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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
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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)
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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
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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
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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-
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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-
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67