Modelling and predictive control of a new injection system for compressed natural gas engines

ARTICLE IN PRESS

Control Engineering Practice 16 (2008) 1216–1230 www.elsevier.com/locate/conengprac

Modelling and predictive control of a new injection system for compressed natural gas engines Paolo Linoa,, Bruno Maionea, Claudio Amoreseb,1 a

Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, via Re David 200, 70125 Bari, Italy b ICMEA Srl, Via Mongelli 11, I-70033 Corato, Italy Received 22 November 2006; accepted 24 January 2008 Available online 17 March 2008

Abstract In internal combustion engines equipped with the common rail injection system the accurate metering of the air/fuel mixture strictly depends on the pressure regulation. Accuracy in metering is difficult to be achieved especially for compressed natural gas (CNG) injection systems, as the gas compressibility makes the fuel delivery process more complex. Since the controller design requires a model of the injection system, this paper presents a physics-based state-space model of an innovative CNG injection system. The model parameters only depend on well defined geometrical data and fuel properties. Comparing simulation and experimental results in different operating conditions validates the model. Further, the proposed model is used for designing a generalized predictive controller for the injection pressure regulation, which is implemented in few steps. Experimental results show the effectiveness of the proposed approach. r 2008 Elsevier Ltd. All rights reserved. Keywords: Injection pressure control; Common rail; Modelling injection systems; Compressed natural gas; Polluting emissions reduction; Predictive control

1. Introduction Because of combustion processes, thermal engines of vehicles release gases and particulate into the atmosphere. Even if the recent application of electronics and control has improved the efficiency of engine operation, the level of the exhaust gases in the atmosphere keeps on rising, as the number of vehicles increases continuously. Recently, to reduce pollutants (CO, NOx , HC) and particulate in the atmosphere, manufacturers have experimented engines employing alternative fuels, e.g. compressed natural gas (CNG) (International Gas Union, 2005; Weaver, 1989). Namely, CNG can keep the emission limits within international prescriptions (e.g. EURO 4 in Europe) and can also restrain the operation costs (McCormick, Corresponding author. Tel.: +39 80 5963851; fax: +39 80 5963410.

E-mail addresses: [email protected] (P. Lino), [email protected] (B. Maione), [email protected] (C. Amorese). 1 Formerly with Centro Ricerche FIAT S.C.p.A., Valenzano Branch, Str. P.le Casamassima Km 3, 70010 Valenzano, Italy. 0967-0661/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2008.01.008

Graboski, Alleman, Herring, & Nelson, 1999). Their further advantages are that the global reserves of methane are estimated to be much larger than the oil reserves (Dyntar, Onder, & Guzzella, 2002) and that natural gases are a resource well distributed worldwide (Kvenvolden, 1999). Hence, even if CNG requires large spaces for high autonomy and for achieving acceptable refuelling distances, few European countries pay attention to the technologies improving performances of the CNG-powered vehicles. As with gasoline engines, CNG engines require a particularly efficient metering to limit the emissions (McCormick et al., 1999). At present, the regulation of the injection pressure of the fuel is based on a mechanical technology which is affected by some limitations yielding inaccurate metering. E.g. with such systems it is not possible to change the injection pressure at will, except in the initial tuning phase; disturbances are poorly compensated; the quality of pressure response to abrupt changes in the injected fuel quantity is poor (Amorese, De Matthaeis, De Michele, & Satriano, 2004; Pan, Li, & Hussain, 1998).

ARTICLE IN PRESS P. Lino et al. / Control Engineering Practice 16 (2008) 1216–1230

Recently, to overcome these difficulties, the FIAT Research Center, Valenzano branch, Italy, experimented an innovative common rail injection system for CNG engines (Amorese et al., 2004; Maione et al., 2004), in which an Electronic Control Unit (ECU) controls the opening/ closing rate of the electro-valve feeding the rail to achieve an effective control of the pressure in the rail and, hence, an accurate metering of the fuel by means of the electronically driven injectors. In addition, this controller can modify the working point of the pressure in the rail, for adapting it to the changes of load, to the request of acceleration, etc. This paper describes a model-based, predictive control of the pressure in common rails of injection systems for GNG engines. Obviously this technique is closely dependent on the assumed process model, which is the main block of the predictive control. In other words, only a model generating accurate output predictions fits the purpose (Qina & Badgwell, 2003; Rossiter, 2003). Unfortunately, there are relatively few papers which describe modelling and control of injection systems for CNG engines. On the contrary, similar problems are frequently considered for diesel engines, so that many methodologies can be transferred to CNG injection systems. In particular, a well known modelling method (Heywood, 1988; Streeter, Wylie, & Bedford, 1998), also used in this paper, splits the system into its main volumes and describes the continuous flow of the fluid through these volumes. However, if the fuel compressibility and the pipeline flexibility are taken into account, the description is often in terms of partial differential equations (PDE) (Catania, Dongiovanni, Mittica, Negri, & Spessa, 1996; Cantore, Mattarelli, & Boretti, 1999; Ficarella, Laforgia, & Landriscina, 1999; Hountalas & Kouremenos, 1998; Kouremenos, Hountalas, & Kouremenos, 1999). These detailed models are useful to diagnose faults and to explore different system configurations and alternative geometrical and functional designs, etc. However, for control problems, they are too complex and contain too large amount of parameters to be useful. Fortunately, a recognized and important property of feedback control is that one can often achieve satisfactory levels of performance by means of relatively simple models, provided that they capture the relevant features of the controller design. In the proposed study, for designing a model-based predictive controller, there is no need to describe all the physics and internal behaviour of the injection process to get a model providing reliable prediction. Namely, only the more significant characteristics can be taken into account leaving most secondary effects un-described mathematically. In this way, without losing essential validity, the injection system can be represented by ordinary differential equations (ODE) (Aespy, Engja, & Skarboe, 1996; Desantes, Arregle, & Rodriguez, 1999; Fuseya, Nishimura, Sato, & Tanaka, 1999; Gauthier, Sename, Dugard, & Meissonnier, 2005; Morselli, Corti, & Rizzoni, 2002), by transfer function (Kuraoka, Ohka, & Ohba, 1989), or state-space description (Lino, Maione, & Rizzo, 2007; Maione et al., 2004). Since

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the simplicity is a focal point, this paper is specifically aimed at defining a simple enough control-oriented model, which at the same time ensures reliable data used to design a predictive control law. All assumptions are clearly specified at each stage of the modelling procedure. Final results are validated by comparison with experimental data. This paper is organized as follows. Section 2 describes the new CNG common rail injection system. Section 3 develops the state-space model of the injection system used in Section 4 to derive the predictive control law for the injection pressure regulation. In Section 5, both experimental and simulation results are presented to validate the proposed model and to evaluate the controller performances. Section 6 gives the conclusions. 2. The CNG injection system The main elements of the CNG injection system of Fig. 1 are a fuel tank storing high pressure gas, a pressure reducer, a common rail and six electro-injectors. The fuel coming from the tank supplies the pressure reducer before reaching the common rail and feeding the electronically controlled injectors. By supplying gas to the intake manifolds, injectors lead to the proper air/fuel mixture. Note that the large volume of the common rail helps in damping the oscillations due to the operation of both pressure controller and injectors. Namely, combining the electronic control of rail pressure with optimum design of the rail volume reduces the pressure oscillations inside the rail and leads to a more accurate fuel metering. More precisely, since the pressure ratio between the intake manifold and the rail pressure is below the so-called ‘‘critical value’’, the flow rate depends only on the rail pressure (see Section 3 for details). Hence, the injected fuel amount can be metered acting on rail pressure and injection timings, which are precisely driven by the ECU. The output of a piezo-resistive pressure sensor inside the rail is processed to drive the injectors and then to close the loop with the pressure reducer consisting of two chambers. The inflow section of the main chamber is variable with the axial displacement of a spherical shutter over a conical seat. The pressure of the second (control) chamber is regulated by a solenoid valve. A piston placed between these chambers varies the shutter opening, which controls the fuel flow rate. When the solenoid valve is opened, the fluid increases the pressure on the upper surface of the piston, which is pushed down together with the shutter. Consequently, more fuel enters the main chamber and forces the pressure to increase. When the solenoid valve is closed, the pressure on the upper side of the piston decreases. Since the direction of the resulting force is upward, the piston moves up and allows a preloaded spring to close the shutter of the main chamber. The solenoid valve, which admits the fluid in the control chamber, encompasses an electromagnet and a moving anchor integral with a spherical shutter. With the

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pressure reference

solenoid valve ECU p1

tank

pressure sensor

control chamber

p2

bypass orifice

common rail

piston rod

main chamber

shutter injectors

pressure reducer p0

pressure sensor

pressure sensor

Fig. 1. Block scheme of the CNG common rail injection system.

non-energized electromagnet, a preloaded spring acting against the hydraulic force takes shutter and anchor closed, and prevents the gas from flowing into the control chamber. As the field circuit of the electromagnet is energized, the magnetic force overcomes the spring preload: the anchor and the poles come together and the pressure force moves the sphere unfolding the supply orifice. With the de-energized circuit, the anchor is forced down and the shutter is pushed against the seat. In this way, varying the duty cycle of the driving current during a control period makes the valve opened and closed in turn, so as to control the chamber pressure. To maintain an equilibrium condition in steady state operation, the fuel in the control chamber is sent to the main circuit through a suitably calibrated bypass orifice (Fig. 1). 3. State-space modelling of the injection system The model of the CNG injection system considers two control volumes, i.e. the control chamber circuit and the common rail circuit (including the main chamber and the common rail). By assumption, in both volumes the pressure distribution is uniform, and the elastic deformations of solid parts due to pressure changes are negligible. In addition, injection and rail pressures are assumed to be equal, so that electro-injectors are not modelled apart, but they are included in the rail circuit as electronic valves. Since the temperature is also considered constant in the whole injection system, the dynamics is completely defined by the pressure variations with time in the control volumes. The tank pressure plays the role of a maintenance input

H m· HK pH, VH, TH

AHK

K

pK, VK, TK

L m· KL pL, VL, TL

Fig. 2. Computation of flows and pressure dynamics in control volumes.

rather than a state variable, as it is almost constant in short time intervals. A measure of the tank pressure is always available on board as it is related to the fuel supply. 3.1. Theoretical issues The schematization of control volumes (H, K, L) in Fig. 1 makes the presentation of the physical relationships among variables easier. So consider the restriction of Fig. 2 sketching a valve or an orifice connecting H and K. Let pH _ HK and pK be the pressures determining the flow rate m through the orifice. Two cases can be considered. In the first one, when pK =pH Xð2=ðg þ 1ÞÞg=ðg1Þ , where g is the heat capacity ratio, the mass flow rate through the restriction is a nonlinear function of pH and pK , given by (Zucrow & Hoffman, 1976) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi "  u  ðgþ1Þ=g # 2=g u 2g pK p _ HK ¼ cd1 pH AHK t . m  K RT H ðg  1Þ pH pH (1) In Eq. (1), AHK is the minimal flow section, perpendicular to flow direction, R the gas constant, T H the absolute

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temperature, cd 1 is an experimental coefficient to account for non-uniform mass flow rates (Mulemane, Han, Lu, Yoon, & Lai, 2004). The second case, when pK =pH oð2=ðg þ 1ÞÞg=ðg1Þ (‘‘critical value’’), is the usual situation considered in this paper. Then the expression of the mass flow rate becomes (Zucrow & Hoffman, 1976) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðgþ1Þ=ðg1Þ g 2 _ HK ¼ cd2 pH AHK m , (2) RT H g þ 1 which is independent of pK (‘‘critical flow condition’’). _ HK and m _ KL are the mass flow rates in and out Now, if m of the container K of volume V K , and rK is the gas density, it holds (see Fig. 2): dðrK V K Þ dr ¼ VK K , dt dt

p1

S1

m· 02 p0

p1

p1

m· 12 p2

S2 hs

LEGEND Flow direction Mechanical motion direction Pressure forces direction

m· 12

p1

m· 01 p0

(3)

p m· 23 2

S0 p3

Fig. 3. Block scheme of the injection system showing the fuel flows and the hydraulic forces involved.

main chamber

hs dsM 90-s

ls d dS p0



p_ K ¼

dsm d0 / 2

RT K _ HK  m _ KL Þ. ðm VK

(4)

The integration of (4) yields the pressure in the generic control volume. Eqs. (1)–(2) are used to model the common rail injection system presented in Figs. 1 and 3. 3.2. Injection system modelling Let p0 , p1 , p2 , p3 be the pressures in the tank, in the control chamber circuit, in the main circuit and in the intake manifold, respectively. To derive the system state equations, pressures p1 and p2 are chosen as state variables. Then p_ 1 and p_ 2 are expressed in terms of the differences between the mass flow rate flowing in and that flowing out of their corresponding volumes, according Eq. (4). Eqs. (1) and (2) are used to express the mass flow rates in terms of the pressures. Note that the flow sections are affected by the mechanical parts motion, which is influenced by the pressures in the control volumes. _ 02 , m _ 12 , m _ 23 are the flows affecting the From Fig. 3, m main circuit. The flow between the tank and the main _ 02 , depends on the instantaneous flow section A02 , circuit, m thus on the shutter and the piston axial displacement hs . The shutter and piston dynamics can be described by applying the Newton’s second law of motion to the forces acting upon each of them. Let hs ¼ 0 when the shutter is closed, and hs 40 when the shutter and the piston move down. To calculate the forces acting on the shutter, by assumption, the pressure gradient p0  p2 is fully applied on the inlet minimal section (Figs. 3 and 4(a)). The force F 0 due to p0 , acting on the lower spherical cap portion of the shutter surface S 0 , can be computed as follows. Since the direction of pressure forces is normal to the shutter surface, the overall pressure force in the motion direction is

p2

ds

s

where possible changes due to mechanical part motions (for example in the main and control chambers) are neglected, without sensibly affecting the model accuracy. Hence the perfect gas law (Streeter et al., 1998; Zucrow & Hoffman, 1976), p ¼ rRT ¼ mRT=V , leads to

Inlet section [mm2]

_ KL ¼ _ HK  m m

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5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

exact

approximated

0

0.1 0.2 0.3 0.4 Shutter displacement [mm]

0.5

Fig. 4. Computation of the pressure reducer inlet section, A02 , and of the pressure forces acting on the shutter: (a) valve geometry; (b) comparison of accurate and approximated computation.

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obtained by integrating the elementary contribution depending on the angle y: dF 0 ¼ p0 dS cos y.

(5)

dF 0 ¼ 12p0 pd 20 sin y cos y dy.

(6)

(7)

180

1 p0 pd 20 sin y cos y dy 90bs 2 1 ¼  p0 pd 20 cos2 bs . (8) 4 The pressure p2 applies on the surface S2 including the piston bottom surface and the upper spherical cap surface of the shutter, so that the resulting force F 2 is the sum of two opposite contributions. By assuming the rod base surfaces negligible, the former contribution is p2 S1 , because the bottom and the top surfaces of the piston are approximately equal. As for F 0 , the latter contribution is obtained by integrating the elementary force p2 dS cos y within the angular interval ½0; 90  bs . Therefore, the overall force due to p2 is

F0 ¼

Z 90bs 1 p2 pd 20 sin y cos y dy F 2 ¼  p2 S 1 þ 2 0   1 ¼  p2 S 1  pd 20 cos2 bs . 4

(9)

Finally, the force F 1 ¼ p1 S1 due to p1 applies on the piston upper surface S 1 . Summing up, if the viscous friction term, the piston and the shutter inertias are neglected in comparison with the large pressure forces, the force balance gives F 0 þ F 1 þ F 2  ks hs  F s0  F c ¼ 0,

(12)

p m p ðd þ d M s Þl s ¼ ð2d 0 cos bs þ hs sin 2bs Þhs sin bs 2 s 2 p ’ d 0 hs sin 2bs . (13) 2

A01 ¼

Hence, the integration within the angular interval ½90  bs ; 180 yields Z

m dM s ¼ d s þ 2l s cos bs ¼ d 0 cos bs þ hs sin 2bs .

Finally, it is straightforward to calculate the inlet section:

Since the elementary surface is d0 1 dS ¼ pd 0 sin y dy ¼ pd 20 sin y dy, 2 2 the substitution of (6) in (5) gives

and the lower base radius is given by

(10)

where ks is the spring constant and F s0 is the spring preload, i.e. the force applied by the spring when the shutter is closed. Finally, the coulomb friction F c is assumed to be constant and evaluated experimentally. Hence, hs is obtained by solving (10), allowing to compute the main circuit minimal inlet section A01 by means of simple geometrical considerations. The flow section A01 , perpendicular to the flow direction, is the lateral surface of a truncated cone (see Fig. 4(a)). If bs is the slope of the conical seat, d 0 the shutter diameter, the cone slant height l s and the upper base diameter d m s are, respectively, ( l s ¼ hs sin bs ; (11) dm s ¼ d 0 cos bs

The last expression on the right side of (13) follows by neglecting the h2s -term: a simplification justified by the small value of hs (Fig. 4(b)). _ 12 between the control and the main Since the flow m circuits is approximately stationary, it can be determined by the following equation (Streeter et al., 1998): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 _ 12 ¼ cd cL A12 ðp  p2 Þ, m (14) RT 1 where cL takes into account the effect of kinetic energy losses in the bypass orifice minimal section A12 and in the pipe cranks. Eq. (14) assumes that no reversal flows occur. In dependence of engine speed and load, the ECU sets the injectors opening time intervals. The whole injection cycle takes place in a 720 interval, with a delay between each injection command that depends on the number of injectors. Since this model neglects the injectors opening and closing transients, only two conditions can occur: injectors closed, or opened with a flow section A23 . This simplification does not introduce a considerable error, as the transients expire within 1 ms, while reduces the system order and computational effort, because it is not necessary to calculate the shutter axial displacement and speed. Since critical flow condition always holds, the injection mass flow _ 23 must be calculated by (2) with AHK ¼ ET inj  A23 , and m where ET inj (injectors energizing time) is the injectors timing, normalized with respect to the injection cycle duration. More in detail, if n is the number of injectors, ET inj is a square signal of amplitude f1; 2; . . . ; ng depending on the number of injectors simultaneously opened, and on the period variable with the engine speed. The pressure dynamics within the control chamber _ 01 , circuit depends on the incoming flow from the tank m _ 12 . and the outgoing flow towards the main circuit m The displacement of the shutter driven by the solenoid _ 01 . Since its inertia is valve regulates the flux m negligible, the inlet section of the valve can be considered completely opened or completely closed, depending on the actual driving current (energized/not-energized circuit). With this assumption, the value of the inlet section can be calculated by using (13) (being the shutter and seat geometries the same as in Fig. 4(a), with the appropriate sizes) and by putting hs ¼ ET sv  hmax , where ET sv (solenoid valve energizing time) is a square signal, assuming values in f0; 1g, representing the valve energizing conditions.

ARTICLE IN PRESS P. Lino et al. / Control Engineering Practice 16 (2008) 1216–1230 Table 1 Model notation Symbol

Description

a1 , b0 , b1 A A, B, C Aði;jÞ , Bði;kÞ H, K, L cij cd , cL M dm s , ds

Coefficients of the family of ARX models Flow section surface across two control volumes ðm2 Þ State, input and output matrices of the linearized model Elements of A and B, respectively Generic control volumes Generic coefficient of the nonlinear state-space model Discharge coefficient and kinetic energy losses coefficient Radiuses of upper and lower bases of the conical inlet section (m) Injectors and solenoid valve normalized energizing times

ET inj , ET sv f Vector nonlinear function of the state-space model F 0 , F 1 , F 2 Pressure forces (N) Coulomb friction force and spring preload (N) F c , F s0 hs Shutter axial displacement (m) ks Spring constant ðN m1 Þ ls Slant height of the pressure regulator conical inlet section (m) m Fuel mass (kg) _ m Flow rate between two control volumes ðkg s1 Þ n Number of injectors p Gas pressure within the control volume ðN m2 Þ R Universal gas constantðJ mol1 K1 Þ S Surface subjected to the action of pressure p ðm2 Þ t Time variable tc , tph Period and hold phase duration of ET sv (s) Injectors opening time interval (s) tj T Temperature (K) u, x, y Input, state and output variables of state-space models u, x Elements of u and x, respectively u, x Input and state vectors of state-space models u, x Vectors of equilibria inputs and states V Volume ðm3 Þ bs Slope of the conical seat (rad) g Heat capacity ratio d Variation operator y Angle (rad) r Gas density ðkg m3 Þ u Engine speed (rpm)

With x1 ¼ p1 , x2 ¼ p2 , u1 ¼ ET sv , u2 ¼ ET inj , Eqs. (3)–(14) can be rewritten in a state space form: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 _ x ðtÞ ¼ c p ðtÞu ðtÞ  c x2 ðtÞ½x1 ðtÞ  x2 ðtÞ; > 1 11 1 12 0 < x_ 2 ðtÞ ¼ c21 p0 ðtÞ½c24 x1 ðtÞ  c25 x2 ðtÞ  c26 p0 ðtÞ  c27  (15) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : c x ðtÞu ðtÞ þ c x ðtÞ½x ðtÞ  x ðtÞ; 22 2

2

23

2

1

2

where cij are constant coefficients (Tables 1 and 3). The system of nonlinear equations (15) completely describes the system dynamics in terms of control volume pressures. 4. A generalized predictive control law for the rail pressure regulation 4.1. Model linearization and discretization To design a controller for the rail pressure, the state equations (15) are linearized around different equilibrium points. Hence each tuning of the controller parameters

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refers to a current working point. This simplification is justified because the control action has to keep the pressure close to a reference value, in dependence of the working conditions set by the driver power request, speed and load. Moreover, with a linearized model it is possible to apply well known linear design techniques. Then the linear continuous-time models are sampled to implement a discrete control with the ECU microcontroller. The linearization of (15) with respect to x and u yields     qf qf _ ¼ dxðtÞ þ duðtÞ ¼ AdxðtÞ þ BduðtÞ, dxðtÞ qx x;u qu x;u (16) where dxðtÞ ¼ xðtÞ  x, duðtÞ ¼ uðtÞ  u, and f is the vector nonlinear function of the state-space model. A and B are, respectively, the state and the input matrices, stemming from linearization. Choosing dx2 ðtÞ as output gives dyðtÞ ¼ CdxðtÞ, where C ¼ ½0 1 is the output matrix. The computation gives the following elements Aði;jÞ and Bði;kÞ of matrices A and B: x2 Að1;1Þ ¼ c12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 x2 ðx1  x2 Þ x1  2x2 Að1;2Þ ¼ c12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 x2 ðx1  x2 Þ x2 Að2;1Þ ¼ c21 c24 p0 þ c23 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 x2 ðx1  x2 Þ x1  2x2 Að2;2Þ ¼ c21 c25 p0  c22 u2 þ c23 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x2 ðx1  x2 Þ

(17)

and Bð1;1Þ ¼ c11 p0 , Bð1;2Þ ¼ 0, Bð2;1Þ ¼ 0, Bð2;2Þ ¼ c22 x2 .

(18)

One of the signals affecting the working condition is the tank pressure, which is approximatively constant within a large time interval. On the contrary, the valve and injectors driving signals, u1 and u2 , respectively, result from modulation of discrete values and cannot be used directly into the model equations to determine the equilibrium pressures. For this reason, instead of the instantaneous values, their mean values within an injection cycle are considered. In particular, the driving current duty cycle is assumed as input signal u1 , while the ratio between the injection time interval and the duration of whole injection cycle, multiplied by the number of injectors, is considered as the input u2 . Given the engine speed u in rpm, the number of injectors n and the injectors opening time interval tj , the input u2 can be computed as u . (19) u2 ¼ n  t j  120 Applying a backward difference method to (16) leads to a 1  2 discrete transfer-function matrix, whose elements

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give the relationship between the output yðp2 Þ and the inputs du1 and du2 , respectively. Considering du1 as the control signal, it is possible to derive a family of AutoRegressive eXogenous (ARX) models suitable for the controller design from the first element of the transferfunction matrix. With some notational abuse, dropping the symbol d and using uðtÞ instead of u1 ðtÞ, and regarding t as the discrete-time variable, yields ð1  a1 q1 ÞyðtÞ ¼ ðb0 q1  b1 q2 ÞuðtÞ,

(20)

where q1 is the shift operator.

J ¼ EfðG~u þ r  wÞT ðG~u þ r  wÞ þ l~uT u~ g,

4.2. Controller design Model predictive control techniques predict the output from a process model and then impress a control action able to drive the system to a reference trajectory (Rossiter, 2003). The control action is applied to systems represented by an AutoRegressive Integrated eXogenous (ARIX) model: Aðq1 ÞyðtÞ ¼ Bðq1 Þuðt  1Þ þ xðtÞ=D,

(21)

where xðtÞ is a zero mean white noise, Aðq1 Þ and Bðq1 Þ are polynomials in q1 of degrees na and nb, respectively, and D is the discrete derivative operator ð1  q1 Þ. The jstep division of 1 by DAðq1 Þ gives the following Diophantine equation: 1 ¼ E j ðq1 ÞAðq1 ÞD þ qj F j ðq1 Þ,

(22)

where E j ðq1 Þ and qj F j ðq1 Þ are, respectively, the quotient and the remainder polynomials of ðj  1Þth and ðj þ naÞth order. By putting together the previous equations it can be calculated the system output in j steps: yðt þ jÞ ¼ F j ðq1 ÞyðtÞ þ E j ðq1 ÞDBðq1 Þuðt þ j  1Þ þ E j ðq1 Þxðt þ jÞ.

(23)

Since the last term in (23) only refers to future noise samples, it cannot be known in advance, so putting G j ¼ E j ðq1 ÞBðq1 Þ gives the following equation of the j-step optimal predictor: ^ þ jjtÞ ¼ G j ðq1 ÞDuðt þ j  1Þ þ F j ðq1 ÞyðtÞ, yðt 1

(24)

1

where G j ðq Þ and F j ðq Þ are polynomials of degrees ðj þ nb  1Þ and na, respectively. Let rðt þ jÞ be the component of yðt þ jÞ depending on known values at time t: rðt þ 1Þ ¼ ½G 1 ðq1 Þ  g10 DuðtÞ þ F 1 ðq1 ÞyðtÞ, rðt þ 2Þ ¼ ½G 2 ðq1 Þ  q1 g21  g20 qDuðtÞ þ F 2 ðq1 ÞyðtÞ, .. . rðt þ jÞ ¼ ½G j ðq1 Þ  qðj1Þ gj;j1      gj0  qj1 DuðtÞ þ F j ðq1 ÞyðtÞ, 1

1

^ þ 1Þ; . . . ; matrix form y^ ¼ G~u þ r, where y^ ¼ ½yðt ^ þ NÞT , u~ ¼ ½DuðtÞ; . . . ; Duðt þ N  1ÞT , r ¼ ½rðt þ 1Þ; yðt . . . ; rðt þ NÞT and G is a lower triangular N  N matrix, whose elements are Gðl; mÞ ¼ gl;lm for lXm, and Gðl; mÞ ¼ 0 for lom (Rossiter, 2003). If w ¼ ½wðt þ 1Þ; wðt þ 2Þ; . . . ; wðt þ NÞT is a sequence of future reference-values, a cost function taking into account the future errors, i.e. the difference between future output and reference values, and the control action variations can be introduced:

(25) ðnbþj1Þ

where G j ðq Þ ¼ gj0 þ gj0 q þ    þ gj;nbþj1 q . Then it is possible to express (24), for j ¼ 1; . . . ; N, in the

(26)

where l is a sequence of weights on future control actions. The minimization of the cost function J with respect of u~ gives the optimal control law for the prediction horizon N: u~ ¼ ðGT G þ lIÞ1 GT ðw  rÞ.

(27)

Since the first element of u~ is DuðtÞ, the current control action is uðtÞ ¼ uðt  1Þ þ gT ðw  rÞ,

(28)

where gT is the first row of ðGT G þ lIÞ1 GT ; at each step the first computed control action is applied and then the optimization process is repeated after updating all vectors. By placing infinite weights on control changes after N U steps, where N U is the control horizon, all next control actions are taken to be equal to uðt þ N U Þ. A higher value for N U results in a more active control action, while a lower value reduces the computational effort, as u~ reduces to a N U  1 vector. Obviously a good choice puts the prediction horizon equal to the system rise time. The above concepts are applied to design a GPC for the rail pressure considering the family of ARX models (20), because it belongs to the class represented by (21). It can be shown (Rossiter, 2003) that the resulting control law becomes DuðtÞ ¼ k1 wðtÞ þ ðk2 þ k3 q1 ÞyðtÞ þ k4 Duðt  1Þ,

(29)

where ½k1 ; k2 ; k3 ; k4  depends on N and N U (Table 2). A simple scheduling strategy based on a sensitivity analysis of linear model coefficients has been implemented in simulation, consisting in what follows. A fixed value of the prediction error is chosen to select the linear model used to compute the controller parameters. Simulation and experimental tests point out that the tank pressure is the main variable affecting controller performances, allowing to set different main families of sub-models. The pressure drop between control chamber and common rail is approximately constant apart the operating condition, thus only x2 is considered for scheduling. Finally, u2 determines the final linear model for control, also taking in mind that Að2;2Þ (and thus u2 ) had a weaker influence on the model coefficients. The complete set of GPC parameters can be computed off-line and stored in a look-up table as a function of the chosen operating conditions. Clearly, increasing the number of operating conditions for the scheduling strategy would require a larger amount of

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memory to store controller parameters, though not increasing the computational effort. Since the GPC law gives the change with respect of the previous control action, it is necessary to integrate (29) to apply the whole input. The control action is bounded in the range [0%, 100%], thus an anti wind-up system to avoid undesired oscillations in the control loop has been introduced. 5. Simulation and experimental results 5.1. Experimental set-up This section describes the validation experiments of the proposed model and the evaluation of the controller Table 2 GPC notation Symbol

Description

AðÞ, BðÞ, EðÞ, F ðÞ, GðÞ Polynomials of ðq1 Þ g Subsidiary column vector for computation of u E Expected value G Matrix composed of elements of GðÞ J Cost function k GPC gains N, N U GPC prediction and control horizons Discrete time shift operator q1 r ¼ ½rðÞ Column vector of known components of y u~ ¼ ½DuðÞ Column vector of future control actions w ¼ ½wðÞ Column vector of future reference values ^ y^ ¼ ½yðÞ Column vector of predicted outputs D Discrete derivative operator l Sequence of weights on future control actions xðÞ Zero mean white noise

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performances. Due to safety reasons, air replaced CNG as test fluid during the experiments, without substantially affecting the fluid-dynamic behaviour. Table 3 compares the values of model parameters for the two cases. Fig. 5 shows the experimental setup consisting of three main subsystems, namely the common rail injection system, a PC systems equipped with a 250 MHz dSPACEs board (ds1104) and a PC system equipped with a National Instruments acquisition board. Moreover, a programmable MF3 development master box took the role of ECU. The test rig replaced the fuel tank with a compressor, providing air at a constant input pressure p0 , and included an injection system with prototype injectors, designed to operate within a 4–20 bar pressure interval, and sending air to a discharging manifold. Two different configurations were considered during tests, using n ¼ 4 and 6 injectors, respectively. The former refers to open loop operations, carried out to collect data for model validation, the latter refers to closed loop operations, to evaluate the controller performances. The system operating conditions were determined by the engine driving shaft speed u (engine speed in the following), the intake manifold air flow, which varies with engine load and power request, the input air pressure p0 and the instantaneous rail pressure p2 . Since the test rig did not include the driving shaft and the intake manifold, the engine speed and load signals were artificially generated by the National Instrument PC system using the labVIEWs software. To inject the proper fuel amount corresponding to the actual working conditions, the ECU imposed the injection timings, in terms of opening time interval tj (as a function of engine load and rail pressure) and frequency n  u=120 (as a function of engine speed), by drawing the appropriate values from a look-up table. Eventually, tj

Table 3 Nonlinear model parameters Symbol c11 c12 c21 c22 c23 c24 c25 c26 c27

Description sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðgþ1Þ=ðg1Þ cd;01 A01 2 gRT gþ1 V1 cd;12 A12 qffiffiffiffiffiffiffiffiffiffi 2 5 RT V1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðgþ1Þ=ðg1Þ cd;02 pd s sin bs 2 gRT gþ1 V2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ðgþ1Þ=ðg1Þ cd;23 A23 2 gRT gþ1 V2 rffiffiffiffiffiffiffiffiffiffi cd;12 A12 2 RT 5 V2 S1 ks S1 1  pd 2 cos2 bs ks 4ks 0 1 pd 2 cos2 bs 4ks 0 F s0 ks

Value (CNG)

Value (air)

2.00

1.53

2.40

1.79

1:70  104

1:30  104

2.47

1.89

8:11  102

6:04  102

2:02  108

2:02  108

1:99  108

1:99  108

3:39  1010

3:39  1010

1:06  103

1:06  103

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Fig. 5. The experimental setup.

could be set independently from the working conditions during open loop operations. The rail pressure reference, which was established by the ECU depending on the engine speed and load, was sent to the dSPACE PC system to drive the solenoid valve by means of a 24 V power driver, both in open loop and closed loop operations. Alternatively, the dSPACE Control Desk software was capable of artificially generating the rail pressure reference bypassing the ECU. The 24 V power driver properly supplied the solenoid valve by performing a peak-hold modulation through a 30 kHz TTL signal (Fig. 5). The resulting driving current, which was applied for a tph time interval to the valve electromagnetic circuit, consisted of a 1.6 ms peak phase followed by a variable duration hold phase, of 6 and 3.5 A amplitudes, respectively. The valve control period tc was defined as the time interval between two leading edges of the current signal, set to 50 and 100 ms during the closed loop and open loop experimental tests, respectively. Finally, the hold phase duration was regulated by varying the tph =tc ratio, namely the control signal duty cycle. During closed loop operations, the control algorithms implemented in the MATLAB/Simulinks environment imposed the signal duty cycle. The dSPACE code generator compiled the Simulink program and then the real-time executable code was downloaded to the board memory. Then the board processor received the feedback from the rail pressure sensor and applied the appropriate control action to the solenoid valve. The signals were processed

using the 16 bit A/D-D/A converters that are integrated in the dSPACE board. During open loop operations, the control signal duty cycle was artificially generated bypassing the controller. 5.2. Model validation The solution of (15) in the MATLAB/Simulink environment enabled the comparison between simulation and experiments. The data necessary for the model validation were obtained in open loop operations with the deactivated pressure controller by properly generating the driving signal of the solenoid valve. With a constant 40 bar input pressure p0 , three cases covering typical operating conditions were analysed: (a) with constant injection timings and varying duty cycle, (b) with both duty cycle and injection timings held constant, (c) with both duty cycle and injection timings varying. Firstly, with constant injectors driving command, the system response to step variations of the duty cycle of the valve driving signal was determined. Fig. 6 compares simulation results with the performed experiments. The operating conditions were a 2400 rpm engine speed and 8 ms injectors exciting time interval, when two pairs of opposite duty cycle step variations were applied (3–9%, 9–3%, 3–12%, 12–3%), at 1.5, 28, 56.8, 82.8 s time instants, respectively. With a positive step variation, the pressure increased in the control chamber (see Fig. 6(a)), making the regulator inlet section to stay open longer.

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simulated

0

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13.6

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0

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12.7 simulated

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80

80.1

80.2 80.3 Time [s]

80.4

80.5

simulated

12.5 12.3 12.1 11.9 experimental

11.7

80

80.1

80.2 80.3 Time [s]

80.4

80.5

Fig. 6. Simulation and experimental results with duty cycle step variations and constant injectors driving signal: (a) control chamber pressure; (b) rail pressure; (c) detail on control chamber pressure; (d) detail on rail pressure.

As a consequence of the larger mean air inflow coming from the compressor, the rail pressure raised (see Fig. 6(b)). Conversely, with a negative step reducing the duty cycle, both the control chamber and the rail pressures fell to lower levels. As shown in Fig. 6(c), the pressure in the control chamber had an oscillating behaviour within the control period: the pressure increased (decreased) when the solenoid valve was open (closed). Shortening the control period could smooth away these pressure variations. The waves amplitude difference between experiment and simulation in Fig. 6(c) is mainly due to model simplifications. In fact, by introducing both the second order dynamics of solenoid valve shutter and piston, better results can be obtained. Obviously, introducing these high order dynamics increases the model complexity, resulting in a higher order controller. On the other hand, the oscillating behaviour should not affect the prediction, which is computed on the base of mean value inputs. Moreover, higher frequency, superimposed rail pressure oscillations due to injections within a cycle occurred (see Fig. 6(d)). In this case, simulation results exhibited wider oscillations than experiments, synchronously with the solenoid valve operations. Modelling assumptions, indeed, considered the reducer main chamber and the common rail as a whole control volume. Consequentially, the decoupling effect of the main chamber volume with respect to

pressure disturbances propagating towards the rail was not taken into account. In addition, the distributed losses along the pipes connecting common rail and pressure reducer were neglected. However, a detailed representation of these phenomena would require a PDE description without adding essential information useful for control. In a second test, duty cycle was held constant (9%), while engine speed and injection timings varied. Figs. 7(a) and (b) depict model output in terms of control chamber and rail pressures. Simulation started from a steady state condition corresponding to a u ¼ 2200 rpm engine speed and tj ¼ 3 ms injectors exciting time interval. At time 4.5 s, a 4000 rpm speed step was applied, and the injection time interval was raised to 11 ms. In these conditions, the current duty cycle was no longer able to maintain the initial rail pressure, because of the increased amount of injected air. Besides, the air flow between main circuit and control chamber made the control chamber pressure to decrease. At time 21 s a complete cut-off was applied, i.e. the injectors were kept closed in the whole injection cycle. So control chamber and rail pressures raised because the air was no more sent to the discharging manifold. Finally, at time 29 s, u and tj were taken back to initial values. Results show that the model approximates the real system behaviour both during the steady state condition and during the cut-off transient, even though it had a smoother dynamics, probably due both to unmodelled nonlinearities

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5

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25

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8 6 4 simulated

2

30

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25

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12

12 simulated

10 8 6 4 2

experimental

simulated

10 8 6 4 2

experimental

0

20 40 60 80 100 120 140 160 Time [s]

7

14

6

12 u (duty cycle) [%]

υ (engine speed) [rpm x 1000]

x2 (rail pressure) [bar]

x1 (control chamber pressure) [bar]

Fig. 7. Simulation and experimental results with a constant duty cycle and when applying step variations of engine speed and injection time interval: (a) control chamber pressure; (b) rail pressure.

5 4 3 2

0

20 40 60 80 100 120 140 160 Time [s]

0

20 40 60 80 100 120 140 160 Time [s]

10 8 6 4 2

1

0

0 0

20 40 60 80 100 120 140 160 Time [s]

Fig. 8. Simulation and experimental results when varying duty cycle and engine speed, with a constant tj : (a) control chamber pressure; (b) rail pressure; (c) engine speed; (d) control signal duty cycle.

(as the pressure discontinuity at time 6 s testifies) and imperfect parameters tuning. Finally, the system behaviour for a constant load (resulting in a 3 ms opening time interval), while varying engine speed and solenoid valve driving signal was evaluated. The engine speed was composed of ramp profiles (Fig. 8(c)), while the duty cycle changed abruptly within the interval ½1%; 13% (Fig. 8(d)). Figs. 8(a) and (b) show the accordance of the resulting dynamics with the expected behaviour. A maximum error of 10% confirms the model validity.

5.3. Controller performances The tuning of the controllers was referred to models linearized at the starting equilibrium point, according to design steps of Section 4. However, the values of the controller parameters was not scheduled during experiments. Namely, the set relevant to initial working condition was only considered, as preliminary results on the test bench were satisfactory. Two different experiments allowed to evaluate the controller performances for different engine

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0

5

10 15 20 25 30 35 40 Time [s]

0

5

10 15 20 25 30 35 40 Time [s]

30 20 10 0

Fig. 9. System response for speed and load ramp variations, in presence of a GPC with N ¼ 5 (0.25 s) and N U ¼ 1 (0.05 s): (a) rail pressure; (b) injectors’ exciting time interval tj ; (c) engine speed; (d) control signal duty cycle.

load and speed, letting the ECU to generate the appropriate injection timings and rail pressure reference, and imposing the rail pressure reference with constant speed and load manually. The first experiment was performed increasing and decreasing ramp profiles of the engine speed and load, and a GPC with a N ¼ 5 (0.25 s) prediction horizon and a N U ¼ 1 (0.05 s) control horizon. The rail pressure reference was provided by the ECU in dependence of the current working conditions set by the National Instrument PC system and had a sort of ramp profile as well. The input air pressure coming from the compressor was always 30 bar. Fig. 9 shows the engine speed accelerating from 1000 to 1800 rpm and then decelerating to 1000 rpm, within a 20 s time interval (Fig. 9(c)). With a ramp slop not exceeding a certain value, the control action guaranteed a good reference tracking (Fig. 9(a), during time intervals [0, 14] and [22, 40]). Starting from time 14 s, owing to the request of a quick pressure reduction the control action closed the valve completely (Fig. 9(d)) by imposing a duty cycle equal to zero. Thanks to injections, the rail pressure (Fig. 9(a)) decreased to the final 5 bar reference value, with a time constant depending on the system geometry. Due to the saturation of the actuation variable the maximum error amplitude could not further decrease. Fig. 9(b) shows the injectors’ exciting time during the experiment.

Analogous working conditions were reproduced in a test where a classical PI controller tuned by a trial and error approach and including a simple anti-wind-up system replaced the GPC (Fig. 10). In this case, the imposed engine speed profile was more favourable than in the previous one (being the speed bounded within the 950–1600 rpm interval, see Fig. 10(c)), so it produced a smaller pressure disturbance due to air injection. Fig. 10(a), clearly shows good tracking performances of the PI controller during the rising transient, and at the same time a considerable undershoot of the rail pressure before to reach the steady state condition, with a large settling time. In the second experiment the controller performances were evaluated by applying a step reference variation taking engine load and speed constant, as well as a constant input pressure from the compressor. As stated earlier, in this experiment, the reference pressure was manually generated, and it was no more set by the ECU depending on the engine speed and load. Figs. 11 and 12 depict the system behaviour in presence of the GPC and the classical PI controllers, respectively. Fig. 11(a) refers to a step reference variation from 5 to 20 bar, while keeping a constant 1460 rpm engine speed (Fig. 11(c)) and a 30 bar input pressure. The ECU reduced the injection time interval from the initial 19–4 ms (Fig. 11(b)), so that the same air amount was injected during a cycle when the rail pressure raised. Fig. 11(d) shows that the control action

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rail pressure

12 8 4

pressure reference

0

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1.6

40

u (duty cycle) [%]

υ (engine speed) [rpm x 1000]

tj (injection time interval) [ms]

x2 (rail pressure) [bar]

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1.4 1.2 1

0

5

10 15 20 25 30 35 40 Time [s]

0

5

10

30 20 10 0

0.8 0

5

10 15 20 25 30 35 40 Time [s]

15 20 25 Time [s]

30

35

40

Fig. 10. System response for speed and load ramp variations, in presence of a standard PI: (a) rail pressure; (b) injectors’ exciting time interval tj ; (c) engine speed; (d) control signal duty cycle.

20 16 pressure reference

12 8

rail pressure

4 0

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1

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2.5

3

50 40 30 20 10 0

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50

1.8

40

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x2 (rail pressure) [bar]

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1

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2.5

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1

1.5 2 Time [s]

2.5

3

0

0.5

1

1.5 2 Time [s]

2.5

3

30 20 10 0

1

0

Fig. 11. System step response for a 1460 rpm engine speed, in presence of a GPC with N ¼ 5 (0.25 s) and N U ¼ 1 (0.05 s): (a) rail pressure; (b) injectors’ exciting time interval tj ; (c) engine speed; (d) control signal duty cycle.

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pressure reference

16 12 8

rail pressure

4 0

3

6

9 12 Time [s]

15

50 40 30 20 10 0

18

2

50

1.8

40

u (duty cycle) [%]

υ (engine speed) [rpm x 1000]

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x2 (rail pressure) [bar]

24

1.6 1.4 1.2 1

0

3

6

9 12 Time [s]

15

18

1229

0

3

6

9 12 Time [s]

15

18

0

3

6

9 12 Time [s]

15

18

30 20 10 0

Fig. 12. System step response for a 1330 rpm engine speed, in presence of a standard PI: (a) rail pressure; (b) injectors’ exciting time interval tj ; (c) engine speed; (d) control signal duty cycle.

during the rising transient was the maximum possible. Moreover, the step response presented a negligible overshoot, and the limited amplitude pressure oscillations close to set-point were basically due to the injection process. Clearly, increasing prediction horizon could result in a sluggish response, while further experiments may show that increasing the control horizon does not result in a better rail pressure behaviour. As for the PI controlled system, the experimental test was performed by applying a 7–15 bar step variation (Fig. 12(a)), considering a constant 1330 rpm engine speed (Fig. 12(c)) and a 100 bar input pressure. Further, the injection time interval was reduced from 40 to 15 ms by the ECU to keep the injected fuel amount constant. Fig. 12(d) shows that owing to the higher input pressure a reduced control effort was necessary to maintain the pressure at set point. Even though conclusive remarks cannot be drawn due to different operational conditions, it seems that the GPC enables a better closed loop dynamics than the PI. In this case, the rise time was considerably longer than the GPC case, and the system response showed a 17% overshoot. However, it must to be underlined that the higher input pressure could have negatively influenced the system overshoot, as more air could enter the control chamber for the same control action, while the higher load, which resulted in a longer tj (Fig. 12(b)), could have increased the system rise time. In the end, the PI

parameters appear unsuited to enable good performances in every working condition. 6. Conclusion The work presented in this paper investigated the feasibility of the electronic control of the injection pressure in a new common rail injection system for CNG engines. Better performances were obtained by taking advantage of some well-established diesel technologies. The main improvements consist in a better fuel metering and in the capability of on-line adaption of the injection pressure to different operating conditions. Even though a direct assessment has not been performed in the present work, other authors proved that performances and pollutant emissions reduction can benefit from a precise control of the injection pressure. To develop a control law for the injection pressure regulation, a model of the injection system in state-space form was presented. The parameters of the model were directly known or easy determined by geometrical data. The comparison of simulation results with experimental data showed that the system dynamics was modelled with a good accuracy and for a wide range of operating conditions. The proposed model was used to design a generalized predictive controller for the injection pressure control. Experimental results confirmed the effectiveness of the approach.

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In conclusion, the model simplicity makes it suitable for on-line use or for inclusion in a complete engine model for emissions control. Moreover, the proposed GPC law seems promising with respect to the classical PI, also in view of the development of nonlinear predictive control strategies. Future developments would also consider the MIMO control of injection pressure and injectors timings under a common control law. References Aespy, V., Engja, H., & Skarboe, L. V. (1996). Fuel injection system design, analysis and testing using bond graph as an efficient modeling tool. SAE Paper 962061. Amorese, C., De Matthaeis, S., De Michele, O., & Satriano, A. (2004). The gaseous fuel option: LPG and CNG. In Procedings of the international conference on vehicles alternative fuel system & environmental protection, Dublin, Ireland. Cantore, G., Mattarelli, E., & Boretti, A. (1999). Experimental and theoretical analysis of a diesel fuel injection system. SAE Paper 199901-0199. Catania, A. E., Dongiovanni, C., Mittica, A., Negri, C., & Spessa, E. (1996). Study of automotive diesel injection-system dynamics under control. SAE Paper 962020. Desantes, J. M., Arregle, J., & Rodriguez, P. J. (1999). Computational model for simulation of diesel injection systems. SAE Paper 1999-010915. Dyntar, D., Onder, C., & Guzzella, L. (2002). Modeling and control of CNG engines. SAE Paper 2002-01-1295. Ficarella, A., Laforgia, D., & Landriscina, V. (1999). Evaluation of instability phenomena in a common rail injection system for high speed diesel engines. SAE Paper 1999-01-0192. Fuseya, T., Nishimura, T., Sato, Y., & Tanaka, H. (1999). Analysis on common-rail injector using bond graph simulation program. SAE Paper 1999-01-2936. Gauthier, C., Sename, O., Dugard, L., & Meissonnier, G. (2005). Modelling of a diesel engine common rail injection system. In Proceedings of the 16th IFAC world congress, Prague, Czech Republic. Heywood, J. (1988). Internal combustion engine fundamentals. New York, USA: McGraw-Hill.

Hountalas, D. T., & Kouremenos, A. D. (1998). Development of a fast and simple simulation model for the fuel injection system of diesel engines. Advances in Engineering Software, 1(29), 13–28. International Gas Union (2005). Global opportunities for natural gas as a transportation fuel for today and tomorrow. Report on study group 5.3 ‘‘Natural gas vehicles (NGV)’’. 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 Paper 1999-01-0527. Kuraoka, H., Ohka, N., & Ohba, M. (1989). Application of H-infinity optimal design to automotive fuel control. In Proceedings of American control conference, Pittsburg, Pennsylvania. Kvenvolden, K. A. (1999). Natural gas hydrate occurrence and issues. Annals of the New York Academy of Sciences, 715, 232–246. Lino, P., Maione, B., & Rizzo, A. (2007). Nonlinear modelling and control of a common rail injection system for diesel engines. Applied Mathematical Modelling, 9(31), 1770–1784. Maione, B., Lino, P., De Matthaeis, S., Amorese, C., Manodoro, D., & Ricco, R. (2004). Modeling and control of a compressed natural gas injection system. WSEAS Transactions on Systems, 3(5), 2164–2169. McCormick, R. L., Graboski, M. S., Alleman, T., Herring, A. M., & Nelson, P. (1999). In-use emissions from natural gas fueled heavy-duty vehicles. SAE Paper 1999-01-1507. Morselli, R., Corti, E., & Rizzoni, G. (2002). Energy based model of a common rail injector. In Proceedings of the IEEE conference on control applications, Glasgow, Scotland. Mulemane, A., Han, J. S., Lu, P. H., Yoon, S. J., & Lai, M. C. (2004). Modeling dynamic behavior of diesel fuel injection systems. SAE Paper 2004-01-0536. Pan, C. P., Li, M. C., & Hussain, S. F. (1998). Fuel pressure control for gaseous fuel injection systems. SAE Paper 981397. Qina, S. J., & Badgwell, T. A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice, 7(11), 733–764. Rossiter, J. A. (2003). Model-based predictive control: A practical approach. New York, USA: CRC Press. Streeter, V., Wylie, K., & Bedford, E. (1998). Fluid mechanics (9th ed.). New York, USA: McGraw-Hill. Weaver, C. S. (1989). Natural gas vehicles—a review of the state of the art. SAE Paper 892133. Zucrow, M., & Hoffman, J. (1976). Gas dynamics. New York, USA: Wiley.