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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 5, SEPTEMBER 1999 555

Analysis and Control of Transient Torque Responsein Engines with Internal Exhaust Gas Recirculation

Anna G. Stefanopoulou,Member, IEEE,and Ilya Kolmanovsky,Member, IEEE

Abstract—In this paper we analyze the nonlinear dynamicbehavior of an internal exhaust gas recirculation system basedon a mean-value model of an experimental automotive engineequipped with a camshaft phaser. We develop a control schemethat adjusts camshaft timing to reduce feedgas emissions whilemaintaining transient engine torque response similar to thatof a conventional engine with zero exhaust gas recirculation.The control scheme consists of a feedforward map that specifiesdesired camshaft timing as a function of throttle position andengine speed in steady state, and a first-order lag that governsthe transition of the camshaft timing to the desired value. Thetime constant of the first-order lag is adjusted based on enginespeed and throttle position.

Index Terms—Dynamic response, emissions, internal combus-tion engines, poles and zeros, pollution control, modeling, torquecontrol.

I. INTRODUCTION

T HE exhaust gas recirculation (EGR) was introduced inthe early 1970’s to control the formation of oxides

of nitrogen (NO ) in internal combustion engines and thusreduce one of the main pollutants in automobile exhausts. Thelimits on acceptable NOemissions from passenger vehiclesare stringently regulated by governments around the world.The recirculated inert exhaust gases dilute the inducted air-fuel charge and lower the combustion temperature whichreduces NO feedgas emissions. Conventionally, exhaust gasrecirculation is accomplished by controlling the exhaust gasthat is supplied from the exhaust manifold to the intakemanifold through a vacuum actuated valve. The EGR controlalgorithm is a simple proportional integral (PI) or proportionalintegral derivative (PID) loop that adjusts the valve positionto the scheduled steady-state value. Exhaust gas recirculationalters the breathing process dynamics and consequently thetorque response. Careful steady-state and transient controldesign is necessary to maintain good engine torque response.For this reason, EGR is typically turned off or is considerablydelayed in transient engine operations, engine warm-up, andidling.

The advances in real-time computing and hardware aremaking possible the application of fully controlled valve

Manuscript received September 3, 1997. Recommended by AssociateEditor, G. J. Rogers. This work was supported in part by the National ScienceFoundation under Contracts ECS-97-33293; matching funds to these Grantswere provided by Ford Motor Co.

A. G. Stefanopoulou is with the Mechanical and Environmental EngineeringDepartment, University of California, Santa Barbara CA 93106 USA.

I. Kolmanovsky is with the Ford Motor Company, Scientific ResearchLaboratory, Dearborn, MI 48121-2053 USA.

Publisher Item Identifier S 1063-6536(99)06450-7.

Fig. 1. Schematic picture of a cylinder with variable valve motion. In thisscheme, the valve overlap occurs later during the intake event. This causesthe induction of the last part of the exhaust gases into the cylinder. Theresulting dilution lowers the combustion temperature and suppresses feedgasNOx emissions.

events. Optimized valve events can improve the gas exchangeprocess and enable control of internal EGR. Variable camshafttiming is an innovative and simple mechanical design approachfor controlling EGR. By retarding the camshaft timing asshown in Fig. 1, combustion products which would otherwisebe expelled during the exhaust stroke are retained in thecylinder during the subsequent intake stroke. This is a methodof phasing the camshaft to control residual dilution and achievethe same results as with the conventional external EGR system.

Achieving exhaust gas recirculation through the exhaustmanifold during the intake stroke is a better way of controllingthe residual mass fraction during transients because 1) weeliminate the long transport delay associated to the exhaust-to-intake manifold path and 2) we bypass the slow dynamicsassociated with the intake manifold filling dynamics. Fasttransient control of the internal exhaust gas recirculation(IEGR) is only limited by the camshaft phasor dynamicsand computational delays. At a first glance, this suggests apossibility of better transient control of feedgas emissions thancan be achieved with the conventional external exhaust gasrecirculation (EEGR). Analysis of the IEGR system in throttledengines shows, however, that the IEGR system interacts withthe slow intake manifold filling dynamics and can cause,in fact, unacceptable engine performance [6]. To improve

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556 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 5, SEPTEMBER 1999

Fig. 2. Control structure.

the engine performance and reject the undesirable effects intorque response additional actuators can be employed, such aselectronic throttle in [2] and air-bypass valve in [3]. However,the authority of the air-bypass valve may not be sufficient tocompensate for torque disturbances due to camshaft motionand electronic throttle is at present not available on mostproduction engines for cost reasons.

In this paper we analyze the nonlinear dynamic behaviorof the IEGR system based on the mean-value model of anexperimental engine equipped with a camshaft phaser [5]. Wedevelop a dynamic camshaft timing schedule that regulatesIEGR while maintaining transient engine torque responsesimilar to an engine with zero EGR. The torque response of anengine with zero EGR provides the benchmark for the engineperformance that we wish to achieve because any level of EGRcan cause severe torque hesitation if not well calibrated.

Modularity of the IEGR control function is another veryimportant requirement for the control design. Modular IEGRcontrol function will allow its rapid implementation in existingreal-time engine controllers. For this reason, the IEGR con-troller architecture that we develop can be seamlessly addedor removed from the powertrain controller depending on theplatform needs. The torque response requirement is achievedwith no need for additional actuator regulating the flow intothe intake manifold. Fig. 2 shows the developed IEGR controlarchitecture. We dynamically schedule the camshaft timing(CT) that regulates IEGR based on the measured throttle angle

and engine speed Briefly stated here, the dynamicschedule consists of 1) the steady-state camshaft timing values(CT) for all throttle and engine speeds and 2) the time constant

of the first-order differential equation that defines thetransient behavior of IEGR from one steady-state point to thenext. The time constant is selected as a function of enginespeed and throttle position as required to meet the torqueresponse requirements.

II. I NTERNAL EGR SYSTEM

The mean-value model of an experimental engine withvariable camshaft timing shows that significant NOreductioncan be achieved by allowing the exhaust valve to remain openfor a longer period of time during the intake stroke. Thisis achieved by increasing the fraction of exhaust gases intothe cylinder. The experimental data in Fig. 3 demonstrate thecorrelation between NOformation and percentage of external

Fig. 3. Effects of external exhaust gas recirculation (EGR) on feedgas NOx:

EGR flow. As the heading in Fig. 3 indicates, the data werecollected using conventional camshaft timing ,stoichiometric air-to-fuel ratio , enginespeed at 1500 r/min intake manifold pressure0.5–0.6 bar ), and spark timing 25–35

. Note here that to demonstrate the correlationbetween EGR flow and NOformation, it is important to fixall other engine variables. Similarly, the experimental data inFig. 4 show the effects of camshaft timing (CT) on the feedgasNO generation in an IEGR engine. Note again that theexperimental data correspond to the same engine conditionsas in Fig. 3 and zero external EGR flowRetarding camshaft timing increases the IEGR1 and that resultsin reduction in feedgas NO. From Fig. 4 it is obvious thatto reduce feedgas NOwe have to ensure engine operationwith maximum CT.

At the same time CT lowers the volumetric efficiency ofthe engine as shown in Fig. 5 allowing less mass of fresh airinto the cylinders. The mean mass flow rate of fresh air intothe cylinders is a function of volumetric efficiency ,the intake manifold air density , the engine displacement

, and engine speed

(1)

The volumetric efficiency measures the effectiveness of anengine’s induction process and is defined as the ratio of massof air inducted in the cylinders over one engine cycle to themass of air contained in engine displacement volume at theintake manifold density [1]. Using the continuity equation forair flow into and out of the intake manifold, and

1The simple relationship “increasing camshaft timing (CT) increases IEGR”is used throughout this paper. A mathematical equation describing the relation-ship between CT and IEGR is not available because it is difficult to measureEGR mass inside the cylinder.


STEFANOPOULOU AND KOLMANOVSKY: ANALYSIS AND CONTROL OF TRANSIENT TORQUE 557

Fig. 4. Effects of camshaft timing (CT) on feedgas NOx:

Fig. 5. Volumetric efficiency as a function of camshaft timing for differentmanifold pressures at 1500 r/min.

the ideal gas law assuming constant air temperature 2 wedescribe the intake manifold pressure

(2)

The air flow into the intake manifold can be computed bythe discharge coefficient at the throttle plate and thepressure difference across the throttle plateUsing the regression equations in [5] we can describe thebreathing process of the IEGR system

(3)2In contrast to the conventional EGR systems, the internal exhaust gas

recirculation system described in this paper maintains constant intake manifoldtemperature for different level of EGR recirculation. This happens because,the exhaust gas recirculation is achieved through the exhaust manifold andnot the intake manifold.

Fig. 6. Effect of camshaft phasing on torque: simulation and experimentalresults.

In-cylinder torque generation can be simply computed usingan empirical function of the mass air flow into the cylinder

, the air-to-fuel (A/F) ratio, the spark timing , andthe engine speed

(4)

Our goal is to schedule IEGR without the detrimental con-sequences to engine torque response shown in Fig. 6. Forsimplicity, we will assume precise A/F ratio and spark control.We will also analyze the effects of IEGR on engine torqueresponse assuming constant engine speed and schedule IEGRas a function of engine speed. With these two assumptions,the engine torque response is determined by the mass air flowinto the cylinder. We use the time-based mean-value enginemodel described in detail in [5] and summarized here with (3)and (4) instead of the crank-based engine dynamics becausethe camshaft timing actuator dynamics and the desired torqueresponse that satisfies driveabilty are treated easier in the timedomain. The prediction ability of the model described by (3)and (4) is demonstrated in Fig. 6, where experimental andsimulation data are plotted.

III. STATIC SCHEDULE

Exhaust gas recirculation decreases the engine torque re-sponse. Fig. 7 shows the normalized torque response for zeroIEGR (dashed line) and for the maximum IEGR (dashed-dot line) that the engine can achieve. The normalized torqueis given as a function of throttle angle for constant enginespeed. To minimize feedgas emissions we need to operate atmaximum IEGR. To ensure maximum torque at wide openthrottle (WOT) we need to reduce IEGR back to zero. Thesolid line in Fig. 7 shows such a smooth transition frommaximum IEGR for part throttle to zero IEGR for WOT. Forvery small throttle angles (small air flow into the cylinders)IEGR deteriorates the combustion stability because of the highlevel of dilution. IEGR, therefore, is scheduled at zero for



Fig. 7. Comparison between static normalized torque response for zeroIEGR, maximum IEGR, and a realizable IEGR scheme.

Fig. 8. The static camshaft scheduling scheme at 2000 r/min.

small throttle angles to maintain combustion stability. Usingthe above guidelines we derive the static camshaft timing

as follows: 1) near idle it is scheduled for idle stabil-ity which requires cam phasing equal to zero; 2) at mid-throttleit is scheduled for emissions which favors fully retarded camphasing; and 3) at WOT it is scheduled for maximum torquewhich requires camshaft timing to be advanced back to 0.The developed camshaft scheduling scheme, shown in Fig. 8,ensures reduction of feedgas emissions under the constraint ofsmooth steady-state torque response for engine speed equal to2000 r/min.

This steady-state map can be derived based on differentstatic optimization schemes depending on static engine per-formance tradeoffs that satisfy design requirements. We chosepedal position and engine speed as inputs to this map toallow modular implementation of the control strategy. Throttleposition is a natural independent variable for the camshaftcontrol scheme. Engine speed, although not independent, is aslowly varying parameter, which implies that it can be safelyused as the second scheduling parameter. The problem that

remains unsolved is the transition characteristics from onestatic IEGR point to the next set point as the steady-state maprequires. To minimize feedgas emissions we want to changecamshaft timing to the optimum position asfast as possible or as fast as the actuator bandwidth allows.

IV. A NALYSIS OF TRANSIENT TORQUE RESPONSE

In this section we analyze the effects of the speed ofresponse of the CT on the transient torque response of theengine. We assume that the IEGR dynamics are first-order,characterized by

(5)

where the time-constant determines the speed with whichCT is changed. As in the previous section, we assume atight A/F ratio and spark control loop which implies thatthe torque response is determined by the air flow response.Fig. 9 shows the torque response of the IEGR engine to aninstantaneous throttle change for the two constant engine speedvalues of 750 and 2000 r/min. When the throttle angle changesinstantaneously at to a new value , the changesto a new value, , according to

(6)

The three different torque profiles in Fig. 9 result for the threedifferent values of If the CT changes fast ( is small)the torque response may exhibit an undershoot. This happensbecause of the tendency of the air flow to decrease with theincrease in If the changes slowly ( is large) thetorque may overshoot. This happens because the air flow firstincreases quickly due to the intake pressure dynamics causedby a larger throttle angle; then the air flow decreases due tothe slowly approaching the desired, larger than the initial,value. From the driver’s perspective both, the undershoot andthe overshoot, are undesirable.

The above intuitive arguments showing that the speed ofadjustment affects the engine torque response can be

supported analytically. In Section IV-A we analyze the airflow response to a throttle step and we confirm analyticallythe intuitive observations that we made above. The responseto a throttle step from an equilibrium can be also viewedas a response to the initial condition corresponding to thisequilibrium. However, if frequent aggressive throttle stepsoccur the system may not have enough time to reach asufficiently small neighborhood of the equilibrium before thenext throttle step occurs. In this situation, the analysis ofresponse to general initial conditions may be more appropriatethan the analysis of the step response. We present this initialcondition response analysis in Section IV-B assuming thatengine speed and throttle are fixed. The linear time invariantsystem that we are using in the following sections is derivedby linearizing (3):



Fig. 9. Torque response (at 750 and 2000 r/min) using different camshaftdynamic characteristics.

(7)

where , for and Linearizationof the dynamic camshaft timing scheme results in

(8)

where depends on throttle angle andengine speed. The gain in contrast to all other gains (’s)that are nonegative may take both positive and negative valuesfor different operating points as described in Section III andshown in Fig. 8.

A. Analysis of Response to Throttle Steps

The transfer function between camshaft timing, throttleposition, and mass air flow into the cylinders is given by

(9)

Substitution of the linear camshaft scheduling scheme

(10)

Fig. 10. Block diagram of the linear IEGR system.

to (10) results in the transfer function from throttle angle tocylinder air flow for the closed-loop IEGR system

(11)

The block diagram of the linear IEGR system is shown inFig. 10. For simplicity we rewrite (11) in terms of its dc gain

, poles and zero

(12)

where

Proposition: The necessary and sufficient condition for thestep response of (12) to be monotonic is

(13)

This result follows from the general conditions for the mono-tonic step response, see [4]. It can also be derived directly bywriting out the step response shown in (12) in the time domain.

We use the notation , ,to indicate the dependence of the poles and zeros ofthe IEGR engine transfer function on the operating pointand the dynamic schedule. For small values of, wehave a nonminimum phase zero if Thisfollows because is positive since

, and the staticschedule ensures for all engine conditions. Asincreases, the zero converges to the origin at a rate equalto The pole converges



Fig. 11. Qualitative cylinder air flow behavior of the IEGR system fordifferent pole zero location.

to the origin slower, at a rate equal to one. Hence, for largevalues of the zero is to the right of the pole , andtorque exhibits an overshoot. Thus the admissible range for

that guarantees monotonic step responses is a boundedinterval that can be computed from (12) and (13).

To summarize and relate the above analysis to the physicalsystem consider the following scenarios. If an aggressiveIEGR dynamic schedule has to be implemented to reducefeedgas emissions, the small time constant might causetorque response to undershoot during step throttle changes(dot-dashed line in Fig. 11). If hardware limitations imposebandwidth constraints in the dynamic IEGR response, thenthe large time constant might cause torque response toovershoot (zero to the right of the largest pole, solid linein Fig. 11), or to respond slowly to the torque demand dueto the dominant slow pole (dashed line in Fig. 11). Theseconclusions hold in the region where the static schedule callsfor cam phasing increase in response to throttle angle increase

In the region of high manifold pressure (normal flow,the static schedule requires i.e., we start

decreasing the dilution to achieve the maximum engine torque(see Section III). In this case it can be verified that admissiblevalues for must be either sufficiently large or sufficientlysmall (see Appendix B).

B. Analysis of Initial Condition Response

Let , ,

The system shown in (7) and (8) with can be written as

(14)

where

(15)

We are interested in the range of values for, andfor which the air flow response of the system in (14) due to theinitial condition is monotonic for To utilize simplenecessary and sufficient conditions for the monotonic responseof the system in (14) we derive a system with identical unitstep response. It can be verified that the response of the system

in (14) to an initial condition

coincides with the step response of the system

(16)

with a zero initial condition for Thetransfer function of the system in (16) is

(17)

where

As in the previous section, we define

and obtain

(18)

The response of the system in (16) to an initial condition

is monotonic if and only if the step

response of the system without-term

(19)

with a zero initial condition, , is monotonic forUsing the pole-zero condition obtained in the

previous section, we require

(20)

Note that and depend linearly on and Hence,the set of , for which the response of the system in(14) is monotonic, , is a union of two conicallyshaped regions

or

(21)



where

The dependence of the set on the operating point is madeclear by including and in the list of arguments. The sets

are positively invariant. That is, any trajectoryof the system in (14) originating in remains in

It turns out that there is a value of , , for which all theinitial conditions result in monotonic responses (see derivationin Appendix A)

(22)

Note that the value of is infinite for the operating pointswhere the flow through the throttle is choking and is indepen-dent of For these operating points, as followsfrom (7).

As an example, we compute the sets forvarious values of For

and we calculate

and

For , the set has a general appearanceshown in Fig. 12. We used a specific value, , togenerate it. The cone region marked by “” corresponds toinitial conditions that result in monotonically increasing airflow while the one marked by “” corresponds to initialconditions that result in monotonically decreasing air flow. As

increases and approaches the value of ,the line becomes more and more horizontal andthe set shrinks. We calculate, For ,the first element of is zero and the lineis horizontal. For the set does not changealthough changes, see Fig. 13. As crosses the valueof 0.067 the set flips over due to the second element ofchanging sign, see Fig. 14. For the set does notchange when changes. For the “flip over” value of 0.067,

and all the initial conditions result in monotonic airflow responses, consistent with (22).

V. TARGET TRANSIENT TORQUE RESPONSE

Using appropriate values of the time constant of the first-order lag we seek to minimize the differences between thecylinder air flow transients of an IEGR engine and an enginewith zero EGR (ZEGR engine). The target dynamic responsecan be specified either by the transfer function from throttle

Fig. 12. The setX0(0:04615; 1300) is a union of two cone regions markedby “+” and “�:”

Fig. 13. The setX0(�ct; 15;1300) for 0:067>�ct> 0:054 is a union oftwo cone regions marked by “+” and “�:”

angle to cylinder air flow for an engine with zero EGR or bythe initial condition response of an engine with zero EGR. Weconsider these specifications in more detail in the remainderof this section.

The transfer function of the ZEGR engine with throttle inputcan be found by linearization of (3), after setting IEGR andhence camshaft timing equal to zero

(23)

Note here that (23) can be derived also from (11) by as-signing equal to zero. We can compare the two dynamical



Fig. 14. The setX0(�ct; 15; 1300) for �ct> 0:067 is a union of two coneregions marked by “+” and “�:”

systems after we rewrite the above transfer function withrespect to its dc gain and pole location

where and as given in (13).Table I summarizes the differences between the closed-loop

transfer functions from throttle to cylinder air flow for theIEGR system and the ZEGR systemwhere

and

As expected, the IEGR engine has considerably smallerdc gain than the ZEGR engine. Its dc gain depends on thestatic camshaft timing schedule during subsonic flow

Although the target response is the ZEGR engine,we cannot match the DC gain of the ZEGR engine and thereis no need to do so. The IEGR engine can meet the torquedemand at higher throttle angle. Doing so will be beneficial to1) fuel economy due to operation in higher manifold pressurewhich results in reduction in pumping losses [7] and 2)engine performance due to faster throttle to torque response athigher intake manifold pressure. Therefore, from the driver’sperspective, the air flow response into the cylinders of theIEGR engine should match the air flow response into thecylinders of the ZEGR engine operating at the high manifoldpressure equilibrium. For this reason the comparison is basedon linearization of ZEGR engine model at the equilibriumpressure corresponding to the IEGR engine model [7]. We

TABLE I

TABLE II

thus assume that the values for , , and forare the same when comparing the transfer functions of thetwo engine models.

Similarly, the initial condition response of the linearizedIEGR model should be compared with the initial conditionresponse of the ZEGR engine model, linearized at the equilib-rium pressure of the IEGR engine model. If is the initialpressure deviation, then, with throttle and engine speed fixed,the initial condition response of ZEGR engine forcoincides with the step response of a system with the transferfunction

(24)

and zero initial condition.Table II summarizes the differences between the initial

condition response of the ZEGR and the IEGR enginewhere

and

The differences in the dynamic characteristics of the twosystems are due to the zero in step response, and ininitial condition response) and the additional pole for theIEGR engine. The zero is the result of the interaction of theIEGR dynamics with the manifold filling dynamics. The poleis due to the the first-order lag of the dynamic schedule.

VI. TIME CONSTANT

In this section we find the time constant that minimizesthe combined effects of the zero and the additional polein the IEGR air flow response. The basic idea is to selectfor which the IEGR transient behavior is similar to the ZEGRengine behavior, a first-order lag with pole at



Fig. 15. The selected time constant�ct for different throttle angles andengine speeds.

The obvious choice of that minimizes the difference ofthe two transfer functions

and

and satisfies the monotonicity constraint in Proposition 13is It leads to pole zero cancelation in

, , thus matching the dynamic behavior (butnot the dc gains) of the linearized IEGR and ZEGR enginemodels in response to throttle steps. Furthermore, it resultsin and matches the dynamic behavior of thelinearized IEGR and ZEGR engine models in response toinitial conditions for Note also that forall initial conditions result in monotonic zero input responseof the IEGR engine, see (22).

In practice, it may not be possible to implement the value ofexactly thereby resulting in inexact pole zero cancelation

and the “flip-over” effect described in Section IV-B. However,this should not be a matter of concern in practice because theresponse for the case of inexact implementation is sufficientlyclose to the one obtained for the exact value of

As we pointed out in Section IV-B, the choice isnot feasible for the operating points for which the flow throughthe throttle is chocked because and consequently isinfinite. Chocked flow through the throttle body occurs duringsmall throttle angle for which is nonegative (see Fig. 8). Inthis region we impose the monotonicity constraint, (13), whichresults in the following condition for :

(25)

In Appendix B, we describe an algorithm that producesspecific values for that satisfy all of the above conditions.The values for obtained by this algorithm for variousoperating points can be interpolated into a nonlinear staticmap (see Fig. 15), so that

(26)

Fig. 16. Nonlinear simulation of 1) the zero EGR system (ZEGR); 2) theIEGR system with the optimized time constant (�ct); 3) the IEGR systemfor a small time constant [IEGR(one cycle)]; and 4) the IEGR system for alarge time constant [IEGR(six cycles)].

Note that the specific procedure that we used to generateis fairly involved computationally because it requires the

linearization of the engine breathing dynamics. However, thesecomputations are off-line and the on-line implementation of(26) is straightforward. A lookup table and linear interpolationare used to interpolate between the discrete values ofascomputed for the discrete set of pairs

VII. RESULTS

In this section we test the nonlinear mapusing a nonlinearsimulation engine model equipped with variable camshafttiming and coupled with vehicle dynamics. Throughout thesimulation we use stoichiometric air-to-fuel ratio

, minimum spark advance for best torque ,and fourth gear at the modeled automatic transmission. Thethrottle steps used represent a series of challenging torquedemands and test the engine behavior over a wide operat-ing range. The ZEGR engine (ZEGR, solid line in Fig. 16)provides the benchmark torque response for evaluating thedynamically schedulled IEGR engine. Fig. 16 also shows thesimulation of the IEGR engine using the developed staticcamshaft schedule and three different transientcharacteristics: 1) time constant determined by the non-



Fig. 17. Dynamic behavior of mass air flow of the four different systems [ZEGR; IEGR(opt), IEGR(one cycle), andIEGR(six cycles)] dur-ing throttle steps.

Fig. 18. Nonlinear simulation of the four different systems[ZEGR; IEGR(opt), IEGR(one cycle), and IEGR(six cycles)]during throttle steps after introducing vehicle dynamics.

linear map dashed line); 2) time constantwas chosen to be equal to one engine cycle for all

operating condition , dotted-dashed line); and3) time constant was chosen to be equal to six enginecycles for all operating condition ( , dotted-

dotted-dashed line). Testing the three different IEGR schemeswill provide a measure of performance degradation whenmemory or computing power constraints impose limitationsin implementing the fully nonlinear map

The differences between the cylinder air flow of ZEGRengine and the IEGR engines are shown in the fifth row ofFig. 16. The subtle differences between the different IEGRdynamic scheduling schemes can be evaluated if we lookcloser in the A, B, and C areas of the nonlinear simulation.

Fig. 17 magnifies the A, B, and C areas of the nonlinearsimulation shown in Fig. 16. At s (plot A) a smallincrease in throttle causes the slow IEGR system to exhibitovershoot and the fast IEGR system to exhibit undershoot inthe cylinder air flow response. The IEGR engine that uses thetime constant as defined by the nonlinear maphas a smoothresponse. A slight undershoot, also present in ZEGR engineresponse, is due to changing engine speed.

At s (plot B) a large throttle increase causes camshafttiming to return to base camshaft timing (0). The IEGRengines that manage to do so at a fast rate (bothand have similar response to the ZEGRengine, whereas the slow IEGR system varies significantlyfrom the target response.

At s (plot C) a decrease in throttle angle requirescamshaft timing to attain its maximum value (see fourth row



of Fig. 16). During this step change, the fast IEGR system andthe slow IEGR system and ,respectively) overshoot or respond slowly to the change intorque demand. The response achieved by the IEGR enginethat uses the nonlinear static map matches the ZEGRdynamic response.

Fig. 18 shows the brake torque (second row) and the vehiclevelocity (third row) of the ZEGR and the three IEGR enginesfor the same throttle steps as in Fig. 16. As expected, the sim-ulated brake torque resembles the cylinder air flow responsejustifying the assumptions made in Section IV.

VIII. C ONCLUSIONS

Extensive simulations of the developed IEGR dynamicscheduling scheme demonstrate its ability to match the dy-namic response of a ZEGR engine. The scheme comprises ofa steady-state map used as set points of a tracking problemand a first-order lag that defines the transition between theoptimal steady-state points. Both the steady-state set point andthe time constant of the first-order lag depend on nonlinearstatic maps of throttle angle and engine speed, and ,respectively. We investigated two other suboptimal dynamicschedules for the IEGR engine with the goal to demonstrate thesubtle but potentially important deterioration of the engine’storque response if is not used.

In future work we will concentrate on the implementa-tion and testing of the developed scheme with the variablecamshaft timing engine mounted on a transient dynamometer.An adaptive algorithm that modifies the static nonlinear maps

and based on crankangleposition measurements will be investigated.

APPENDIX IDERIVATION OF

For

Define which results, accordingto (15), in

Using positiveness of the gains we show that

Hence, and

APPENDIX IITIME CONSTANT SCHEDULE

The algorithm used is described as follows:

if

otherwise(27)

When is small (but nonzero) we can achieve polezero cancelation but the resulted and are small when

compared to In this case inexact cancelation of the polewith the zero might result in unacceptable slow response.

Therefore, equilibrium for which should be treatedsimilar to the equilibrium for which In the algorithmused for the simulations shown in Section VII, we chose

In practice the pole-zero cancelation may not be feasibledue to actuator bandwidth constraints. In particular duringoperation in high intake manifold pressures the bandwidthrequirements can be restrictive. In this case we can relaxthe objective of designing the IEGR engine controller tomatch the dynamic behavior of the ZEGR engine, and imposethe monotonicity constraint in (13). In the region of highmanifold pressure (normal flow, the static schedulerequires i.e., we start decreasing the dilution to achievethe maximum engine torque (see Section III). The equivalentconstraint on is

(28)

where andIt is easy to verify thatand By

choosing we ensure monotonic behavior withdominant pole As a result the throttle-to-torque response of the IEGR engine is slower that thethrottle-to-torque response of the ZEGR engine.

ACKNOWLEDGMENT

The authors wish to acknowledge Prof. J. Freudenberg andProf. E. Gilbert for the helpful discussions and K. Buttsfor the vehicle simulation model; T. Leone and M. Seamanfor the impeccable experimental data; and E. Badillo forthe invaluable comments on driveability and engine perfor-mance.

REFERENCES

[1] J. B. Heywood, Internal Combustion Engine Fundamentals. NewYork: McGraw-Hill, 1988.

[2] C. Hsieh, A. G. Stefanopoulou, J. S. Freudenberg, and K. R. Butts,“Emission and drivability tradeoffs in a variable cam timing SI enginewith electronic throttle,” inProc. 1997 Amer. Contr. Conf., Albuquerque,NM, 1997, pp. 284–288.

[3] M. Jankovic, F. Frischmuth, A. G. Stefanopoulou, and J. A. Cook,“Torque management of engines with variable cam timing,”IEEE Contr.Syst. Mag., vol. 18, pp. 34–42, Oct. 1998.

[4] S. Jayasuriya and J.-W. Song, “On the synthesis of compensators fornonovershooting step response,”ASME J. Dynamic Syst., Measurement,Contr., vol. 118, pp. 757–763, 1996.

[5] A. G. Stefanopoulou, J. A. Cook, J. S. Freudenberg, and J. W. Grizzle,“Control-oriented model of a dual equal variable cam timing sparkignition engine,”ASME J. Dynamic Syst., Measurement Contr., vol. 120,pp. 257–266, 1998.

[6] A. G. Stefanopoulou and I. Kolmanovsky, “Dynamic scheduling ofinternal exhaust gas recirculation systems,”Proc. IMECE 1997, DSC-vol. 61, pp. 671–678.

[7] R. A. Stein, K. M. Galietti, and T. G. Leone, “Dual equal VCT—Avariable camshaft timing strategy for improved fuel economy andemissions,” SAE Paper 950975.



Anna G. Stefanopoulou(S’93–M’96) received theDiploma degree in 1991 from the National TechnicalUniversity of Athens, Greece, the M.S. degree inin naval architecture and marine engineering in1992 and the M.S. and Ph.D. degrees in electricalengineering and computer science in 1994 and 1996,respectively, all from the University of Michigan,Ann Arbor.

She is presently Assistant Professor at the Me-chanical Engineering Department at the Universityof California, Santa Barbara. Her research interests

are multivariable feedback theory, control architectures for industrial applica-tions, and powertrain modeling and control.

Dr. Stefanopoulou is Vice-chair of the Transportation Panel in ASMEDSCD and a recipient of a 1997 NSF CAREER award.

Ilya Kolmanovsky (S’94–M’95) received the M.S.degrees in Aerospace engineering and mathematicsin 1993 and 1995, respectively, and the Ph.D. degreein aerospace engineering in 1995, all from theUniversity of Michigan, Ann Arbor.

He is presently a technical Specialist with FordResearch Laboratory, Dearborn, MI, conducting re-search on advanced powertrain modeling and con-trol.


analysis and control of transient torque response in ...annastef/papers_vvt/pz.pdf · analysis and...

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