Control Engineering Practice 20 (2012) 421–430

Contents lists available at SciVerse ScienceDirect

Control Engineering Practice

0967-06

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/conengprac

Model predictive control of HCCI using variable valve actuation andfuel injection

Nikhil Ravi a,n, Hsien-Hsin Liao a, Adam F. Jungkunz a, Anders Widd b, J. Christian Gerdes a

a Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, United Statesb Department of Automatic Control, Lund University, Lund, Sweden

a r t i c l e i n f o

Article history:

Received 23 August 2010

Accepted 10 December 2011Available online 9 January 2012

Keywords:

HCCI

Model predictive control

Actuator constraints

Split injection

61/$ - see front matter & 2011 Elsevier Ltd. A

016/j.conengprac.2011.12.002

esponding author.

ail address: nikhil.ravi@us.bosch.com (N. Rav

a b s t r a c t

Control of work output and combustion phasing on a Homogeneous Charge Compression Ignition

(HCCI) engine is essential to realize the benefits of superior efficiency and emissions. This paper

presents a model predictive control approach for cycle-by-cycle control of HCCI while respecting

constraints on actuators that might exist on a production implementation. The strategy is based on a

physical model developed in previous work and uses valve actuation and split fuel injection to achieve

the control objectives. In addition, it considers constraints on air–fuel ratio, ensuring that the system

stays away from very lean or rich regions. Simulation and experimental results show that the controller

works well over a range of conditions, and demonstrate the potential of this approach as a practical

cycle-by-cycle control strategy for HCCI.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Homogeneous Charge Compression Ignition (HCCI) enginespresent significant opportunities for reduced emissions andhigher efficiencies. However the lack of a direct combustiontrigger such as a spark makes the control of the phasing ofcombustion a challenging task. In addition, cycle-to-cycledynamics that are induced by the exhaust gases used to initiatecombustion make closed-loop control imperative for practicalHCCI implementation.

Different inputs have been considered for control of HCCI –these include control of the intake air temperature (Haraldsson,Tunestal, Johansson, & Hyvonen, 2004; Martinez-Frias, Aceves,Flowers, Smith, & Dibble, 2000); compression ratio (Christensen,Hultqvist, & Johansson, 1999); external exhaust gas recirculation– EGR (Kang, Chang, Chen, & Chang, 2009; Shi, Cui, Deng, Peng, &Chen, 2006); internal EGR through variable valve actuation (Law,Kemp, Allen, Kirkpatrick, & Copland, 2001; Shaver, Roelle et al.,2005); and dual-fuel mixtures (Bengtsson, Strandh, Johansson,Tunestal, & Johansson, 2004; Olsson, Tunestal, Haraldsson, &Johansson, 2001).

While each of the above strategies have been shown to beuseful for controlling HCCI, they each come with their ownchallenges when considered for practical implementation onmass-produced engines. Thermal inertia makes intake heating

ll rights reserved.

i).

more difficult to use for fast control during transients, and istherefore not suitable for cycle-by-cycle control. Varying thegeometric compression ratio on an engine real-time is a challen-ging proposition. The thermal effect of external EGR is limited dueto the heat loss that occurs along the EGR path. Dual-fuelstrategies would require changes in gas pump infrastructure, aswell as consumer refueling behavior. Internal EGR with variablevalve actuation (VVA) can be extremely powerful in terms of itseffects on combustion. In previous work (Ravi et al., 2010), suchcycle-by-cycle control of HCCI using a flexible VVA system anddirect injection of fuel was demonstrated. Cycle-by-cycle controlof HCCI was seen to offer two key benefits—it enabled accuratetracking even during fast transients, and could reduce the CoV ofcombustion at operating points that were otherwise highlyvariable, thereby extending the useful HCCI operating range.The major drawback of this approach, however, is that typicalhardware used to achieve variable valve actuation (such as camphasers) cannot achieve the kind of bandwidth needed for controlon every engine cycle, nor can they be used for controlling eachcylinder separately. Therefore it is necessary to develop a controlstrategy that can achieve cycle-by-cycle and cylinder-individualcontrol of HCCI while only using a production cam phaser system,and other inputs available on current production vehicles.

One promising actuator in such a scenario is fuel injec-tion—the timing and quantity of fuel injection can be controlledwith a direct injection system, and can have a strong effect onHCCI combustion. This is particularly true when HCCI is achievedwith exhaust recompression, where the exhaust valve is closedearly in the exhaust stroke to trap and recompress a portion of the

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430422

hot combustion products. When fuel is injected into the cylinderduring this recompression process, it can undergo reactions thatcan significantly influence the phasing of the main HCCI combus-tion (Hiraya, Kakuho, Urushihara, & Ito, 2002; Song & Edwards,2008). In previous work (Ravi et al., 2012), the authors havepresented simple control strategies that use the timing of a smallpilot injection during recompression to control the phasing ofcombustion in HCCI. The main advantages of using fuel injectionas a control knob stem from the fact that production vehiclestoday come equipped with direct injection systems that wouldallow cycle-by-cycle and cylinder-individual control of fuel injec-tion quantity and timing. However the authors have demon-strated in previous work (Ravi et al., 2012) that fuel injectiontiming has only a limited range of influence.

Fuel injection and valve actuation are therefore both powerfulknobs that can be used to control HCCI—but they both come withtheir own limitations from the standpoint of cycle-by-cyclecontrol over a broad operating range. This paper therefore pre-sents an approach that combines these inputs as a first steptowards a strategy for cycle-by-cycle control of HCCI that can berealized with actuators that currently exist on production cars.The approach is based on a model predictive control (MPC)strategy that enables coordinated control of fuel injection quan-tity, injection timing and valve timings while considering con-straints that exist on each of these actuators. Though MPC hasbeen used for HCCI control in the past (Bengtsson, Strandh,Johansson, Tunestal, & Johansson, 2006; Widd, Tunestal, &Johansson, 2008, 2009), it has largely been used to set absolutelimits on actuator range as well as constrain outputs such as therate of pressure rise. Imposing rate constraints on valve actuationand the use of injection timing, both of which are central to thework presented here, have not been considered.

In addition to considering actuator constraints, it might also bedesirable to constrain other variables. One such variable consideredin this work is the air–fuel ratio within the engine cylinder. As someof the main benefits of HCCI are related to lean operation, it isdesired to keep the air–fuel ratio away from the rich region. Olssonet al. (2002) show that there is a risk of producing high levels of NOxif the mixture is too rich. Very lean combustion is also not desirableas it can have a detrimental impact on combustion stability—thelean limit for air–fuel ratio can be linked to allowable limits in the

90 100 110 120 132.6

2.7

2.8

2.9

3

3.1

3.2

NM

EP

(bar

)

90 100 110 120 139

9.510

10.511

11.512

12.513

Tota

l fue

l inj

ecte

d (m

g)

T

Fig. 1. Effect of fuel q

CoV of NMEP. Xu et al. (2007) show that for any given intake valvetiming, there is a range of air–fuel ratios where combustion isconsistent and the phasing of combustion falls within an acceptablerange. Therefore the controller presented here attempts to constrainair–fuel ratio within a finite range.

The control strategy is based on a physical model of the systemdeveloped in previous work, and is complemented by an estima-tor that uses two measurements – combustion phasing and themeasurement from an exhaust oxygen sensor – to estimate themodel states. The controller is tested both in simulation and on anHCCI engine testbed. Results show that the controller is effectivein satisfying all the control objectives—the work output andcombustion phasing are tracked accurately over a wide range,actuator constraints are respected, and the air–fuel ratio ismaintained within respectable bounds.

2. Physical model for HCCI with existing actuators

2.1. Effects of inputs in HCCI

Before delving into an overview of the physical model, it isuseful to understand the effects of the various inputs currentlyavailable for control of the HCCI process. In particular, theireffects on two specific outputs are of interest—the phasing ofcombustion (defined in terms of CA50, the crank angle locationwhere 50% of the energy from combustion has been released) andthe work output (measured in terms of the net mean effectivepressure during an engine cycle, NMEP).

The model focuses on an exhaust recompression HCCI engine withvariable valve actuation and direct fuel injection. A split injectionstrategy is used—most of the fuel (quantity determined by thecontroller) is injected at a fixed timing at the end of recompression,while a 1 mg pilot injection is injected at a variable timing (againdetermined by the controller) during recompression. This strategy hasbeen shown to be extremely useful in control of work output andcombustion phasing in previous work (Ravi et al., 2012).

Therefore the main inputs available for control are

1.

0

0ime

uan

total moles of fuel injected, nf ,k;

2. cylinder volume at intake valve closure, VIVC,k;

140 150 160 170

140 150 160 170(s)

tity on NMEP.

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430 423

3.

cylinder volume at exhaust valve closure, VEVC,k and 4. pilot injection timing, uth,k.

361

362

363

364

365

366

367

CA 50

(CA

D)

The total fuel quantity has a strong influence on the net workoutput or NMEP as shown in Fig. 1 (data obtained from one cylinderof a multi-cylinder HCCI testbed)—more fuel injection leads to moreenergy release and therefore a higher NMEP. Changing the fuelquantity has little impact on combustion phasing on the sameengine cycle, which is much more strongly dependent on themixture temperature and oxygen concentration. However theinherent dynamics of the process ensure that changing fuel quantityon cycle k influences combustion phasing on cycle kþ1 because ofits effect on the total energy release on cycle k which in turn affectsthe mixture temperature on cycle kþ1.

The intake and exhaust valve timings influence the relativequantities of fresh air and trapped exhaust, thereby influencingthe phasing of combustion. For example, retarding the exhaustvalve closing time (EVC) reduces the quantity of trapped exhaust,thereby lowering the sensible energy of the reactant mixture onthe next engine cycle and retarding the phasing of combustion.This is confirmed by the data presented in Fig. 2, which shows aretarding of CA50 with a step increase in EVC timing.

Finally, fuel injection during recompression is seen to have asignificant effect on the phasing of combustion, due to reactionsthat the fuel can undergo in the moderately high temperatureconditions that exist during recompression (Song & Edwards,2008). The nature of these reactions can be quite complex anddepends on factors such as the air–fuel ratio, but their net effect isto advance the phasing of combustion. Even an 1 mg pilotinjection during recompression can have a significant effect, asseen in Fig. 3—an earlier injection gives the fuel a longerresidence time during recompression to undergo reaction, andconsequently leads to an earlier combustion phasing.

-50 -40 -30 -20 -10 0 10 20 30 40359

360

Pilot injection timing (CAD)

Main injection timing: 60 CADMain injection quantity: 9mgPilot injection quantity: 1mg

Fig. 3. Variation of combustion phasing (CA50) with pilot injection timing.

2.2. Control model

In order to capture the various physical effects described in theprevious section, a five state discrete-time nonlinear model hasbeen developed in previous work (Ravi et al., 2012, 2010). A briefoverview is presented here.

380 390 400 410355

360

365

370

375

380 390 400 410635

640

645

650

655

EV

C (C

AD

)

Tim

CA

(C

AD

)50

Fig. 2. Effect of EVC

Fig. 4 shows a graphical summary of the nonlinear model withreference to a typical in-cylinder pressure trace during HCCI. Thestates, outputs and inputs of this model are given by

xk ¼ ½½O2�s Ts ½f �s VIVCsKth�

Tk , yk ¼ ½CA50 NMEP�Tk

uk ¼ ½nf VEVC VIVC uth�Tk

The states are defined in terms of the oxygen and fuelconcentrations in the reactant mixture (½O2�s and ½f �s), the mixturetemperature (Ts), cylinder volume at intake valve closure (VIVCs

)and a combustion threshold value (Kth) which captures theinfluence of the pilot injection timing. These states are definedat a fixed location ys after IVC during the compression stroke, asthe characteristics of this reactant mixture are what influencecombustion. This location is chosen here as ys¼300 CAD (60 CADbefore TDC-combustion). Every engine cycle (indexed by k),therefore, is assumed to begin at this location.

The overall nonlinear model can be expressed as

xkþ1 ¼ Fðxk,ukÞ

420 430 440 450 460

420 430 440 450 460

e (s)

timing on CA50.

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430424

yk ¼ GðxkÞ ð1Þ

where the functions F and G are obtained by breaking down asingle HCCI cycle into several distinct processes, each of whichcan be modeled based on simple thermodynamic assumptions.

The phasing of combustion is captured through an integratedArrhenius model based on global reaction kinetics. The reactionrate for the global reaction is given by

RR¼ AtheðEa=RuTÞ½fuel�a½O2�b ð2Þ

where Ea is the activation energy, Ath is a pre-exponential factor,Ru is the universal gas constant and a and b are constants.Integrating this global Arrhenius rate equation from ys to thestart of combustion gives

ZRR dt¼

Z ysoc

ys

AtheðEa=RuTÞ½fuel�a½O2�b

o dy ð3Þ

Combustion is assumed to begin when the integral in Eq. (3)crosses Kth, a pre-determined threshold. This expression can thenbe used to calculate the phasing of combustion, CA50.

Changing the intake and exhaust valve timings influences theoxygen concentration and temperature of the reactant mixture inthe above equation, while changing the total fuel quantity

0 10 20 30 40640

642

644

646

648

EV

C (C

AD

)

0 10 20 30 40 507

7.5

8

8.5

9

9.5

10x 10−3

O2

stat

e (k

mol

/m3 )

Cycle

Fig. 5. Comparison of control and simulati

Crank angle

45

40

35

30

25

20

15

10

5

Pre

ssur

e (b

ar)

Fig. 4. Nonlinear control model—states, inputs and outputs.

influences the fuel concentration. The primary effect of a varyingpilot injection timing is modeled as a change in the Arrheniusthreshold value, Kth. The rationale behind this is that the reactionsduring recompression lower the overall threshold for the maincombustion reaction. Therefore the injection timing input, uth,directly affects the Arrhenius threshold state, Kth.

The relationship between the states and the other output ofinterest, NMEP, can be derived from the model based on the in-cylinder pressure at different points during the cycle. However,the work output is essentially determined by the quantity of fuelinjected into the cylinder when the engine is operating at thedesired combustion phasing. Therefore the NMEP here is consid-ered to be only a function of the fuel state.

NMEP¼ k½f � ð4Þ

Model parameters are tuned based on comparing the state andoutput predictions with a more complex continuous-time simula-tion model of HCCI developed by Shaver, Gerdes, Roelle, Caton,and Edwards (2005). The model is tuned based on data from onecylinder of a multi-cylinder engine. Parameterization and valida-tion results have been presented in previous work (Ravi et al.,2010, 2012). As an example, Fig. 5 shows the response of thecontrol model to a step change in EVC, along with the response ofthe simulation model, showing a very good dynamic matchbetween the two. Also plotted is the response of a linearizedversion of the model obtained through an analytical linearizationaround a nominal operating point. The linear state space model isgiven by

xkþ1 ¼ AxkþBuk

yk ¼ Cxk ð5Þ

where A, B and C are matrices that are functions of the operatingcondition about which the system is linearized.

The conceptual structure of these matrices is shown in Eq. (6),where X represents a non-zero value (determined by the parti-cular operating condition). The actual numerical values of these

0 10 20 30 40356

358

360

362

364

366

CA

50 (C

AD

)

0 10 20 30 40 50630

640

650

660

670

Tem

pera

ture

sta

te (K

)

Cycle

Continuous Simulation

Nonlinear control model

Linear control model

on models over a step change in EVC.

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430 425

matrices are provided in Appendix A.

½O2�s

Ts

½f �sVIVCs

Kth

26666664

37777775

kþ1

¼

X X X X X

X X X X X

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

26666664

37777775

½O2�s

Ts

f� �

s

VIVCs

Kth

26666664

37777775

k

þ

X X X 0

X X X 0

X 0 0 0

0 0 X 0

0 0 0 X

26666664

37777775

nf

VEVC

VIVC

uth

266664

377775

k

CA50

NMEP

� �k

¼X X X X X

0 0 X 0 0

� �½O2�s

Ts

f� �

s

VIVCs

Kth

26666664

37777775

k

ð6Þ

As seen, the last three states have no dynamics of their own,but are purely dependent on the inputs on the previous cycle(respectively, nf, VIVC and uth, shown by the Xs in the last threerows of the B matrix). The oxygen and temperature states,however, depend on all the states on the previous cycle, as wellas the first three inputs. It is assumed here that they do notdepend on the final input, uth—or in other words, that theinjection of a small quantity of pilot fuel during recompressionhas a much more significant effect on the Arrhenius thresholdthan on the oxygen concentration or temperature of the finalreactant mixture after IVC.

This model can now be used as the basis for the developmentof a control strategy that allows tracking of load and phasingwhile maintaining desired constraints on the inputs. There areseveral ways in which this can be done—here, a model predictivecontrol approach is adopted. MPC allows the explicit specificationof rate and range constraints on inputs. These are both importanthere, as the rate of change of valve timings is constrained by thehardware limits of cam phaser systems, while the range withinwhich the pilot injection timing influences CA50 is limited. MPCprovides a convenient framework to specify these constraints aswell as constraints on outputs such as air–fuel ratio, while at thesame time allowing tradeoffs between different actuators to bespecified with cost-function weights.

One concern with MPC is the computation time required,especially with long time horizons. However the HCCI processtypically has little dependence on mixture conditions more than2–3 engine cycles earlier, which allows a short time horizon. Thiscan be understood from the perspective of the trapped exhaust onany engine cycle—the relative quantity of this exhaust thatremains trapped within the engine cylinder on successive enginecycles rapidly decreases and becomes insignificant after 3–4engine cycles. Therefore there is little to be gained with largeprediction and control horizons. Widd et al. (2009) confirm thisby testing predictive controllers with different time horizons onan HCCI engine, and conclude that an increase in time horizonbeyond 4–5 cycles has no discernible effect on performance,while adding a significant amount of computation time. ThereforeMPC is quite easy to implement in HCCI.

3. Framework for cycle-by-cycle control with actuator andoutput constraints

The main objectives of HCCI control considered in this paperare the tracking of desired load and combustion phasing trajec-tories. Based on a linearization of the control model presentedearlier, it is easy to incorporate this control problem in a standardMPC framework.

The system constraints considered by the controller are

1.

Absolute limits on the cylinder volume at intake and exhaustvalve closure.

2.

Absolute limits on the timing of the pilot injection, such thatthe region of saturation in Fig. 3 is avoided.

3.

A limit on the maximum allowable rate of change of the valvetimings.

4.

Limits on the maximum and minimum allowable air–fuelratio.

The first three constraints involve the control inputs and socan be easily defined while formulating the MPC optimization.Constraints on AFR in this work are modeled as constraints on theratio of oxygen and fuel in the reactant mixture. This is necessarybecause a conventional definition of air–fuel ratio does nottranslate directly to the oxygen–fuel ratio in exhaust recompres-sion HCCI because of the presence of trapped exhaust gases. AsHCCI combustion is typically lean, these trapped gases have someoxygen—therefore the final oxygen–fuel ratio before combustionis determined both by the quantity of inducted air, as well as thequantity and oxygen concentration of trapped exhaust.

Therefore lambda (l) is defined as the ratio of the actualoxygen–fuel ratio and the stoichiometric oxygen–fuel ratio for theparticular fuel under consideration. Therefore

l¼½O2�=½F�

ð½O2�=½F�Þstoich

ð7Þ

Constraints on AFR therefore are considered equivalent toconstraints on l.

AFRminrAFRrAFRmax()lminrlrlmax ð8Þ

For gasoline fuel modeled as C7H13, the stoichiometric reactionequation is given by

C7H13þ10:25O2þ38:54N2-7CO2þ6:5H2Oþ38:54N2 ð9Þ

Then, based on Eq. (9), if lrlmax, then

½O2�

½F�r10:25lmax ) ½O2��10:25lmax½F�r0 ð10Þ

Similarly

lZlmin ) ½O2��10:25lmin½F�Z0 ð11Þ

Therefore the bounds on the oxygen–fuel ratio can be trans-lated to equivalent bounds on linear combinations of two of thesystem states—oxygen and fuel concentrations.

Based on the above discussion, the output (y) vector isextended to be

y¼ ½CA50 NMEP yAFR,min yAFR,max�T ð12Þ

where

yAFR,min ¼ ½O2��10:25lmin½F�, yAFR,max ¼ ½O2��10:25lmax½F�

The overall MPC problem then is defined as follows:Minimize

JðkÞ ¼XHp�1

i ¼ 0

JCA50ðkþ i9kÞ�CA50,desðkþ i9kÞJ2Q1

þXHp�1

i ¼ 0

JNMEPðkþ i9kÞ�NMEPdesðkþ i9kÞJ2Q2

þXHu�1

i ¼ 0

JDuðkþ i9kÞJ2Rþ

XHu�1

i ¼ 0

Juthðkþ i9kÞ�uth,ref J2QuþXHp�1

i ¼ 0

rJEJ2

subject to

xkþ1 ¼ AxkþBuk

yk ¼ Cxk

ufuelZ0

2

2.5

3

3.5

NM

EP

(bar

)

DesiredActual

360

365

370

CA 50

(CA

D)

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 1001.4

1.6

Lam

bda

Engine cycle

ActualBounds

Fig. 6. MPC implemented in simulation with no constraint on l-outputs.

10

11

12

13

Fuel

qua

ntity

(mg)

640645650655660

EV

C (C

AD

)

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100−40

−20

0

20

40

Pilo

t inj

ectio

ntim

ing

(CA

D)

Engine cycle

ActualReference

Fig. 7. MPC implemented in simulation with no constraint on l-inputs.

2

2.5

3

3.5

NM

EP

(bar

)

DesiredActual

360

365

370

CA 50

(CA

D)

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 1001.4

1.6

Lam

bda

Engine cycle

ActualBounds

Fig. 8. MPC implemented in simulation with constraint on l-outputs.

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430426

Duevc,minrDuevc,krDuevc,max

uevc,minruevc,kruevc,max

uth,minruth,kruth,max

yAFR,min,kZ0�E

yAFR,max,kr0þE ð13Þ

The cost function in this optimization has five terms. The first tworepresent costs on output tracking errors—in CA50 and NMEP respec-tively. The third is a cost on control effort. The input cost matrix R isset such that only two of the three available inputs are used—theintake valve is kept fixed and not controlled. The fourth term in thecost function imposes a cost on deviations of the pilot injectiontiming input from a reference—this cost ensures that all things beingequal, an input trajectory that maintains the pilot injection timingcloser to a reference (set at the middle of its range) will be preferred.Therefore this term serves to achieve some form of mid-rangingaction as described in previous work (Ravi et al., 2012) so as to keepthe injection timing in a range where it has maximum controlauthority. However by incorporating this into the cost function, it isensured that if necessary, the injection timing can deviate from thereference value—either to lower the cost of output deviations, or toensure constraint satisfaction. The cost Qu is set at a value muchsmaller than the cost Q on the output error, thereby giving priority tooutput tracking over mid-ranging.

The last term of J(k) imposes a cost on E, which is a slackvariable. This slack variable essentially softens the last twoconstraints, which represent the lower and upper bounds on las described earlier. Softening these constraints is necessary todeal with the possibility of infeasibility—a condition that mightoccur, for example, with an unexpected large disturbance, inwhich case it might be impossible to keep the plant within thespecified constraints. Introducing the slack variable E ensures thatthe constraints can be crossed occasionally, but only if reallynecessary. The relative allowance given for constraint violationcan be controlled with the weight r—a higher value of r makesthe constraint relatively ‘‘hard’’ to violate.

The other inequalities represent constraints on the actuators—

absolute limits on EVC and injection timing, as well as a rate limiton the EVC. These inequalities are not softened with the slackvariable as these represent hardware constraints and thereforeshould not be violated under any circumstances.

The prediction horizon Hp is set at three time steps, while thecontrol horizon Hu is set to be 1—as was explained earlier, these aresufficient to capture all the relevant dynamics in HCCI. This frame-work can also be extended to include other constraints, such asconstraints on the rate of pressure rise within the cylinder that havebeen applied in other MPC approaches (Bengtsson et al., 2006). Fornow, the focus is on constraining the rate of change of valve timings,and the range of allowable injection timing and air–fuel ratio.

The controller is tested on the continuous time simulationmodel (presented by Shaver, Gerdes et al., 2005) against whichthe control model was validated. Figs. 6 and 7 show the outputand input trajectories with the predictive controller without anyconstraints on l, while Figs. 8 and 9 show results from the sametest repeated with constraints on l. The controller in each test isswitched on after 10 engine cycles. As seen, in both cases, thecontrol outputs (NMEP and CA50) are tracked well in steady state.However in the first case the constrained output l does not staywithin the desired bounds, while in the second it remainsconstrained. Also, the controller in the second case slows downthe response to a step change in desired NMEP by changing thefuel quantity command more slowly, as it recognizes that a stepchange in fuel quantity would lead to constraint violation. The

10

11

12

13Fu

el q

uant

ity (m

g)

640645650655660

EV

C (C

AD

)

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100−40

−20

0

20

40

Pilo

t inj

ectio

ntim

ing

(CA

D)

Engine cycle

ActualReference

Fig. 9. MPC implemented in simulation with constraint on l-inputs.

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430 427

other inputs can then be used to ensure that l stays within thedesired ranges on every engine cycle. The tracking of CA50

remains very similar to the earlier case because of the ability ofthe pilot injection timing to respond quickly to track phasing.

Note that the l here relates to the overall oxygen–fuel ratio inthe mixture before combustion as described earlier. The lowerand upper bounds here are set somewhat arbitrarily at 1.6 and1.7. Around the nominal operating condition, these boundscorrespond roughly to inducted AFR l’s (ratio of inducted air–fuel ratio to the stoichiometric air–fuel ratio) of about 1.25 and1.32 respectively. These are chosen to illustrate the controller’sresponse to constraints and not to reflect exact limits in practice.

The EVC input responds over several cycles to step changes inthe desired conditions, while the injection timing responds moreinstantly, due to the constraint on the rate of change of the EVCinput. Note that without the additional constraint on l, a mid-ranging effect is seen with the injection timing going to itsreference value over several cycles. However once the l con-straint is added, the injection timing input remains close to itsreference value, but does not track it exactly. This is to beexpected since it is not possible to meet the air-fuel ratioconstraint with the fuel injection in the middle of its range.

3600−3600

5

10

15

20

25

30

35

40

Crank angle

Pre

ssur

e (b

ar)

Fig. 10. Location in engine cycle where measurements are available.

4. Estimator structure

Two of the states, oxygen concentration and temperature, cannotbe directly measured and so need to be estimated. In previous work(Ravi et al., 2010, 2012) both states have been estimated based on asingle measurement of combustion phasing. However an exhaustoxygen sensor can give an independent measurement of the oxygenconcentration in the exhaust gas. If the oxygen sensor is located veryclose to the port, this measurement can serve as a good estimate ofthe oxygen concentration in the trapped exhaust within the enginecylinder and can be used to improve the estimation.

A wide band oxygen sensor typically gives a measurement of thefraction of oxygen in the exhaust gas. This measurement can becompared to an estimate of the fraction of oxygen in the trappedexhaust gas obtained from the control model. As the model stepsthrough each of the distinct stages that occur during an engine cycle,it is easy to obtain an expression for the percentage of oxygen in theexhaust gas as a function of the states that can be included in theoutput equation of the linear model. However including a measure-ment of rO2

in the estimation scheme presents one hurdle which

arises from the fact the measurements of CA50 and rO2are available at

different points in the engine cycle. This can be understood byconsidering the relationship between estimation and control. Typi-cally a Kalman filter estimator involves two steps:

1.

Time update, where the state estimate is updated based on themodel and the applied inputs.

xk9k�1 ¼Axk�19k�1þBuk�1 ð14Þ

2.

Measurement update, where the state estimate is updatedbased on the measured outputs.

xk9k ¼ xk9k�1þMðyk�Cxk9k�1Þ ð15Þ

where M is the Kalman filter gain matrix. The state estimate xk9k

can then be used to determine the control input uk for cycle k. It istherefore essential that all the measurements yk be obtainedbefore the control input can be determined.

However, as shown in Fig. 10, it is only the measurement of CA50

that becomes available before the inputs need to be applied—ameasurement of exhaust oxygen fraction, rO2

is only available after

the exhaust process, as the measurement occurs outside the enginecylinder. Based on this observation, the Kalman filter update isperformed in three distinct steps, with the measurement updatesplit into two steps.

1.

Time update, where the state estimate is updated based on themodel and applied inputs.

xk9k�1 ¼Axk�19k�1þBuk�1 ð16Þ

2.

Combustion phasing measurement update, where the stateestimate is updated based on the measured CA50:

xk9k,partial ¼ xk9k�1þMCA50ðCA50,meas,k�CCA50

xk9k�1Þ ð17Þ

3.

Oxygen sensor measurement update, where the state estimate isfurther updated based on the oxygen sensor measurement.

xk9k ¼ xk9k,partialþMO2ðrO2,meas,k

�CO2xk9k�1Þ ð18Þ

MCA50and MO2

are the Kalman gain vectors for each of theoutputs, CA50 and rO , while CCA and CO represent the rows of

2 50 2

7

7.5

8

8.5x 10−3

Oxy

gen

stat

e (k

mol

/m3 )

5 10 15 20 25 30 35 40 45 50

5 10 15 20 25 30 35 40 45 50630

635

640

645

650

Engine cycle

Tem

pera

ture

sta

te (K

)

Estimated state with 1 measurementEstimated state with 2 measurementsActual state

Fig. 11. Comparison of estimator performance in simulation with one vs. two

measurements.

1.5

2

2.5

3

3.5

NM

EP

(bar

)

MeasuredDesired

350

355

360

365

370

CA 50

(CA

D)

x 10−3

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430428

the C matrix corresponding to these two outputs. The third step ofthis estimation is actually computed on the following time step,once the oxygen sensor measurement is obtained. Therefore thebest estimate available to the controller is xk9k,partial and uses onlyone of the two measurements. However this is not a significantproblem, as the second measurement still serves to update theestimate before the next time step.

Fig. 11 shows a comparison of the performance of twoestimators – one that uses just the measurement of combustionphasing and one that also uses the measurement of the exhaustoxygen fraction – in simulation. These responses are plotted for astep increase in fuel quantity, while all other inputs are kept fixed.The estimation is started after 10 engine cycles. The stateestimates plotted here are the estimates used by thecontroller—therefore for the single measurement case the esti-mate is xk9k, while for the case with two measurements it isxk9k,partial. As seen, with two measurements the accuracy ofestimation is significantly improved and the estimate tracks theactual state value very closely. Apart from the steady state match,the estimator based on two measurements also captures thedynamic response much better, as seen when the step changein fuel mass is applied after 30 engine cycles.

It should be noted that this estimation strategy assumes ameasurement from a wide-band oxygen sensor in the exhaustport of each engine cylinder. If only a single oxygen sensormeasurement is available (such as from a sensor in the exhaustmanifold) the estimator will need to be modified, and possibly anexhaust manifold model will need to be included. However thisanalysis does serve to underline the benefits of an estimationscheme that takes into account both the phasing of combustionand the characteristics of the exhaust stream in determining thestates.

1500 2000 2500 3000 3500 4000 4500 5000−8−6−4−2024

y AFR

,min

and

yA

FR,m

ax

Engine cycle

yAFR,min (should be ≥ 0)

yAFR,max (should be ≤ 0)

Fig. 12. NMEP-phasing MPC implemented in experiment without AFR constraints—

outputs.

5. Experimental implementation

The estimator and MPC scheme are implemented on a fourcylinder General Motors L850 gasoline engine modified to runHCCI. The engine runs on a compression ratio of 12:1. A commonrail direct injection system is used to inject fuel directly into eachcylinder. The testbed is equipped with a fully flexible VVA systemthat can be used to actuate the intake and exhaust valves on eachcylinder independently. A Kistler 6125 piezoelectric sensor

provides in-cylinder pressure information. The sensor has anatural frequency of 70 kHz, and is sampled at 10 kHz (approxi-mately 1 sample/CAD at 1800 rpm)—this provides pressure datathat is accurate enough to calculate CA50 on each engine cycle.The exhaust oxygen concentration is measured by a Bosch LSUbroadband lambda sensor fitted directly in the exhaust port. AnETAS LA4 lambda meter samples the data from the lambda sensorat a frequency of 200 Hz. This is fast enough to get an accuratemeasure of oxygen concentration in the exhaust port, which isused as a proxy for the oxygen concentration in the trappedexhaust inside the cylinder. There is no additional filtering ofeither the CA50 or the exhaust oxygen concentration data beforeuse for estimation/control.

An explicit representation of the model predictive controller asdescribed by Bemporad, Borrelli, and Morari (2002) is obtainedand implemented on the engine through Matlab’s hybrid controltoolbox. The explicit representation takes the form of a piecewiselinear and continuous state feedback control law. This is advanta-geous, as the online computation is reduced from a complexquadratic optimization to a simple linear evaluation. Additionally,the stability and performance of the explicit representationremains identical to the original model predictive controller.

The state estimates on a particular cycle are updated based onmeasurements of CA50 and the exhaust oxygen sensor on thatengine cycle (and not averages over previous cycles). Theseestimates are then used by the MPC scheme to determine theappropriate inputs for the next engine cycle.

Figs. 12 and 13 show the output and input trajectories on oneof the four cylinders of the engine over a series of step changes indesired load and phasing. This optimization is run without the lconstraints (the constraints on yAFR,min and yAFR,max in Eq. (13))active. Both work output and phasing are tracked fairly accu-rately, including over step changes in load as large as 1 bar atconstant phasing (around 3400 cycles). There is some steady stateerror seen at higher loads around 3 bar—this can be attributed tothe fact that a single model is used over the entire range, andthere is no integral action to correct for linearization errors oversuch a wide range. The maximum CoV of NMEP during this test isabout 2.8%.

Also plotted in the bottom plot of Fig. 12 are the outputsyAFR,min and yAFR,max, representing the constraints on l. The lower

6

8

10

12

14Fu

el q

uant

ity(m

g)

640

650

660

EV

C (C

AD

)

1500 2000 2500 3000 3500 4000 4500 5000−30

−20

−10

0

10

Pilo

t inj

ectio

ntim

ing

Engine cycle

Fig. 13. NMEP-phasing MPC implemented in experiment without AFR constraints—

inputs.

1.5

2

2.5

3

3.5

NM

EP

(bar

)

MeasuredDesired

350

355

360

365

370

CA 50

(CA

D)

1500 2000 2500 3000 3500 4000 4500 5000−8

−6

−4

−2

0

2

4x 10

y AFR

,min

and

yA

FR,m

ax

Engine cycle

yAFR,min (should be ≥ 0)

yAFR,max (should be ≤ 0)

Fig. 14. NMEP-phasing MPC implemented in experiment with AFR constraints—

outputs.

68

101214

Fuel

qua

ntity

(mg)

640

650

660

EV

C (C

AD

)

1500 2000 2500 3000 3500 4000 4500 5000−30−20−10

010

Pilo

t inj

ectio

ntim

ing

Engine cycle

Fig. 15. NMEP-phasing MPC implemented in experiment with AFR constraints—

inputs.

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430 429

and upper limits chosen for l here are lmin ¼ 1:6 and lmax ¼ 2, aslightly wider range than in simulation so as to give the controllersome flexibility. Based on the optimization problem presented inEq. (13), it is desired to keep yAFR,minZ0 and yAFR,maxr0. Thiswould ensure that 1:6rlr2. However it is seen here that theconstraints are exceeded significantly several times during thetest. This is not surprising since the constraints on yAFR,min andyAFR,max are not active here. In particular, large excursions are seenany time step changes in NMEP are commanded. This can beattributed to the step change in fuel quantity—as there are noconstraints on l, the fuel quantity very quickly ramps up/down tothe new value desired, while the pilot injection timing alsochanges quickly to maintain the desired combustion phasing.Over several cycles this input is brought back to its referencevalue by a slowly changing EVC.

The same test is then run with the constraints on yAFR,min andyAFR,max activated—this significantly changes the behavior, as seenin Figs. 14 and 15. The tracking of the two control outputs, NMEP

and CA50, remains as accurate—however the new steady state inNMEP is reached a little more slowly, especially during large stepchanges, such as around 3400 cycles. This is because the con-troller recognizes the constraint on yAFR,min and so slows downfuel changes to prevent it from going below zero. Additionally, thepilot injection timing no longer tracks its reference value—whileit remains close, there are significant deviations seen, especially athigher loads (between 3400 and 4500 cycles). Consequently theconstraints on yAFR,min and yAFR,max are met most of the time, andonly at very high loads is there a slight violation of the yAFR,minZ0constraint, implying that lo1:6. This is not surprising, as there isa tradeoff between output tracking and constraint satisfaction,and with such a high load demand it becomes impossible tosatisfy both requirements. Therefore it is only when somethinginfeasible is demanded of the controller that it allows a violationof the output constraint. One point to be noted though is thateven this constraint violation can be avoided by increasing theweight r on the slack variable in the optimization, E. This wouldlead to poorer output tracking when faced with an infeasibledesired output trajectory. Therefore the relative tradeoff betweenaccurate output tracking and constraint satisfaction can be set ata desired value by changing the value of this weight in the costfunction. The maximum CoV of NMEP in this case is 2.3%, which isslightly reduced as compared to the test without l constraints.

6. Conclusion

The control framework presented in this paper shows greatpromise as a practical tool for cycle-by-cycle HCCI control.Variable valve actuation and fuel injection were both recognizedas powerful control knobs in HCCI—however each of these inputscomes with its own limitations. The framework presented hereaims to explicitly specify these actuator constraints that representreal physical limitations on production engines. This is donethrough framing the control objectives in terms of a modelpredictive control problem. In particular, the range limitationsof the pilot injection timing input, and the speed of responselimitations of a cam phaser system can both be accounted for.Additionally, MPC allows the constraining of outputs such as theair–fuel ratio. It was seen that constraints on the air–fuel ratio,expressed through the oxygen–fuel ratio in the reactant mixture,could be modeled as constraints on a linear combination of themodel states. Simulation results show that the controller satisfiesall the desired objectives, tracking desired output trajectorieswhile meeting actuator restrictions and staying within the

N. Ravi et al. / Control Engineering Practice 20 (2012) 421–430430

specified bounds on air–fuel ratio. When implemented on anexperimental HCCI testbed, the performance is very similar tothat in simulation, with good output tracking and constraintsatisfaction.

The work presented in this paper therefore demonstrates thepossibility of developing a cycle-by-cycle controller for HCCI thatrespects constraints that exist on production engines. The MPCframework presented here is one such approach, and representsan important step towards practical and robust control strategiesfor HCCI. As a next step in that direction, this work will need to beexpanded to consider control of multiple cylinders within a singleMPC framework. For this it would be necessary to expand theframework to explicitly include interactions between cylinders.This might require the inclusion of an intake or exhaust manifoldmodel, especially if exhaust oxygen sensors were not available oneach exhaust port. Also of interest would be a control strategythat uses a single cam phaser for all the engine cylinders (andtherefore independent valve actuation on each cylinder would beeliminated) while using fuel injection as a cylinder-individualcontrol knob to balance differences. Other aspects that will needto be considered in some form include emissions and efficiency,both of which could potentially be optimized within thisframework.

Appendix A. Linearized model used for controllerdevelopment

The linear model that is used as a basis for the control resultspresented in this paper was obtained by linearization around anominal HCCI operating point. The numerical values of the matrixelements in this model are given in Eq. (A.1).

½O2�s

Ts

f� �

s

VIVCs

Kth

26666664

37777775

kþ1

¼

0:64 0:22 �0:47 �0:13 �0:02

�0:07 �0:07 0:14 �0:43 0:06

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

26666664

37777775

½O2�s

Ts

f� �

s

VIVCs

Kth

26666664

37777775

k

þ

0:05 �1:79 2:74 0

�0:02 0:33 �0:04 0

1 0 0 0

0 0 1 0

0 0 0 1

26666664

37777775

nf

VEVC

VIVC

uth

266664

377775

k

CA50

NMEP

� �k

¼�0:08 �0:76 0:02 0 0:07

0 0 1 0 0

� �½O2�s

Ts

f� �

s

VIVCs

Kth

26666664

37777775

k

ðA:1Þ

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