A MODEL BASED CONTROL OF COMPRESSED NATURAL

GAS INJECTION SYSTEMS

Bruno Maione, Paolo Lino, Alessandro Rizzo

Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, via Re David 200, 70125 Bari, Italy

Keywords: Modeling injection systems, Common Rail, CNG, Pressure control.

Abstract: Low fuel consumption and low emissions are key issues in modern internal combustion engines design. For

this reason, an effective on-line control of the injection process requires the mathematical equations

describing the system dynamics. The inherent nonlinearities make the modeling of the fuel-injection system

hard to accomplish. Moreover, it is necessary to trade off between accuracy in representing the dynamical

behavior of the most significant variables and the need of reducing complexity to simplify the controller

design process. In this paper we present a second order lumped parameters model of a Compressed Natural

Gas injection system for control system synthesis and analysis. Based on the proposed model, we propose a

generalized predictive controller to regulate the injection pressure, which guarantees good performances and

robustness to modeling errors.

1 INTRODUCTION

Today it is a widely accepted opinion that

performances of internal combustion engines strictly

depend on fuel injection dynamics and metering of

air/fuel mixture (Heywood). Owing to a better

control on air/fuel ratio, the innovative Common

Rail injection system remarkably reduces noxious

emissions, consumptions and noise in Diesel

engines, while improving efficiency and available

power (Maione, 2004a). These goals are achieved by

setting the injection pressure to a fixed value, while

controlling injection timings electronically for

different operating conditions.

It is also well known that, if compared to liquid

fuels, t

he Compress Natural Gas (CNG) reduces

polluting emissions of CO, NOx, HC and particulate

of internal combustion engines, and guarantees their

better efficiency, thanks to its good antiknock

properties (Weaver). However, greater difficulties in

metering make the use of CNG less worthwhile.

This drawback can be overcome by applying the

Common Rail technology to CNG engines and by

using the injection control to improve performances.

However, improving the controllers design process

requires a quite accurate model for predicting the

system behavior.

Injection system models for Diesel

engines are

mainly based on three different approaches. The

straightest one is founded on fluid-dynamic

simulation packages like AMESIM, which

encompasses libraries of mechanical components,

and requires precise knowledge of the system

geometrical data (Mulemane). Although the

resulting models provide an accurate representation

of system dynamics, which is appropriate for

mechanical design, they are not in the form of

mathematical equations useful for control purposes.

Different classes of models descend from

identification processes based on real data. They

guarantee a good prediction of the system behavior

if nonlinear functions are exploited (Maione,

2004b). Finally, some injection system models are

based on equations describing the physics

underlying the process. Basically, this approach

leads to Partial Differential Equations or high order

representations, which are certainly not suitable for

control purposes (Cantore), (Kouremenos), (Maione,

2004a).

However, to the best authors knowledge, there is

lack

of studies carried out for modeling and

controlling gaseous fuel injection systems. In this

paper we propose a simple lumped parameters

model describing only main fluid-dynamic

phenomena of the CNG injection system. It is in the

form of a second order state space representation

suitable for designing controllers of the rail pressure.

Moreover, we also stress that tuning the parameters

of the model requires a minimal set of the system

geometric data. Finally, on the basis of this model

132

Maione B., Lino P. and Rizzo A. (2005).

A MODEL BASED CONTROL OF COMPRESSED NATURAL GAS INJECTION SYSTEMS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 132-137

DOI: 10.5220/0001158401320137

Copyright

c

SciTePress

Figure 1: Block scheme of the CNG Common Rail

injection system

we show how to design a Generalized Predictive

Controller (GPC) for the rail pressure which

parameters directly descend from the model

equations (Rossiter).

2 STATE SPACE MODELING OF

THE CNG INJECTION SYSTEM

The main elements of the CNG injection system are

a fuel tank, storing high pressure gas, a controlled

pressure regulator, a common rail and four electro-

injectors. The regulator reduces the pressure of the

fuel supplied by a tank, and sends it to the common

rail, feeding the electronically controlled injectors.

Then the injectors send the gas to the intake

manifolds to obtain the proper air/fuel mixture

(Figure 1).

The large volume of the common rail helps in

damping the oscillations due to the operation of both

pressure regulator and injectors. So it ensures a

constant pressure as requested by a correct metering

of the injected fuel. In fact, the injection flow only

depends on rail pressure and injection timings,

which are precisely driven by the Electronic Control

Unit (ECU). The output signal of a pressure sensor

inside the rail is processed to close the control loop.

The pressure regulator consists of a main

chamber with a variable inflow section, which

depends on the axial displacement of a spherical

shutter over a conical seat, and of a control chamber,

whose pressure is regulated by a solenoid valve. A

piston between the two chambers provides the seal

for the main valve shutter. The equilibrium of the

applied forces determines piston and shutter

dynamics (Smith) (Figure 1). In particular, the

control chamber pressure is regulated by varying the

driving current duty cycle (d.c.) among a control

period, making the valve opened and closed in turn:

in this way is possible to control the fuel flow from

the tank towards the rail. Finally, to maintain an

equilibrium condition in steady state operation, the

fuel in the control chamber is sent to the main circuit

through an high resistance orifice.

To model the CNG injection system we consider

two control volumes having a uniform, time varying,

pressure distribution, i.e. the regulator control

chamber and the rail circuit. We consider the tank

pressure as an input rather than a state variable as its

measure is always available on board as it is related

to the fuel supply. Furthermore, it is likely to assume

equal injection and rail pressures, so that electro-

injectors are not modeled apart, but included in the

rail circuit as control electronic valves. Finally, we

assume a constant temperature in the whole injection

system, so that the system dynamics is completely

defined by the pressure variations in the control

chamber and the rail circuit.

Continuity equation and perfect gas law

(Zucrow) lead to the state equations of control

volumes. In particular the perfect gas law is:

pmRTV=

(1)

where p is the control volume pressure, R the gas

constant, T the temperature and m the fuel mass

stored in the instantaneous volume V. We can

neglect possible volume changes due to mechanical

part motions (for example in the control chamber)

without sensibly affecting the model accuracy.

Hence the derivative of (1), immediately gives the

continuity equation:

(

in out

RT

pmm

V

=−

&&&

)

(2)

where

in

and

out

are the input and output mass

flows, which sum has to be equal to the overall mass

change in the control volume. Integrating the

equation (2), after the evaluation of mass exchanges,

yields the pressure in the generic control volume.

m

&

m

&

By considering mass flows through control

chamber and regulator inlet orifices as isentropic

transformations and by applying momentum

equation, we get the following equations, depending

on the output/input pressure ratio r = p

o

/p

i

(Zucrow):

21

2

1

in d in

k

kk

kRT

mcA rr

k

ρ

+

=⋅−

−

⎡

⎤

⎢

⎥

⎣

⎦

&

(3)

A MODEL BASED CONTROL OF COMPRESSED NATURAL GAS INJECTION SYSTEMS

133

1

1

2

1

in d in

k

k

mcAkRT

k

ρ

+

−

=⋅

+

⎛⎞

⎜⎟

⎝⎠

&

(4)

where A is the outlet section surface, ρ

in

is the intake

gas density and k is the gas elastic constant.

Equation (3) holds if r > 0.5444 and refers to

subsonic speed flows, while equation (4) holds if

r ≤ 0.5444 and refers to sonic speed flows. The

effect of non-uniformity of the mass flow rate is

accounted for by a discharge coefficient c

d

.

The pressure regulator inlet flow section A

r

is the

lateral surface of a truncated cone and depends on

shutter and piston axial displacement h

s

(i.e. the cone

height):

(

)

[

]

0.5 sin 2 sin

rs s s s

Ad h h

s

β

π

β

⋅=+ ⋅

(5)

where β

s

is the slope of the conical seat and d

s

is the

minimal seat diameter. The shutter and piston

dynamics are determined by applying the Newton’s

second law of motion to the forces acting upon each

of them. If we neglect the viscous friction term, the

piston and the shutter inertias due to the large

hydraulic forces, we can write the force balance:

0

si si s s so c

i

pA kh F F−+−=

∑

(6)

where p

si

is pressure acting on the to surface A

si

, if

we assume that pressure gradients are applied to the

flow minimal section. Moreover, k

s

is the spring

constant, F

so

is the spring preload, i.e. the force

applied when the shutter is closed. Finally, F

c

is the

coulomb friction. Hence, we get h

s

from equation (6)

and then A

r

from (5).

The shutter displacement of the electro-hydraulic

valve regulates the flux incoming in the control

chamber. As its inertia is negligible, we assume that

the inlet section can be completely opened or closed,

depending on the actual driving current

(energized/not-energized circuit), and calculated

using the equation (5), with h

s

= {0, h

max

}.

Since the flow between control chamber and rail

circuit can be considered stationary, it is determined

by the following equation (Zucrow):

()

out d L out in out

mccA pp

ρ

=

&

−

(7)

where c

L

takes into account the effect of kinetic

energy losses in the nozzle minimal section A.

Equation (7) assumes that no reversal flows occur.

The injectors opening time intervals are set by

the ECU, in dependence of engine speed and load.

The whole injection cycle takes place in a 720°

interval, with a 180° delay between each injection

command. Since in this model we neglect the

injectors opening and closing transients, we express

the injectors flow section as ET·A

inj

, where ET

(Energizing Time) is a square signal with a variable

period and equal to 0 or 1 depending on injection

timings. This simplification does not introduce a

considerable error, while reduces the system order

and computational effort. As critical flow condition

always holds, the injection mass flow has to be

calculated applying equation (4).

Equations (1)-(7) can be rewritten in a state

space form:

(

)

(

)

(

)

() () ()

[]

() () () ()

[

()

]

() ()

() () () ()

[]

11112

12 2 1 2

22111122

31 4 22 2 3

23 2 1 2

xt aut u t

axtxtxt

xt aut bxt bxt

b u t b a x t u t

+a t x t x t x t

=⋅+

−⋅−

=⋅−+

−

−− +

⋅−

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

&

&

(8)

where a

ii

are constant coefficients. The set of inputs

and state variables is:

()

[]

,

T

cc rail

xt p p=

, (9)

()

[]

,

T

tank

ut p d.c.,ET=

where p

cc

, p

rail

and p

tank

are the control circuit,

common rail and tank pressures respectively. The

system of non linear equations (8) can be solved

given the inputs and the initial conditions, and

completely describes the system dynamics in terms

of control volume pressures.

3 A GENERALISED PREDICTIVE

CONTROL LAW FOR THE

RAIL PRESSURE

REGULATION

Model Predictive Control techniques are based on

the idea of predicting output from a system model

and then to impress a control action able to drive the

predicted output to a reference trajectory (Rossiter).

We assume that the system is represented by an

ARIMAX (AutoRegressive Integrated Moving

Average eXogenous) model:

(

)

(

)

()()

11

() 1Aq yt Bq ut t

ξ

−−

⋅

=⋅−+∆

(10)

ICINCO 2005 - ROBOTICS AND AUTOMATION

134

Figure 2: The GPC scheme for the rail pressure control

where u(t), y(t), and ξ(t) are the control action, the

system output and a zero mean white noise

respectively, A(q

-1

) and B(q

-1

) are polynomials in the

shift operator q

-1

, and ∆ is the discrete derivative

operator (1-q

-1

). The corresponding j-step optimal

predictor is (Rossiter):

()

()

()

()

()

1

1

ˆ

|1

j

j

yt jt G q ut j

F q y t

−

−

+= ⋅∆+−

+⋅

+

(11)

where G

j

(q

-1

) and F

j

(q

-1

) are polynomials in the shift

operator q

-1

. Let f(t+j) be the component of y(t+j),

which only depends on known values at time t. We

can express (11), for j=1, …, N, in matrix form as

ˆ

y

=Gu+f

%

, where ŷ = [ŷ(t+1), …, ŷ(t+N)]

T

,

ũ = [∆u(t), …, ∆u(t+N-1)]

T

, and f = [f(t+1), …,

f(t+N)]

T

and G is a lower triangular N×N matrix. If

w = [w(t+1), w(t+2), …, w(t+N),]

T

is a sequence of

future reference-values, we introduce a cost function

taking into account the future errors:

()()

{

}

T

T

JE

λ

=− −+Gu + f w Gu + f w u u

%%%%

)

where λ(j) is a sequence of weights on future control

actions. The minimization of the cost function J with

respect of ũ gives the optimal control law for the

prediction horizon N:

()

(

1

TT

uGGIGwf

λ

−

=+ −

%

(12)

As the first element of ũ is ∆u(t), the current control

action is:

() ( )

(

)

T

1ut ut=−+gw-f

(13)

where g

T

is the first row of (G

T

G+λI)

-1

G

T

; at each

step the first computed control action is applied and

then the optimization process is repeated after

updating all vectors.

We apply the above concepts to design a GPC

for the rail pressure. We assume the d.c. as control

variable and the rail pressure itself as output

respectively. As the GPC law gives the change with

respect of the previous control action, it is necessary

to use an integrator to get the whole input to be

applied. Since this signal is bounded in the range

[0, 100%], we have introduced an anti wind-up

system to avoid undesired oscillations in the control

loop. To tune the GPC for the rail pressure, the

proposed model is linearized considering different

equilibrium points. Linearization is justified by the

aim of the control action to keep the pressure close

to a reference value, in dependence of the working

conditions, set by the driver power request, speed

and load. From the state space linearized models we

derive a transfer function representation that is

finally discretised by a first order holder, leading to

a family of ARX models in the following form:

(

)

(

)

(

)

()

11

101

11aq y t b bq u t

−−

+

⋅=+ ⋅−

The GPC control that derives is:

(

)

(

)

(

)

(

)

(

)

1

123 4

1ut kwt k kz yt k ut

−

∆

=++ +∆−

where [k

1

, k

2

, k

3

, k

4

] depends on N and N

U

, and the

related control scheme is depicted in Figure 2.

4 SIMULATION RESULTS

We have carried out extensive simulations in

MATLAB/Simulink environment to evaluate the

effectiveness of the proposed approach, considering

different operating conditions, in terms of speed,

load and rail pressure.

We have performed a first set of test to check the

model effectiveness to predict the system behavior.

Firstly, we have considered constant engine speed

and load, resulting in constant injectors driving

command, and constant tank pressure. Then we have

evaluated the system response to d.c. step variations.

Figure 3 shows the simulation results for a 40 bar

tank pressure, 2400 rpm engine speed and 8 ms

injectors exciting time interval, when two opposite

6% d.c. variations are applied, the first one starting

from a 3% value, occurring at 1.5 s and 28 s time

instants respectively. When applying the first step

variation, a pilot circuit pressure increment occurs,

causing the regulator inlet section to stay open

longer. As a consequence, the larger mean fuel

inflow coming from the tank raises the rail pressure.

Conversely, when the d.c. is reduced by the second

step, the pilot pressure drops and the rail pressure

diminishes too. A picture magnification points out

A MODEL BASED CONTROL OF COMPRESSED NATURAL GAS INJECTION SYSTEMS

135

the pilot and rail pressures oscillating behavior

within the 100 ms control period: the pressure

increases when the solenoid valve is opened, while

decreases when it is closed. Further simulations may

show that shortening the control period attenuates

these pressure variations.

Fig. 4 depicts model output for constant 40 bar

tank pressure and 9% d.c. and a varying injectors

driving signal. Simulation starts from a steady state

condition corresponding to a 2200 engine speed and

3 ms injectors exciting time interval within the

injection cycle. At time 4.5 s we have applied a 4000

rpm speed step, and raised the injection time interval

to 12 ms, so that the applied d.c. is no longer able to

maintain the initial rail pressure, because of the more

injected fuel amount. Besides, the fuel flow between

main and pilot circuit causes the pilot circuit

pressure to decrease. At time 21 s we have applied a

complete cut-off, i.e. we have kept the injectors

closed in the whole injection cycle: pilot and rail

pressures rise because the fuel is no more sent to the

intake manifolds. In conclusion, we observe that the

accordance of the resulting dynamics with the

expected behavior shows the model validity.

A second set of tests investigates the GPC

performances. To this end, we consider a 100% set-

point variation, to evaluate the rail pressure

response. We have tuned all the tested controllers

referring to models linearized at the starting

equilibrium point. Figures 5 and 6 show the rail

pressure dynamics when the system is controlled by

a GPC with N = 5 (0.5s) and N = 15 (1.5s) prediction

horizons respectively, and a N

U

= 1 (0.1s) control

horizon for both cases. We have also compared the

linear and nonlinear model responses with the above

controllers. Clearly, increasing prediction horizon

results in a sluggish response, while considerably

decreases pressure overshoot, which is strongly

desirable. Further simulations may show that

increasing the control horizon does not result in a

better rail pressure behavior.

Figures 7 and 8 compare the GPC with a

standard PI controller, tuned according the Ziegler-

Nichols rules. To evaluate the controllers robustness

to model uncertainties, we have tuned them by

considering a 3 bar rail working pressure (Figure 7).

Then we have assumed a different operating

condition (Figure 8), holding the same parameters.

We have considered the linear system response for

the GPC, since it almost coincides with the nonlinear

one. Compared with the PI controller, the proposed

regulator grants lower pressure overshoot and

oscillation amplitude. PI parameters are tuned for a

narrow working range, while the fact that the

dynamic performances of the GPC are independent

from the set-point demonstrates the superiority and

robustness of such control approach.

5 CONCLUSIONS

In this paper we have presented a simple lumped

parameters control-oriented model of a CNG

injection system. The model equations describe the

main fluid-dynamic phenomena and require a

minimal set of geometric data. By using the model

equations, we have designed a linear Generalized

Predictive Controller to regulate the injection

pressure, and then we have compared its

performances with those obtained with a standard PI

controller. The proposed controller structure is

simple enough for on-line computation and

simulation results validate the control approach.

Future work will concern a narrow model validation

through lab tests and the implementation of a

nonlinear control strategy.

REFERENCES

Cantore, G., Mattarelli, E., Boretti, A., 1999. Experimental

and Theoretical analysis of a Diesel Fuel Injection

System. SAE Technical Paper 1999-01-0199.

Heywood, J., 1988. Internal Combustion Engine

Fundamentals, McGraw-Hill. New York, 1988.

Kouremenos, D. A., Hountalas, D.T., Kouremenos, A.D.,

1999. Development and Validation of a Detailed Fuel

Injection System Simulation Model for Diesel

Engines, SAE Technical Paper 1999-01-0527.

Maione, B., Lino, P., De Matthaeis, S., Amorese, C.,

Manodoro, D., Ricco, R., 2004a. Modeling and

Control of a Compressed Natural Gas Injection

System, WSEAS Transactions on Systems, Issue 5,

Vol.3, pp. 2164-2169.

Maione, B., Lino, P., Rizzo, A., 2004b. Neural Network

Nonlinear Modeling of a Common Rail Injection

System for a CNG Engine, WSEAS Transactions on

Systems, Issue 5, Vol.3, pp. 2282-2287.

Mulemane, A., Han, J.S., Lu, P.H., Yoon, S.J., Lai, M.C.,

2004. Modeling Dynamic Behavior of Diesel Fuel

Injection Systems, SAE Technical Paper 2004-01-

0536, 2004.

Rossiter, J.A., 2003. Model-Based Predictive Control: a

Practical Approach, CRC Press. New York.

Smith, C.A., Corripio, A.B., 1997. Principles and Practice

of Automatic Process Control, Wiley & Sons. New

York, 2

nd

Edition.

Weaver, C. S., 1989. Natural Gas Vehicles - A Review of

the State of the Art. In Gaseous Fuels: Technology,

Performance and Emissions SP-798, Society of

Automotive Engineers, Inc., Warrendale, PA.

Zucrow, M., Hoffman, J., 1976. Gas Dynamics, John

Wiley & Sons. New York.

ICINCO 2005 - ROBOTICS AND AUTOMATION

136

0 10 20 30 40 50

2

4

6

8

10

12

Time [s]

Pressure [bar]

Control Pressure

Rail Pressure

Figure 3: Control and rail pressures for duty cycle step

variations and constant engine speed and injectors ET

0 5 10 15 20 25 30

2

3

4

5

6

7

8

Time [s]

Pressure [bar]

Pressure Reference

Rail Pressure

(linear model)

Rail Pressure

(nonlinear model)

Figure 5: Rail pressure dynamics when the system is

controlled by a GPC with N = 5 (0.5s) and a N

U

= 1 (0.1s)

0 5 10 15 20 25 30

2

3

4

5

6

7

8

Time [s]

Pressure [bar]

PI Controller

GPC

(linear model)

Reference

Figure 7: System step responses when controlled by a PI

regulator and a GPC with N = 15 (1.5s) and N

U

= 1 (0.1s)

0 5 10 15 20 25 30

6

7

8

9

10

11

12

13

Time [s]

Pressure [bar]

Control Pressure

Rail Pressure

20.05 20.1 20.15

20

20.

2

Figure 4: Control and rail pressures for engine speed and

injectors ET step variations and constant duty cycle.

0 5 10 15 20 25 30

2

3

4

5

6

7

8

Time [s]

Pressure [bar]

Rail Pressure

(linear model)

Pressure Reference

Rail Pressure

(nonlinear model)

Figure 6: Rail pressure dynamics when the system is

controlled by a GPC with N = 15 (1.5s) and N

U

= 1 (0.1s)

0 5 10 15 20 25 30

5

6

7

8

9

10

11

Time [s]

Pressure [bar]

PI Controller

GPC

Reference

(linear model)

Figure 8: System step responses when controlled by a PI

regulator and a GPC with N = 15 (1.5s) and N

U

= 1 (0.1s),

assuming model uncertainties

A MODEL BASED CONTROL OF COMPRESSED NATURAL GAS INJECTION SYSTEMS

137