SAE TECHNICAL PAPER SERIES
2003-01-1054
An Algorithm for the Calibration of Wall-Wetting Model Parameters Alessandro di Gaeta and Stefania Santini Dipartimento di Informatica e Sistemistica, Università degli Studi Napoli Federico II, Napoli, ITALY
Luigi Glielmo Dipartimento di Ingegneria, Universitàdel Sannio in Benevento, ITALY
Ferdinando De Cristofaro, Carlo Di Giuseppe and Antonello Caraceni Elasis SCpA Control Systems Department, Napoli, ITALY
Reprinted From: Electronic Engine Controls 2003 (SP-1749)
2003 SAE World Congress Detroit, Michigan March 3-6, 2003 400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A. Tel: (724) 776-4841 Fax: (724) 776-5760 Web: www.sae.org
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of SAE. For permission and licensing requests contact: SAE Permissions 400 Commonwealth Drive Warrendale, PA 15096-0001-USA Email:
[email protected] Fax: 724-772-4028 Tel: 724-772-4891
For multiple print copies contact: SAE Customer Service Tel: 877-606-7323 (inside USA and Canada) Tel: 724-776-4970 (outside USA) Fax: 724-776-1615 Email:
[email protected] ISSN 0148-7191 Copyright © 2003 SAE International Positions and opinions advanced in this paper are those of the author(s) and not necessarily those of SAE. The author is solely responsible for the content of the paper. A process is available by which discussions will be printed with the paper if it is published in SAE Transactions. Persons wishing to submit papers to be considered for presentation or publication by SAE should send the manuscript or a 300 word abstract of a proposed manuscript to: Secretary, Engineering Meetings Board, SAE. Printed in USA
2003-01-1054
An Algorithm for the Calibration of Wall-Wetting Model Parameters Alessandro di Gaeta and Stefania Santini Dipartimento di Informatica e Sistemistica, Universita` degli Studi Napoli Federico II, Napoli, ITALY
Luigi Glielmo ` Dipartimento di Ingegneria, Universitadel Sannio in Benevento, ITALY
Ferdinando De Cristofaro, Carlo Di Giuseppe and Antonello Caraceni Elasis SCpA Control Systems Department, Napoli, ITALY Copyright c 2003 SAE International
ABSTRACT Spark-ignited engines equipped by a three-way catalyst require a precise control of the air fuel ratio fed to the combustion chamber. A stoichiometric mixture is necessary for the proper working of the catalyst in order to meet the legislation requirement. A critical part of the air fuel ratio control is the feed-forward compensation of the fuel dynamics. Conventional strategies are based on a simplified model of the wall-wetting phenomena whose parameters are stored in off-line computed look-up tables. Unfortunately, errors in the parameters calibration over the whole engine map deteriorate the control performances in terms of emissions.
to let the catalyzed reactions proceed simultaneously with a satisfactory efficiency, the mixture fed to the cylinder has to be stoichiometric. In current technology for gasoline cars, the signal of an oxygen sensor (λ-sensor), placed in the exhaust pipe, is used as a feed-back control signal for the fuel injection system in order to ensure stoichiometry of the mixture [3] (see figure 1).
In this paper an automatic procedure for a rapid and efficient identification of the wall-wetting parameters is presented. The whole procedure has been experimentally tested on a vehicle by using a test bench. Using the identified parameters values, a significant reduction in the air fuel ratio excursion has been achieved during rapid throttle transients with respect to the same vehicle equipped by a commercial ECU with resident engine maps computed by traditional calibration activity. Moreover, the algorithm can be also on-line used to improve air-fuel ratio control performances. INTRODUCTION Automotive regulations require the tight control of pollutant emissions. In particular, carbon monoxide (CO), oxides of nitrogen (NOx ) and hydrocarbons (HC) emissions have to be reduced. To this aim, in commercial vehicle a three-way catalytic converter (TWC) is used to post-treat engine exhaust and fulfill law requirements. The TWC chemically enables the removal of CO, NOx and HC, but
Figure 1: After-treatment system. Working on the fuel injection, the control goal is to maintain the air fuel ratio (A/F) constant in spite of rapid air transients depending on driver commands. Unfortunately, only some of the sprayed fuel immediately feeds the cylinder, while the rest contributes to the formation of a fuel puddle in the intake runner. This physical phenomenon is commonly known as wall wetting [9]. The limitate bandwidth [2], due both to plant and sensors dynamics, and
open loop variable delays, makes the feed-back action ineffective to compensate this phenomenon during transients. Thus, an appropriate model based open-loop scheme has to be designed (see figure 2). The effectiveness of the control strategy is strongly related to model accuracy.
Figure 3: Schematics of the fuel dynamics. puddle evaporates through time, thus participating to the mixture formation process and entering the cylinder. All the process mainly depends on the mass of the puddle, manifold pressure, wall and valve temperatures and backflow phenomena [4]. Figure 2: A/F control scheme. Fuel path dynamics depend on engine operating conditions (engine speed, air manifold pressure, coolant temperature, etc.). Current automotive control strategies employ simplified dynamic models of the engine behavior based on off-line tuned look-up tables (engine maps). The tuning of the feed-forward compensation is crucial to limit A/F shooting: indeed if they are not well captured, the unknown fuel dynamics can cause undesirable excursion, during throttle opening and closing, and so they are directly responsible of pollution. Usually the parameter values are determined by an experimental optimization process based on a trial-and-error approach. This approach is time consuming, expensive for industries and not efficient. In order both to improve calibration effectiveness and to save time, in this paper a procedure for the wall-wetting parameters identification is presented. A least squares method (LSM) is used to identify the unknown wall-wetting parameters along those transients not well compensated by A/F control resident in a commercial ECU. The control strategies was not modified but only engine maps have been updated with parameters identified by the proposed procedure. Experiments show that, using the engine map obtained by identification, a sensible reduction in the A/F excursion has been achieved with respect to the engine map stored in the ECU. FUEL FILM DYNAMICS The mixture formation process is well known as the source of the A/F problems and therefore will be only briefly introduced here. In figure 3, the main characteristics of the fuel dynamics are summarized. The fuel is sprayed in the intake runner. Some of the fuel immediately feeds the cylinder, while the rest contributes to the formation of a fuel film on the intake manifold wall and/or on the valve body. Liquid from the
Usually, the engine behavior is described by a Mean Value Engine Model (MVEM) for control applications (see, for example [6] and references therein). The goal of this modeling technique is to represent the dynamics of the overall engine behavior on a time scale of several engine events with an overall accuracy of 1 to 3 percent. In other words, the model has to be phenomenological, consistent, compact, with just a minimum number of fitting parameters, so as to be easily adaptable to different engines. Following this approach, the simplest MVEM wall wetting model can be written according to Aquino [1] (see also figure 3): 1 ˙ fi m ˙ f f = − mf f + X m τ 1 ˙ fi m ˙ f c = mf f + (1 − X)m τ
(1a) (1b)
where mf f is the fuel mass of the ’puddle’, m ˙ f i is the injected fuel mass flow rate, m ˙ f c is the mass flow rate of the fuel entering the cylinder; X is a parameter representative of the fraction of injected fuel feeding the puddle; τ parameter describes the evaporation from the puddle. Notice that, at low temperatures, some of the liquid fuel from the puddle directly enters in the combustion chamber and τ can be chosen as representative of this phenomenon too. The normalized fuel air ratio φ entering the cylinder is φc = S
m ˙ fc , m ˙ ac
(2)
where S is the stoichiometric A/F, and m ˙ ac is the incylinders air mass flow rate. X and τ parameters depend on many factors [1, 5, 9] such as manifold pressure, manifold and valve temperature, engine speed, puddle dimensions, fuel composition and volatility. Only some of them can be directly measured such as manifold pressure, engine speed and coolant temperature. The calibration activity has to ensure robustness against all these various dependencies, but this causes a less-than-optimal behavior in emission reduction during transients (see figure 4).
mands, cannot be directly measured, but it can only be estimated. In current technology vehicle it is determined by using the speed-density equation based on manifold pressure and engine speed measurements [7, 8]. Often, in practical cases, an off-set in the in-cylinder air mass flow rate estimate is observed. These offsets can be taken into account by considering a corrective factor, represented by a multiplicative constant, which ties the actual in-cylinder air mass flow rate to its estimate as ˆ˙ ac . m ˙ ac = K m
(7)
The K-factor will be estimated by the procedure presented in this paper; it could be used to correct the speeddensity map resident in the ECU. DELAY IN THE FUEL PATH
Figure 4: Normalized air fuel ratio excursion due to improper compensation of fuel dynamics. Graphics,obtained by simulations, refer to a parameter mismatch of 10%, 20%, 30% (simulation results). In this paper we propose a calibration algorithm that identifies the X and τ parameters. The procedure is based on the angle version of model (1) dmf f (θ) dθ
= −
m ˙ f c (θ)
=
1 X m ˙ f i (θ) (3a) mf f (θ) + 6N (θ)τ 6N (θ)
1 mf f (θ) + (1 − X)m ˙ f i (θ) τ
(3b)
where θ is crankshaft angle [deg], mf f (θ) = mf f (t)|t=t(θ) , R θ t(θ) = 0 6Ndµ(µ) and N is the engine speed [rpm]. The discrete angle model, by using the Euler method on equation (3), is 1 m ˙ f c (i + 1) = m ˙ f c (i) + [m ˙ f i (i + 1) − m ˙ f i (i)] + H W m ˙ f i (i) − m ˙ f c (i) + . (4) H N (i) where the sample angle θc is 180◦ and the parameters H, W are related to X and τ parameters as H=
1 , 1−X
W =
θc H. 6τ
(5)
This model relates the in-cylinder fuel mass flow rate m ˙ fc to the injected fuel mass flow rate m ˙ f i . In the feed-forward action of A/F control the desired normalized fuel air ratio is related to the in-cylinder fuel mass flow rate as (see figure 2) m ˙ ac m ˙ fc = φcDES , (6) S where φcDES is the desired fuel air ratio, set-point for the A/F control. Notice that, on actual commercial vehicles, the quantity of air entering the cylinder, depending on driver’s com-
Fuel air ratio cannot be measured at the cylinders inlet, but only at the exhaust pipe upstream TWC. Thus, the fuel air ratio evaluated upstream the oxygen sensor, φs , is a delayed version of the in cylinder one φs (t) = φc (t − tD ).
(8)
where tD includes injection, combustion and transport delays from the exhaust valve to the oxygen sensor (see [2], and references therein, for more details): tD = tinj + tburn + ttrans ,
(9)
where the injection delay tinj is sum of computation duration, injection duration and injection timing; the combustion delay, tburn , depends on open timing of the exhaust valves; and the transport delay, ttrans , depends on engine velocity, exhaust air mass flow rate, temperature and pressure, exhaust manifold geometry. Relation (9) can be rewritten in the discrete angle domain as φs (i) = φc (i − d),
(10)
where d is the integer number nearest to 6N tD /θc . SENSOR DYNAMICS Experiments show that parameter identification can be affected by sensor dynamics. For this reason, we have to introduce explicitly in the plant also a sensor model. In the neighborhood of stoichiometry, the dynamic behavior of the oxygen sensor can be well modelled as a linear first order system with unit gain φm = −
1 1 φm + φs , τs τs
(11)
where φm is the measured fuel air ratio, τs [sec] is the sensor time constant. In the angle domain the model (11) can be easily rewritten as dφm (θ) 1 1 =− φm (θ) + φs (θ). dθ 6N (θ)τs 6N (θ)τs
(12)
Choosing the same sample angle θc and let ρ = θc /(6τs ) [deg/sec], equation (12) can be discretized by Euler
method
with
φm (i + 1) =
1−
ρ N (i)
φm (i) +
ρ φs (i). N (i)
The sensor inverse model is obviously N (i) N (i) φs (i) = 1 − φm (i) + φm (i + 1). ρ ρ
(13)
(14)
PARAMETERS ESTIMATION The calibration algorithm furnishes an estimate of the parameters X and τ ; moreover it provides a parameter K for a correction of the in-cylinder air mass flow rate estimation. In order to use LSM a regressor model, linear in the unknown parameters, will be now introduced. By substituting equations (2) and (7) in equation (4), a relation among φc , injected fuel, engine speed and air estimate can be obtained as φc (i) + w(i)φc (i − 1) = γ1 z1 (i)φc (i − 1) + (15) +γ2 z2 (i) + γ3 z3 (i)
ψ1 (i) = z1 (i − d − 1) × (21) N (i − 2) N (i − 2) φm (i − 1) + 1 − φm (i − 2) × ρ ρ ψ2 (i) = z2 (i − d − 1) ψ3 (i) = z3 (i − d − 1). Taking into account model uncertainties and measurement errors through an error term, e(i), the final regressor model can be written as y(i) = ΨT (i)γ + e(i), where ΨT (i) = [ψ1 (i) ψ2 (i) ψ3 ], γ = [γ1
(22) γ2
γ3 ]T .
By equation (22), under the hypothesis of white noise, the optimal model predictor is given by [10] yˆ(i|γ) = ΨT (i)γ,
(23)
while the prediction error is e(i|γ) = y(i) − yˆ(i|γ).
(24)
where ˆ˙ ac (i − 1) m , ˆ˙ ac (i) m ˆ˙ ac (i − 1) m z1 (i) = − , ˆ m ˙ ac (i)N (i − 1) w(i) = −
m ˙ f i (i) − m ˙ f i (i − 1) S, ˆ˙ ac (i) m m ˙ f i (i − 1) z3 (i) = S, ˆ˙ ac (i)N (i − 1) m
z2 (i) =
(16a)
The optimal γ vector minimize n
V (γ) =
(16b) (16c)
W , H
γ2 =
1 , KH
γ3 =
(16d)
YT
W . (17) KH
Ω(n)
By evaluating equation (15) in (i − d) and taking into account the delay, we obtain
y(i) = γ1 ψ1 (i) + γ2 ψ2 (i) + γ3 ψ3 (i)
(19)
where N (i − 1) φm (i) + (20) y(i) = ρ N (i − 2)w(i − d − 1) − N (i − 1) + 1+ φm (i − 1) + ρ N (i − 2) + 1− w(i − d − 1)φm (i − 2) ρ
[y(1) · · · y(n)], T Ψ (1) · , · = · ΨT (n)
=
The solution of problem (25) is a least-squares estimation of the γ parameters given by
φs (i) + w(i − d)φs (i − 1) = γ1 z1 (i − d)φs (i − 1) + (18) +γ2 z2 (i − d) + γ3 z3 (i − d). We now insert equation (14) into (18), and evaluate the whole expression in (i − 1), so as to write the following linear regressor model
(25)
where n is the number of measurement instants. Let us now define
and γ1 =
1X 2 ei (γ). n i=1
γˆ = (ΩT Ω)−1 ΩT Y.
(27)
Parameters X, τ and K can be directly found from γ1 , γ2 and γ3 as H=
γ3 ; γ1 γ2
γ3 ; γ2 1 X =1− ; H W =
γ1 ; γ3 Hθc τ= . 6W K=
(28) (29)
Notice that a problem can arise in choosing the delay d. In this work, an average delay in function of the only engine speed has been considered. Better results could be achieved by repeating the presented optimization procedure for different values of the delay, for example varying d from 3 to 10 according to experimental exploration.
RESULTS Experiments were conducted on a vehicle (FIAT Punto 1240cm3 ) test bench with computerized control facility for all ECU signals and engine maps. Here we show some results obtained in different engine working points from 1500 [rpm] to 3500 [rpm] with pressure jumps of about 500 [mbar] related to throttle opening and closing (see table 5). All experiments were performed in thermal steadystate conditions with coolant temperature at 90◦ C. The normalized fuel air ratio (φm ) was measured by a linear λ-sensor (HORIBA).
Figure 6: Performance indexes. As is the gray area.
Figure 5: Table of engine working points selected during experiments. Experiments were performed by using the following procedure: 1. one starts from a certain working point characterized by an intake manifold pressure and engine speed (Pm0 , N0 , table 5) 2. a non smooth closing/opening of the throttle is performed so as to obtain a final engine working point (Pmf , N0 , table 5)
Figure 7: Over shooting ξo . Gray: commercial ECU; stripes: identification strategy. • tr recovery time, defined as the time necessary to definitively re-enter the ±3% threshold;
3. all necessary measures are acquired 4. the optimal estimation of the wall wetting parameters is obtained 5. parameters are stored in look-up tables 6. point 1, 2, 3 are repeated 7. strategy performances are evaluated In order to define acceptable excursions of φm , a threshold of about ±3% has been considered. The performance indexes used to evaluate the identification efficiency are (see also figure 6): • ξo over-shooting percentage, defined as the maximum excursion of φm with respect to 1 value; • ξu under-shooting percentage, defined as the minimum excursion of φm with respect to 1 value; • ξmax maximum shooting percentage, defined as the maximum between ξo and ξu ;
• As area subtending the elongations during recovery. The proposed calibration algorithm induces a significant reduction in the φm excursion during rapid throttle transients with respect to the same vehicle equipped by a commercial ECU with resident engine maps computed by ”traditional” calibration activity. In particular, for all the selected working points the algorithm induces a sensible percentage improvement for the maximum obtainable shooting (ξmax ). As it can seen from figure 9, reductions range from 10% to 75%. Shooting ξo and ξu are reduce respectively from 30% to 80% and from 4% to 75%, as it can be seen from figures 7, 8. Only in one working point, closing the throttle at constant engine speed of 1500 [rpm] the shooting is getting worse, but in this conditions the subtended area As is reduced of about 90% (see figure 11) and the recovery time improves of about 80% (see figure 10). Good results have been also achieved with the recovery time. They are percentage from 80% up to 100% till 3200 [rpm]. Increase between 10% and 15% has been detected during the throttle valve closing.
Figure 8: Under shooting ξu . Gray: commercial ECU; stripes: identification strategy.
Figure 10: Recovery time tr . Gray: commercial ECU; stripes: identification strategy.
Figure 9: Maximum shooting ξmax . Gray: commercial ECU; stripes: identification strategy. Figure 11: Area subtended As . Gray: commercial ECU; stripes: identification strategy. CONCLUSIONS In this paper an algorithm for the automatic calibration of the wall-wetting model parameters has been presented. By this way it will be possible both to achieve good results independently from the calibrator experience and to save time.
Moreover, the approach can be also used on-line to adapt the air-fuel ratio feed-forward control parameters during tip-in tip-out.
The algorithm has been experimentally tested. Parameters values obtained by identification procedure allowed a significant reduction in the A/F excursion during transients with respect to the same vehicle equipped by a commercial ECU with resident engine maps computed by ”traditional” calibration activity.
[1] Aquino, C. F., ‘Transient A/F Control Characteristics of the 5 Liter Fuel Injection Engine,’ SAE paper 810494, 1981.
The algorithm is also robust with respect to uncertainties that could be present in the in-cylinder air mass flow rate prediction. Indeed it furnishes a parameter for the correction of the air estimation.
REFERENCES
[2] Chevalier, A., C. W. Vilgid and E. Hendricks, ‘Predicting the Port Air Mass Flow of SI Engines in Air/Fuel Ratio Control Applications,’ SAE paper 2000-010260, 2000. [3] Kiencke, U., and L. Nielsen, Automotive Control Systems, Springer, 2000.
[4] Haluska, P., and L. Guzzella, ‘Control-oriented modeling of mixture formation phenomena in multi-portinjection SI gasoline engines’, SAE paper 980628, 1998. [5] Hendricks E., T. Vestrholm, P. Kaidantzis, P. Rasmussen, M. Jensen, ‘Nonlinear Transient Fuel Film Compensation (NTFC),’ SAE Paper 930767, 1993. [6] Hendricks, E., ‘Engine Modelling for Control Applications: A Critical Survey,’ Control and Diagnostic in Automotive Applications, 1996, pp. 357–368. [7] Hendricks, E., A. Chevalier, M. Jensen, and S. C. Sorenson ‘Modelling of the Intake Manifold Filling Dynamics’ ; SAE Technical Paper 960037, 1996. [8] Chevalier A., C. W. Vigild, and E. Hendricks ‘Predicting the Port Air Mass Flow of SI Engines in Air/Fuel Ratio Control Applications’, SAE Technical Paper 2000-01-0260, 2000. [9] Heywood J.B., Internal Combustion Engine Fundamentals, McGraw-Hill International Editions, 1988. [10] Ljung, L., System Identification: Theory for the User, Prentice Hall Inc., Englewood Cliffs, New Jersey, 1987.
