Hybrid control of a gasoline direct injection engine.pdf

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Proceedings of the 38 T hAO l 10:40

Conference on Decision & ControlPhoenix, Arizona USA December 1999

Hybrid Control of a Gasoline Direct Injection Engine

Maria Druzhinina Ilya Kolmanovsky Jing SunFord Research Laboratory Dearborn Michigan 48121-2053.

Abstract

This paper describes an automotive control problem

where switching is an essential feature of the opera-

tion of a direct injection spark ignition engine. The

engine operates in two distinct combustion modes with

different emissions and torque characteristics: the ho-

mogeneous mode and the stratified mode. The control

system must be capable of changing the combustion

mode and the air-to-fuel ratio of the engine rapidly

and without any noticeable torque disturbance to the

driver. A hybrid control scheme is described in this

paper to control the mode transitions in this engine

and its operation is illustrated with simulations on a

mean-value model of the engine.

1 Introduction

Hybrid and switching systems are common in power-

train control applications [2]. In this paper we describe

a case study based on a gasoline direct injection strati-

fied charge (DISC) spark ignition engine, see Figure 1.

This engine can combust fuel in two distinct modes cor-

responding to either a stratified fuel-air mixture (strat -

ified mode) or a well-mixed homogeneous fuel-air mix-

ture (homogeneous mode). With stratified combus-

tion? the DISC engine is able to operate at extremely

lean overall air-to-fuel ratios (up to 50:l as compared

to 14.64:l for stoichiometric operation of conventional

PF I engines). The stratification is achieved by inject-

ing fuel diwctly into the engine cylinder late in the com-

pression stroke, and enhanced by an elaborate cylinder

head and piston bowl design and charge motion control.

As a result, zin ignitable mixture is formed near the

spark plug, although the overall in-cylinder air-tefuel

ratio is extremely lean. At higher air-to-fuel rat io the

intake manifold pressure is higher and pumping losses

are reduced, thereby leading to improved fuel economy

and reduced carbon dioxide emissions. See [l,41 for

more information on the operation of stratified chargeengines.

Typically, stratified operation is limited to low and

part-load engine operating conditions. This is because

the intake manifold pressure is limited by atmospheric

pressure, hence the (overall) air-to-fuel ratio decreases

On leave from the Institute for Problems of Mechanical En-gineering, Russian Academy of Sciences.

Figure I: Direct injection stratified charge engine.

as he load on the engine increases. The decreased air-

to-fuel ratio results in increased levels of smoke and

hydrocarbons. That and the fact that at similar values

of the intake manifold pressure the stratified combus-

tion mode is actually less efficient than the homoge-

neous combustion mode prevents utilization of strati-

fied combustion mode at higher loads. Also at higher

engine speeds stratified combustion mode is not fea-

sible due to insufficient time for mixing and breath-

ing. Consequently,at higher speed and load conditions

the engine is operated in the homogeneous combustion

mode with the air-to-fuel ratio lower than that in the

stratified mode or sometimes rich of stoichiometry. Thefuel is still injected directly into the cylinder, but early

in the intake stroke to ensure a well mixed homoge-

neous charge. The torque and emission characteristics

in the homogeneous combustion mode are distinctly

different from the stratified combustion mode, see e.g.

[4], thereby resulting in a truly hybrid plant to be con-

trolled.

Under lean operating conditions, the conventional

three-way catalyst (TWC) oxidizes hydrocarbon (HC)

and carbon monoxide (CO) emissions but has a very

low conversion efficiency for oxides of nitrogen (NOx)

emissions. One technique to t rea t NOx is to incor-

porate a lean NOx trap (LNT) in the exhaust system

after the TWC. This device accumulates NOx duringlean operation, but as it is being filled up its trapping

efficiency gradually decreases to zero. Hence the LNT

is periodically purged of stored NOx in order to regen-

erate its capacity in such a manner th at the stored NOx

(pollutant) is converted to nitrogen and carbon diox-

ide. The operation at a rich air-to-fuel ratio to purge

the LNT is referred to as the purge operation while

the nominal lean operation is referred to as he normal

0-7803-5250-5/99/ 10.00 1999 IEEE 2667



operation. Consequently, the transition between strat-

ified combustion and homogeneous combustion modes

may be initiated not only when the engine torque de-

mand increases but also when there is a need to purge

the LNT although the engine torque demand is small.

The mode transitions have to be accomplished in a

manner that does not create a disturbance to the vehi-

cle that is noticeable by the driver. At typical steady-

state highway cruise conditions the purge operation

may last 2-3 sec. for every 60 sec. of normal opera-

tion. The control system must ensure a constant en-

gine torque torque value T = Td, while the engine is

going through this rapid transition from normal oper-

ation to purge operation and back. Other performance

objectives for this transition include the minimization

of the transition time and avoidance of spikes in tran-

sient NOx and HC emissions.

The objective of this paper is to develop a control

scheme that accomplishes the transitions and supportsthe desired value of the engine torque throughout th e

transition. The controller has a hybrid structure, with

a high level l hns i t ion Gove rno r (used to determine

the combustion mode and the set-points) and the low

level Coordinated Feedback Controller. The Coordi-nated Feedback Controller is designed using the Speed-

Gradient (SG) approach [3] to coordinate the spark

timing and the throttle inputs. The third subsystem,

the Fueling Cont rol ler, ensures the desired value of th e

engine torque throughout the transition.

2 Engine Model

A constant engine speed, zero EGR model is assumed

for this study. The rationale for these assumptions is

that the engine speed is varying slowly when the vehi-

cle is in gear and that the EGR valve is typically closed

during the mode transitions. The intake manifold fill-

ing dynamics are of the form:

where pl is the intake manifold pressue, ‘Ldthis the elec-

tronic throttle position, N is the engine speed value,

Wth is the mass flow rate through the throttle, Wc,l is

the engine intake mass flow rate. The functions Wth

and Wcylare nonlinear functions of the intake manifold

pressure. They are obtained by regressing the engine

static mapping data, see[4]. he function Wth depends

linearly on u t h which is (modulo nonlinear transforma-

tions) is the electronic throttle position.

The engine torque, T , depends on the engine fueling

rate, Wf, engine spark timing, b , and intake manifold

pressure, pl . The functional dependence is different for

stratified and for homogeneous combustion:

T = T, (Wf , l ,Wc r 6,N ) if stratified mode,

T = Th (Wf , P I ,Wcyl,6,N ) if homogeneous mode.

(2)The expressions (2) can be obtained by regressing

steady-state engine mapping data. Specifically, thebrake torque value in (2) is a sum of friction torque

(quadratic in N , linear in P I ) , pumping torque (linear

in p l , quadratic in N ) and indicated torque ( h e a r in

Wf, quadratic in the deviation of the spark from the

maximum brake torque (MBT) spark, where the MBT

spark depends on the air-to-fuel ratio and N ) , see [4].

The engine air-to-fuel ratio is defined as

Wc,l

WfA = - .

The feasible intervals of the ai r- tdue l ratio are differ-.

ent for the stratified combustion (A E d8) nd for the

homogeneous combustion (A E d h ) but these intervals

do overlap, A, n d h # 0.

3 Control Design

The control architecture consists of a higher level Tran-

sition Governor and a lower level Coordinated Feed-

back Controller. The Coordinated Feedback Controller

is used near the desired operating point to drive the

throttle and the spark timing inputs in response to

the set-points generated by the Transition Governor.

The Transition Governor drives spark timing and throt-

tle inputs during the rapid transient phases and also

decides on when to initiate a switch from the strati-

fied combustion mode to the homogeneous combustion

mode. The Transition Governor also utilizes another

degree of freedom often available but frequently ne-

glected in hybrid designs - resetting the state of the

dynamic controller. The Fueling Controller is used

to support the desired value of the engine torque. In

what follows, we first describe the Fueling Controller,

then the Coordinated Feedback Controller and then the

Transition Governor.

3.1 f iel ing Controller

To deliver the desired value of the torque output, T d

we invert the torque functions (2) so that the fueling

rate value is generated according to

wf = ~ , T d , p l , W c y l , b ,) ,

wf = Ah Td,pl,wcyl ,N ) ,

for stratified mode,

for homogeneous mode.

In either case, Wf achieves the torque d u e of Td for

the given p l , 6 , Wcyl,N . Both p l and N are mea-

sured while Wcyl s estimated according to the %peed-

density” equation, Wcyl= ko N)p lN I T , . Here the in-

take manifold temperature TI is either measured or z s-

timated.

(3)

2668



3.2 Coordinated Feedback Controller

To render the system affine in control, we augment an

integrator t o the spark timing input, i.e.

We introduce a vector of engine states

z = l p IT ,

and engine controls

TU = [ U t h 2161

The engine operating point is defined by the value of

the engine speed N and the desired engine torque T d :

Then the engine dynamics as represented by 1) and

(4) can be summarized by the equation

i: = F z ,U, ).

The controller derivation assumes th at the fueling rate

is generated according to (3).

The controller is designed using the Speed-Gradient

(SG) method [3] which is reviewed in the Appendix.

This method achieves the convergence to zero of the

following objective function:

Q z , w ) = I + Q z +&3 + 0,

Q = O*~'YI(WC,~A d W f ) ' ,

Qz = 0.5'Yz(pi i , d ) ' , Q 3 = 0 . 5 ( 6 d ) 2 ,

where A d , P l , d , 6 d are the commanded d u e s of the

air-to-fuel ratio, intake manifold pressure and spark-

timing that are functions of the demanded value of the

engine torque Td, he value of the engine speed N and

the desired combustion mode. Th e weights 71, 2 , 7 3

are used to shape the closed loop system transient re-

sponse. Specifically, these weights can be adjusted t o

speed up th e response of some of the variables relative

to the other.

In accordance with the SG method and assuming that

~ t )w is constant, we first calculate a time deriva-

tive of Q along the trajectories of the closed loop sys-

tem:

The derivative of w with respect to U ( speed-

gradient ) is

If we consider the evolution of Q over a discrete time

interval [t, + At ] we have

Q t+At) Q t )+Q t ) A t ,

and, hence, to minimize Q t+At) we select u t ) n the

direction of minus the speed-gradient , .e. -@(z, w).

Note tha t because F depends linearly on U,@ does not

depend on U. We explicitly calculate the entries of the

vector Q(z,w) as follows

where A = A, for the stratified combustion mode and

A = A h for the homogeneous combustion mode.

We select t o force Q t o decay along a descent direc-

tion, i.e.

21= u d n*(z,w , 5 )

where II > 0 is a 2 x 2 matrix of constant gains and

Ud is the feedforward of desired values for the engine

inputs, ud = [u th , d OIT Here 'LLth,d is the feedforward

of the thr ott le position. The controller (5) is referred

to as a Proportional Speed-Gradient Controller (SG-

P). Another controller choice is a Proportional-plus-

Integral Speed-Gradient (SG-PI) Controller of the form

U = Ud - n@(Z, r * 2 S ) , W ) d S , 6 )I,where II > 0, I? > 0 are 2 x 2 matrices of constant

gains. Th e additional freedom of th e integral control

can be exploited t o shape th e closed loop transient re-sponse, e.g. to facilitate the purge of the LNT where it

is desirable that the air-to-fuel ratio falls slightly lower

than t he steady-state value during the transient ; this

ensures a faster TWC breakthrough. The implementa-

tion of the SG-PI controller is possible without knowing

precisely the value of th e feedforward te rm Ud, ee the

Appendix.

The closed loop stability requirement in 131 imposes

a restriction on the weights 71 12, 7 3 . To verify lo-cal asymptotic stability, they must be selected so that

w ( z , u d , w ) < 0 at least for all z Xd close to z d ,

where Z d is the vector of equilibrium states of the en-

gine corresponding to ud and w. The verification of

these stability conditions has been done numerically.

The specification of the controller has been done in con-

tinuous time. The actual implementation of the SG-P

2669



and SG-PI controllers is accomplished in discretetime

with appropriate low-pass filters applied to the mea-

sured signals. T he implementation of SG-PI controller,

in general, requires an additional antiwindup compen-

sation tha t avoids performance deterioration due to ac-

tuator saturation. These aspects are standard for dig-

ital implementation of controllers specified in continu-

ous time.

3.3 Transition Governor

The desired combustion mode, Pd , is a function of the

engine operating point w = [N,alT and at a given time

instant t may be different from the present combustion

mode, p t ) . The variables p , p d can take two discrete

values: p = 0 corresponds to the stratified combustion

mode and p = 1 corresponds t o the homogeneous com-

bustion mode. The desired air-to-fuel ratio, Adr spark

timing 6 d , intake manifold pressure p1,d and the de-

sired throttle position ?&h,d are functions of w and Pd

and may change with time.

The Tkansition Governor utilizes either th e SG-P or theS G P I Coordinated Feedback Controller locally, only

at engine operating conditions close to steady-state.

Specifically, the Coordinated Feedback Controller is

used whenever the current state z t ) = [p1 t ) ,6 t )JTis inside a capture zone defined by

r c o ( w , P ) = tz = lpl,qT: (s ,w) 5 Q ( W , P ) I .

The trajectories generated by SG-P controller or by

SG-PI controller with zero initial integrator state have

the property tha t if they sta rt in rc0w,p)hen they re-

main in reow, p ) as ong asw, p remain constant. This

invar iance property allows to avoid chattering when

switching into and out of the mode where the SG-P or

the SG-PI controllers are active and that is the mainreason while we use Q in the definition of the capture

zone. By selecting a sufficiently small value for c - we

can always ensure that the capture zone is within the

domain of att ract ion of the closed loop system. In ad-

dition, certain pointwisein-time state and control con-

straints can be enforced. In our case we have been able

to find a single value of Q that works for all w and p

in the range of interest.

We define two functions that are used to calculate

the air-to-fuel ratio in stratified combustion mode,

X , p l t ) , 6 t ) , w t ) ) , nd the air-to-fuel ratio in the

homogeneous combustion mode, Xh(p1 ( t ) , t ) , w t ) ) .Note that these functions depend on 6 because we gen-

erate the fueling rate by th e controller (3) that involves

the spark timing input. The objectives of minimiz-

ing transient emissions and fuel consumption trans-

late into the minimization of the performance func-

tion J , (PI,,w) in stratified combustion mode and

J h ( P 1 , 6 , w) in homogeneous combustion mode. For

example, J may reflect our objective of minimizing

HC emissions (that are high for stratified combustion)

while J h may reflect our objective of minimizing NOx

emissions (that are high for homogeneous combustion

around the air-to-fuel ratio value of 16).

The Transition Governor has a discrete state, p , that

takes values 0, 1,2,3. Each sta te value corresponds to

a particular way of operating the engine:

e p = 0: The engine is operated in the stratified

combustion mode with spark timing and throt tle

input governed by the SG-PI Coordinated Feed-

back Controller with the set-points , 6d and

p l , d and th e desired throttle position Uth ,d .

e p = 1: The engine is operated in the stratified

combustion mode with the throttle commanded

to the fully closed position if p l ( t ) > P l , d ( t )

or fully open position if p l ( t ) < p l , d ( t ) . The

spark timing input 6 t ) s selected (within feasible

range) t o minimize J , ( p l t ) , 6 t ) , (t )) subject to

the constraint X = X,(pl(t),6(t),w(t)) A,.

0 p = 2: The engine is operated in the homoge-

neous combustion mode with the throttle com-

manded to the fully closed position if p l ( t ) >p l , d t ) or fully open position if p l ( t ) < p l , d ( t ) -

The spark timing input 6 t ) is selected (within

feasible range) so that Jh(p1 t ) , d t ) , t ) ) s min-

imized while x = X h ( p l ( t ) , 6 ( t ) , w ( t ) ) E d h .

e p = 3: The engine is operated in the homoge-

neous combustion mode with spark timing and

thro ttle input governed by the Coordinated Feed-

back Controller with the set-points ,& andp1,d

and th e desired throttle position u t h , d .

Let the present sampling time instant be t , while the

previous sampling time instant be denoted by t-1. The

specification of th e transition between various valuesof

p can now be easily done by the following rules:

If p t - 1) = 0:

e If p t 1)= 1:

If p d ( t ) = 1, check if there exists a spark

value 6aw, lsuch that PI t ) , S w , 1 ,w t ) )E

d h . If so, set p t ) =:2, else p t ) = 1.

If p d ( t ) = 0, determine a spark value

a S w , 2 that minimizes ldsw,2 dl while

IP1 t),S,,,2IT E r,,(w(t),O). If no such

2670



Ssw,a exists set p t ) = 1, otherwise set range, we switch to to p = 2. Finally, we switch to

p t ) = 0, S t ) = dSw,aand reset the inte- p = 0 when the intake manifold pressure rises suffi-

gral state of the SG-PI controller to zero. ciently high and with an appropriately selected value

of the spark timing, is within an appropriately defined

capturing zone.

Note th at in the above scenario it can happen t hat the

the value of P d remains equal to zero, if large changes in

w(t ) render the present state outside of the capturing

zone. In thi s case, we basically engage th e bang-bang

controller in intake manifold pressure corresponding to

If p t 1)= 2:

If p d t ) = 0, check if there exists a spark

Sw,3 such that '*(pl t ) ,Sw939 w(t)) Econtroller switches from p = 0 to I.1 = 1 even thoughA,. If so, set p t ) = 1,else p t ) = 2.

If P d t ) = 1, determine a spark value

6 s w 0 , 4 that minimizes I ~ s w , ~sdl while

bi(t), s w s ] ~ rcO(w( t ) ,1). If no such

Ssw ,4 exists, set p t ) = 2; otherwise set L1 = 1.

p t ) = 3, b t ) = bsw]4 and the integral state

of the SG-PI controller to zero.

If p t 1)= 3:

Note that in practical implementation of the switching

out of the capture zone condition we use a threshold

A > 0. We do this as an additional precaution to

avoid chattering and t o account for discrete-time im-

plementation errors. In the transitional states p = 1or p = 2) the spark timing is selected based on the

one-dimensional optimization of J , and Jh. This opti-

mization can be (approximately) accomplished on-line

or off-line by a search over a few grid points in the

feasible range of th e spark timing input.

4 Simulation Results

To illustrate the workings of this controller suppose,

for example, that initially p = 0, p = 0 (stratified com-

bustion) while the t arge t operation has just changed

to purge as defined by the appropriate values of the

set-points and pa = 1. Then the Transition Gover-

nor swithes t o p = 1 and the thrott le is closed. At

each sampling time instant it then attempts to find a

value of the spark timing, 6 = 6,w,1, such th at the esti-

mated air-to-fuel ratio in the homogeneous combustion

regime falls within the feasible range for the homoge-

neous regime. If such a value of spark timing can be

found, we switch from p = 1 o p = 2. The switch from

p = 2 o , = 3 is prompted if at the present time in-

stant t , the intake manifold pressure p l ( t ) s such tha t

there exists a spark timing value 6 = 6 ew , 4 such that

the pair p1 t ) , ,w,4) is inside the capture zone. Simi-

larly, if initially p = 3, = 1 while pd = 0 we switch to

p = 2 and then, when a spark value daw,4 can be found

such tha t the estimated air-to-fuel rat io if we switch to

the stratified combustion regime is within the feasible

The simulated closed loop responses for the transition

from the normal operation to the purge operation are

shown in figures 2-3. The transition from the normal

operation to the purge operation is requested and starts

at t = 0.2 sec. The engine torque and engine speed (40Nm, 2000 rpm) are constant throughout this transition.

The air-to-fuel ratio changes from 35 o 14 within 0.5

sec.

2000 rpm, 40 N-m

a

00 0.1 0.2 0.3 0.4 0.5 08 0.7 0.8 0.9 1

-30 . .. .. .. .. .. .. . .. ... ...... ... ..-ro 20

10 -

0 01 0 2 03 0 4 0 5 0 8 0 7 0 8 0 9 I

Time [S

Figure 2: Time histories of Mode lkansition Governor

State p, throttle position U t h and spark timing,

6. Setpoints corresponding to p d = 1 are shown

by the dashed lines.

5 Appendix: Speed-Gradient Feedback Laws

Here we demonstrate that the SG-PI controller rejectsunmeasured additive constant input disturbances. Al-

though this is an easy observation (and is a basic prop-

erty of linear systems with integral control) for SG-PI

controllers we have not found it in the book [3] r other

literature with which we are familiar.

For the specific case here we consider a version of the

Speed-Gradient algorithm [3] hat deals with the sys-

267 1



tm

.... ............... .. .. .. .. .. .. .. .. ........x . .

1 4 011 12 013 0:4 015 016 017 ; 019

0.S

I0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I

0.6

Time [SI

Figure 3: Time histories of intake manifold pressure p l

(kPa), air-to-fuel ratio A , and fueling rate W j

g p s ) . Set-points corresponding to Pd = 1 are

shown by the dashed lines.

tem &ne in control

i:= f z)+g s ) u . (7)

The objective is t o ensure the convergence of t he state

trajectories of the system (7) to the desired equi-

librium z, E R” corresponding to a constant in-

put 21 E R“ which satisfies f ze)+ g Z e ) U e = 0,

g z e ) # 0. With U = U , the equilibrium x is assumed

to be asymptotically stable. The SG control design

proceeds using a designer specified objective function

which is smooth scalar radially unbounded and such

that Q x ) = Q z - 2,) > 0 if z 2 6 , Q 0 ) = 0 . The

parameters of the function Q must be selected so thatthe stability condition in [3] is satisfied. Here we as-

sume a more restrictive form of the stability condition,

that is

L I Q ~ ) L g Q z ) u e I P ( Q ( z ) )

where p q ) > 0, p 0 ) = 0, is a continuous function.

The Speed Gradient Proportional-plus-Integral (SG-

PI) Controller has an additional feedforward term U,:

U = U , I L g Q x t ) ) T L g Q z s ) ) T d s , (8)I’where r = rT> 0, II = IIT > 0 are gain matrices. The

control law (8) can be rewritten in a more convenient

equivalent form:

U = U I L g Q z t ) ) T 8 , = r L g Q z t ) ) T ,9)

where 8 E Rm are the integrator states.

It will be shown that the implementation of the SG-PI

controller (9) is possible without knowing precisely the

value of the feedforward term u e since the integrator

states of SG-PI controller provide the means of adap-

tation to the values of the set-points th at are used for

feedback. Indeed, we introduce a constant vector dis-

turbance w E Rm into (9) so that

U = Ue n L g Q ) T+8 +W , = - I ’ L gQ)T , (10)

and choose the following Lyapunov function:

1V = Q ( z 2, + ( f 3 +w)TI’-l (0 +W 0.

Calculating it s time derivative along th e trajectories of

the system (7), (9) we obtain:

V = V z Q ) T f z ) g z ) U e .-g z ) n L g Q ) T++ g z ) e +w ) )+ e+w r - l e =

= L f Q z )+ L g Q z b e - L g Q ) T n L g Q ) ,

and

Consequently, the closed loop system trajectories

z t ) , e t )are bounded and, according to LaSalle’s in-

variance theorem, converge to th e largest invariant set

M of (7), (9) contained in E = { q8) Rn+ml Q x

2, = 0 , L g Q = 0 } , hat is, the set where V = 0. It is

clear tha t for any trajectory in M we have z= 2, and

x = 0. Setting x = x and x = 0 in (7) we get

f x e ) + g x e ) U e - I I L g Q ) T + 8 + w ) = 0, V(z,8) E M .

Recalling that ue is the feedforward of t he values of

the set-points x E R“, .e. satisfies th e equality

f ( z e ) + g ze )ue E 0 , we get g(z,)(8 +W ) = 0 on M.

Since g(ze) 0, the largest invariant set M in E has

expression for M and the convergence of the system

trajectories z t ) , e t ) o M proves that the SG-PI con-troller ensures z + x e even when the feedforward value

of U , that is consistent with the value of the desired

equilibrium ze is not known precisely.

form M = {@ , e ) E Rn+ml = z,, 8 = - w } . This

References

[l] Anderson, R.W., Yang, J., Brehob, D., Vallance,and Whiteaker, R.M., “Understanding the thermody-

namics of direct injection spark ignition (DISI) com-

bustion systems: An analytical and experimental in-

vestigation,” SAE paper 962018.[2] Butts, K., Kolmanovsky, I., Sivashankar, N.,

Sun, J., “Hybrid systems in automotive control applica-

tions,” in Control Using Logic-Based Switching, edited

by Morse, S., Springer, 1997.[3] F’radkov, A.L., Adaptive Control an Large-scaleSystems, Nauka, Moscow, 1990, (in Russian).

[4] Sun, J., Kolmanovsky, I., Brehob, D., Cook, J.,Buckland, J., and Haghgooie, M., “Modelling and con-

trol problems for gasoline direct injection engines,”

Proceedings of 1999 IEEE Conference on Control Ap-

plications, Hawaii.

67

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Hybrid control of a gasoline direct injection engine.pdf