engine afr model

A Model-based SI Engine Air Fuel Ratio Controller

Kenneth R. Muske

Department of Chemical Engineering, Villanova University, Villanova, PA 19085

James C. Peyton Jones

Department of Electrical and Computer Engineering, Villanova University, Villanova, PA 19085

Abstract

A novel linear state-space model predictive con-troller for SI engine air fuel ratio control is pre-sented and demonstrated over a range of engineoperation. The linear model-based controller isan analytical controller that does not require on-line optimization. Time-varying delay compen-sation is adapted based on the measured enginespeed. A Kalman filter is used to estimate themodel and unmeasured disturbance states.

1. Introduction

An essential component in the reduction of auto-motive tail-pipe emissions is the use of three-wayautomotive catalysts in exhaust after-treatmentsystems. A three-way automotive catalyst re-duces engine emissions by both oxidizing un-burned hydrocarbons and carbon monoxide andreducing nitrogen oxides contained in the pre-catalyst engine exhaust. In order for the three-way catalyst system to catalyze both oxidationand reduction reactions, the engine must be cy-cled between lean (excess oxygen) and rich (ex-cess fuel) operation. When the engine is oper-ated lean, the excess oxygen in the pre-catalystexhaust gas is stored in the catalyst throughchemisorption with the cerium oxides containedin the catalyst preventing lean (nitrogen oxide)tailpipe emissions. When the engine is operatedrich, oxygen is released from the catalyst pre-venting rich tailpipe emissions by oxidizing thehydrocarbons and carbon monoxide.

A common operating practice for three-waycatalyst systems is to cycle the pre-catalyst airfuel ratio across stoichiometric at a frequency de-termined during engine calibration or obtainedthrough relay feedback from a relay type post-catalyst heated exhaust gas oxygen (HEGO) sen-

sor [1]. A more sophisticated control strategy isto maintain the catalyst oxygen level near thestoichiometric equilibrium state in order to max-imize the time required to reach either rich orlean tailpipe emission breakthrough when sub-ject to disturbances. Linear control strategies ofthis type have been proposed based on H∞ [2],LQR [3], and IMC [4] design methods. Nonlinearmodel-based control formulations following thisstrategy include [5], [6], and [7]. A key require-ment for these control strategies is good servoand regulatory control of the engine air fuel ratio.This loop is typically the secondary controller forthe primary catalyst emission controller within acascade control structure. Poor control by thissecondary loop will result in poor primary loopcontrol resulting in increased tailpipe emissions.

The engine air fuel ratio is controlled by ma-nipulating the fuel injector pulse width. This isthe length of time the fuel injectors are openedto supply fuel to the engine. In this work, theair fuel ratio is measured by a wide-ranging uni-versal exhaust gas oxygen (UEGO) sensor up-stream of the catalytic converter. The difficul-ties in achieving tight air fuel ratio control arethe rather long (in comparison to the dominanttime constant) and time-varying nature (due tochanging engine speed) of the time delay in thepre-catalyst UEGO sensor response. Previouswork in linear air fuel ratio controllers has con-sidered LQG [8], H∞ [9], Smith predictor [10],and adaptive [11] approaches. Nonlinear con-trollers include sliding mode control [12], [13],and neural network approaches [14], [15], [16].In this work, we propose an adaptive, linear,state-space, analytical model predictive controlformulation for air fuel ratio control. Analyticalmodel predictive controllers have a closed-form

Proceedings of the 2006 American Control ConferenceMinneapolis, Minnesota, USA, June 14-16, 2006

ThB14.4

1-4244-0210-7/06/$20.00 ©2006 IEEE 3284

solution that does not require online numericaloptimization [17]. This aspect is critical for anair fuel ratio controller. Time delay compensa-tion is adapted based on the measured enginespeed. The result is a model predictive controlalgorithm with time-varying time delay compen-sation that can easily execute within the timescale necessary for air fuel ratio control using astandard engine control module (ECM).

2. Process Model

Air fuel ratio is defined as the ratio of the airmass flow rate to the fuel mass flow rate

λ =1

Kλ

mO2

mf(1)

where λ = 1 represents a stoichiometric air fuelratio, mO2

is the inlet oxygen mass flow rate tothe engine, mf is the fuel mass flow rate, and Kλ

is the stoichiometric air fuel ratio times the oxy-gen mass fraction of air. Because we are manip-ulating the fuel flow rate, it is more convenientto express the exhaust gas composition in termsof the equivalence ratio φ defined as

φ = 1/λ (2)

When φ > 1, the mixture is rich and the cor-responding pre-catalyst engine exhaust is defi-cient of oxygen. When φ < 1, the mixture islean and the corresponding pre-catalyst engineexhaust gas contains excess oxygen.

Although there is significant complexity in thesystem dynamics due to the effects of fuel pud-dling, manifold wall wetting, and the intake man-ifold hydrodynamics [18], the predominant dy-namic characteristic of the UEGO sensor re-sponse to a change in the fuel injector pulse

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φ

Figure 1: Sensor step response at 1800 rpm.

width closely resembles a FOPDT response. Fig-ure 1 shows the response to step changes in thebase fuel rate (injector pulse width) for a 2L FordI-4 engine at 1800 rpm. The response for 3500rpm is shown in Figure 2. As shown in the fig-ures, the response is closely approximated by aFOPDT process with time-varying dynamics, atime delay on the order of the dominant timeconstant, and a noisy sensor measurement.

The amount of fuel injected into the intakemanifold is roughly proportional to the pulsewidth of the fuel injectors. The engine con-trol module (ECM) computes a base fuel rateintended to maintain stoichiometric combustionbased on an estimate of the engine air flow.Feedforward adjustments to the base fuel rateare also made by the ECM to account for mea-sured disturbances. The model-based controllerin this work maintains the pre-catalyst exhaustgas equivalence ratio at a specified target tra-jectory around stoichiometry by manipulating amultiplier to the base fuel flow rate as computedby the ECM. Unlike many of the previously citedreferences, we do not attempt to determine thebase fuel flow rate. The intent of this controlleris to perform feedback control by correcting thebase fuel flow rate determined by the productionengine management system and not to replicatethese calculations.

To illustrate this approach, we choose to modelthe system as a first order linear process

xk+1 = Axk + Buk−d (3)y = Cx (4)

where the model states include a first order con-tribution from the base fuel multiplier u, with a

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φ

Figure 2: Sensor step response at 3500 rpm.

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time delay of d sample periods, and a constantoutput step disturbance state p

x =

[xφ

p

], A =

[aφ 00 1

], B =

[bφ

0

]

and the output is determined with C = [ 1 1 ].The delay time in sample periods is computed

from the number of engine revolutions requiredfor the exhaust gas to travel from the cylinders tothe UEGO sensor and the current engine speed

d = 60N(R)RΔt

(5)

where R is the current engine speed in rpm, Δt isthe sample period in sec, and N(R) is the numberof engine revolutions which may be a functionof engine speed. Implementation of the time-varying delay is through an array of past inputmoves and the corresponding number of enginerevolutions since that input was applied. Thenumber of engine revolutions is updated at eachsample period under the assumption that the en-gine speed was constant at the current measuredvalue over the previous sample period. When thenumber of engine revolutions matches the num-ber required to reach the sensor, the correspond-ing input move is used in the model prediction.

3. Model Predictive Controller

The model predictive controller in this work is alinear state-space implementation of [19] that in-corporates a time-varying deadtime. A Kalmanfilter is used to estimate the current model andunmeasured disturbance state of the system.The covariance matrices used to compute the fil-ter gain are determined from engine data usingthe technique discussed in [20]. This techniqueallows a more computationally efficient outputstep disturbance model to be employed withoutthe performance penalty associated with outputstep disturbance models if the true disturbanceenters through the state of the system. Themodel state at the current sample period is

xk|k−1 = Axk−1|k−1 + Buk−d (6)xk|k = xk|k−1 + L(zk − Cxk|k−1) (7)

where d is the current delay time, zk is the cur-rent equivalence ratio measurement, and L is thefilter gain determined on a delay-free basis.

The model predictive control algorithm de-termines a constant base fuel multiplier ateach sample period that results in the model-predicted equivalence ratio attaining a referencetarget at a specified future time. The specifiedfuture time is a tuning parameter referred to asthe coincidence time. The reference target is ob-tained by defining a first-order reference trajec-tory between the current equivalence ratio set-point and the model predicted equivalence ratioafter the time delay. The model predicted stateafter the time delay is

xk+d−1|k = Ad−1xk|k +d−1∑j=1

Aj−1Buk−j (8)

where uk−j are the stored past inputs at the pre-vious sample periods. The model predicted stateat the coincidence time c sample periods in thefuture is computed assuming that the input atthe current sample period uk remains constant.

xk+c|k = Ac−d+1xk+d−1|k +

⎛⎝c−d∑j=0

Aj−1B

⎞⎠uk

yk+c|k = Cxk+c|k(9)

The reference trajectory is a first-order dy-namic response between the predicted equiva-lence ratio after the time delay and the currentsetpoint. The reference target c sample periodsin the future, yref

k+c, is determined as follows

yk+d−1|k = Cxk+d−1|k, α = e−3cΔtTc (10)

yrefk+c = (1 − α)ysp

k − αyk+d−1|k (11)

where yspk is the desired equivalence ratio set-

point, cΔt is the coincidence time c sample pe-riods in the future, and Tc is the closed-loop re-sponse time. The closed-loop response time isthe tuning parameter that determines the speedof response of the controller. The single futurecontrol move is determined by setting yk+c|k =yref

k+c and solving for the constant base fuel ratemultiplier uk.

uk =(1 − α) ysp

k +(αC − CAc−d+1

)xk+d−1|k

C

⎛⎝c−d∑j=0

AjB

⎞⎠(12)

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Figure 3: 1000 RPM and 30% throttle.

4. Controller Example

We demonstrate the controller performance us-ing a nonlinear mean value engine model similarto that presented in [18]. A series of first orderdynamic models, in which K is the gain and τ

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is the time constant, were identified using thismean value model at various engine operatingconditions as shown in Table 1. The time delayranged from 0.3 sec at 1000 rpm to 0.08 sec at4000 rpm. Based on this analysis, the first ordermodel with K = 1.1 and τ = 0.28 is employed.

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Figures 3 through 8 present the closed-loopcontrolled and manipulated variable servo re-sponse for a series of step changes to the setpoint.The controller sample period is Δt = 0.1 sec andthe controller tuning parameters are c = 6 andTc = 1.8 sec. As shown in these figures, the con-troller provides good closed-loop control over therange of engine operating conditions. Althougha first order model is used in the controller, themean value model dynamics are second order dueto the combined contribution of first order wallwetting/fuel puddling and first order air flow dy-namics. The engine data in Figures 1 and 2 sug-gests that one of these dynamic lags is dominant.

Throttle 1000 2500 4000

Position rpm rpm rpm

5%K =1.17τ =0.31

K =1.20τ =0.28

–

30%K =1.08τ =0.31

K =1.10τ =0.28

K =1.12τ =0.26

70%K =1.08τ =0.31

K =1.09τ =0.29

K =1.10τ =0.27

Table 1: First order dynamic response models.

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In addition to the mismatch between the ac-tual and controller model order, fractional delaytimes are an additional source of model error inthis application. Because the discrete control al-gorithm implicitly assumes that the time delayis an integer multiple of the sample period, thefractional part of the delay time must either beneglected or treated as a full sample period. Thiseffect can be reduced with a smaller sample pe-riod, however, there is a practical minimum limitto a sample period that can be implemented.

5. Conclusions

We have presented an analytical model predic-tive controller for SI engine air fuel ratio control.The time-varying delays inherent in this systemare accounted for by adapting the time delay inthe model based on the engine speed. The re-sults of this study suggest that good performanceis possible over a wide range of engine operationusing a single linear model. Future work includeson-line implementation of this algorithm alongwith the possibility of adapting the model dy-namics in addition to the time delay.

Acknowledgments

Support for this work from the National ScienceFoundation under grant CTS-0215920, Ford Mo-

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tor Company, Johnson Matthey, and ExxonMo-bil is gratefully acknowledged.

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