Model-based cold-start speed control scheme for spark ignition engines

Control Engineering Practice 18 (2010) 1285–1294

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Model-based cold-start speed control scheme for spark ignition engines Jiangyan Zhang a,, Tielong Shen a, Riccardo Marino b a b

Department of Engineering and Applied Sciences, Sophia University, Kioicho, Chiyoda-ku, Tokyo, Japan Electronic Engineering Department, The University of Rome ‘‘Tor Vergata’’, Rome, Italy

a r t i c l e in fo

abstract

Article history: Received 10 April 2009 Accepted 20 January 2010 Available online 12 February 2010

This paper presents a model-based control scheme to the cold-start speed control in spark ignition (SI) engines. The multi-variable control algorithm is developed with the purpose of improving the transient performance of the starting engine speed: the control inputs are the fuel injection, the throttle and the spark advance (SA), while the engine speed and the air mass flow rate are the measured signals. The fuel injection is performed with a dual sampling rate system: the cycle-based fuel injection command is individually adjusted for each cylinder by using a TDC (top dead center)-based air charge estimation. The desired performance for speed regulation is achieved by using a coordinated control of SA and throttle operation. The speed error convergence of the closed loop system is proved for simplified, second-order model with a time-delay, and the robustness with respect to parameter uncertainties is investigated. The performance and the robustness with respect to modeling uncertainties of the proposed control scheme are tested using an industrial engine simulator with six cylinders. & 2010 Elsevier Ltd. All rights reserved.

Keywords: SI engine Cold starting Speed control Transient performance Model-based analysis

1. Introduction Speed control for internal combustion engines is not a new topic in the engine control community. As it is well-known, idle speed control, which involves engine speed regulation, disturbance rejection and transition into the idling, is one of the fundamental control specifications for engine management. The core of speed control problems is to improve the performance of speed responses in any operating condition, since it has significant impacts on optimizing the vehicle features such as fuel economy, emission and vibration. There is therefore a continuous interest in investigating speed control problems for combustion engines. Several approaches have been provided such as PID control, H1 control (Hrovat & Sun, 1997), fuzzy logic (Thornhill, Thompson, & Sindano, 2000), sliding mode control (Puleston, Spurgeon, & Monsees, 2001; Vesterholm & Hendricks, 1994) and others reported in the cited references; nevertheless the precise speed control for SI engines during transient operation modes is still a challenging issue (Ohata, Kako, Shen, & Ito, 2007; Thornhill et al., 2000) in particular during the cold starting transient mode (Ohata et al., 2007). Recently, many research results on the cold-start issues of SI engines have been reported. Sampson and Heywood (1995), Cheng, Wang, Zhuang, and Wu (2001), Yamada, Gardner, Bruno, Zello, and Santavicca (2002) and Liu, Li, and Deng (2007) provide

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E-mail addresses: [email protected] (J. Zhang), [email protected] (T. Shen), [email protected] (R. Marino). 0967-0661/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2010.01.010

precise descriptions for the dynamic characteristics during cold starting transient: experimental results show that engine performance in terms of combustion reliability, speed transition and emission is significantly affected by instantaneous changes of engine state from cold to warm-up. On the other hand, it is known that SI engine is a complex physical system with multi-actuation control inputs, fuel injection, throttle and SA, which play different roles on engine performance (Hrovat & Sun, 1997; Liu et al., 2007). Therefore, many control strategies have been presented to improve engine performance during cold starting: great efforts were paid on air–fuel ratio (A/F) control (Eng, 2007; Leisenring, Samimy, & Rizzoni, 1996; Ma, Yurkovich, & Dudek, 2007, 2008) by adjusting fuel injection, and a proposed quick warm-up method was reported in Ueno, Akazaki, Yasui, and Iwaki (2000) in which the throttle and spark timing are the control inputs. However, most of aforementioned investigations focused on the emission performance. Indeed, in internal combustion engines, a very large fraction of emissions is produced during the cold starting phase. On the other hand, the transient performance of starting speed becomes an important issue in the recent advanced powertrains such as in hybrid electrical vehicles and in idling stop vehicles. Especially, the transient speed performance during the cold staring phase is a challenging issue since the thermal characteristics of engines change dramatically. Recently, to put this issue on the spot, the Research Committee on Advanced Powertrain Control Theory of the Society of Instrument and Control Engineers (SICE) in Japan proposed a benchmark problem of cold-start speed control for internal combustion engines (Ohata et al., 2007) and an industrial simulator of a sixcylinder gasoline engine was provided for validation. Some

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challenging results have been reported in the invited session of the IFAC World Congress, Seoul (see Kitazono, Sugihira, & Ohmori, 2008; Ogawa & Ogai, 2008; Tomohiko & Yoshikazu, 2008; Zhang, Shen, & Marino, 2008). As it is well-known, the starting process of an engine is roughly divided into four phases: cranking, unstable combustion, combustion stabilization and steady idling (Eng, 2007; Sampson & Heywood, 1995). The constraint of unstable combustion makes the regulation of the speed response during the second and the third phases and their coordination a challenging task. A great number of control-oriented engine models, called the mean-value models, have been presented in the last decades for advanced model-based control approaches (Cho & Hedrick, 1989; Guzzella & Onder, 2004). However, the mean-value models are usually developed to describe the behavior of engines at a pseudo-static operation mode. For the cold starting mode, these models are not accurate enough to describe the dramatically changed engine dynamics. The objective of this paper is to provide a solution to the challenging speed control problem during cold starting phase. The proposed scheme is based on control-oriented engine models and it includes control actions which take into account the specific physics during cold starting. First, the fuel injection is scheduled during the initial phase on the basis of the air charge estimation which is obtained using the mean-value models with timevarying parameters. Second, to obtain rapid acceleration followed by quick convergence to a specified idle speed, proportional feedback laws with constant and speed-dependent gains, respectively, are introduced for throttle and SA: they are coordinated since their actions are scheduled depending on the engine speed error. This scheduling consists of two steps: at the first step, the control action is performed by SA with a speed error proportional feedback control, and at the second step, the control action will be passed to throttle operation with a nonlinear feedback gain. Moreover, the system is a multi-rate sampling control system. The fuel injection is a cycle-based sampling control with fixed phase delay in crank angle, while the air charge estimation for each individual cylinder is provided with the TDC-based sampling rate. The control loop which coordinates the throttle and SA is designed in continuous time domain. The convergence and the robustness of the closed loop system in the second step are investigated on the basis of a simplified second-order mean-value model with a time delay in which the engine speed and the manifold pressure are the state variables. The performance and the robustness with respect to parameter and modeling uncertainties are tested on a detailed industrial engine simulator which involves 46 state variables and is used by automotive companies. The remainder of the paper is organized as follows. The engine model and the cold-start speed control problem are described in the next section. Section 3 introduces the presented cold-start speed control scheme. The proposed control approach is validated on the engine simulator provided by the SICE benchmark problem, and simulation results are shown and discussed in Section 4. Finally, concluding remarks are given in Section 5.

2. System modeling and problem formulation A simplified physical structure of a four-stroke SI engine is shown in Fig. 1, where only one cylinder is sketched for simplicity. In the engine with multi-cylinders, the torque to drive the crankshaft rotational motion is provided by each cylinder serially along crank angle, and the torque generated in each cylinder during its own expansion stroke is determined by individual air charge and fuel injection during the corresponding induction stroke. The behaviors of air charge and fuel injection are

Electronic Control Unite

Engine speed

Fuel Spark Air mass Throttle flow rate openning injection advance Fuel injector

Spark

Air Intake manifold

Throttle

Intake port fuel puddle

Command

Starting Motor

Fig. 1. Schematic representation of SI engine.

influenced by the air inlet path and fuel path of the engine system, respectively. Thus, the dynamics related to engine speed should be basically divided into three parts: air intake dynamics, fuel injection dynamics and crankshaft rotational dynamics. Typically, the control inputs are chosen as throttle opening, fuel injection and SA. For the problem discussed in this paper, the measurement signals for online control include crank angle, engine speed and air mass flow rate entering the intake manifold, respectively. Due to the above-mentioned complexities, control algorithms for the engine system are usually developed based on mean-value models, which ignore the characteristics of each individual cylinder and capture the average features of engine physics. In the following, a brief review of the engine models (see Cho & Hedrick, 1989; Guzzella & Onder, 2004; Heywood, 1988; Moskwa, 1988) is given, and the uncertain characteristics in the models during cold starting operation mode are discussed. Due to the thermal uncertainties, mean-value models cannot handle the behavior of engine during cold starting. However, as it will be shown in a later section, these models with a slight modification might be effectively exploited for cold starting control. ¨ First, many researches (see Chevalier, Muller, & Hendricks, 2001; Guzzella & Onder, 2004; Stotsky, Egardt, & Eriksson, 2000, for example) point out that an adequate description of air inlet path behavior should be given by considering adiabatic intake manifold. Under the assumption that the intake manifold temperature is constant and equal to the atmospheric temperature Ta [K], the air intake dynamics can be characterized by the following equation (Stotsky et al., 2000), which comes from the differentiation of the ideal gas law: p_ m ¼

RT a _ i m _ oÞ ðm Vm

ð1Þ

where R [J/(kg K)] is the gas constant, and pm [Pa] and Vm [m3] denote the air pressure and the volume of the manifold, _ i ðÞ denotes the air mass flow rate through throttle respectively. m entering the manifold and can be modeled by a nonlinear function used to describe air flow passing through an orifice (Heywood, 1988), i.e. pa _ i ¼ s0 ð1cosfÞ pffiffiffiffiffiffiffiffi m RT a

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi p c m pa

ð2Þ

where s0 [m2] denotes the throttle area, f ½rad is throttle opening, pa [Pa] is the atmospheric pressure, and cðÞ is

J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

sZ



2

k=ðk1Þ

kþ1

otherwise

_ o ðÞ in (1) denotes the air mass flow rate leaving the manifold m into the cylinders, which should be modeled as (2) according to the principle of air flow passing through an orifice with respect to the pressure ratio of pm and the corresponding cylinder pressure _ o will be switched according to the inlet pci ði ¼ 1; . . . ; NÞ. Hence, m phases of each individual cylinder. A common mean-value model _ o has been given according to Eq. (3) by means of the for m volumetric efficiency coefficient, Z, which measures the breathing performance of the engine (Chevalier et al., 2001; Heywood, 1988) _o¼ m

ra Vc Z opm 4ppa

ð3Þ

where ra [kg m  3] denotes the atmospheric density and Vc [m3] denotes the total cylinder volume of an engine. Actually, the effectiveness of the model (3) is ensured under pseudo-static modes, and the value of Z typically depends on engine speed, manifold pressure and temperature of cylinders, etc. (Chevalier et al., 2001; Heywood, 1988). Since these values are varying dramatically during cold starting, especially during the first few engine cycles, it is difficult to determine the volumetric efficiency precisely. For cold starting operation mode, another uncertainty in the mean-value models is the variation of the manifold temperature which can be modeled using the conservation law (Chevalier et al., 2001; Guzzella & Onder, 2004), and leads to a complex dynamic model of the manifold. In this paper, model (1) is used for control design purposes. Second, the following model, the so-called wall-film model, is widely used to represent the amount of fuel mfc [kg] injected into each cylinder during the induction stroke (Aquino, 1981) ( _ f ¼ wmf þ euf m ð4Þ mfc ¼ wmf þ ð1eÞuf in which mf [kg] denotes the fuel mass entering the intake port per induction stroke, uf [kg] is the fuel injection command. e represents the fraction of fuel deposited on the inlet port and w is the inlet port time constant. In the model (4), the parameters e and w are significantly affected by engine temperature (Eng, 2007; Guzzella & Onder, 2004). Consequently, it is difficult to obtain an accurate model (4) to describe the fuel path dynamics during engine cold starting. Finally, crankshaft rotational dynamics is obtained from Newton’s law as follows: _ ¼ te tf Jo

800

ð5Þ

where J [kg m2] denotes the inertia moment of the crankshaft, o ½rad=s denotes engine speed, and te ½N m and tf ½N m represent the engine torque and friction torque, respectively. As mentioned before, the torque of an engine with multicylinders is serially generated; an accurate description for engine torque is given by an event-based discontinuous mathematical model, which will cause difficulties in control design. Generally, the following mean-value computation of engine torque can be found in Cho and Hedrick (1989) _ ðttd Þ m  f ðlÞ  fs ðus Þ te ¼ a  o oðttd Þ l

where td is the intake-to-power delay, which is a function of o, i.e. td C p=o, fl ðlÞ A ½0; 1 and fs ðus Þ A ½0; 1 denote the normalized influences of the A/F l and the SA us on the mean-value engine torque, respectively, and a [Nm/(kg rad)] represents the maximum torque capacity when fl ðlÞ ¼ fs ðus Þ ¼ 1. Cold starting is a typical transient operation mode of SI engines, which as pointed out includes four phases. Crankshaft is initially driven to have a low constant rotational speed by the starting motor. Along the crank angle, the engine strokes of each cylinder follows. After the first ignition occurs in a cylinder, the driven torque on the crankshaft will be shifted from the starting motor to engine. This will cause an acceleration of crank rotational velocity to achieve stable idle speed. In other words, the basic requirement of engine starting is fast acceleration followed by quick convergence to a target idle speed. However, as mentioned above, the significant characteristic of engine during cold starting is that many engine parameters change during the starting evolution. Following the transition of engine speed, manifold pressure will be down from the atmospheric pressure to a static value which is extremely far from its initial value (Eng, 2007; Sampson & Heywood, 1995). Moreover, the cold manifold and cylinder wall will be warmed-up due to the combustion in the cylinders. All of these changes will affect the performance of engine cold starting and, as mentioned above, will cause great limitations to use the mean-value engine models to describe the behaviors during cold starting. The purpose of this paper is to provide a cold-start speed control scheme which satisfies a pre-specified engine speed performance; the design specification is given by the SICE benchmark problem of engine cold starting control (Ohata et al., 2007), i.e. the target idle speed 650 rpm with transient performance constraints indicated by the shadowed zone in Fig. 2, is considered for a SI engine with six cylinders. The starting motor provides an initial speed of 2507 50 rpm until a cylinder gets a successful combustion, but the maximum cranking time is 1.5 s. Clearly, before addressing the starting speed control, successful combustion should be guaranteed first which depends on the A/F in each cylinder. The uncertain air mass flow rate (3) and fuel path dynamics (4) will be an unavoidable obstacle to achieve this goal. Furthermore, even if the A/F is under control to ensure successful combustion, open-loop control, which keeps throttle and SA to be equal to the constant values determined by nominal model parameters according to static operation mode, cannot provide satisfactory transient speed performance. Fig. 3 shows a response of the engine speed and the manifold pressure during cold starting obtained on a full-scale engine simulator (Ohata et al., 2007), where the desired stable speed is set at 650 rpm and control signals are throttle opening at f ¼ 5:21 and SA ¼ 201. In this case, the starting speed shows an extremely large overshoot

Engine speed (rpm)

defined by 8  ðk1Þ=k > > 2=k 2k > ð1sÞ ; s > < k1 cðsÞ ¼  ðk þ 1Þ=ðk1Þ > 2 > > ; >k : kþ1

1287

600

± 50 rpm

Speed stable

650 rpm 400

200

Motoring stop 0

0

1

1.5

2

3

4

Time [s] ð6Þ Fig. 2. Image of cold-start speed with desired transient performance.

5

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150

Fuel path control

speed

100

1000 50

pressure

0 0

1

2

3

4 5 Time [s]

6

7

8

0

... uf6 mi

Supervisor

Engine speed (rpm)

2000

Intake manifold pressure (KPa)

uf1 Fuel injection controller

Spark controller

us

Throttle controller

uth

Engine system (6 cylinders)

Speed regulation Fig. 4. Block diagram of cold starting control system.

Fig. 3. Response of engine speed and manifold pressure during cold start with constant control inputs.

which is more than twice the required one, and therefore a long settling time which is also more than twice the required value. This is due to the over torque generation during the first few cycles. To suppress the undesired torque generation, delicate torque management is required during the cold starting stage: fine air charge estimation and fuel injection are recommended for individual cylinders so that the A/F l remains in the interval ½lmin ; lmax  for successful combustion. The next section will introduce a cold starting control scheme which includes a TDCscaled air charge estimation, a fuel injection control method for each individual cylinder fuel path and a scheduled control strategy for throttle opening and SA adjusting law.

3. Control scheme A SI engine with six cylinders is targeted to demonstrate the effectiveness of the proposed multi-variable control scheme. The structure of the proposed control scheme is sketched in Fig. 4, where uth denotes the normalized control signals of throttle opening, us denotes SA and ufi ði ¼ 1; . . . ; 6Þ denote the fuel injection commands for each cylinder. As shown in the figure, the control system consists of three main parts. (i) Fuel path control. The proposed control law, that delivers the fuel injection commands to each individual cylinder, is designed based on the air charge estimation under the A/F constraint condition. It is constructed by TDC-based air charge estimation with sampling period Tc ¼ 2p=ð3oÞ [s] (i.e. 1201 in crank angle), and cycle-based fuel injection with sampling period Ts ¼ 4p=o [s] (i.e. 7201 in crank angle). (ii) Speed regulation loop. The control laws for the two actuators throttle opening and SA are designed with a coordinated scheduling. To reject the over torque generation, SA is used during the acceleration stage of engine starting, and then the control will be definitely switched to the throttle opening action. This control law is performed in continuous time domain. (iii) Supervisor block. As mentioned above, the proposed control system is a multi-input, multi-sampling rate system, and the speed control loop is with dual actuators. All necessary switching signals for the multi-sampling rates will be delivered by this supervisor block.

Actually, in automotive engines, A/F should be stoichiometric for the requirements of optimal combustion and emission: especially emission performance has been the main issue for cold starting control for a long time. However, this paper will focus on the speed performance during the cold starting operation stage, i.e. the target is to obtain a desired transient performance of engine speed during cold starting, where A/F is not required to be strictly stoichiometric but it is required to remain within the constraint condition for firing in each cylinder. The influence caused by inaccurate A/F on speed performance is treated as uncertainty in the torque generation. On the other hand, from the strategies applied to the speed control of warm-up engines (Hrovat & Sun, 1997; Thornhill et al., 2000), it is known that regulation of SA provides a fast actuation on engine torque generation, which, however, is limited. A much larger control action on engine torque generation can be obtained by throttle operation to control the air inlet path: its disadvantage is the slow characteristic due to the intake-to-power delay. The scheduling control policy is motivated by these observations. The details of the control scheme are now given. 3.1. Fuel path control The objective of fuel path control is to give a proper fuel injection command cyclically such that the A/F constraint condition can be guaranteed during the combustion phase in each cylinder. To this purpose, the fuel dynamics (4) discretized with cycle-based sampling rate Ts is used, i.e. ( mfi ðk þ 1Þ ¼ ð1wÞmfi ðkÞ þ eufi ðkÞ ð7Þ mfci ðkÞ ¼ wmfi ðkÞ þ ð1eÞufi ðkÞ where i ¼ 1; . . . ; 6 and k denotes the sampling sequence. It is known that the A/F control in SI engines is usually based on the estimation of air charge into the cylinders (e.g. Grizzle, Cook, & Milam, 1994; Leroy, Chauvin, & Petit, 2009). With the estimated air mass, the needed amount of fuel to be injected into the corresponding cylinder is determined on the basis of the inverse dynamics of (7) with a desired A/F (ld ). In steady state, air charge into each cylinder can be calculated approximately _ o and engine speed according to the air mass flow rate m o (Grizzle et al., 1994). But, this method cannot work well during cold starting due to the dramatic speed change. In the following, a simple approximate calculation method is used to provide an estimation of the air charge into individual cylinders per cycle.

J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

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_ o in the considered case is Since the air mass flow rate m unmeasurable, an open-loop observer is introduced which is based on the air inlet path dynamics. The isothermal air inlet path model (1) and (3) is used to construct the observer shown as follows: 8 RT a > _ ^_ o Þ > _ i m ðm < p^ m ¼ Vm ð8Þ : > > ^_ o ¼ ra Vc Zop^ : m m 4ppa where the initial condition p^ m ð0Þ is the atmospheric pressure pa.

Fig. 5. Structure of fuel path control system.

Remark 1. Error dynamics between real air intake dynamics (1) and observer (8) is p_~ m ¼ c~ ðoÞp~ m

0 (i.e. φ = 0)

where p~ m ¼ p^ m pm and c~ ðÞ ¼ RT a ra Vc Z=ð4Vm ppa Þ  o. Under the assumption of constant manifold temperature, the observation error p~ m will converge to zero, since c~ ðoÞ 4 0 for any speed value o. Remark 2. In (8), constant volumetric efficiency, that is effective in the steady-state idling operation mode is used for simplicity. Therefore, according to the description of the coefficient Z in ^_ o given by observer (8) is an Section 2, the air mass flow rate m approximate estimation during the transient starting operation mode, and a parameter scheduling is suggested to improve the ^_ o . accuracy of air mass flow rate estimation m Moreover, it is known that the air and fuel are inducted into each cylinder synchronously during the induction stroke. Therefore, the air charge estimation should be achieved at each TDC timing so that the fuel injection command is able to meet the A/F constraint condition. However, the real air charge of each cylinder is obtained after the whole induction stroke. The following estimation method for the air charge into each cylinder denoted ^_ o given by the observer (8) ^ cyl is proposed with m by m ^_ o ðlT c Þ  t^ T ðlÞ ^ cyl ðlÞ ¼ m m dc

ð9Þ

where l denotes the sampling sequence of the TDC-based sampling rate Tc, and t^ Tdc ðlÞ ¼ 2p=ð3oðlT c ÞÞ is a predicted time of the induction stroke with o measured at each TDC timing. With the estimated air charge (9), the injected fuel for the i-th cylinder is calculated as follows: mfci ðkÞ ¼

^ cyl ð6ðk1Þ þiÞ m

ð10Þ

ld

Here, the relation between k and l is k ¼ fix((l  1)/6)+ 1. The inverse dynamics of the sampled model (7) is used to obtain the fuel injection command, i.e. ufi ðkþ 1Þ ¼ Aufi ðkÞ þ Bmfci ðkÞ þ Cmfci ðkþ 1Þ

ð11Þ

where A¼

weð1wÞð1eÞ ; 1e

B¼

1w ; 1e

C¼

1 1e

Furthermore, to compensate the influence of engine temperature on the fuel dynamics (4), the coefficients e and w are scheduled during the first few cycles. The developed fuel path control algorithm is summarized in Fig. 5. 3.2. Control design for throttle and SA Speed regulation is obtained by a coordinated control between throttle operation and SA adjusting: the control system is a multiinput single-output with us and uth as inputs and o as output, respectively. The presented approach is motivated by the typical

ωr

Reference ω d model

+ -

uth

us P controller

ω

eω

Σ

0≤ eω ≤ 50 rpm & eω < 0

Feedback control Feedforward compensation

+ Mbt

+ u th

Σ

us

Fig. 6. Logic structure of cold-start speed control.

transient response given in Fig. 3, where it can be seen that there are two phases: the first phase is the rapid acceleration stage while the second phase is the idle speed regulation. In the following, a reference model-based control approach is presented for cold-start speed regulation. The reference model is introduced to generate a desired speed trajectory od ðtÞ during engine starting. Driven by the idle speed or , the model which is chosen as a first-order system, begins to work as soon as a cylinder is successfully ignited at t ¼ t0, i.e.

o_ d ðtÞ ¼ sðod or Þ; t Z t0 ;

od ðt0 Þ ¼ oðt0 Þ

ð12Þ

where s is a positive constant. With the reference speed trajectory od , feedback control is carried out by SA and throttle opening, respectively. The switching logic of the coordinated control algorithm is as depicted in Fig. 6. Controller us is turned on at t ¼ t0 and turned off definitely at t ¼ t1 when the speed error eo : ¼ ood satisfies the following condition: 0 r eo ðt1 Þ r 50 rpm

and

e_ o ðt1 Þ o0

ð13Þ

and at t ¼ t1, controller uth is turned on. In the acceleration stage, us is chosen as a proportional feedback controller with an additional term us0(t) which will be designed on the basis of manifold pressure estimation given by the observer (8) and the speed reference model (12): ( uth ðtÞ ¼ 0; ð14Þ t0 rt r t1 us ðtÞ ¼ ks eo þ us0 ðtÞ; where ks is a given proportional gain and us0(t) is still to be designed. During this stage, the goal is to reduce the excessive engine torque generation and to suppress the speed response overshoot, which is caused by the high manifold pressure pm. Hence, from physical consideration, the throttle opening uth is set at 0 (i.e. f ¼ 0) during the acceleration stage. In order to achieve a lower torque generation, us0(t) is designed as follows. In the engine literature, the relation between SA relative to Mbt (minimum SA for best torque) and engine torque

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J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

generation (6) can be modeled by the following normalized function (Cho & Hedrick, 1989; Moskwa, 1988) fs ðus Þ ¼ ½cosðus Mbt Þ2:875

ð15Þ

in which for a special real engine, slight modification of model coefficient 2.875 may be needed. Furthermore, the mean-value description for engine friction torque in (5) is given as (Cho & Hedrick, 1989)

tf ¼ Do þ D0

o_ ¼ ct pm ðttd Þ½cosðus Mbt Þ2:875 D oD 0

ð17Þ

where the coefficients are defined as follows: ara Vc Z ; 4J ppa

D¼

D ; J

D0 ¼

D0 J

From the reference model (12) and the crank rotational model (17), the SA control component us0(t) is designed by solving for us0 the following equation: ct p^ m ðtÞ½cosðus0 ðtÞMbt Þ2:875 z  ðD od ðtÞ þ D 0 Þ ¼ z  sðod ðtÞor Þ ð18Þ where p^ m ðtÞ is obtained from the observer (8) and z A ð0; 1Þ is a given coefficient such that the torque generation under a given us0 is smaller than the real value. Furthermore, from physical considerations, the spark timing should be retarded from Mbt to avoid combustion instability. Note that speed error eo o 0 during the initial stage, hence the proportional gain ks should be set in (14) as ks 4 0. As soon as the speed error satisfies the condition (13), the engine torque generation is decreasing due to the thin air in the manifold and throttle should be opened as quickly as possible to provide the necessary air. Hence condition (13) determines the switching from SA control action to throttle control action. Meanwhile, the engine is considered to be in a warm-up state after the acceleration stage, and it is thought as the operational phase of idle speed stabilization. The feedback control is shifted from SA adjusting to throttle operation. The control signal uth is related to the throttle opening angle f defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi pm uth :¼ ð1cosfÞ c ð19Þ pa and the isothermal air intake dynamics can be written as p_ m ¼ cp1 uth cp2 pm o

ð20Þ

where the coefficients are defined as follows: cp1

D or þ D 0 ct

pm ¼

ð22Þ

the feedforward compensation u*th is given as uth ¼

cp2 pm od cp1

ð23Þ

ð16Þ

where D and D0 are constants. Under the proposed fuel path control algorithm (11), the A/F influence on engine torque generation is supposed to be minor and negligible during the idle speed regulation stage, i.e. fl ðlÞ ¼ 1. Then, crank rotational dynamics (5) can be written using (15) and (16) as

ct ¼

the following equations:

RT a s0 pa ¼  pffiffiffiffiffiffiffiffi ; Vm RT a

cp2

RT a ra Vc Z ¼  Vm 4ppa

In this case, uth is constructed by combining a feedforward compensation to ensure a rapid response and a feedback control for the steady-state accuracy. The inputs for idle speed stabilization are designed as ( uth ¼ uth þ kt ðoÞeo ; ð21Þ t 4 t1 us ¼ Mbt ; where u*th denotes the feedforward compensation and kt ðÞ denotes a feedback gain function. Based on the equilibrium ðor ; pm Þ of nominal engine models (17) and (20) which satisfies

3.3. Convergence and robustness analysis As shown in Section 3.2, the control authority is definitely switched to throttle at t ¼ t1. Therefore, the speed error convergence must be guaranteed for this control loop. Under the feedback controller (21), the aim is to choose a feedback gain kt ðoÞ such that the starting speed can converge to the desired idle speed. The convergence is shown by proving the asymptotic stability of the error dynamic system consisting of the engine dynamics (17) and (20) with the reference model (12) and the control law (21) 8_ e r ¼ ser > > > < e_ o ¼ ðDsÞer Deo þ ct ep ðtt Þ d   ð24Þ cp2 pm > > > e_ p ¼ kt ðoÞ eo cp2 oep : cp1 where er :¼ od or and ep :¼ pm pm . Obviously, system (24) is a time-delay system, where the intake-to-power delay td ð¼ p=oÞ is time-varying depending on engine speed o. Here, for simplicity, td is assumed to be a constant, i.e. td ¼ p=or . Then, the proof for the convergence of the idle speed regulation is performed by using the Lyapunov– Krasovskii stability theorem (Hale & Lunel, 1993). Denote xt ¼ ½ert eot ept T . The notation xt represents the function in the space defined by Cr ¼ fxjx : ½0; r-Rn g, where r 40 is a constant. x(t td) ¼ xt(td), td A ½0; r denotes the value of function xt at td. To analyze the stability of system (24), a candidate of Lyapunov–Krasovskii functional is chosen as follows: Z g g 1 1 t 2 Vðxt Þ ¼ 1 e2r þ 2 e2o þ e2p þ e ðsÞ ds ð25Þ 2 2 ttd p 2 2 where g1 and g2 are given by

g1 ¼

1 þB 2s

ð26Þ

with any constant B 40, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 D þ D 2½ct2 þ ðDsÞ2 e g2 ¼ ct2 þ ðDsÞ2

ð27Þ

with a given e satisfying 0oeo

D

2

2½ct2 þðDsÞ2 

ð28Þ

In addition, it is noted that cp2 o 4 1 under the allowable operation modes of the engine. The following convergence result can be obtained. (See Appendix A for the proof.) Proposition 1. For any given desired idle speed or and s 40, if the feedback gain kt ðoÞ is designed in (21) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c p p2 m kt ðoÞ ¼ rðtÞ 2eð2cp2 o1Þ þ ð29Þ cp1 with rðtÞ any continuous function satisfying jrðtÞj o1, 8t Z t1 , then for any initial condition xt1 A Cr , the controlled system (24) is asymptotically stable at the origin, and in particular o-or as t-1.

J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

During engine starting, Proposition 1 shows that for the idle speed regulation stage, the convergence of the desired speed can be guaranteed by using the feedback gain (29), which depends on nominal values of engine parameters. To show the robustness of the feedback control system, the case of feedback gain (29) with mismatched value of real parameters is discussed. In this case, it will be shown that the speed convergence can be achieved by using a new coefficient rðtÞ (denoted as r0 ðtÞ) in (29). With the fact that for a given e according to (28), a positive number q can be found such that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eð2cp2 o1Þ Z q ð30Þ holds, the following extended result of Proportion 1 is obtained with the same Lyapunov–Krasovskii functional candidate (25). (The proof is shown in Appendix B.) Proposition 2. Consider the controlled system (24). For any given desired idle speed or and s 4 0, if the feedback gain kt ðoÞ is designed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kt ðoÞ ¼ r0 ðtÞ 2eð2cp2 o1Þ þM 0 ð31Þ with the constant M0 satisfying the following condition:    0 cp2 pm  M   r Dm  cp1 

ð32Þ

with a known positive number Dm o q, and the continuous function r0 ðtÞ satisfying the following condition: jr0 ðtÞj o1

Dm q

;

8t Z t1

ð33Þ

then the origin is asymptotically stable and the engine speed o converges to the idle speed or from any initial condition xt1 A Cr .

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SICE benchmark problem (Ohata et al., 2007). A detail description for the simulator can be found in Ohata et al. (2007). The system parameters of the models (1)–(3) are listed in Table 2; the nominal parameters, which are obtained by using model identification techniques based on the simulator data under steady-state operation mode, are ct ¼ 2:5  103 , D ¼ 0:3, D 0 ¼ 35:89, cp1 ¼ 1:736  107 , cp 2 ¼ 0.0479, e ¼ 0:1 and w ¼ 0:01. In the simulation, the starting of engine relies on the successful combustion in the cylinders that is constrained by the A/F condition. In the proposed control scheme, this is guaranteed by the fuel injection control given by (8)–(11). In order to compensate the fuel requirement of the cold engine, a scheduling for the fuel path parameters e and w is used in the first three cycles as shown in Table 3. In the speed regulation loop, the time coefficient of the reference model (12) is set at s ¼ 15. In the acceleration stage, the proportional feedback gain in (14) for us(t) is chosen as ks ¼ 0.07 which is an empirical value from the simulation testings, the us0(t) given in (18) turns out to have negligible variations so it is simply set at us0 ¼ 101. As the feedback control shifts from SA to throttle at t ¼ t1, the nominal case is considered first. According to condition (28), the advisable e in the feedback gain kt ðoÞ in (29) for the throttle control (21) belongs to the interval ð0; 2:0  104 Þ. Choosing e ¼ 1:2  104 and the design parameter rðtÞ ¼ 0:8, the responses recorded from the simulator are reported in Figs. 7–9. Fig. 7 shows the response of the open-loop observer (8) for the estimation of air mass flow rate leaving manifold into the ^_ o follows the cylinders. It can be seen that the estimated m Table 2 Engine parameters in simulation. R pa

287

4. Simulation verification

ra

1:01  105 298.15 1.1837

s0

3:5  103

Simulations are performed using the engine simulator provided by the SICE benchmark problem. It is an industrial engine model that consists of the engine block, the control block and the starting motor block. The basic engine specifications are listed in Table 1. The engine system in the simulator has six cylinders, 13 inputs (throttle angle, fuel injection and SA of each cylinder), two outputs (engine speed and the air mass flow rate through throttle into the manifold). The thermal and fluid dynamics of each cylinder, of the inlet pipes and of the intake manifold are individually modeled; the stochastic features in the combustion phase are also taken into account; the model parameters are adjusted so that the experimental dynamic behavior is reproduced. It is a hybrid system which combines continuous and discrete dynamical systems including a 46th-order model. Moreover, in order to simulate the cold starting behavior, the initial water temperature of the model is set at room temperature at 298.15 K. It should be noted that the simulator is used by automotive companies to perform hardware in the loop tests for engine electronic control unit (ECU). It is also used to validate the cold starting control strategies which were proposed to solve the

Vm

6:0  103

Vc

3:6  103 1 14.5 20 deg.

Ta

ld Mbt

Table 3 Parameter compensation for fuel path.

w e

X 10

6 86 86 9.8 43 146.65

k¼1

k¼2

k¼3

k¼4



0.08 0.5

0.6 0.1

0.2 0.1

0.01 0.1

 

Air mass flow rate into cylinders

-5

1 [kg/s]

Table 1 Basic engine specifications. Number of cylinders Bore [mm] Stroke [mm] Compression ratio Crank radius [mm] Connection rod length [mm]

Z

0 Estimated Measured -1 0

0.2

0.4

0.6 Time [ s]

0.8

1

Fig. 7. Estimated air mass flow rate into the cylinders with open-loop observer (8).

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J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

x 10

-5

Fuel injection

20 u f1 u f3 u f5

6 [kg]

Air-fuel ratio

4

15 10

0

5

#1 #3 #5

2

0 -5

0

1

2

3

5 0

4

1

Time [s]

2

3

4

5

Time [s]

Fig. 8. A/F control result with proposed fuel injection control law.

Engine speed 1000

from (32). Finally, by the condition (33), jr0 ðtÞj o 0:99. With r0 ðtÞ ¼ 0:3, the speed convergence with desired transient performance can be observed from the response curve in Fig. 10. Moreover, to verify the control scheme, the case with another idle speed value or ¼ 750 rpm is tested while the control parameters maintain the values computed for or ¼ 650 rpm. The effectiveness is demonstrated by the simulation result in Fig. 11. Considering that in the control scheme for cold starting, both the fuel path control and speed regulation loop are constructed based on the mean-value models, simulation is conducted to test the tolerance of the control scheme with respect to parameter fluctuations. Consider the volumetric efficiency coefficient Z used in the open-loop observer (8) as an example. The simulations are performed with three different values for Z in the observer (8). The comparison results shown in Fig. 12 indicates that the coefficient Z in the observer affects the estimation of air mass flow rate, the A/F control and engine speed clearly, but, in a certain Engine speed

600

1000

400

r d 

200 0

0 SA

1

2

3

800 [rpm]

[rpm]

800

4 5 Throttle opening

400

r d 

200

5

20

600

0

0

2

3

4 5 Throttle opening 5

1

2

3

4

5

0

[deg.]

0

[deg.]

20

us φ -20

1 SA

[deg.]

[deg.]

0

0

Time [s]

us



Fig. 9. Cold-start speed control result (curves of o in black, blue and green are with e ¼ 1:2  104 , 1:9  104 , and 0:5  104 , respectively). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

-20

0 0

1

2

3

4

5

Time [s] Fig. 10. Robustness test result.

Engine speed

1000

[rpm]

800 600 400

r d 

200 0 0

1

2

3

SA

4 5 Throttle opening 6

20 10 2

0

[deg.]

4 [deg.]

simulated value closely during the initial period. Based on this estimation, the fuel injection commands for each cylinder are obtained. The A/F output curves shown in Fig. 8 indicate that the proposed fuel injection algorithm for each individual cylinder guarantees the combustion condition. Moreover, from the curves of the fuel injection commands, it can be seen that in the first few cycles, the required fuel quantity is almost five times the steady state fuel injection. After the engine being started successfully, the starting speed response under the coordinated control between throttle and SA is shown in Fig. 9. The speed responses when different e values in the feedback gain are chosen are also shown in Fig. 9; it can be seen that the controller with the chosen feedback gain guarantees the speed convergence. Moreover, these results shown in Figs. 8 and 9 indicate that the specifications of the SICE benchmark problem are met. According to Proposition 2, if the system parameters are uncertain, the convergence of engine speed during idle speed regulation stage can be also guaranteed by using a modified design parameter r0 ðtÞ. To test the robustness, the following case is considered as an example. The system parameter cp1 in the feedback gain function (29) is changed to 0:9  1:736  107 . Choose M0 ¼ 0. Considering that engine runs with o Z 200 rpm, let q ¼ 1:55  102 based on (30) with e ¼ 1:2  104 . Moreover, there exists proper Dm ð oqÞ since in this case, Dm ¼ 0:1  103

us 

-10

0 0

1

2

3

4

5

Time [s] Fig. 11. Cold-start speed control result with desired stable speed or ¼ 750 rpm.

J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

x 10

depends on the temperature condition significantly, especially at the cold condition and at an almost warm-up condition (in this case, the warm-up temperature is 358.15 K).

-5

ˆ o (t) m η=1 η=0.8 η=0.78

[kg/s]

1.0

5. Concluding remarks

0 80 A/F - #1 60 40 20 0 1000

[rpm]

ω(t)

500

0 0

1

3

2

5

4

Time [s] Fig. 12. Simulation results with different Z in observer (8).

Engine speed 900 800 700

[rpm]

600 500 400 300 200 100 0

0

1

2

3

4

1293

For the SI engines during cold starting operation mode, a control scheme that focuses on improving the transient performance of the starting speed is presented. In contrast to the speed control problems for a warm-up engine, cold-start speed control involves challenging issues due to the dramatic variations of the engine dynamics. The control system for cold starting engines is a multi-input, single-output system: the aim of cold-start speed control is to achieve quick acceleration and fast convergence at a target idling operation mode. The open loop control does not meet the specifications given by the SICE benchmark: this motivates the design of feedback controls. A model-based feedback control scheme is presented to tackle the cold-start speed control. For the combustion event management, a discrete time control with a dual sampling rate is proposed: the cycle-based fuel injection command is delivered with a TDC-based air charge estimation provided for each individual cylinder. Then, it is shown that if the combustion can be guaranteed by adjusting fuel injection, a coordinated feedback control algorithm which acts on SA in the acceleration stage and on the throttle operation for idle speed regulation can improve the starting speed performance. Furthermore, the convergence of the engine speed in the idle speed regulation stage is discussed using the Lyapunov–Krasovskii stability theorem, since the speed control system has a time delay. The proposed control scheme is tested using the simulator provided for the SICE benchmark problem. It is shown that the control scheme satisfies the speed control performance specified by the SICE benchmark problem. Finally, it should be noted that currently used cold starting strategies are mainly based on the scheduling of the actuators such as throttle opening, spark advance and A/F at gas mixing point. Cheng et al. (2001) provided an experiment-based study which focuses on seeking a good scheduling of A/F and SA by experiments: comparing the presented cold starting scheme with these results show that the idling speed is regulated at a much lower level with a better transient performance and the A/F of each cylinder is constrained to a much smaller range. Furthermore, as a result, the A/F is constrained in a strict range, even though this work does not address the emission control problem during cold starting which is widely investigated in automotive control community. The unpredictable influences caused by other unsteady state factors, such as oil and water temperatures, need to be investigated in further works through experiments on the considered automotive engines.

5

Time [s] Fig. 13. Simulation results with different initial water temperature (curves in black, blue and green are with Tw0 ¼ 298.15, 323.15 and 348.15 K). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

range for Z (roughly speaking for Z A ½0:8; 1 according to simulation testing), the control scheme can work effectively. Finally, simulation tests are conducted at other initial water temperatures, which mean that the engine is started under certain warm conditions if the room temperature for the above tests is referred to be the cold condition. Using the proposed control scheme with the same design parameters, the speed responses shown in Fig. 13 indicate that starting speed performance

Acknowledgments The authors would like to acknowledge the Toyota Motor Corporation for the financial and technical supports in this research.

Appendix A. Proof of Proposition 1

Proof. Calculating the time derivative of the Lyapunov–Krasovskii functional (25) along the trajectory of (24) gives V_ ðxt Þ ¼ g1 se2r g2 ðDsÞer eo þ g2 ct eo ep ðttd Þg2 De2o cp2 oe2p

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J. Zhang et al. / Control Engineering Practice 18 (2010) 1285–1294

  cp2 pm 1 1 1 þ kt ðoÞ eo ep þ e2p  e2p ðttd Þ r  ð2g1 s1Þe2r 2 2 2 cp1 1 1 2  ð2cp2 o1Þe2p  ½2g2 D½ct2 þ ðDsÞ2 g22 e2o 2  2 cp2 pm þ kt ðoÞ eo ep cp1 This, in view of conditions (26) and (27), yields   c p 1 1 2 o1Þe2p þ kt ðoÞ p2 m eo ep ¼  xT Q ðoÞx V_ ðxt Þ r Be2r ee2o  ð2cp2 cp1 2 2

ð34Þ where 2 2B 6 6 0 6 Q ðoÞ ¼ 6 6 4 0

0 2e   cp2 pm  kt ðoÞ cp1

3 0   cp2 pm 7 7  kt ðoÞ 7 cp1 7 7 5 2cp2 o1

Taking condition (29) into account, a sufficiently small l 4 0 can be found such that xT Q ðoÞx Z lJxJ2

ð35Þ

where J  J denotes the Euclidean norm. This implies that there exists a continuous nondecreasing function lJxJ2 such that V_ ðxt Þ r lJxJ2 Moreover, it is clear that for the candidate of Lyapunov–Krasovskii functional (25), there exist continuous nondecreasing function mi ðsÞ 40; s 40 and mi ð0Þ ¼ 0 (i ¼ 1,2) such that

m1 ðJxJÞ rVðxt Þ r m2 ðJxt Jc Þ where J  Jc denotes the norm defined by Jxt Jc :¼ sup0 r td r r fJxðtd ÞJg. Hence, the asymptotic stability of system (24) at the origin follows by the Lyapunov–Krasovskii functional stability Theorem 2.1 in Hale and Lunel (1993). & Appendix B. Proof of Proposition 2 Proof. For feedback gain kt ðoÞ given in (31), the following condition can be obtained immediately in view of (32):      kt ðoÞ cp2 pm  ¼   c  p1qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    0 r 2eð2cp2 o1Þ þ M 0  cp2 pm  r jr0 j 2eð2cp2 o1Þ þ Dm   cp1 ð36Þ Taking (33) and (30) into account one obtains    qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    kt ðoÞ cp2 pm  o 1 Dm 2eð2cp2 o1Þ þ Dm o 2eð2cp2 o1Þ  cp1  q ð37Þ This guarantees that there exists a sufficiently small l 4 0 such that the matrix Q ðoÞ in (34) with kt ðoÞ given in (31) satisfies the condition (35). Therefore, the proof is achieved with the same argument used in Proposition 1. &

References Aquino, C. (1981). Transient a/f control characteristics of the 5 liter central fuel injection engine. SAE Technical Paper 810494. Cheng, Y., Wang, J. X., Zhuang, R. J., & Wu, N. (2001). Analysis of combustion behavior during cold-start and warm-up process of SI gasoline engine. SAE Technical Paper 2001-01-3557. ¨ Chevalier, A., Muller, M., & Hendricks, E. (2001). On the validity of mean value engine models during transient operation. SAE Technical Paper 2000-01-1261. Cho, D., & Hedrick, J. K. (1989). Automotive powertrain modeling for control. Transactions of the ASME, 111, 568–576. Eng, J. A. (2007). Cold-start hydrocarbon emission mechanisms. In Technologies for near-zero-emission gasoline-powered vehicles, SAE international (pp. 79–120). Grizzle, J. W., Cook, J. A., & Milam, W. P. (1994). Improved cylinder air charge estimation for transient air fuel ratio control. In Proceedings of the American control conference (pp. 1568–1573), Baltimore, USA. Guzzella, L., & Onder, C. H. (2004). Introduction to modeling and control of internal combustion engine systems. Berlin: Springer. Hale, J. K., & Lunel, S. M. (1993). Introduction to functional differential equations. New York: Springer. Heywood, J. B. (1988). Internal combustion engine fundamentals. New York: McGraw-Hill. Hrovat, D., & Sun, J. (1997). Models and control methodologies for IC engine idle speed control design. Control Engineering Practice, 5(8), 1093–1100. Kitazono, S., Sugihira, S., & Ohmori, H. (2008). Starting speed control of SI engine based on extremum seeking control. In Proceedings of the 17th IFAC World congress (pp. 1036–1041). Leisenring, K., Samimy, B., & Rizzoni, G. (1996). IC engine air/fuel ratio feedback control during cold start. SAE Technical Paper 961022. Leroy, T., Chauvin, J., & Petit, N. (2009). Motion planning for experimental air path control of a variable-valve-timing spark ignition engine. Control Engineering Practice, 17(12), 1432–1439. Liu, Z., Li, L., & Deng, B. (2007). Cold start characteristics at low temperatures based on the first firing cycle in an LPG engine. Energy Conversion and Management, 48, 395–404. Ma, Q., Yurkovich, S., & Dudek, K. P. (2007). Cylinder air rate prediction during engine start and crank-to-run transition. In Proceedings of the American control conference (pp. 3686–3693), New York, USA. Ma, Q., Yurkovich, S., & Dudek, K. P. (2008). Predictive fuel dynamic control during engine start and crank-to-run transition for port fuel injected gasoline engines. In Proceedings of the American control conference (pp. 994–1001), Seattle, USA. Moskwa, J. J. (1988). Automotive engine modeling for real time control. Ph.D. dissertation, Massachusetts Institute of Technology, USA. Ogawa, M., & Ogai, H. (2008). The cold starting control of engine using large scale database-based online modelling. In Proceedings of the 17th IFAC World congress (pp. 1030–1035). Ohata, A., Kako, J., Shen, T., & Ito, K. (2007). Benchmark problem for automotive engine control. In SICE annual conference (pp. 1723–1726), Kagawa, Japan. Puleston, P. F., Spurgeon, S., & Monsees, G. (2001). Automotive engine speed control: A robust nonlinear control framework. In Proceedings of IEE control theory application (vol. 148(1), pp. 81–87). Sampson, M. J., & Heywood, J. B. (1995). Analysis of fuel behavior in the sparkignition engine start-up process. SAE Technical Paper 950678. Stotsky, A., Egardt, B., & Eriksson, S. (2000). Variable structure control of engine idle speed with estimation of unmeasurable disturbances. Journal of Dynamic Systems, Measurement, and Control, 122, 599–603. Thornhill, M., Thompson, S., & Sindano, H. (2000). A comparison of idle speed control schemes. Control Engineering Practice, 8(5), 519–530. Tomohiko, J., & Yoshikazu, H. (2008). Physical-model-based control of engine cold start via role state variables. In Proceedings of the 17th IFAC World congress (pp. 1024–1029). Ueno, M., Akazaki, S., Yasui, Y., & Iwaki, Y. (2000). A quick warm-up system during engine start-up period using adaptive control of intake air and ignition timing. SAE Technical Paper 2000-01-0551. Vesterholm, T., & Hendricks, E. (1994). Advanced nonlinear engine speed control systems. In Proceedings of the American control conference (pp. 1579–1580), Baltimore, USA. Yamada, T., Gardner, D. V., Bruno, B. A., Zello, J. V., & Santavicca, D. A. (2002). The effects of engine speed and injection pressure transients on gasoline direct injection engine cold start. SAE Technical Papers 2002-01-2745. Zhang, J., Shen, T., & Marino, R. (2008). Model-based cold-start speed control design for SI engines. In Proceedings of the 17th IFAC World congress (pp. 1042–1047).