Modeling, Analysis and Simulation of a Gasoline Direct Injection System

Copyright © IFAC Advances in Automotive Control Salemo, Italy, 2004

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MODELING, ANALYSIS AND SIMULATION OF A GASOLINE DIRECT INJECTION SYSTEM

E. Alabastri*, L. Magni*, S. Ozioso*, R. Scattolini@, C. Siviero·, A. Zambelli(*) Dipartimento di Informatica e Sistemistica

Universita di Pavia Via Ferrata 1,27100 Pavia, Italy

($) Dipartimento di Elettronica e Informazione Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected]

r-)

Magneti Marelli Powertrain Via Timavo 33, 40134 Bologna, Italy

Abstract: In this paper a physical dynamic model of a Gasoline Direct Injection (GD!) Common Rail system is developed and validated with experimental data. The model is used to study different system configurations. Specifically, the placement of the pressure sensor is investigated in terms of the observability of the pressure wave inside the rail, and the influence of the geometry of the pipe connecting the high pressure pump and the rail on the pressure variations is examined. The analysis is performed by resorting both to a simplified lumped linearized model and to the theory of distributed parameter systems applied to the original mass and momentum equations. Copyright © 2004 IFAC Keywords: Automotive, modeling, distributed parameter systems, sensor placement, observability.

1. INTRODUCTION

design and performances, a different and complex piston head design. From an engine control point of view, it's easy to realize the main task is to maintain the pressure almost constant inside the common rail system at the desired value, together with the use of electronic control of the injection parameters. The combination of hardware improvements and correct control strategies definition should allows for an extremely precise control of fuel supply so as to obtain a very favorable power/fuel consumption ratio with reduced pollutants emissions. However, many issues related to the design and implementation of GD! systems have still to be investigated. Among them, a fundamental problem concerns the definition of the geometry of the system, in terms of the relative position of the common rail and of the pipe connecting it to the high pressure pump, as well as of the position of the pressure sensor inside the rail. These elements influence the form of the pressure wave inside the rail, the possibility to measure its

Gasoline Direct Injection (GD!) engines are going to have a widespread diffusion for their inherent advantages in terms of fuel economy and environmental impact. The most relevant improvement expected from car makers are related to the possibility of limiting the stoichiometric air/fuel ratio only in the zone nearby the ignition coil electrodes and generating a lean mixture otherwise, with promising benefits in term of specific fuel consumption, the lower combustion chamber temperature with lower knock risks also at full load and increased compression ratio, the expectation of reducing pumping work by removing the throttle valve, the possibility of increasing air/fuel ratio control by injecting directly in the combustion chamber. The most technological efforts to rich these goals are mainly the necessity of a high pressure gasoline circuit, a new and innovative injectors

273

It basically works as a small tank which supplies the four injectors providing sequentially the fuel to the cylinders with frequency equal to the double of the frequency of the pump. The connection pipe is connected at one end of the rail, while the pressure sensor is placed at the opposite end (see again Fig.l). In the following this geometry will be called the nominal configuration. It is worth noting that, contrary to what happens in HDI (High pressure Direct Injection) systems, there is not any pressure regulating valve on the rail , but its discharge is only due to the injectors. The role played by the pressure regulating valve on rail is now taken by the pump inlet valve. The pump intake valve, controlled in Duty-Cycle, provides the rail with the correct amount of fuel strictly necessary to guarantee injections and to recover component dispersion. In so doing, both the waste of energy required to rise the pressure with a fuel quantity that will after be discarded and the rise in tank temperature caused by returned fuel are avoided.

characteristics and, in turn, the design of efficient strategies for pressure control. In this paper the considered system and the available experimental data are fIrst presented in Section 2. In Section 3 a distributed parameter model of the common rail and of the injection pipe is derived, discretized and linearized to obtain a more tractable lumped parameter system, which is validated with real data. Then the model is used to exploit the effects of different sensor and actuator locations on the pressure wave and on its measure (Section 4). Finally, in Section 5, the theory of distributed parameter systems (R.F. Curtain, et al., 1995) is used to verify in a rigorous framework the obtained results and to derive simple and reliable reduced order models useful for the controller synthesis. 2. THE SYSTEM AND THE MEASURED

VARIABLES 2.1 Geometry and components.

2.2 Operating point and available signals.

The GDI common rail injection system is composed by two parts: the low and the high pressure circuits. The analysis reported in this paper focuses on the high pressure circuit, schematically depicted in Fig. I, and formed by a high pressure pump, the connection pipe among pump and rail, the rail itself, the injectors, a pressure sensor and a safety valve. The high pressure pump, endowed with one pumping element, is operated by the camshaft, whose rotational speed is half the crankshaft rotational speed, through a two lobes cam. Due to this layout, the frequency of fuel pumping inside the connection pipe coincides with the engine rotational frequency, expressed in RPM (Revolution Per Minute) . This impulsive and discontinuous fuel feeding is the main cause of strong pressure oscillations inside the connecting pipe and the rail. The amount of fuel pumped into the rail is determined by controlling in duty cycle the intake valve of the pump. The connection pipe is a horizontal, rectilinear pipe made by steel with length Lp=O.6m, internal and external diameters Dpi=O.006m and Dpe=O.OOBm respectively. The rail is a horizontal, rectilinear pipe made in aluminium with length Lp=0.41m, internal and external diameters Di=O.014m and De=O.022m respectively and volume V=60cc.

Fool from tank (iow"",",,)

--+1 ~

In the operating point considered, the initial pressure inside the rail is 66bar while the crankshaft speed is equal to 4500RPM, which corresponds to a pumping frequency of 75Hz, while the injection frequency is 150Hz. The pumping and injection flow rates, computed with suitable models starting from the recorded command signals of the pump and the injectors, are shown in Fig. 2, while the measured pressure inside the rail is shown in Fig. 3.

\

11 11 11 11 11 11

i· 0

3

I I I

I I I

~

U

n

I

"11

11 11 11 11 11 11

"" "" 1\

,

-

~

\

I

I

,

n

iI

11 11 11 1\ 11

11 11 11

,

I

11 11

iI

.....

-It'IftctOrJ

I ~

iI

-11

11 11 11 11 11 1\

Fig. 2. Pumping and injection fuel flows.

Rail~

Connection pipe

~

~

~

f

f

f

~\ High pressure pump

f

=r-

1/

Injectors

Fig. 1. Schematic representation of the high pressure circuit.

274

\

Safety valve

Pressure Sensor

.

..

8p

ax =

WIth the correspondIng mcrements -

au = u

i =l...N and -

ax

the

contain

au =u

ax

-u ) 1 - , if the cell does not dl of

coordinate

-u· I

+U. In}

I

1

1-

Pi+) - Pi dl

.=.....:..'-'---=-.!..

an

injector,

otherwise, where

Uinj

or

is the

dl

injected fuel flow rate. Then, the lumped parameter model of generic cells is: ~19

119 .0" '9.0211903119 C' ·""'- " "'''C· · g.0'7 119.[J3 ' '9.09 11 9.'

dp. (p . I- P .) j3{p.) (ui+uini-ui_l) _I+U 1+( ~ =0 dt i d l l + Kf3(pi) dl

1(5)

1+

Fig. 3. Measured pressure inside the rail.

I)

du, (ui +Uini -Ui_l ) I (Pi+1 - Pi) 32vcin 0 - +u. + +--U· = dt' dl p dl

3. THE DYNAMIC MODEL

D/'

In this Section, a physical distributed parameter, nonlinear model based on coupled mass and momentum partial differential equations (PDE) is ftrst presented for a generic horizontal pipe. This model has been used to describe both the connection pipe and the rail. Then, it has been discretized along the spatial coordinate x and linearized to obtain lumped parameter models more tractable for simulation and analysis.

i=l , ... ,N (2) with the boundary conditions PNd=PN, u(O)=u m , Um being the fuel flow rate from the connection pipe. The simulated pressure computed with model (2) and N=32 cells is compared in Fig. 4 with the experimental transient; it is apparent that the model accurately describes the main dynamic phenomena inside the rail. The computed Mean Square Error (MS£) is 4.11ge10 Pal.

3.1 The physical nonlinear model.

-

nonh",ar modtl

--- UPtntrlt",.1

an

An isentropic, steady-state and laminar flow in a horizontal pipe with constant section A can be described with the following mass and momentum equations (p. Thomas, 1999; L.A. Catalano, et aI., 2002): j3(p) au = 0 ax (I + Kj3(p)) ax au +u au +..!... ap +32vcin U = 0 at ax pax D.2

l

ap +u ap + at

(1) 119.025

" 9.03

119.035

"9.CA

119.G6S

119.05

t(. )

1

Fig. 4. Measured and computed pressure - lumped nonlinear model.

where u, p and p are the speed, density and pressure respectively, x and t are the spatial and temporal coordinates, f3(p) is the Bulk modulus which depends on the pressure, vcin is the kynematic viscosity and K is a coefficient which depends on the elasticity of the rail. Model (I) can be used to describe, with the appropriate boundary conditions, both the connection pipe and the rail. However, for brevity, only the rail will be studied in detail although most of the following developments have been performed also for the derivation and simulation of a dynamic model of the connection pipe.

3.3 The lumped linearized model.

The simulation of model (2) is computationally demanding, and the model is not suitable for a subsequent analysis of its main characteristics. Hence, a simple yet effective procedure consists of neglecting the nonlinear quadratic terms and assuming that the Bulk module is constant in the considered operating range. This second hypothesis is justifted by its small sensitivity with respect to pressure variations . The model of the single cell is then given by:

3.2 The lumped non linear model. A classical way to study distributed parameter models is to resort to a spatial discretization, so that the rail can be seen as composed by N cells, including the injectors. Assume now that the i-th cell, with length dl=LIN, is described by model (1), substitute the partial derivatives with respect to x

dPi + j3 (ui +uinj -Ui-l) = 0 dt (1 + Kj3) dl

j

dUi + 1 (Pi+l - Pi) + 32vcin u. = 0 dt p dl D2 I I

275

i=l, ... ,N (3)

In Fig. 5 the simulation results obtained with the linear model (3) are compared to the experimental data and the transients provided by the nonlinear model (2). Remarkably, the use of the linear model in simulation does not deteriorate significantly the achievable results; as also confinned by the value of the MSE=4.126e10 Pal.

the connection pipe at x=O and the sensor at x=L. All the results achieved have shown that the pressure transient inside the rail is almost insensitive to the injectors positions, as shown in Fig. 6 where the pressure computed in the nominal configuration is compared to that obtained by fictitiously placing all the injectors in x=O, that is by assuming that all the inlet and outlet flows are placed at the initial section of the rail.

. 10'

-

......

.a ~J ' ~ors '" pO ~ GI'I'Mn ~

cGnfguralion

6.2

6.1 6

~

~'~ 19~OI~5~'~'9~ .W~~~~~--~=-~

Fig. 5. Measured and computed pressure - lumped linear model. Fig. 6. Measured and computed pressure - all injectors placed at x=O.

The set of eqs. (3) defines a linear system with 2N states, given by the pressures and the fluid speeds inside the cells, which can be given a standard state space formulation. However, since the N-th speed is always equal to zero due to the imposed boundary conditions, the order can be reduced to 2N-1 ; the obtained final system has been used to study the main characteristics of the rail.

4.3 Influence of the coupling between the connection pipe and the rail

In simulation, the coordinate x of the coupling between the connection pipe and the rail has been varied to analyze the sensitivity of the pressure wave with respect to different system configurations, see e.g. Fig. 7 where the coupling is placed in the middle of the rail (x=Ll2) .

4. ANALYSIS OF THE SENSOR AND ACTUATOR POSITION INSIDE THE RAIL In order to develop an efficient control strategy, a number of fundamental questions have to be addressed, such as the analysis of the dependence on time of the pressure wave inside the rail, the effect on the system dynamics of different couplings between the connection pipe and the rail, the position of the sensor, the geometry of the injectors.

t

Fig. 7. Connection pipe coupled with the rail at x=Ll2.

4.1 Temporal analysis of the pressure wave.

The analysis has been performed assuming that there are not burbles in the proximity of the coupling and that the inlet flow is equally distributed at the right and left hand sides of the coupling section. This hypothesis is justified by the previous results on the instantaneous constancy of the pressure along the rail. The obtained results, compared to those achieved in the nominal configuration (x=O), have shown that in any configuration the average value of the pressure remains constant along the spatial coordinate, while the peak to peak value increases with the distance from the coupling with the connection pipe. This is illustrated in Fig. 8 which shows the peak to peak pressure variations at the coordinates of the four injectors when the coupling with the connection pipe is at x=O. In view of these results, and recalling that a primary goal of the system is to maintain the pressure

It is well known that, letting X be the propagation velocity, the wavelength of a non-stationary wave with frequency j; is ).=xlfr (e.T.A. Johnk, 1975). Recalling that the fundamental frequency due to the pump is 75Hz, it easily turns out that the wavelength inside the rail is about 12m. This value, compared to the length of the rail (Lp=0.41m) allows to conclude that in the injectors' position there is no phase shift of the pressure wave inside the rail. 4.2 Influence of the injectors positions.

The analysis of different configurations of the injectors have been performed in simulation by modifying their relative positions with respect to those in the nominal configuration, while maintaining

276

constant in the rail to guarantee an unifonn behaviour of the injectors, it could be advisable to select the configuration of Fig. 7.

---

,. ,.

,.,

.

_...... Fig.

10. Poles and configuration.

zeros

in

the

nominal

t{l )

Fig. 8. Peak to peak pressure variations at the injection points.

5. THE LINEAR DISTRIBUTED PARAMETER MODEL OF THE RAIL The analysis of the lumped parameter model (3) has shown that in the system here considered the connection pipe plays a crucial role in the defmition of the overall system dynamics. However, it is also of interest to study with more detail the rail alone; in fact, it can be readily verified that the use of a shorter connection pipe or the simple insertion of a diaphragm between the manifold and the rail almost completely decouples the two dynamics. Then, in the following the distributed parameter model (1) of the rail is considered to obtain a simulation result more accurate than those provided by the lumped parameter models (2), (3). Indeed, the accuracy of models (2), (3) clearly depends on the number N of cells used to discretize the rail : small values of N lead to poor models, while for a large number of cells the model can be difficult to examine. Moreover, the analysis of the distributed parameter system (1 ) allows one to draw some important conclusions on the dependence of the observability of the pressure modes on the sensor position inside the rail. Concerning system Cl), the results of Figs. 5, 6 show that: (a) the nonlinear terms appearing in model (1) can be ignored and the Bulk module can be assumed to be constant, Cb) an equivalent system configuration with all the inlet and outlet flows at x=O can be considered without detriment of the model perfonnance. For these reasons, the following linear distributed parameter model can be considered:

4.4 Influence of the sensor location.

A fundamental requirement in the design of any control strategy concerns the observability of the system under control (T. Kailath, 1980), that is the possibility to estimate the system state from available measures . In the problem here considered, the observability property can simply be checked by verifying that no pole-zero cancellations occur in the transfer function from the fuel pumped in the connection pipe and the pressure measured by the sensor, which can be easily computed from model (3) used to describe both the connection pipe and the rail with the appropriate boundary conditions. This analysis has been repeated for different positions of the sensor inside the rail, in all the cases it has been verified that there are no pole-zero cancellations, so that in the considered system the relative position of the sensor does not influence the possibility to efficiently control the pressure. The position of the pole and zeros in the complex plane corresponding to the configuration with the connection pipe connected at one end of the rail (x=O) and the sensor placed at the middle (x=Ll2) is shown in Fig. 9, while the position of the poles and zeros in the complex plane corresponding to the nominal configuration is shown in Fig. 10. Note in particular that in the nominal configuration there are no zeros, so that any observability problem is automatically solved.

: (I~P): = 0 +..!... at 1 ax +

(4)

Op + 32vcin u = 0

Ou

p

D~I

with the boundary conditions u(O,t)= um-Iu inj . System (4) can be given the standard fonn (see [3]): Z(t) = Az(t) z(O) = zo { P z(t) = u(t) (5)

--

where z(t, x)

Fig. 9 Poles and zeros with the manifold connection at x=O and sensor at x=Ll2.

=(P(t,X) ], u(t,x)

A=[0

_J..~

pOx

277

-1 +~,8 ~l _ 32vcin D?

and fJ = (0 l}5x, b;. being the Kroneker delta. The eigenvalues associated to system (5) are }..o=O and 2

--=-± 2..

)' n =

2

c 2 _ 4 bn ,,2 ,

pL 2

2

n=1,2, ... ,

The results above are also useful to analyze the observability of the rail, expressed in terms of the absence of pole-zero cancellations, as function of the sensor placement. As the system poles are already known, the following zeros of the system have been computed with symbolic computational tools.

(6)

fJ 32vcin where b = - - - and c = - - 2 - ' For ease of I+KfJ

Di

Zm=-~±.!. c2_4b(2m-ll~

notation A.", n8, represents the n-th pair of complex conjugate eigenvalues. Associated to the eigenvalues, there are the following orthonormal eigenvectors

.!.

4

=(H} tP. o

=

If

(2 s~LcOS

[ s -n L

tr

2 4

.

SIO - -

L

The knowledge of the eigenvalues and eigenvectors allows one to compute (R.F. Curtain, 1995) the transient of the pressure due to the input u1olt)=u(O,t), given by:

p(l ,x) = P. -

x=2(n-m)+I L 2n

n"O I

(8)

°

f 4X~L ~cos ( nJrX ) fe).' (H )u,o/(s)ds + n Lo

f

(9)

REFERENCES

The transient of p computed with (9) is compared to the experimental data in Fig. 11 . Notably, associated to model (9) one has MSE3 =3.31elO Pal, much smaller than the values MSE, and MSEl provided by the discretized models (2), (3).

P. Thomas, (1999). Simulation of Industrial Process. Butterworth-Heinemann. L. A. Catalano, V. A. Tontolo, A. Dadone (2002). SAE. Dynamic Rise ofPressure in the Common Rail Fuel Injection System. Paper 2002 - 01 0210, pp. 19-25. R.F. Curtain, HJ. Zwart (I995). An introduction to Infinite - Dimensional Linear Systems Theory. Springer - Verlag. T. Kailath, (1980). Linear Systems. Prentice Hall. e.T.A. Johnk, (1975). Engineering electromagnetic field and waves. N.Y. John Wiley and sons.

POE modr - - ."P4'I'W'*IIald •• 69 :-

6 B~ 67~

66

6.

63 6.2 .

61

~ 19 . 01 5

(II)

The analysis reported in this paper has investigated the influence on the pressure wave of the geometry of the system. It has also been shown that the placement of the pressure sensor inside the rail must be chosen with care in order to guarantee the observability of the pressure wave from the available measure.

.~" ,.0

b ' + - u,o, (s)ds L 0

, n"?m

6. CONCLUSIONS

where f(x) =pi + 1 and X is a free design parameter to be properly tuned in the validation phase. Dropping in (8) the negligible terms: p(t,x):; Po +

, meZ (la)

Then, when the rail can be viewed as an independent system, the sensor placement must be selected with care in order to guarantee that the pressure oscillations can effectively be observable from the measured signal. Notably, for the coordinate specified by (ll), the cosinuisodal terms in (9) are equal to zero, that is some oscillatory components do not influence the pressure in the points where observability is lost.

.~Df(x)~.,dx]~.,,[Um (I) +

+ An j/"(I-S)U 101 (s)ds]

L

L-x

where x is the spatial coordinate corresponding to the sensor position. Now, comparing (6) and (IO), it is easy to conclude that there is a pole-zero cancellation, that is a loss of observability, provided that, for any pair (n,m) of positive integers associated to the complex conjugate system poles and zeros, the sensor is placed at

(ntrx) ] L (ntrx)' neZ (7)

,d.{A" + c)

{L2

11 9.035

Fig. 11 Measured and computed pressure-distributed parameter model.

278