1

Variable Camshaft Timing Engine ControlA. G. Stefanopoulou, J. S. Freudenberg, J. W. Grizzle

Abstract| Retarding camshaft timing in an engineequipped with a dual equal camshaft timing phaser reducesthe unburned hydrocarbons (HC) and oxides of nitrogen(NOx) emitted to the exhaust system. Apart from thispositive e�ect to feedgas emissions, camshaft timing cancause large air-to-fuel ratio excursions if not coordinatedwith the fuel command. Large air-to-fuel ratio excursionscan reduce the catalytic converter e�ciency and e�ectivelycancel the bene�ts of camshaft timing. The interactionbetween the camshaft timing and the air-to-fuel ratio re-sults in an inherent tradeo� between reducing feedgas emis-sions and maintaining high catalytic converter e�ciency.By designing and analyzing a decentralized and a multi-variable controller, we describe the design limitation asso-ciated with the decentralized controller architecture andwe demonstrate the mechanism by which the multivariablecontroller alleviates the limitation.

Keywords|Multivariable Feedback Control, Air-to-FuelRatio, Emissions, Pollution Control, Internal CombustionEngines.

I. Introduction

Optimization and real-time control of cam timing inan engine equipped with a dual equal camshaft timingphaser can potentially reduce the unburned hydrocarbons(HC) and oxides of nitrogen (NOx) emitted to the ex-haust system ([20], [12]). In particular, by retarding thecam timing, combustion products which would otherwisebe expelled during the exhaust stroke are retained in thecylinder during the subsequent intake stroke. The contri-bution of this diluent to the mixture in the cylinder re-duces the HC and NOx feedgas emissions. On the otherhand, the diluent a�ects the fresh air mass charge intothe cylinders, thus altering the torque response and act-ing as a disturbance in the A=F loop. The e�ect of camtiming to torque response is undesirable due to potentialdrivability issues. Moreover, the e�ect of cam timing onA=F is undesirable due to potential degradation of cat-alytic converter e�ciency. A mathematical model of anexperimental engine equipped with a variable cam timingmechanism (VCT) has been derived in [17]. The VCTengine was shown to have very strong interactions amongthe cam timing (CAM ), the air-to-fuel ratio (A=F ), andthe torque (Tq) response.In this paper we design a model-based controller that

coordinates variable camshaft timing and fuel charge in

Work was supported by the National Science Foundation undercontract ECS-96-31237 (Grizzle), ECS-94-14822 (Freudenberg), andECS-97-33293 (Stefanopoulou); matching funds to these grants wereprovided by Ford Motor Co.

A. G. Stefanopoulou is with the Mechanical and EnvironmentalEngineering Department, University of California, Santa Barbara

J. S. Freudenberg and J. W. Grizzle are with the Control SystemsLaboratory, Department of Electrical Engineering and ComputerScience, University of Michigan

an internal combustion engine to reduce feedgas emis-sions, regulate air-to-fuel ratio (A=F ) at stoichiometry,and maintain torque response similar to that of a conven-tional (�xed cam timing) engine.

Despite the various studies of variable valve schemes([3], [5], [9], [10], [12], [20], and references therein) there isno systematic investigation of a control scheme which canachieve improved overall engine performance. The di�-culties lie in three areas: (i) the performance objectivesare interrelated and impose severe tradeo�s in the controldesign, (ii) measurements of the performance variables areunavailable or largely delayed due to cost or technologi-cal limitations, and (iii) implementation and calibrationfavor decentralized controller architectures that constraincontroller design. These three di�culties exist in manyindustrial control applications and there are no well ac-cepted guidelines that will ensure a satisfactory solution.For the variable cam timing engine, issue (i) arises fromthe fact that minimization of feedgas emissions, smoothengine torque response, and tight A/F regulation at sto-ichiometry are con icting objectives due to their subsys-tem interactions. Issue (ii) arises because real-time mea-surements of torque and feedgas emissions are not cur-rently available in conventional vehicles. Furthermore,A=F measurement is largely delayed imposing bandwidthlimitations in the feedback loop.

Issues (i) and (ii) are currently addressed only in steady-state using o�-line optimization of static engine mappingdata. The control problem is usually de�ned as a track-ing problem with set-points provided by steady-state op-timization. The feedback control problem is to track thesteady-state cam command with a speci�ed speed of re-sponse and regulate A=F at stoichiometry. Conventionalfeedback control design practice in the automotive indus-try is to calibrate multiple single-input singe-output con-trol loops to achieve individual subsystem performance.Such an approach results in decentralized controller de-velopment satisfying the third requirement above (issue(iii)). On one hand, this approach allows e�cient organi-zation of the engineering task of implementation. On theother hand, this approach often results in less than opti-mal system performance, especially when, the overall sys-tem is calibrated by de-tuning a subsystem controller toavoid unintentional excitation of another subsystem. Thedesign of decentralized controllers can potentially imposea large calibration burden in time and e�ort for highlycoupled systems such as the VCT engine [2].

In this paper we concentrate on the interactions be-tween the cam timing and the A/F response. This in-teraction results in an inherent tradeo� between reduc-

2

ing feedgas emissions and maintaining high catalytic con-verter e�ciency. By designing and analyzing a decentral-ized and a multivariable linear controller, we describe thedesign limitation associated with the decentralized con-troller architecture and we demonstrate the mechanismsby which the multivariable controller alleviates the limita-tion. We �nally simplify the fully multivariable controllerto a lower triangular form with equivalent closed-loop per-formance. The results are based on linear analysis of thesubsystem interactions at one operating point and eval-uated using a nonlinear simulation model. This type ofanalysis is valuable because it indicates that a multivari-able controller, at least at this operating point, gives per-formance enhancements.

Several other simpli�cations of the VCT engine controlproblem are employed. We do not consider external ex-haust gas recirculation (EGR) because variable cam tim-ing increases the internal exhaust gas recirculation (dilu-tion) and can potentially eliminate the need for externalEGR. Spark timing is assumed to be scheduled at min-imum spark advance to achieve best torque (MBT). Wealso ignore the e�ects of the cam timing on the charac-teristics of back ow and temperature of the surface wherethe fuel is injected. In general, variable camshaft tim-ing allows operation at higher intake manifold pressures,reduces the back ow through the intake valve and canconsequently reduce the uncertainty in the fuel evapora-tion rate. Our inability to measure (or robustly infer fromother measurements) torque and feedgas emissions limitsus to \o�-line" design of the cam timing loop bandwidth.We de�ne the cam timing bandwidth that achieves a rea-sonable tradeo� between torque performance and aver-aged feedgas emissions. The problem of transient torquecontrol for an engine equipped with VCT is addressedin [19]. The potential of using additional actuators suchas drive-by-wire throttle or an air-bypass valve in torquemanagement of a VCT engine is investigated in [7], [8].

This paper is organized as follows. Nomenclature isde�ned in Section II. In Section III, we formulate theVCT engine control problem. The performance tradeo�between torque performance and feedgas emissions is de-scribed in Section IV. This tradeo� leads to the derivationof the static cam timing schedule, Section IV-A, and thebandwidth of the cam timing loop, Section IV-B. Thecam timing and A/F feedback control design are derivedin Section V. The A/F controller consists of a feedforwardfuel command that depends on an air charge estimationwhich is described in Section VI; and a feedback fuel com-mand that depends upon a linear exhaust gas oxygen sen-sor (UEGO) which is described in Section VII. In SectionsVII-A and VII-B we show that by allowing the fuel com-mand to depend on cam timing we achieve tighter A/Fcontrol at the expense of more complex controller archi-tecture. The VCT engine controller design is evaluatedusing a simulation of the nonlinear VCT engine model;simulations results are shown in Section VIII. Finally,

our conclusions are summarized in Section IX.

II. Nomenclature

A=F air-to-fuel ratioA=Fcyl : air-to-fuel ratio in the cylinderA=Fexh : air-to-fuel ratio at the UEGO sensorA=Fstoic : stoichiometric air-to-fuel ratio

c coe�cients on physical equations(with various subscripts)controller termscommand when used in subscripts

CAM camshaft timing (degrees)CAMc : commanded camshaft timingCAMa : actual camshaft timingCAMm : measured camshaft timingCAMdes : desired camshaft timing

Fc fuel command (gr/event)fw feedforward (as an index)fb feedback (as an index)K, or k static gains derived after linearization_m mass air ow rate (gr/s)

_m� : mass air ow rate through the throttle body_mcyl : mass air ow rate to the cylinder

m mass (gr)ma : mass air charge (gr/event)

MAF mass air ow measuredusing a hot wire anemometer

N engine speed (rpm)P pressure (bar)

Pm : manifold pressure (bar)Po : atmospheric pressure (bar)

R speci�c gas constant (J/KgK)T temperature

or torque (it is made clear through the context)Vm manifold volume (m3)�T fundamental sampling time interval (s)� throttle angle (degrees)� time constant in lowpass �lters (s)

III. Overview of the Controller Architecture

In this section we formulate the control problem. Dur-ing rapid throttle changes imposed by the driver we needto control cam timing and fuel charge to regulate A=F atstoichiometry and minimize feedgas NOx and HC emis-sions, under the constraint of monotonic brake torque re-sponse. Feedgas emissions and torque response cannotbe used as feedback signals because they are not cur-rently measured in the vehicle (stringent emission stan-dards might require the introduction of such sensors inthe future). Figure 1 shows the input and output signalsof the VCT engine as they are used in the control design.The control problem at hand, shown in Figure 1, has threeperformance variables (we can lump NOx and HC in onevariable) and two control variables.

CAMc

A/Fego

CAMmθm

MAF

Fuel

θ

VCTEngine

TqNOxHCA/Ftwc

}}control

signals

performancevariables

measurements{Fig. 1. Input-output relationship for the VCT engine control prob-

lem.

Cam timing is primarily used to reduce the feedgasemissions. Although dynamic measurements of feedgas

3

emissions are not available, there is a simple rule to follow:\retarded cam timing reduces feedgas emissions" [17]. Wecannot, however, operate always in retarded cam becauseretarded cam timing a�ects the dilution in the cylinders,which in turn alters the torque response. Therefore thecam timing loop must satisfy a reasonable tradeo� be-tween reducing feedgas emissions and maintaining mono-tonic torque response. We design the steady-state camtiming schedule based on throttle angle measurement.The resulting static cam timing schedule allows operationin maximum retarded cam when combustion stability andadequate torque can be achieved. Based on this staticcam timing schedule, we de�ne a tracking problem for thecam timing loop. The bandwidth of the cam timing loopis determined o�-line to avoid undesirable excitation totransient torque response. Detailed analysis and design ofthe cam timing loop bandwidth can be found in [19].

The bandwidth in the A=F feedback loop is determinedby the delay associated with the A=F measurement at alinear exhaust gas oxygen (UEGO) sensor, A=Fexh signal.Throttle and cam timing alter the air charge and there-fore act as a disturbance in the A=F loop. The fuel com-mand cannot reject this disturbance based on the delayedA=Fexh signal, so a two degree of freedom A=F controllerwill be used.

The resulting controller structure, shown in Figure 2,consists of a steady-state cam timing schedule, a feedfor-ward fuel controller and a multiple-input multiple-output(MIMO) feedback controller. During throttle changes,the feedback controller will be tracking cam timing tothe desired cam timing position, and maintain A=F atstoichiometry. The interaction between the cam timingand A=F loops suggests the development of a fully mul-tivariable feedback controller. On the other hand, twoPI feedback loops (decentralized controller) would ensureindependent development of the new feature, cam tim-ing. Here, the motivation arises from the fact that if thesimpler controller structure results in satisfactory perfor-mance, then addition of the variable cam timing mecha-nism to a conventional automotive engine does not requirecompletely new software development and calibration pro-cedures, but merely involves calibration of (i) the PI gainsin the A=F loop and (ii) the bandwidth of the cam phasingdynamics (see Figure 3).

CAMc

FcΣ

Cam TimingSchedule MIMO

Controller

VCT Engine

Feedforward Fuel

Σ

Σ

Fc

fb

fw

A/Fstoich

CAMd

..

θ

θm

θm

A/Fstoich

MAF

A/Fego

CAMm

Fig. 2. Overview of the MIMO controller structure.

CAMc

FcΣ

Cam TimingSchedule

VCT Engine

Feedforward Fuel

Σ

Σ

Fcfb

fw

A/Fstoich

CAMd

..

θ

θm

θm

A/Fstoich

MAF

A/Fego

CAMm

SISO Control

SISO Control

Fig. 3. Controller structure when cam timing and A=F control aretwo separate SISO controllers.

IV. Torque Requirements

The control requirement is to maintain brake torqueresponse similar to the conventional engine at constantengine speed during rapid throttle changes. Engine speedis a slowly varying parameter and can be assumed con-stant for rapid changes in throttle position imposed bythe driver during rapid acceleration demands. Simulationsshowed that varying engine speed provides a dampeningfactor in the engine power response. Hence, imposing con-straints upon the engine torque response is a conservativeway of addressing drivability requirements.In the next two sections we brie y describe the design

speci�cations for the cam timing loop. The static feed-forward cam timing schedule is determined based on itse�ect on the steady-state torque response; the bandwidthof the cam timing loop is determined based on its e�ecton the transient torque response.

A. Feedforward Cam Timing Schedule

The static cam timing schedule is designed o�-line basedon the static e�ects of cam timing on engine torque re-sponse. Figure 4 (left) shows the torque response fora conventional cam timing (CAM = 0 degrees|dashedline) and maximum retarded cam timing (CAM = 35degrees|dashed-dot line) versus throttle angle at 2000RPM. To minimize feedgas emissions we need to operateat maximum cam retard. To ensure maximum torque atwide open throttle (WOT) we need to advance cam tim-ing back to the base cam timing. The solid line in Figure4 (left) shows a smooth transition from fully retarded tobase cam timing for part throttle to WOT. For very smallthrottle angles (low load), cam timing does not a�ect thestatic torque response1, but it deteriorates the combustionstability because of the high level of dilution.Cam timing, therefore, is scheduled at base cam timing

for small throttle angles to avoid increase of unburned HCand to maintain combustion stability. Hence, the steady-state cam phasing, CAMdes = G(�m; Nm), that minimizesfeedgas emissions while maintaining smooth steady-statetorque response is scheduled as follows: (1) near idle it

1Cam timing does not a�ect the steady-state ow through thethrottle body during sonic conditions. Detailed discussion of thenonlinear engine behavior due to the two distinct regimes of owthrough the throttle body can be found in [18].

4

is scheduled for idle stability which requires cam phas-ing equal to zero; (2) at mid-throttle it is scheduled foremissions which favors fully retarded cam phasing; and(3) at wide open throttle (WOT) it is scheduled for max-imum torque which requires cam phasing to be advancedback to 0 degrees. The developed cam scheduling scheme,shown in Fig. 4 (right), ensures reduction of feedgas emis-sions under the constraint of smooth steady state torqueresponse for constant engine speed.

Throttle Angle (degrees)

10 20 30 400 50

Nor

mal

ized

Tor

que

0.2

0.4

0.6

0

0.82000 RPM

Throttle Angle (degrees)

10 20 30 400 50

Cam

Pha

sing

(de

gree

s)

10

20

0

30

2000 RPM

Fig. 4. Comparison between static torque response using the con-ventional cam phasing (CAM = 0 degrees), the designed camscheduling scheme, and the fully retarded cam phasing. Thedesigned cam scheduling scheme is shown in the right �gure.

B. Transient Cam Timing

The speed of cam timing changes during throttle stepscan dramatically a�ect the brake torque response and evencause undershoot during throttle tip-in. This behavioris consistent with the results in [16], where it is shownthat the torque response of the VCT engine during coor-dinated throttle and cam steps can be described by a non-minimum phase system. It is known that the undershootin a system with a non-minimumphase zero will increase ifwe require a short settling time, and thus fast speed of re-sponse (pp. 15, [15]). Therefore, reduction of the feedgasemissions and good dynamic torque response cannot beachieved independently: good torque response favors slowcam phasing to the new set points, but this might de-grade the feedgas emissions which require fast cam tran-sients. The following simulations illustrate a compromisebetween the torque response and emissions, and will beused to de�ne the cam phasing control loop bandwidth.Figure 5 illustrates cam phasing and torque response to

step changes in throttle position at 750, and 2000 rpm (leftand right plots respectively). In all simulations, stoichio-metric A=F and MBT spark timing were used. There-fore the torque response in Fig. 5 is only a function ofair charge and internal exhaust gas recirculation (or camphasing). For simplicity, the closed-loop cam phasing dy-namics are described by a low-pass �lter between the de-

sired and the commanded cam phasing. Figure 5 showsthe cam phasing and torque response in closed loop withtime constants of 0.5, 1.0 and 1.5 engine cycles. Goodtorque response is a critical requirement at low enginespeeds, and a time constant of 1 engine cycle was con-sidered su�cient for 750 RPM. At 2000 rpm the interac-tion between torque and cam phasing loop is considerablyweaker than the interaction at lower engine speeds, and atime constant of 1 engine cycle is once again adequate forthe cam phasing controller. Thus, the bandwidth of thecam phasing controller is scheduled in this paper as a func-tion of engine speed to correspond to 1 engine cycle in or-der to achieve a reasonable torque response over the entireoperating regime. An extended analysis of the selectionof the optimal cam phasing time constant as a function ofthrottle position and engine speed can be found in [19].

0.5 1 1.50 2

Thr

ottle

(de

gree

s)

2

3

4

1

5

750 RPM

0.1 0.2 0.3 0.4 0.5 0.6 0.70 0.8

Thr

ottle

(de

gree

s)

8

9

10

11

12

7

13

2000 RPM

0.5 1 1.50 2

Cam

Pha

sing

(de

gree

s)

5

10

15

20

0

25

0.1 0.2 0.3 0.4 0.5 0.6 0.70 0.8

Cam

Pha

sing

(de

gree

s)

5

10

15

20

25

30

0

35

Seconds

0.5 1 1.50 2

Tor

que

(Nm

)

20

40

60

80

0

100

Seconds

0.1 0.2 0.3 0.4 0.5 0.6 0.70 0.8

Tor

que

(Nm

)

20

40

60

80

0

100

slow

fast fast

slow

slow

fastfast

slow

Fig. 5. Torque response (at 750 and 2000 RPM) using di�erentlow-pass �lters in the cam phasing dynamics.

V. Emissions Requirements

In the VCT engine, changes in the throttle positionare followed by changes in the cam phasing based on thescheduling scheme shown in Fig. 4. The desired camphasing reduces feedgas emissions by regulating the dilu-ent in the cylinders. The contribution of this diluent tothe mixture in the cylinder a�ects the breathing process,and consequently the mass charge in the cylinders, whichin turn a�ects the air fuel ratio (A=F ) in the cylinder mix-ture. This makes the A=F response highly coupled withthe cam phasing activity. Cam phasing is used to regu-

5

late feedgas emissions during changes in throttle position,but through its interaction with A=F , it also a�ects thecatalytic converter e�ciency.

We will achieve reduction of tailpipe emissions if we re-duce feedgas emissions at equivalent catalytic converter ef-�ciency. Therefore, to improve feedgas emissions we needto regulate cam timing to its steady-state value as fast aspossible while maintaining A=F at stoichiometry. Unfor-tunately the long delay (810 degrees) in the A=F mea-surement associated with the combustion-exhaust strokeand the transport delay in the exhaust manifold imposes abandwidth limitation on the A=F loop. If the disturbanceto the A=F loop caused by the cam activity has high fre-quency content (i.e., beyond the achievable bandwidth ofthe A=F controller), then the disturbance cannot be re-jected. In this case, it is a common technique to slow downthe cam phasing signal, i.e., to de-tune the subsystem thatcauses the high frequency disturbance. This alternative,although consistent with current design practice, entailsloss of the potential emissions bene�ts of the VCT engine.

Hence, the dynamic performance tradeo� is between (a)the feedgas emissions that the catalytic converter mustprocess and (b) the e�ciency of the catalytic converter(which is a function of A=F excursions from stoichiom-etry). Due to the interaction between the cam timingloop and the A=F loop, we cannot simultaneously min-imize (a) and maximize (b); this is because maximumcatalytic e�ciency requires that A=F be held perfectlyat stoichiometry, which in turn rules out moving the camrapidly to reduce feedgas emissions. A dynamic model ofthe catalytic converter e�ciency could help specify a rig-orous tradeo� between the two bandwidths, because, afterall, the ultimate goal is to minimize tailpipe emissions [1].Here, for simplicity we selected the bandwidth of the camphasing loop that satis�es the torque requirements, oneengine cycle (720 degrees).

To achieve good A=F control during rapid throttle andcam movements we utilize the commonly used two de-gree of freedom A=F controller topology modi�ed for thevariable cam timing engine [6]. The A=F control loopconsists of a feedforward term that adjusts the fuel com-mand based on the measured mass air ow (MAF ) andthe measured cam position (CAMm), and a feedback termthat regulates the fuel commandbased on the signi�cantlydelayed A=F signal from the EGO sensor.

VI. Feedforward Design

The feedforward term is based on the estimation of thecylinder pumping mass air ow rate (d_mcyl) from the mea-sured cam phasing (CAMm), the estimated manifold pres-

sure (cPm), and the engine speed

d_mcyl = P (CAMm; cPm; N ) (1)

The estimated manifold pressure (cPm) is calculated as:

d

dtcPm = Km(�

d

dtMAF +MAF � d_mcyl) (2)

where we have used the dynamics of the mass air ow me-ter : � d

dtMAF +MAF = _m�. To eliminate the derivative

on the air ow measurement, we use the variable

� = cPm�Km �� �MAF . This yields the estimated cylindermass air ow rate :

ddt� = Km(MAF � d_mcyl)cPm = �+Km � � �MAF

d_mcyl = P (CAMm; cPm; N )

(3)

For the estimation of the mass air charge into the cylinders(cma), we assumed uniform ow during the fundamentalengine event (�T = 120

n�N, where n, the number of cylinders

equals 8, and N is the engine speed in rpm):

cma = d_mcyl ��T = d_mcyl �15

N(4)

The feedforward fuel command is given by Fcfw = dma

14:64.

VII. Feedback Design

In this section we design a linear feedback controllerthat tracks the desired cam timing as de�ned by the staticcam scheduling scheme, and maintains A/F at stoichiom-etry (see Fig. 2). The controller design considerations are:(a) There is an 810 degree delay in the A=F process. At2000 rpm, this translates into a time delay of 0.0675 sec.The A=F bandwidth should not exceed 7.5 rad/sec sinceby using a Pad�e approximation for the 0.0675 sec delaywe have the deleterious e�ects of a non-minimum-phasezero approximately at 15 rad/sec. (b) The required timeconstant for the cam phasing dynamics is 720 degrees (1engine cycle). At 2000 rpm this corresponds to a timeconstant equal to 0.06 sec, which translates into a camphasing closed-loop bandwidth equal to 17 rad/sec.We linearized the model at a throttle position equal to

9.33 degrees, cam phasing equal to 10 degrees, and enginespeed equal to 2000 rpm. This operating point lies in thetransition region on the cam phasing scheduling schemeshown in Fig. 4. The delays are represented by 1st and2nd order Pad�e approximations. The linear model has9 states and 2 integrator states are introduced to ensurezero steady-state error in tracking the desired cam timingand the stoichiometric A/F. Figure 6 shows the Bode gainplots of the plant linearized at 2000 RPM. Cam phasing ismeasured in degrees, A=F is dimensionless, and the fuelcommand is scaled so that a unit deviation in fuel causesa unit deviation in the A=F signal. The plant has a lowertriangular form, i.e., there is no interaction between thefuel command and the cam phasing loop, since fuel chargea�ects the system downstream of the breathing process.

6

CAM

A/Fexh

CAMc

Fc=

10.1

-5

-10

-15

-20

-25

-30

-35

0

10010 1000 10.1

-20

-40

-60

-80

-100

0

10010 1000

10.1

-40

-60

-80

-100

-120

-20

10010 1000 10.1

0

-10

-20

-30

-40

10

10010 1000

21

11

22

p

pp

Fig. 6. Bode gain plots of the linearized plant.

In Figure 6 we can see the interaction term (p21) be-tween the cam phasing control signal and A=F measure-ment. The feedforward portion of the A/F controler de-scribed in Section VI ensures decoupling of the cam phas-ing and the A=F in steady-state, but allows high frequencyinteraction. The peak of the interaction term occurs at 20rad/sec while we require the cam phasing activity to rollo� after 17 rad/sec. Therefore, the control signal gener-ated to force the cam phasing to track a command inputwill also produce a transient response in the A=F loop; ine�ect, the cam loop acts as a disturbance to the A=F loop.Furthermore, there is a 9�T sec delay (shown in Fig. 7)that is located downstream of the disturbance from thecam loop to the the A/F loop. We thus see that we havea two-input two-output (TITO) system with strong inter-action between the two loops and a bandwidth limitationdue to the sensor delay.

1 T 6 T 3 T

Controller

CAMmCAMc

CAMerror

A/Ferror

A/Fstoic

A/Fexh

-

-

CAMdes

Fc

A/Fcyl

A/Ftwc

A/Fexh

Fig. 7. Block diagram of the linearized plant.

The simplest approach to design the feedback controlleris to ignore the interaction between the two loops and de-sign two SISO feedback systems (see Fig 3). It is wellknown, however, that multivariable controllers managedesign tradeo�s due to subsystem interactions more suc-cessfully than do decentralized controllers [13], [4]. In[11] it is mentioned that multivariable design techniquesreduce the interaction between subsystems even withouthaving taken explicit steps to do so. For this reason inthe following subsections we design a multivariable anda decentralized controller and study their impact on theemissions performance.

A. TITO and 2 SIS0 Design

A multivariable controller was designed to track the de-sired cam phasing as described in Sec. IV-A and maintainA=F at stoichiometry. The controller schedules the fueland cam phasing commands based on the cam phasingposition measurement and the A=F signal from the EGOsensor. The LQG/LTR methodology was used because itprovided a straightforward way to meet the performancerequirements discussed in Sec. VII, and has been studiedextensively for its robustness properties [11]. Robustnessissues in VCT design arise because the VCT engine modelused for design does not include fuel puddling dynamicswhich is considered as input uncertainty in the fuel path.Also, there is a signi�cant uncertainty in the cam phaserdynamics due to aging. A comprehensive study of robust-ness requires more extensive modelling of these and otheruncertainties and is beyond the scope of the present paper.A few design iterations yielded the decentralized and

the multivariable controllers, the Bode gain plots of whichare illustrated in Figure 8. Appendices B and C containdetailed description of the two controllers. Both controllerdesigns achieve the bandwidth requirement in the camphasing loop, and provide adequate speed of response inthe A=F loop.

CAMerror

A/Ferror=

CAMc

Fc

110.1

30

20

10

0

-10

-20

-30

40

10010 1000

DEC

MIMO

110.1

0

-10

-20

-30

-40

10

10010 1000

110.1

-20

-25

-30

-35

-40

-45

-15

10010 1000 110.1

30

20

10

0

-10

-20

40

10010 1000

7

the two di�erent controller architectures. The A=F devia-tions for the multivariable control scheme are signi�cantlybetter than those corresponding to the decentralized con-trol scheme. Implementing the multivariable controllerthus seems to be bene�cial. The questions we must ad-dress are: How did the multivariable controller manageto reject the A=F disturbance faster than the decentral-ized controller? In which way did the multivariable con-troller reduce the interaction between the two loops? Inthe next section we identify the mechanism by which themultivariable controller achieves smaller A=F excursionsduring cam phasing transients.

A/F

15

14.8

14.6

14.4

14.2

14

15.2CAM SSteps

CA

M

25

20

15

10

5

0

-5

-10

30

CA

Mc

20

15

10

5

0

-5

-10

25

Seconds

86 10

D E C M I M O u l a t i o n d u r i n g c a m p h a s i n g c o m m a n d s .

B . T I T O a n d 2 S I S 0 A n a l y s i s

W e b e g i n b y d e s c r i b i n g a d e s i g n l i m i t a t i o n p r e s e n t w i t h d e c e n t r a l i z e d c o n t r o l . C o n s i d e r t h e d e c e n t r a l i z e d c o n t r o l

s y s t e m i n F i g u r e 1 0 . T o p o l o g i c a l l y , t h e C A M l o o p a c t s a s a n o u t p u t d i s t u r b a n c e t o t h e A = F loop. As noted in

Section V, there is noin teraction fromthe A=F looptotheCAMloop.Denotethesensitivit yandcomplemen tarysensitivit y

functionsforeac hloopb y s

ii

( s )=(1+ p

ii

( s ) c

ii

( s ))

� 1

andt

ii

( s )=1 � s

ii

( s ), i =1 ; 2.Thenthetransferfunctiondescribingtheclosed-loop A=F responseisgiv enb y

A=F

exh

( s ) = t

22

( s ) A=F

stoic

( s )+ s

22

( s ) p

21

( s ) c

11

( s ) s

11

( s ) CAM

des

(s)

|

8

p11

p22

p21

CAMm

A/Fexh

-

c11CAMc

A/Fstoichc22

CAMdes

Fc

-

Fig. 10. Block diagram of the decentralized control scheme.

p11

p22

p21

CAMm

A/Fexh

-

c11

CAMc

A/Fstoichc22

CAMdes

Fc

-

c12

c21

Fig. 11. Block diagram of the fully multivariable scheme.

feedforward path from the cam phasing error to the fuelcommand used to control A=F . The feedforward term(c21) sends information to the fuel command about thecam phasing error, and this allows faster response duringcam phasing transients. The disturbance imposed on theA=F loop by a command issued to the cam phasing loopis shown in the following equation :

A=Fexh(s) = (p21(s)c11(s) + p22(s)c21(s))CAMerror (s)(6)

Note here, that the same disturbance for the decentralizedcontroller (see Figure 10) is given by :

A=Fexh = p21(s)c11(s)CAMerror (s) (7)

The multivariable controller can potentially reduce thecoupling between the two subsystems by choosing theterm (c21) such that

j p21(j!)c11(j!) + p22(j!)c21(j!) j<j p21(j!)c11(j!) j(8)

It is possible to interpret the action of the MIMO con-troller as partially decoupling the A=F response from theCAM loop. Indeed, setting the feedforward term equal to

c21(s) =�c11(s)p21(s)

p22(s)(9)

achieves zero closed-loop interaction from CAM to A=F .An alternate representation of the perfect decoupler (9) isdepicted in Figure 13. With this topology, the CAM andA=F loops become completely decoupled, and the two re-maining controller parameters, c11 and c22, may be chosenindependently. This controller design may be prone to ro-bustness problems, since the term (c21) is canceling theundesired disturbance by inverting the signal along thepath of the plant interaction.

p11

p22

p21

CAMm

A/Fexh

-

c11

CAMc

A/Fstoichc22

CAMdes

Fc

-

c21

Fig. 12. Block diagram of the simpli�ed multivariable controlscheme.

p11

p22

p21

CAMm

A/Fexh

-

c11CAMc

A/Fstoichc22

CAMdes

Fc

-

-p21p22

Fig. 13. Block diagram of the decoupling controller.

In practice, there is no need to achieve perfect decou-pling. In Figure 14, we see that MIMO control reducesonly the peak in the closed-loop response from CAM com-mands to A=F measurements. Indeed, at lower frequen-cies, the integral action in the A=F loop achieves zerosteady state error despite the interaction with the CAMloop. At higher frequencies, on the other hand, the CAMloop rolls o� and thus does not produce a response inA=F .As we see in Figure 14, the MIMO controller merely re-duces the peak due to the interaction, thus attenuatingthe e�ect of the CAM loop upon A=F without achievingtotal decoupling.

CAM

A/Fexh= CAMdes

A/Fstoich

10.1

-10

-20

-30

-40

-50

-60

-70

0

10010 1000 10.1

-20

-40

-60

-80

-100

0

10010 1000

DEC

MIMO

10.1

-20

-40

-60

-80

-100

0

9

upon A=F .

VIII. Simulation Example.

In this section simulations of the nonlinear plant withthe linear controllers are presented. The simulation inFig. 15 shows that the simpli�ed multivariable controllerresults in better A/F control than the decentralized con-troller even during large throttle steps. Thus, the per-formance improvements due to the simpli�ed multivari-able feedback controller can potentially be realized with-out elaborate gain scheduling.

Figure 16 illustrates the performance improvementsthat the VCT engine can achieve when compared to con-ventional engine (�xed cam phasing). It shows the simu-lated response of the conventional engine (�xed cam phas-ing) and the VCT engine to a square wave in commandedthrottle at 2000 rpm. The sequence of throttle commandsis 9.0 degrees (nominal) to 7.2 degrees to 12.0 degreesand back to 7.2 degrees and then to 9.0 degrees. Thecorresponding cam phasing set-points are 3 degrees to 10degrees to 35 degrees (maximum) and back to 3 degreesand then to 10 degrees. The conventional engine schemehas �xed cam at 0 degrees.

The resulting torque response of the VCT engine is keptas responsive as the conventional engine in acceleration.During the abrupt deceleration at the 5th second of thesimulation, the torque response of the VCT engine has anabrupt transient behavior which might entail drivabilityproblems. It also indicates that the dash-pot system hasto be calibrated taking cam timing into account duringrapid tip-outs.

The transient A=F response of the VCT engine is sim-ilar to the A=F response of the conventional engine satis-fying the equivalent catalytic converter e�ciency require-ment. Therefore the emission improvement of the VCTengine over the conventional scheme can be demonstratedby the feedgas NOx and HC emissions. Using the inte-grated area de�ned by the NOx and HC emission curvesas a crude measurement of engine emissions, we can es-timate a possible reduction of 10% in NOx and 20% inHC during that period. Moreover, the engine operates athigher manifold pressure, which may reduce the pumpinglosses and provide an improvement in fuel economy.

IX. Conclusions.

Variable cam timing fundamentally a�ects both A/Fand torque response. To achieve satisfactory performance,the engine controller must take these interactions into ac-count. In this paper we analyze the implication of theseinteractions in the control design parameters and con-troller stucture. We formulate the cam timing problem asa set-point tracking problem with closed-loop bandwidththat achieves a reasonable tradeo� between torque perfor-mance and averaged feedgas emissions. This bandwidthhowever, allows high frequency interactions between the

Cam

(de

gree

s)

35

30

25

20

15

10

5

0

40

Tor

que

(Nm

)

100

80

60

40

20

0

120

A/F

15.5

15

14.5

14

13.5

16

NO

x (g

/KW

-h)

20

18

16

14

12

22

S

20

HC

(g/

KW

-h)

10

8

6

4

2

0

12

Seconds

64 8

PI

MIMO

Fig. 15. Simulation response of the multivariable (MIMO) and thedecentralized (2 PI loops) control scheme.

cam timing loop and the A/F loop and favors a multivari-able feedback controller design.

In an e�ort to achieve e�cient controller developmentfor the VCT engine, we investigate the relative utility ofa decentralized controller architecture versus a multivari-able control strategy. We show that a fully multivariablecontroller is not necessary. In fact, we simpli�ed the 2x2multivariable controller by eliminating the cross-couplingterm that modi�es the cam command based on A=F .On the other hand, we show that allowing the fuel com-mand to depend upon the cam phasing results in smallerA=F transients. The simpli�ed lower triangular controllerstructure will be used in future work to schedule the lin-ear time-invariant controller over a wide range of engineoperation. Finally, nonlinear simulations were used in thispaper to demonstrate the potential performance improve-ments achieved by the variable cam timing engine.

X. Acknowledgment.

The authors wish to thank Je�rey A. Cook and KennethR. Butts of Ford Motor Company, Scienti�c Research Lab-oratory for the guidance and constant support throughoutthis project.

10

Man

if. P

res.

(ba

r)

0.6

0.5

0.4

0.3

0.2

0.7

Tor

que

(Nm

)

100

80

60

40

20

0

120

A/F

16

15.5

15

14.5

14

13.5

16.5

NO

x (g

/KW

-h)

22

20

18

16

14

24HC

(g/

KW

-h)1

0864201

2

S

econds

64 8

V C TF i x e d C a m u l a t i o n r e s p o n s e o f t h e c o n

v e n t i o n a l (

C AM = 0) and

the V CT engine.

Appendix

I. V CT Engine Model

The lo w-frequency , phenomenological engine mo del

used for the con trol dev elopmen

t is brie y describ ed here.

F orthespeci�cfunctionsandv alues,see[17].

11

II. Multivariable Controller

The TITO controller was designed to track the de-sired cam phasing (value from the scheduling map) andmaintain A=F at stoichiometry by controlling fuel andcam phasing command using the cam phasing positionmeasurement, the A=F signal from the EGO sensor, andthrottle position measurement. We linearized the modelat a throttle position equal to 9.33 degrees, cam phasingequal to 10 degrees, and engine speed equal to 2000 rpm.This operating point lies in the transition region on thecam phasing scheduling scheme shown in Fig. 4. Thedelays are represented by 1st and 2nd order Pad�e approx-imations. The linear model has 9 states. The state spacerepresentation of the linearized model is given by :

_x(t) = Ax(t) + Bu(t) + Brr(t)

y(t) = Cx(t) +Du(t)

where

x =

26666666666664

Pm (manifold pressure)A=Fexh (at the UEGO sensor)X (air charge estimator state)

MAF (mass air ow)CAMm (measured cam phas.)CAMa (actual cam phas.)

Fdel (delayed fuel)A=F (at the cat. conv.)A=Fcyl (at the cylinder)

37777777777775;

u =

�CAMc (cam phas. command)

Fc (fuel command)

�r = [� (throttle position)] , and

y =

�A=Fexh (A/F measurement)CAMm (cam phas. measured)

�:

During changes in throttle position, it is important tomaintain A=F at stoichiometry and maintain zero track-ing error in the cam timing. This is accomplished by aug-menting the state vector with the integral of the error inthe A=F ( _q1 = A=Fstoic�A=Fexh), and the integral of theerror in the cam timing ( _q2 = CAMdes � CAMm). Theaugmented input and state vector are:

r̂ =

24 �A=FstoicCAMdes

35 , and x̂ =

�xq

�:

The controller feedback gain u = �Kx̂ = [�K1 �K2 ] [ xq ]is found by solving the LQR problem. The weighting ma-trices Rxx and Ruu used in the minimization of the costfunction J =

R1

0(x̂0Rxxx̂+ u0Ruuu) dt are:

Rxx = diag([0:364; 1500; 0:3; 18; 1000; 1000; 0:9;

: : :1500; 1500; 70000; 5000])

Ruu = diag([1; 1])

The observer gains are derived from a Kalman �lter de-sign by minimizing the covariance between the actual and

the estimated values of the state. The real symmetricpositive semi-de�nite matrix representing the intensitiesof the state noises Qxx, and the real symmetric positivede�nite matrix representing the intensities of the measure-ment noises Qyy are assumed diagonal. The Kalman �ltergain is adjusted so that the LQG controller can asymp-totically approach the robustness properties of the LQRdesign. Loop transfer recovery in the input is employedand the chosen constant values for the covariance matricesQxx, and Qyy are :

Qxx = 0:001 � diag([1; 100; 1; 1000;100;100; 0:9;100;100])

Qyy = 100 � diag([1; 1]):

III. Decentralized Controller

The decentralized control design involves de�ning thetwo single-input single-output feedback loops that satisfythe design speci�cations. The transfer function describingthe dynamics in the cam phasing loop (p11 term) is givenby :

CAM =�0:01348(s� 2000)

s+ 26:96| {z }p11(s)

CAMc : (10)

The transfer function describing the dynamics in the A=Floop (p22 term) is given by :

A=Fexh =(s� 133:34)(s2� 88:8081s+ 2633:67)

(s+ 14:286)(s+ 133:33)(s2+ 88:8081s+ 2619:44)| {z }

p22(s)

Fc :

(11)

To meet the design speci�cations on the cam phasing loopthe controller was chosen to be :

CAMc =0:3425(s+ 26:96)

s| {z }c11(s)

CAMerror : (12)

A few design iterations resulted the following controller inthe A=F feedback loop :

Fc =0:57(s+ 10)

s| {z }c22(s)

A=Ferror : (13)

Both controller designs achieve the bandwidth require-ment in the cam phasing loop, and provide adequate speedof response in the A=F loop. The bandwidth speci�ca-tions for the two loops essentially �x the bandwidths ofthe diagonal elements of the controller independently ofthe controller structure. As a result, the diagonal ele-ments of the two controllers are approximately identical(see Fig. 8).

12

References

[1] E. P. Brandt and J. W. Grizzle, Dynamic Modeling of a Three-Way Catalyst for SI Engine Exhaust Emission Control, to ap-pear in IEEE T-Control Technology.

[2] J. A. Cook, W. J. Johnson, \Automotive Powertrain Control:Emission Regulation to Advanced Onboard Control Systems",Proceedings of the American Control Conference, Seattle, pp.2571-2575, June 1995.

[3] A. C. Elrod and M. T. Nelson, \Development of a VariableValve Timing Engine to Eliminate the Pumping Losses Associ-ated with Throttled Operation," SAE Paper No. 860537, 1986.

[4] J. S. Freudenberg, J. W. Grizzle and B. A. Rashap, \A Feed-back Limitation of Decentralized Controllers for TITO Sys-tems, with Application to a Reactive Ion Etcher," Proc. 1994Conference on Decision and Control, pp. 2312-2317, Orlando,1994.

[5] C. Gray, \A Review of Variable Engine Valve Timing," SAEPaper No. 880386, 1988.

[6] J. W. Grizzle, J. A. Cook and W. P. Milam, \Improved Tran-sient Air-Fuel Ratio Control using Air Charge Estimator,"Proc. 1994 Amer. Contr. Conf., Vol. 2, pp. 1568-1572, June1994.

[7] S. C. Hsieh, A. G. Stefanopoulou, J. S. Freudenberg, and K. R.Butts, \ Emission and Drivability Tradeo�s in a Variable CamTiming SI Engine with Electronic Throttle," Proc. 1997 Amer.Contr. Conf., pp. 284-288, Albuquerque, 1997.

[8] M. Jankovic, F. Frischmuth, A. G. Stefanopoulou, and J. A.Cook, \Torque Management of Engines with Variable CamTiming," IEEE Control Systems Magazine, vol. 18, pp. 34-42,Oct. 1998.

[9] T. G. Leone, E. J. Christenson, and R. A. Stein, \Comparisonof Variable Camshaft Timing Strategies at Part Load," SAEPaper No. 960584, 1996.

[10] T. H. Ma, \E�ects of Variable Engine Valve Timing on FuelEconomy," SAE Paper No. 880390, 1988.

[11] J. M. Maciejowski, Multivariable Feedback Control, Addison-Wesley Publishing Company, 1989.

[12] G.-B. Meacham, \Variable Cam Timing as an Emission ControlTool," SAE Paper No. 700645, 1970.

[13] M. Morari and E. Za�riou, Robust Process Control, PrenticeHall, 1990.

[14] F. G. Shinskey, Process Control Systems, John Wiley &Sons,1989.

[15] M. M. Seron, J. H. Braslavsky, and G. C. Goodwin, Funda-mental Limitations in Filtering and Control, Springer-Verlag1997.

[16] A. G. Stefanopoulou and I. Kolmanovsky, \Dynamic Schedul-ing of Internal Exhaust Gas Recirculation Systems," Proc.IMECE 1997, DSC-Vol. 61, pp. 671-678, Sixth ASME Sym-posium on Advanced Automotive Technologies, Dallas, 1997.

[17] A. G. Stefanopoulou, J. A. Cook, J. S.Freudenberg, J. W.Grizzle, \Control-Oriented Model of a Dual Equal VariableCam Timing Spark Ignition Engine," ASME Journal of Dy-namic Systems, Measurement and Control, vol. 120, pp. 257-266, 1998.

[18] A. G. Stefanopoulou, J. W. Grizzle and J. S. Freudenberg, \En-gine Air-Fuel Ratio and Torque Control using SecondaryThrot-tles," Proc. 1994 Conf. on Decision and Control, pp. 2748-2753,Orlando, 1994.

[19] A. G. Stefanopoulou and I. Kolmanovsky, \Analysis and Con-trol of Transient Torque Response in Engines with Internal Ex-haust Gas Recirculation," IEEE Transactions on Control Sys-tem Technology, to appear.

[20] R. A. Stein, K. M. Galietti and T. G. Leone, \Dual EqualVCT| A Variable Camshaft Timing Strategy for ImprovedFuel Economy and Emissions," SAE Paper No. 950975, 1995.

Anna G. Stefanopoulou obtained herDiploma (1991, Nat. Tech. Univ. of Athens,Greece) and M.S. (1992, Univ. of Michigan,U.S.) in Naval Architecture and Marine Engi-neering. She received her M.S. (1994) and herPh.D. (1996) in Electrical Engineering andComputer Sc. from the University of Michi-gan. Dr. Stefanopoulou is presently an Assis-tant Professor at the Mechanical EngineeringDept. at the University of California, SantaBarbara. Dr. Stefanopoulou is Vice-chair of

the Transportation Panel in ASME DSCD and a recipient of a 1997NSF CAREER award. Her research interests are multivariable feed-back theory, control architectures for industrial applications, andpowertrain modeling and control.

James S. Freudenberg received BS de-grees in mathematics and physics from Rose-Hulman Institute of Technology, and the MSand PhD degrees in electrical engineeringfrom the University of Illinois in 1982 and1985. Since 1984 he has been on the faculty ofthe University of Michigan, where he is cur-rently an Associate Professor in the ElectricalEngineering and Computer Science Depart-ment. Dr. Freudenberg received a Presiden-tial Young Investigator Award, and is a past

Associate Editor of the IEEE Transactions on Automatic Control.His research interests are in the theory of fundamental design limi-tations, and in applications to automotive powertrain and semicon-ductor manufacturing control.

A recent picture of Dr. Freudenberg can be found in the Decem-ber 1997 issue IEEE Transactions on Automatic Control.

Jessy W. Grizzle received the Ph.D. inelectrical engineering from The University ofTexas at Austin in 1983. Since September1987, he has been with The University ofMichigan, Ann Arbor, where he is a Profes-sor of Electrical Engineering and ComputerScience. He is a past Associate Editor ofthe Transactions on Automatic Control andSystems & Control Letters, served as Publi-cations Chairman for the 1989 CDC, and in1997 was elected to the Control Systems So-

ciety's Board of Governors. Dr. Grizzle's major research interestslie in the �eld of control systems. Since his doctoral work, he hasinvestigated theoretical questions in nonlinear systems, where hehas concentrated on discrete-time problems and observer design.He has been a consultant in the automotive industry since 1986,where he jointly holds several patents dealing with emissions reduc-tion through improved controller design. In 1992, along with K. L.Dobbins and J. A. Cook of Ford Motor Company, he received thePaper of the Year Award from the IEEE Vehicular Technology So-ciety. Since 1991, he has worked with an interdisciplinary team ofresearchers at the University of Michigan on applying systems andcontrol techniques to improve the operation of plasma-based mi-croelectronics manufacturing equipment. Dr. Grizzle was a NATOPostdoctoral Fellow from January to December 1984; he receiveda Presidential Young Investigator Award in 1987, the University ofMichigan's Henry Russell Award for outstanding research in 1993,a College of Engineering Teaching Award, also in 1993, and waselected to Fellow of the IEEE in 1997.