Control of Air-to-Fuel Ratio of Spark IgnitedEngine using Super Twisting Algorithm

Ahmed Yar , Aamer Iqbal BhattiDepartment of Electronic Engineering

Mohammad Ali Jinnah UniversityIslamabad, Pakistan

ahmedyar@gmail.com, aib@jinnah.edu.pk

Abstract—Mean Value Engine Model of Spark Ignited InternalCombustion Engine (fourth order) suitable for the control ofAir-to-Fuel ratio has been developed. Air-to-Fuel ratio plays akey role in engine operation concerning numerous aspects ofthe performance. These include fuel efficiency, smooth operation,components of engine exhaust, torque output and many others.Aforementioned are aspects of performance which are prettilyaffected by degraded Air-to-Fuel ratio both during transientand steady state operation. Deviation of air-to-fuel ratio fromits ideal value merely occurs due to difference in required massflow rate of fuel which would, in such case, not be in properproportion. For the purpose; super-twisting algorithm (HOSM)based controller is developed. Self convergent and inherent robustnature of super twisting algorithm proves to be advantageousin control of Air-to-Fuel ratio. Performance of the controller isanalyzed against variation in different parameters of the fuelsystem and corresponding change in actuation of the controlleris discussed.

I. INTRODUCTION

As soon as electronic control was introduced in enginemanagement systems, new horizons were opened in the fieldof controls and diagnostic systems associated with internalcombustion engines. Forcing functions behind creation andevolution of electronic control/ diagnostic systems were in-creasing demands of system/ fuel efficiencies, strict environ-ment legislations and improved operational reliability. Underthese motivations, internal combustion engine evolved fromcrude form to present form, electronically controlled withnumber of closed and open control loops with capability ofon-board diagnostics.

Significant role of air-to-fuel ratio has attracted a majorresearch to the issue of its control; resultantly it acted asalways open doors for researchers and industry in internalcombustion engines.

Pioneering work on the issue of air-to-fuel ratio startedfrom Aquino [1]. Air-to-fuel ratio characteristics of a 5litrecentrally fuel injected dynamometer engine were studied.However [1] has considered throttle generated transients asa main focus of study in excursions of air-to-fuel ratio.The effects of throttle body and manifold temperature wereinvestigated and a comparison of speed-density and mass-flow metering strategies was made. Arise et al. [2] focusedon experimental identification and validation of Recurrentneural networks (RNN) models for air-fuel ratio estimation andcontrol in spark ignited engines. [2] proposed suited training

procedures and experimental tests to improve precision andgeneralization of RNN in predicting air-fuel ratio transientsfor wide range of operation scenarios. It is shown that RNNcarry sufficient ability to reproduce the target patterns withsatisfactory accuracy. [2] reach upto the implementation of realtime RNN based controller, inverse neural network controlleracting on injection time to limit air-to-fuel ratio excursionfrom stoichiometric ideal value. Huajin et al. [3] has takenthe engine as SISO and MIMO plant for the control of air-to-fuel ratio problem. Techniques of Globally Linearizing Control(GLC) and conventional first order sliding mode (FOSM) isapplied to the engine. It shows that GLC can give trackingfor the engine model without uncertainty and disturbance;however, it fails to keep good tracking performance whenuncertainty is considered. In contrast, controller employingsliding mode [3] is able to achieve the target, which showsthat engine system with proposed sliding mode controller hasgood robustness to uncertainty and disturbance. HongmingXu [4] has considered variations in air-to-fuel ratio withina 16-valve port-injection spark-ignition engine and examinedthese as a consequence of rapid transients in load at constantspeed with fuel injection controlled by the production engine-management system and by a custom-built controller. Thepurpose was to minimize excursions from stoichiometry bythe use of a controller to impose an injection strategy, guidedby results obtained with the production management system.The strategy involved a model that takes account of manifoldfilling and the delays in transport of fuel from the injectors tothe cylinder. The results showed that the excursions in air-fuelratio from stoichiometry were reduced from more than 25%to 6%.

Symbol DescriptionPman Intake Manifold PressureVman Intake Manifold VolumeTman Intake Manifold Temperaturemai Mass Flow Rate of Air Entering the Intake Manifoldmao Mass Flow Rate of Air Leaving the Intake ManifoldCD Coefficient of Discharge (throttle plate)ai Generic Constantτf Fuel Evaporation Constantε Fuel Separation Constant

TABLE ISYMBOLS AND THEIR DESCRIPTION

978-1-4673-4451-7/12/$31.00 ©2012 IEEE

Fig. 1. A) Cam Shaft B) Valve Cover C) Intake Valve D) Intake Port E)Engine Head F) Coolant Jackets G) Engine Block H) Oil Pan I) Oil Sump J)Spark Plug K) Exhaust Valve L) Exhaust Port M) Piston N) Connecting RodO) Big End Bearing P) Crank Shaft

II. PROBLEM FORMULATION

Model development of the system is meant to start froma generalized Mean Value Engine Model (MVEM) whichcaptures the dynamics of process from theory of one di-mensional flow and thermodynamical processes occurring inengine. Final stage is to reach upto a generalized non-linearsystem of the form of Eq (1)

x = f(x) +Guy = h(x)

(1)

Where u is actuation, as per standard nomenclature of controlsystem and h(x) being output bears to be Air-to-Fuel ratio.

Among others, MVEM developed by Hendircks et al. [5]carries a few advantages. One might be its parametric naturewhile other can be its claim of adequate level of accuracy.MVEM is applied to engine control.

Whole engine model is formed by the integration ofthe models of three interconnected subsystems. Subsystemsnamely: air dynamics, fuel delivery and torque productionsubsystem.

Air intake system indicated in Fig 2, owns two dynamicallydeveloping variables. Those are Intake Manifold Pressure(Pman) and Intake Manifold Temperature (Tman). Based onemptying and filing model, Eqs (2) and (3) present stateequations of the engine intake manifold [6].

dPman

dt=

γR

Vman

× (maiTa − maoTman) (2)

dTman

dt=

RTman

PmanVmanCv

[Cp(maiTa−

maoTman)− CvTman(mai − mao)] (3)

Where mai (in the region of choked flow [7]) and mao as afunction of ω, Pman and Tman are as per Eqs (5) and (6)

mai =√

2γ/(γ − 1)RTa × PaCD(1− cos(α)) (4)

πD2

4×

√γ − 1

2γ

γ

γ + 1

γ+1

γ−1

= a1 × (1− cos(α)) (5)

and

mao =Pmanω × Vdηv4πR× Tman

= a2 ×Pmanω

Tman

(6)

Pman thus evolves dynamically depending on α, ω andTman.

Model of the fuel system considers an input mass flowrate of fuel mai (which depends on injector fuel flow rateand duration of pulse width to the injector). Which afterbearing the dynamics of fuel subsystem and associated variabledelays flows to the engine cylinder with mass flow rate mfo.mfo is basically summation of two components of fuel flow.Components are namely 1) fast fuel flow mfo1 2) slow fuelflow mfo2. mfo1 is ε times the total fuel injected that is readilyvaporized, mixed with air flowing from intake manifold toengine cylinder. Whereas mfo2 contributes to formation offuel film.

mfo = mfo1 + mfo2

mfo1 = mfi × ε(7)

With a supposition mfm being the total mass of fuelresiding in intake manifold in the form of film (1 − ε)mfi

is part of injected fuel added to fuel film while mfo2 is massevaporating from film.

mfm = (1− ε)mfi − mfo2 (8)

As described by [5]

mfo2 ∝ mfm

mfo2 =1

τfmfm

mfo2τf = mfm

(9)

Simplification of Eqs (7),(8) and (9) give the final expres-sion for fuel dynamics

mfo = εmfi +1

τfmfi −

1

τfmfo (10)

Other than moment of inertia rotational dynamics of crankshaft are govern by break torque. Which is torque producedby power stroke minus all losses in torque.

ω = Tb/Je (11)

For break torque(Tb), indicated torque (Ti) and other associ-ated torque losses formulations are described by Eq (12).

Tb = Ti − (Tf + Tp + TL)Ti = imep× Vd/4πTp = f(Pman)Tf = f(ω)TL = engine load torque

(12)

Whereas indicated mean effective pressure (imep) is for-mulated on the basis of thermodynamics of Otto cycle i.e.adiabatic compression of the charge followed by constantvolume heating [8].

imep =Pman ×

[Vc+Vd

Vc

] [P3

P2− 1

] [(Vc+Vd

Vc)γ − 1

](γ − 1)

[Vc+Vd

Vc− 1

] × ηotto

(13)For which P3 bears the role of peak pressure after constantvolume heating and P2 is pressure after adiabatic compression.

Eqs (2), (3),(10) and (11) define a 4th order MVEM. Torepresent the system model in general form described by Eq(1), parameters are chosen in a way (Pman, Tman, ω, mfo) =(x1, x2, x3, x4) and u = mfi. Nonlinear system gets the formin Eq (14)

f(x) =

⎡⎢⎢⎣x1(t)x2(t)x3(t)x4(t)

⎤⎥⎥⎦

=

⎡⎢⎢⎢⎣

a3k − a4x1x3

a5

[a6k

x2

x1− a7x2x3 − a8k

x22

x1+ a9x2x3

]1

Je[a10x1(a11(1 + a12f(x1, x2, α)))− 1]− T

−x4

τf

⎤⎥⎥⎥⎦

(14)

where

k = 1− cos(α)

T = Tf + Tp + TL

Where input and output of the SISO system in Eq (14) aredefined by Eqs (15) and (16) respectively

u = εu+1

τf× u (15)

h(x) = λ = a12 ×x1x3

x2x4

(16)

III. CONTROLLER DESIGN

For the purpose to achieve ideal air-to-fuel ratio, controllerneeds to adjust the input mass flow rate of fuel (mfi); soas ideal proportion for combustion between mass of air andfuel is maintained in charge induced in engine cylinder duringeach intake cycle. Thus objective is to keep the error minimum

Fig. 2. Engine System with Required Controller Structure

during transients and steady state engine operation. Eq (16)defines output of the system

h(x) = a12 ×x1x3

x2x4

In order to investigate the relative degree of the system, whichis required to be unity for the application of Super TwistingAlgorithm [9], output equation is differentiated once w.r.t time

h′(x) = a12 ×x2x4 [x1x3 + x3x1] + x1x3 [x2x4 + x4x2]

(x2x4)2(17)

Presence of x4 in 1st time derivative of output equation Eq(17) clearly reflects that when input and output are taken asdefined in section II, spark ignited engine come out to have arelative degree of one. Resultantly Super Twisting Algorithmcan be applied.

Since objective of the controller is that the value of air-to-fuel ratio sticks close to that of the ideal stichiometric valuei.e.

mao

mfo

= 14.7

λ =Current Air-to-Fuel Ratio

Ideal Air-to-Fuel Ratio

=mao/mfo

14.7h(x) = λ (18)

Since λ = h(x) is output of the dynamical system modeldeveloped in Sec II, and it has to stay around unity then errorsignal can be defined in a fashion as in Eq (19)

e = λ− 1 (19)

For the design of hyper plane or sliding surface, if error ischosen as hyper plane then at convergence

s = e = λ− 1

= 0

⇒ e = 0 ⇒ λ = 1 (20)

Fig. 3. Simulation Schematics

Structure of a super twisting algorithm based controller ispresented in Eq (21)

u = −(u1 + u2)

u1 = −λc

∣∣s∣∣ρ sgn(s)u2 = −Wsgn(s)

u = λc

∣∣s∣∣ρ sgn(s) + ∫Wsgn(s)dt

ρ = 0.5 for super twisting [9]

(21)

Where λc and W are adjustable controller gains, also stabilityand finite time convergence do depend on values of theseparameters [9]. Which are to be tuned for the system.

IV. SIMULATION RESULTS

Model of an engine system described in Sec II and controllerdescribed in Sec III is implemented in MALTAB/ Simulinkusing S-Function for an engine 3.8litre. Fig (3) presents thesimulation schematics. Table II summarizes the simulationparameters for controller and engine fuel system.

Simulation runs the system at idle (idle throttle position)for 20sec; at which a step in throttle angle α is applied to actas a source of transient.

Parameter Valueλc 0.00001W 0.001ε 0.53τf 0.22

TABLE IISIMULATION PARAMETERS

Fig (4) shows normalized air-to-fuel ratio (λ) for controllerperformance under normal plant parameters. Super twistingbased controller adjusts mfi such that λ adheres to unity. Fig(5) presents typical twisting of phase portrait around originwhen super twisting algorithm is applied.

Performance of the controller is analyzed under two cases1) performance under controller parameter variation, whichput up a compromise over chattering and speed of responsein transient 2) performance of the controller against variationin parameters of the fuel delivery system. Possible reasonsand corresponding controller action against such variations isshown.

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

λ (W = 0.001 λc = 0.00001)

Time (Sec)

λ

λ = Current AFR/Ideal AFR

τf = 0.22 & ε = 0.53

Fig. 4. Normalized Air-to-Fuel Ratio

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−5

0

5

10

15

20Phase Plane

Error (e)

de/d

t (ed

ot)

Fig. 5. Twisting around Origin

0 20 40 60 80 100−5

0

5

10Speedy Response Compromised Chattering

Time

Err

or

W = 0.001, λc = 0.00001

0 20 40 60 80 100−5

0

5

10Attenuated Chattering Compromised Speed of Response

Time

Err

or

W = 0.0001, λc = 0.000001

Fig. 6. Sliding Surface for Different Controller Gains

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8λ (CASE − I)

Time (Sec)

λ

λ = Current AFR/Ideal AFR

τf = 0.30 & ε = 0.53

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8λ (CASE − II)

Time (Sec)

λ

λ = Current AFR/Ideal AFR

τf = 0.22 & ε = 0.62

Fig. 7. Performance for Variation in Fuel System Parameter

18.5 19 19.5 20 20.5 21

1.5

2

2.5

3

3.5

4

Time (Sec)

PW

(ra

dian

s)

Injector Pulse Width

Injector PW for CASE − IInjector PW for CASE − II

Fig. 8. Injector Pulse Width (PW)

Transient speed of response and level of chattering comeout to be compromised on each other, an increased speed ofresponse give rise to chattering and reduced chattering resultin slow response in transient. Optimality lies mid way, Fig 6presents error surface for two different set of controller gainparameters (λc and W)

Fig (7) shows performance of the controller against variationin the parameters of fuel system, which result in changedfuel dynamics. These changes could be a result of eitherenvironmental change, temperature change of the body ofintake manifold or either due to slight changes in gasolinegrade. Controller action against aforementioned variations forboth cases is shown in Fig (8).

V. CONCLUSION

Model of Spark ignited engine suitable for control of air-to-fuel ratio is presented, coupled with tuned super-twistingalgorithm based controller. Which adjusts mfi for the controlof air-to-fuel ratio. Robustness of the controller against param-eter variation is also discussed. Controller is showing stableresponse to steady state and transients, with excellent responsefor mean values of system parameters and robustness againstparameter variation.

Future work contains formulation of a MIMO model of anengine system suitable for the application of Super TwistingAlgorithm.

REFERENCES

[1] C. Aquino, “Transient a/f control characteristics of the 5 liter central fuelinjection engine,” SAE Paper 810494, February 1981.

[2] M. Sorrentino, I. Arsie, C. Pianese, G. Rizzo, D. Iorio, and Silvana,“Recurrent neural networks for air-fuel ratio estimation and control inspark-ignited engines,” Proceedings of the 17th IFAC World Congress,vol. 17(1), November 2008.

[3] H. Tang, L. Weng, Z. Y. Dong, and R. Yan, “Engine control designusing globally linearizing control and sliding mode,” Transactions of theInstitute of Measurement and Control, vol. 32(2), pp. 225–247, April2010.

[4] H. Xu, “Control of a/f ratio during engine transients,” SAE paper 01-1484,1999.

[5] E. Hendricks and S. C. Sorenson, “Mean value modeling of spark ignitionengines,” SAE Paper 900616, 1990.

[6] L. Guzzella and C. Onder, Introduction to Modeling and Control ofInternal Combustion Engine Systems. ETH Zurich: Springer, 2004.

[7] Y. A. Cengel and M. A. Boles, Thermodynamics: An EngineeringApproach. McGRAW HILL, 2008.

[8] W. W. Pulkrabek, Engineering Fundamentals of the Internal CombustionEngine. New Jersey: Prentice Hall, 2003.

[9] W. Perruquetti and J. P. Barbot, Sliding Mode Control in Engineering.New York: Marcel Dekker, 2002.