution which was verified by the criteria• A true solution must satisfy its bound con-

straints. The deviations δ should be betweenthe bounds of -0.2 and +0.2, and Θoffset > 0 tobe physically meaningful.

• The corrected engine speed from a true solutionmust fit the measured speed.

In column 6, ”singular” indicates that the Jaco-bian is singular and computation stops. The othercase ”∆ ≤ ∆min” implies that the current trust-region radius ∆ is less or equal to ∆min = 1.0e-6.In this case, further computation will hardly ben-efit and computation terminates. Each componentof initial x holds the number given in column 1. Incolumn 2, we use the number

• 1 for Newton’s method,



• 2 for Newton’s method with line search, and

• 3 for Newton’s method with trust-region.

Newton’s method and Newton’s method with linesearch work better than Newton’s method withtrust-region for this nonlinear system of equations.With negative initial points the methods do not workproperly. The methods mostly succeed if the initialpoints are larger than or equal to zero. We noticethat the methods work well when the initial pointslie between 0 and 0.2.

6 CONCLUSION

It is unlikely that a single variant will always givethe best performance for different varieties of identi-fication problems. Newtons method for the identi-fication problem of sensor wheels demonstrates bet-ter performance than its variants. Key issues arethe selection of initial points for the methods. The-ory has little to say about such matters, but poorchoices lead to poor performance. The model of sen-sor wheels can be improved by considering the effectof crankshaft torsional vibrations.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the people form IAVGmbH, Germany for their collaboration in this work.

REFERENCES

1. Cruz-Peragon, F., Jimenez-Espadafor, F.J., Palomar,J.M., and Dorado, M.P. (2008). Combustion Faults Di-agnosis in Internal Combustion Engines Using Angu-lar Speed Measurements and Artificial Neural Networks.Energy Fuels, 2972–2980.

2. Fehrenbach, H., Hohmann, C., Schmidt, T., Schultal-bers, W., and Rasche, H. (2002). Kompensation desGeberradfehlers im Fahrbetrieb. MTZ journal, 588–591.

3. Heath, M.T. (2002). Scientific Computing: An Introduc-tory Survey. McGraw-Hill.

4. J. E. Dennis, Jr. and Schnabel, R.B. (1996). NumericalMethods for Unconstrained Optimization and NonlinearEquations. Siam.

5. Kawahara, N. (2008). Automotive Applications.Compre- hensive Microsystems, ScienceDirect, 369–390.

6. Kelley, C.T. (2003). Solving Nonlinear Equations withNewtons Method. Siam.

7. Lawson, C.L. and Hanson, R.J. (1999). Solving LeastSquares Problems. Siam.

8. Nocedal, J. and Wright, S.J. (1999). NumericalOptimiza- tion. Springer.

9. Patel, R. (2008). Comparison Between Un-Throttled,Sin- gle and Two-Valve Induction Strategies Utilizing

Both Port and Direct Fuel Injection; Emissions, Heat-Release and Fuel Consumption Analysis. SAE TechnicalPapers.

10. van Romunde, Z. and Aleiferis, P. (2009). Effect of Op-erating Conditions and Fuel Volatility on Developmentand Variability of Sprays from Gasoline Direct-InjectionMulti-Hole Injectors. Atomization and Sprays.

11. Zheng, J., Cao, S., Wang, H., and Huang, W. (2006).Hy- brid genetic algorithms for parameter identifica-tion of a hysteresis model of magnetostrictive actuators.Compre- hensive Microsystems, ScienceDirect, 369–390.

AUTHOR BIOGRAPHIES

Muhammad Nomani Kabir received his PhD in ComputerScience at the University of Braunschweig, Germany. He is anAssistant Professor at the Department of Computer Science,Taibah University in KSA. His research interests include is-sues related to numerical methods, embedded systems andcombustion engines.

Yasser Alginahi earned a Ph.D. in Electrical Engineer-ing from the University of Windsor, Ontario, Canada, and aMaster of Science in Electrical Engineering from Wright StateUniversity, Ohio, U.S.A. He is a licensed Professional Engi-neer, Ontario, Canada, since September 2006 and is a seniormember of IEEE and IACSIT since 2010. He is on the edi-torial board of several international journals. He published abook entitled Document Image Analysis and he has over 35journal and conference publications for his credit. He workedas the Principle Investigator on many funded research projectsat Taibah University and several other projects supported byother organizations. His current research interests are Mod-eling and Simulation, Numerical Computation and DocumentImage Analysis.

Professor Dr Munir Ahmed is a professional memberof the Institution of Engineering and Technology (MIET),United Kingdom (UK). He is completing his DProf - Doc-tor in Professional Studies (Computer Communications En-gineering - Information Security) with Middlesex University,London, UK in June 2012. He holds permanent positions asProfessor of Computer Networks and Security Engineering,Chairman of Advisory Board and Director of Research at Lon-don College of Research, Reading, UK. He works for TaibahUniversity, Saudi Arabia as Professor of Computer Networksand Communications Engineering on contractual basis sinceSeptember 2006. His areas of research are Wireless SensorNetworks, Routing Protocols, Network Architectures and In-formation Security. Professor Ahmed is an author or co-author6 books and has above 250 research activities including papersand articles in international journals and conferences; techni-cal manuals, workshops and presentations in industrial milieu.

Abdisalam Issa-Salwe obtained his PhD in InformationSystem at Thamas Valley University, London, UK. He is anAssistant Professor at the Department of Information Systemof Taibah University, KSA. His research interests are in In-formation Systems Man- agement, Knowledge Management,Information architecture, embedded System and numericalMethods.


real time systems: pa rameter identification forspeed sensor wheels

Documents