ISG hybrid powertrain: a rule-based driver model .../file/VSD_2009... · Vehicle System Dynamics...

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Vehicle System Dynamics Vol. 48, No. 3, March 2010, 301–337 ISG hybrid powertrain: a rule-based driver model incorporating look-ahead information Shuiwen Shen a *, Junzhi Zhang b , Xiaojiang Chen a , Qing-Chang Zhong c and Roger Thornton a a Ricardo UK Ltd, Cambridge, UK; b State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, P.R. China; c Department of Electrical Engineering and Electronics, University of Liverpool, UK (Received 4 June 2008; final version received 29 January 2009; first published 13 May 2009 ) According to European regulations, if the amount of regenerative braking is determined by the travel of the brake pedal, more stringent standards must be applied, otherwise it may adversely affect the existing vehicle safety system. The use of engine or vehicle speed to derive regenerative braking is one way to avoid strict design standards, but this introduces discontinuity in powertrain torque when the driver releases the acceleration pedal or applies the brake pedal. This is shown to cause oscillations in the pedal input and powertrain torque when a conventional driver model is adopted. Look-ahead information, together with other predicted vehicle states, are adopted to control the vehicle speed, in particular, during deceleration, and to improve the driver model so that oscillations can be avoided. The improved driver model makes analysis and validation of the control strategy for an integrated starter generator (ISG) hybrid powertrain possible. Keywords: driver model; ISG hybrid vehicle; discontinuous powertrain torque 1. Introduction Hybrid technologies are widely recognised as alternative solutions to the ever increasing demand for more efficient and less polluting vehicles at present. An HEV, compared with a vehicle powered by an ICE alone, has the potential of saving extra fuel owing to the following reasons. First, the ICE rated power is significantly reduced as the electric motor can provide additional power required for propelling the vehicle. Second, the ICE can be stopped during idling and its restarting by the electric motor is smooth and quick. Next, the vehicle kinetic energy can be (partly) recaptured with regenerative braking, which can then be used to assist the ICE at a later stage. Finally, the use of the electric motor gives rise to a chance for optimising the ICE operation. Commercialising a hybrid vehicle is inevitably a complicated engineering process with full of challenges. First, the power demands from drivers can be satisfied by either an ICE *Corresponding author. Email: [email protected] ISSN 0042-3114 print/ISSN 1744-5159 online © 2010 Taylor & Francis DOI: 10.1080/00423110902803465 http://www.informaworld.com Downloaded By: [Loughborough University] At: 15:37 19 October 2010

Transcript of ISG hybrid powertrain: a rule-based driver model .../file/VSD_2009... · Vehicle System Dynamics...

Vehicle System DynamicsVol. 48, No. 3, March 2010, 301–337

ISG hybrid powertrain: a rule-based driver modelincorporating look-ahead information

Shuiwen Shena*, Junzhi Zhangb, Xiaojiang Chena, Qing-Chang Zhongc

and Roger Thornton a

aRicardo UK Ltd, Cambridge, UK; bState Key Laboratory of Automotive Safety and Energy,Tsinghua University, Beijing, P.R. China; cDepartment of Electrical Engineering and Electronics,

University of Liverpool, UK

(Received 4 June 2008; final version received 29 January 2009; first published 13 May 2009 )

According to European regulations, if the amount of regenerative braking is determined by the travelof the brake pedal, more stringent standards must be applied, otherwise it may adversely affect theexisting vehicle safety system. The use of engine or vehicle speed to derive regenerative braking isone way to avoid strict design standards, but this introduces discontinuity in powertrain torque whenthe driver releases the acceleration pedal or applies the brake pedal. This is shown to cause oscillationsin the pedal input and powertrain torque when a conventional driver model is adopted. Look-aheadinformation, together with other predicted vehicle states, are adopted to control the vehicle speed, inparticular, during deceleration, and to improve the driver model so that oscillations can be avoided.The improved driver model makes analysis and validation of the control strategy for an integratedstarter generator (ISG) hybrid powertrain possible.

Keywords: driver model; ISG hybrid vehicle; discontinuous powertrain torque

1. Introduction

Hybrid technologies are widely recognised as alternative solutions to the ever increasingdemand for more efficient and less polluting vehicles at present. An HEV, compared with avehicle powered by an ICE alone, has the potential of saving extra fuel owing to the followingreasons. First, the ICE rated power is significantly reduced as the electric motor can provideadditional power required for propelling the vehicle. Second, the ICE can be stopped duringidling and its restarting by the electric motor is smooth and quick. Next, the vehicle kineticenergy can be (partly) recaptured with regenerative braking, which can then be used to assistthe ICE at a later stage. Finally, the use of the electric motor gives rise to a chance for optimisingthe ICE operation.

Commercialising a hybrid vehicle is inevitably a complicated engineering process withfull of challenges. First, the power demands from drivers can be satisfied by either an ICE

*Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online© 2010 Taylor & FrancisDOI: 10.1080/00423110902803465http://www.informaworld.com

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or an electric motor or the combination of these two. It is therefore an over-actuated system[1], which introduces an additional degree of freedom. The performance, in particular, fueleconomy, of a parallel HEV system largely depends on how this additional degree of free-dom is resolved. Unlike a prototype vehicle in which analytical approaches such as optimalcontrol [2,3], dynamics programming [4,5] and equivalent consumption minimisation strat-egy [6] are often applied, the production vehicle adopts more practical means like rule-basedstrategies [7], which are simple and require little or no analytical efforts but involve mas-sive calibration works because many parameters have to be tuned. Second, needless to say,developing a robust and tolerant energy management strategy to split the driver demandsand to govern the power flow between the engine and the electric motor is fundamental andinevitable. However, the following are more important: (1) developing a rechargeable highvoltage battery with necessary hardware and software to predict and govern its states, whichshould be functioning regardless of the operational environments it may have to face as longas the vehicle is still functioning; (2) developing one or more electric machines that are con-trolled via reliable power electronics to provide power or torque as required, which is highlydynamic in reality; (3) decoupling the dependency from several accessory devices (such asthe power steering system) on the engine so that the engine can be shut down during idlingor when the power required from the engine is sufficiently low, thus very inefficient; (4) opti-mising or redeveloping the engine to allow less dynamics and more efficient operation suchas running on Atkinson cycles [8], which cannot be implemented in conventional vehiclesbecause the driveability is compromised otherwise. Finally, fuel economy is not the onlyconcern during the development stage, meeting stringent emission and safety regulations,without sacrificing the driveability, NVH and gradability, is equally important but more time-consuming. Although the rapid growth of electronics in automobiles, especially the growthof embedded software, increases the complexity of vehicle control system development at anincredible pace, the design cycle of new functions in the area of safety, fuel efficiency andcomfort is by contrast to be shortened because of the extreme time-to-market pressure. Thenew functions must meet stringent standards at a competitive cost, and quality and reliabilityremain a paramount concern. The confluence of these factors unavoidably drives automotiveelectronics design methodology to a model-based approach [9]. Traditionally, any problemassociated with the design is largely identified when a prototype ECU and a test vehicle ora test bench are available. Iteration in such a way is inevitably very time-consuming, thusunacceptable. A time-efficient solution is the automation of the verification and validation inall development phases. MiL, SiL and PiL methods allow the early detection of the designdeficiencies while HiL makes the automation and optimisation of the verification and valida-tion possible. However, all these simulation methods are only viable if the vehicle model hasthe required degree of details, and most essentially if the model is verified and available at thetime of validation. Hence, it has never been more critical to develop an appropriate model atthe very early stage of the development than now.

Given the facts stated earlier, we are not surprised to see that ISG1 have been chosen bymany passenger car manufacturers as their mainstream type the of hybrid vehicle development,targeting to make appropriate compromises among complexity, cost and benefits, and amongfuel economy, emission and driveability. Ricardo UK Ltd. has recently completed severalHEV projects and we plan to systematically address the following issues associated with theparticular ISG powertrain development in a series of papers:

(1) The torque management strategy to coordinate the torque between ICE and ISG. Thestrategy adopted is practical and has quite a significant number of parameters needing tobe calibrated. Comparison between analytic and practical approaches for coordinating thepowertrain torque will also be made.

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(2) Developing a sub-optimal fuel consumption control method that can be used online andrequires no or little knowledge about future driving cycle conditions.

(3) HEV modelling. The model of an HEV2 is needed to validate and calibrate the torquemanagement strategy. Since most of the conventional vehicle models can still be appliedto HEVs, the attention will be paid particularly to the driver model, as discussed in thisstudy.

(4) Improvement of EMS and idle speed control to allow the coordination of HCU for fastand slow torque delivered to the powertrain across the ICE and the ISG. In addition, theconventional idle speed control may not be able to handle the ISG hybrid system becausethe hybrid controller can request generation during idling if the traction battery SoCis sufficiently low. Inappropriate compensation of this torque will lead to a driveabilityproblem.

(5) Impact of ISG hybridisation on automated gearbox shift strategies. The shift strategy usedin conventional vehicles is to select an appropriate gear ratio according to the engineand the vehicle conditions so that the powertrain operation can be optimised. Becauseof the presence of the ISG system, this shift strategy may not be able to optimise thepowertrain operation any more and its modification is essential on the basis of the systemlevel performance analysis.

(6) Machine and battery modelling and control. Traction motors and traction batteries are notpresent in conventional powertrains. Machines and batteries play a significant part in thetorque split and in the SoC control.

The focus of this study is on the driver model of HEVs. The key to avoiding the stringentbrake safety regulation is that the travel of brake pedal must not be used in deriving the regen-erative braking torque. However, the use of the engine or vehicle speed inevitably introducesa discontinuous powertrain torque with respect to (w.r.t.) the pedal input, especially when theacceleration pedal is released or when the brake pedal is applied. Because discontinuity cannotbe handled by a conventional driver model, oscillations in driver pedal inputs and the drivelinetorque during deceleration will arise. Thus, the use of the model-based approach to analyse thevehicle performances and to validate the control strategy will be limited if the driver model isnot improved. In this article, a rule-based driver model incorporating look-ahead informationis proposed. The proposed driver model only manages to trace a target vehicle decelerationif it has the full ability of doing so. Otherwise, by employing look-ahead information, thedriver model adopts a controllable vehicle deceleration which may be different from the targetvehicle deceleration.

The rest of the article is organised as follows. To provide a general context for the seriesof papers mentioned earlier, an overview of the ISG vehicle together with the HiL systemunder study is described in Section 2. The discontinuity problem caused by ISG regenerativebraking strategy is presented in Section 3, followed by a novel driver model with look-aheadinformation proposed in Section 3 to solve the discontinuity problem. The performance of theHEV system incorporating the proposed driver model is assessed in Section 4, primarily usinga HiL system. Conclusions and possible future research directions are stated in Section 5.

2. Overview of ISG vehicle systems

An overview of the ISG hybrid vehicle under study is given at first and then the HiL systemfor control strategy validation and verification is described briefly. After modelling the hybridpowertrain, a simplified version of the HCU and powertrain controllers is proposed to facilitatethe investigation of a driver model.

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2.1. The control system architecture of an ISG powertrain

The components of an ISG hybrid powertrain and the interaction among them are illustrated inFigure 1. It consists of a gasoline engine ICE, a permanent magnetic synchronous ISG, a high-voltage battery, a transmission system including a clutch (to decouple/re-couple the ISG andthe ICE from the rest of the powertrain when necessary), an automated manual transmissionhaving five fixed ratios3 and an axle with a final drive and a differential drive (but the effectof the differential is not modelled), and a vehicle containing the effect of the tyre, whichconverts the thrusting torque into propelling forces, and the effect of air drag and rollingresistances. The control system for an ISG hybrid powertrain, as illustrated in Figure 1, consistsof an HCU, a BMS, an MCU, an EMS and a TCU. These control units communicate witheach other through a CAN bus, which is not shown in Figure 1. The control system alone isnot designed to have the full control of the vehicle speed. Instead, a driver model (to mimicthe behaviour of a driver) is placed in the loop to produce the necessary pedal input to regulatethe vehicle speed on the basis of vehicle and powertrain states.

The HCU plays a supervisory role in the control system. It receives pedal input from thedriver model and interprets it as a driver torque demand.4 It then splits the torque demandinto ICE torque T ice and ISG torque T isg. HCU also determines when and which gear to shift,which is the primary task of TCU originally. A simplified HCU control strategy is presentedin Section 2.4 for validating the driver model to be proposed in Section 4.

The BMS, MCU, EMS and TCU are considered as the actuators of the HCU. The EMS andthe MCU receive torque commands from HCU, while the TCU receives gear ratio commandand the BMS receives battery connection and disconnection command. It is assumed that thedynamics of the ICE and the ISG to implement the torque and the TCU to complete the gearshift are much quicker than that of the vehicle speed response. Therefore, these dynamics areconsidered only when investigating the control of the corresponding subsystems.

In order for the HCU torque request to be fulfilled in such a way that the engine efficiencyand emission is optimised, the EMS governs the ignition, the air/fuel ratio and the amountof the air for the engine by using three feedback loops, namely an anti-knock control loop,a lambda control loop and an electrical throttle control loop [10,11,12]. The EMS splits thetorque into slow and fast channels. The fast torque is implemented via the change of ignitionangle while the slow torque is managed by appropriate air charge. Since the machine responseis reasonably quick nowadays, the fast torque of ISG hybrid vehicle must be coordinatedbetween the ICE and the ISG. Another task of the EMS is to manage the engine speed duringidle. The idle speed control is different from that of conventional vehicles as it must cope withthe ISG generation during idle, in which the torque could be sufficiently high.

Figure 1. ISG hybrid powertrain: control architecture.

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The MCU converts the HCU torque request into a winding current request at a rotatingvector space and controls insulated-gate bipolar transistor devices in PWM manner to supplypower to the permanent magnetic ISG. The vector control [13,14] of the winding current isvery difficult during flux weakening and may lead to instability or converges to a completelydifferent winding current level. A practical solution of the MCU flux weakening control willnot be addressed in this paper, but the MCU is merely regarded as an actuator that performsthe HCU torque request instantaneously.

The BMS estimates the SoC of the battery and reports this value to the HCU. It is then usedto determine the ISG torque. The commonly used charging and discharging current integralmethod for the estimation will inevitably accumulate the measure noise and current drift. Thisproblem is intensified when the battery open-circuit voltage cannot be used for resetting theestimation if its correlation with SoC is less obvious. Normally, a model-based SoC methodshall be employed for the SoC estimation [15,16], but in this paper a commonly used SoCestimation method is adopted in a simple battery model.

Unlike the conventional vehicle, the gear shift strategy becomes the task of HCU but theshift quality control remains the task of TCU. In an HEV with an electric motor mounted atthe output shaft side of the gearbox, the vehicle driveability is significantly affected. However,the ISG hybridisation only has a slight impact on the shift quality control. In this case, the sumof the ISG torque and the ICE torque have to be regulated during the shift, which, togetherwith a proper clutch engagement control, improves the vehicle driveability. The TCU uses asimilar algorithm to control the vehicle launch. Although the shift quality and vehicle launchcontrol do not have much difference from that of conventional vehicles, the vehicle launchcontrol is briefed in Section 2.4 for the reason of validating the driver model.

2.2. The HiL system under study

The use of HiL systems to validate and verify the control strategy is vital for HCU development[17]. One of the critical aspects of any HiL system is the development of an appropriateplant model that is validated. The corresponding plant model of the HCU includes two majorsubsystems: the powertrain model, including ICE, ISG, battery, transmission and vehicle, andits controller, including BMS, EMS, MCU and TCU. The HiL system under study is shownin Figure 2. It includes a dSPACE simulator containing the HCU plant model (the rack on theright-hand side, RHS), an HCU (the small box on the table next to the dSPACE simulator)and a computer/monitor. The HCU communicates with the dSPACE simulator using the samehardware interface as if it were communicating with the rest of the real vehicle control system.Figure 3 illustrates the interaction among the HCU plant model, the vehicle control model andthe HCU strategy model.

There are two purposes in using a HiL system. One is that a validated plant model allowsthe early detection of the strategy defects, tuning of control parameters, evaluation of thestrategy robustness by varying the plant model parameters and by simulating extreme vehicleconditions that may be highly difficult to produce in reality, and the other is that a validatedHCU can be used to assess the plant model if necessary. For most of the cases in this seriesof papers, the HiL system is used for the first purpose. However, in this article, it is usedfor the second purpose. The HCU, powertrain controller and the powertrain model togetherform a software package. As previously mentioned, the powertrain controller (EMS, BMS,MCU and TCU) are assumed to perform HCU commands instantaneously and accurately inthis study; only the powertrain model, HCU and clutch engagement control model used inthis study are described in the following subsections. Unless otherwise stated, the simulationresults presented in this article are obtained by using this software package on the HiL system.

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Figure 2. The HiL system under study.

Figure 3. The interaction among the models.

2.3. Modelling of the ISG hybrid powertrain

After making the following assumptions,(1) the energy losses at the gear box, the final drive and the battery, the stiffness of the driving

shaft, and the gravitational load of the vehicle on a slope are not modelled,(2) the engine internal loss is compensated by the engine combustion torque at idle speed

control and it acts as a source to slow the vehicle down when the fuel cut strategy isactivated.

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Figure 4. Powertrain and vehicle model.

The ISG hybrid powertrain model can be simplified as illustrated in Figure 4. It consists of amodel of batteries to estimate its SoC, a combined engine/ISG model with the lumped inertiaof Je = J isg + J ice, a vehicle with the lumped mass of mv =m + Jwh/R2, which is linked to thelumped engine/ISG model via a clutch, a gear box and the final drive.

The engine and ISG, and the vehicle are governed by the following equations:

Jeωice = (Tice + Tisg) − Tclt, (1)

mvv = rvTclt − (Fload + Fbrk), (2)

where rv is a ratio between the engine angular speed ωice and the vehicle speed v, thus rvT clt

is a force, F load is the vehicle resistance, modelled as

Fload = Fair + Froll

= 12ρCdAv2 + frollmg,

(3)

Fbrk is the mechanical brake force

Fbrk = F maxbrk β, (4)

and T clt is the clutch toque. It is modelled, using the Karnopp’s approach [18], to overcome theproblem with zero sliding velocity detection and to avoid switching between different stateequations for stick and slip, as

Tclt ={

Tc sgn(�ω), |�ω| > δ,

min(Tstick, Tcfs) sgn(Tstick) otherwise,(5)

where T c is the clutch demanding torque from TCU, fs = 1.2 is the static friction coefficient,�ω is the slipping speed of the clutch defined as

�ω = ωice − rvv, (6)

and

Tstick = mv(Tice + Tisg) + rvJe(Fload + Fbrk)

mv + r2v Je

is the clutch stick torque to maintain the clutch speed �ω close to zero if its magnitude is lessthan that of the break-away torque (or static friction torque) T cfs. The clutch torque has thefunction of reducing its slipping speed (see [19] for detailed analysis of this effect).

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For given ISG torque and speed, the battery SoC is estimated based on the batterycharging/discharging power Pbatt [20]

Pbatt ={

Tisgωiceηchg for generation,

Tisgωice/ηdis for motoring,(7)

where ηchg and ηdis are the respective charging and discharging efficiencies of the ISG. Thevariation of the SoC is

˙SoC = − 1

CbattPbatt. (8)

Since the motoring torque T isg > 0 discharges the battery while the generation torque T isg < 0charges the battery, the negative sign in the above equation is essential.

Equations (1), (2) and (8), having three states ωice, v and SoC, and four inputs β, T ice, T isg

and T c, represent a simplified model of the hybrid powertrain. The last three variables are theoutputs of the powertrain controller (see the next subsection for more details) and the brakepedal input β is the output of the driver model to be built in Section 3.

2.4. HCU and clutch engagement control

The core functions of the HCU are: (i) to interpret the driver pedal input α as the driverdemanding torque, considering the capability of both the ISG and the ICE; (ii) to split thedemanding torque between ICE and ISG so that the vehicle efficiency is improved withoutadversely affecting the driveability; and (iii) to determine the regenerative braking. It is beyondthe scope of this study to discuss the design of advanced HCUs. A simple version of the HCU,capturing the above-mentioned functions and being sufficient for the evaluation of the drivermodel, is presented here.

2.4.1. Interpretation of the driver pedal input

The driver demanding torque is a function of both the driver pedal position and the enginespeed i.e.

T drvdem = (α, ωice)(T

maxisg + T max

ice ).

For the HiL system under study, this relationship is shown in Figure 5(a) as thin solid lines.As it is nonlinear, the vehicle is more sensitive to 10–60% pedal inputs at low engine speedwhereas more sensitive to 30–80% pedal inputs at high engine speed. The introduction ofthe nonlinear interpretation gives more stable and easier vehicle operation. For conventionalvehicles with the same interpretation function this relationship is shown as dash-dot linesin the same figure for comparison. For instance, the hybrid vehicle has a more consistentsensitivity to the pedal input throughout the whole engine speed range, but it is not true for theconventional vehicle. Owing to the characteristics of the combustion engine, the engine torquecan increase with engine speed for widely opened throttle (WOT) in a certain speed region.Because of this, the rapid change of the engine power may make the control of conventionalvehicles difficult. Also shown in the figure are the road loads mapped to the engine side atdifferent gear ratios. Clearly, for a given pedal input (e.g. α = 20%), the negative slope of thedrive demanding torque w.r.t. the engine speed will make the engine settle down to a morestable speed provided that the driver demanding torque is implemented with no sacrifice. Forinstance, the engine speed will settle down to around 3150 rpm for α = 20% if the gearboxratio is set to the 3rd. Furthermore, the negative slope also determines the robustness around

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Figure 5. Pedal interpretation and engine characteristics.

and the velocity converging to this speed. It is this kind of speed-control-alike function thatunderlines the interpretation of the pedal.

2.4.2. Splitting the torque between ICE and ISG

Another function of the HCU is to split the driver demanding torque T drvdem. One way is to

make the engine work on or close to the so-called E-line [21], which is a collection of theengine operating points in its speed–torque coordination at which the engine fuel economy ismaximised for a given engine power requirement.5 This method of splitting the demandingtorque concentrates on optimising the engine operation only. The use of the ISG is targeted tocompensate the difference between the driver demanding torque and the engine demandingtorque. Thus, care must be taken when applying this simple version of the hybrid controlstrategy as it does not intend to guarantee the optimum operation of the hybrid powertrainsystem. Nevertheless, the strategy shall provide similar fuel benefits because the engine is theprimary and ultimate energy source to propel the vehicle. Figure 5(b) illustrates the BSFCcharacteristics and the E-line of the engine under investigation.

Furthermore, the HCU must also provide the functions to ensure the battery is sustainablethroughout the drive cycles. Consequently, the determination of the ISG torque must take theSoC into account.

The proposed torque split strategy is represented as follows:

Tice = min(max(T elice(ωice)ϒ(SoC), T dem

drv − T maxisg ), T max

ice ), (9)

Tisg = max(min(T drvdem − Tice, T max

isg ), −T maxisg ). (10)

The first term T elice(ωice)ϒ(SoC) in Equation (9) plays a critical part in determining the

engine torque, in which T elice is the engine torque on the E-line as shown in Figure 5(b) while

the function ϒ(SoC) is for SoC control:

ϒ(SoC) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

T maxice

T elice

+(

1 − T maxice

T elice

)eklm(SoC−SoClm) when SoC < SoClm,

ekhm(SoChm−SoC) when SoC > SoChm,

1 otherwise,

(11)

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where SoClm and SoChm are, respectively, the low and high boundary of the SoC. The engineis targeted to work on the E-line when SoClm < SoC < SoChm. This is also a SoC region whereSoC control does not come into force. On the other hand, the engine has to be away fromthe E-line when SoC > SoChm or SoC < SoClm. The engine torque reduces significantly onceSoC > SoChm, leading to an increase of the ISG torque and in turn discharging the battery.Similarly, the engine torque shall increase if SoC < SoChm, resulting in less discharging oreven charging the battery. The parameters klm and khm in Equation (11) regulate the speed ofthe engine torque departing from the E-line. The careful tuning of these parameters is essentialto get an acceptable SoC control performance.

The second term T demdrv − T max

isg in Equation (9) ensures the driver demanding torque issatisfied. It plays a particular role when the torque availability is significantly reduced becauseof low SoC. T max

isg and T maxisg are different, the former being the ISG torque capability considering

both the ISG and the battery conditions while the latter only taking ISG conditions into account.The following equations relate T max

isg to T maxisg

T maxisg =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

T maxisg when SoC ≥ SoChm

hl ,

T maxisg

SoC − SoClmhl

SoChmhl − SoClm

hl

when SoChmhl > SoC ≥ SoClm

hl ,

0 otherwise,

(12)

where SoChmhl and SoClm

hl are the hard limits of the battery and SoChmhl > SoClm

hl . The batterycapability is penalised when SoC < SoChm

hl , and it may even not be able to provide any powerif SoC < SoClm

hl .Finally, how HCU determines ISG generation torque is discussed at the beginning of

Section 3 during regenerative braking. The strategy consists of Phase 1 when the gas pedal isreleased and Phase 2 when the brake pedal is applied.

2.4.3. Clutch engagement control

Part of the roles of TCU is to regulate the vehicle start process via an appropriate clutchengagement control. The following two factors must be considered when determining theclutch torque: (i) the driver demands; and (ii) the engine speed. The former is to start the vehiclein the way the driver requires and the latter is not to stall the engine. Thus, the following strategyis proposed:

Tc = 1

2[T drv

dem + PI (�ω)][

2

πarctan(ωice − ωidle) + 1

], (13)

where the term T drvdem represents the driver demands and the term PI(�ω) acts as a PI regulator

to force the clutch engagement in which slipping speed �ω is defined in Equation (6). Thelast term 2/π arctan(ωice − ωidle) + 1 reduces the clutch torque if the engine has a tendencyto stall.

3. The need of an improved driver model: to eliminate oscillations in the pedal inputand powertrain torque

Regenerative braking converts the vehicle kinetic energy into electric energy. It is a criticalfunction of HEVs since most of the energy used by the motoring assistant function, to boost

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vehicle acceleration and/or to improve engine efficiency, comes from the regenerative braking.However, how much the vehicle kinetic energy could be recovered by regenerative brakingdepends on how the braking pedal information could be applied and how the braking torqueis distributed between mechanical and electrical brake. Three regenerative braking strategiesare described here for comparison:

(1) The brake pedal input is used to determine the regenerative braking torque [8]. Theregenerative braking is only limited by the ISG machine torque capacity.

(2) The brake pedal input is used to determine the regenerative braking torque. In additionto that the regenerative braking is limited by the ISG machine torque capacity, it is alsolimited by the allowed depth of discharge (DoD) of the battery, which is 10% in this study.

(3) The engine speed, instead of the brake pedal input, is used to determine the regenerativebraking torque.6 Furthermore, the regenerative braking torque is firstly applied whenthe driver releases the acceleration pedal (referred to as Phase 1 hereinafter) and moreregenerative braking torque is applied when the driver depresses the braking pedal (referredto as Phase 2 hereinafter). Both the limitations of the ISG machine torque capacity andthe DoD of the battery apply.

For the HiL system under study, the ISG machine torque capacity (dashed line) and theregenerative braking torque as a function of the engine speed (solid lines) for strategy REG3are shown in Figure 6.As just mentioned, the torque must be limited to the ISG machine torquecapacity (dashed line) as well as the limit of the battery DoD, which is not shown in Figure 6.The battery life reduces exponentially with the increase of its DoD. Therefore, this constraintis as equally important as the ISG torque capacity. The rapid decrease of the magnitude of theregenerative braking torque in low engine speed region is necessary to prevent the engine fromstalling. It is worth mentioning that these torques are shown in absolute values. In general,ISG motoring torque is defined as positive while generation torque (including regenerativebraking) as negative.

Table 1 shows the main energy components of the HiL system under study during NEDCcycles, which represents a typical vehicle. Eair and Eroll are the energies used for overcomingthe respective air and rolling resistances while Eloss

eng is the energy lost because of the engine

Figure 6. Regenerative torques.

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Table 1. Vehicle energy distribution during NEDC cycles.

Parameter Value Unit

Eair 1511 (kJ)Eroll 2705 (kJ)Eloss

eng 2832 (kJ)Ekn 1918 (kJ)Ebrk 950 (kJ)

friction. Ekn is the sum of the vehicle kinetic energy before applying brake or releasing gaspedal and Ebrk represents the mechanical brake energy. Clearly, the kinetic energy Ekn ismore than twice the brake energy Ebrk, implying that more than half of the vehicle kineticenergy is lost in the engine brake, air and rolling resistance. Consequently, the regenerativebraking could not recover more kinetic energy than those lost in mechanical braking, i.e. Ebrk.Thus, any design to reduce the engine friction loss, air and rolling resistance is as equally (ifnot less) important as to improve the vehicle fuel economy. Figure 7 shows the energy (inkJ) distribution between conventional mechanical braking and regenerative braking for theabove-mentioned regenerative braking strategies. It is clear that:

• Not all of the vehicle kinetic energy Ekn = 1918 kJ is recoverable. The energy lost dueto the rolling resistance, the air drag and the engine friction during the braking cannot berecuperated by the regenerative braking.

• The energy lost due to the mechanical braking Ebrk = 950 kJ represents a chance for beingrecovered by the regenerative braking.

• With a 10 kW ISG and 10% DoD of a 1.22 kW h battery, more than 60% of the mechanicalbraking energy can be recovered. The actual figures of the regenerative braking for REG1,REG2 and REG3 are 685, 676 and 565 kJ, respectively.

• Although REG1 and REG2 are better than REG3, the difference is not significant.

Figure 7. Energy distribution between mechanical (dark grey) and electrical braking (light grey).

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Vehicle System Dynamics 313

The regulation number 13H [22] of United Nations Economic Commission for Europeidentifies two types of regenerative braking for M1 category vehicles7:

• The regenerative braking system is not part of the vehicle service braking system.• The regenerative braking system is part of the vehicle service braking system.

For Category B, the regulations on the vehicle stability and safety are much stricter, whichis beyond the scope of this study. As a result, Category A is selected for the ISG hybrid vehicledevelopment in this work. There are numerous detailed requirements in the regulations but itis believed that the amount of regenerative braking cannot be influenced by the brake pedalposition for Category A, i.e. REG1 and REG2 strategies cannot be applied. One of the conse-quences is that the energy that could be recovered by the regenerative braking will be reduced,according to Figure 7. Another consequence is that the torque at wheels exhibits discontinu-ities w.r.t. the pedal input, as illustrated in Figure 8(a). The first discontinuity arises when thedriver releases the acceleration pedal when regenerative braking Phase 1 applies, followed byanother discontinuity triggered by regenerative braking Phase 2 if the driver applies the brakepedal. However, these discontinuities disappear if the engine speed is relatively low, as shownin Figure 8(b). This is because the engine idle speed control is activated and the regenerativebraking is inhibited. Otherwise, it may brake the engine. Thus, these discontinuities appear aslong as the engine idle speed control is not triggered. It is worth noting that, in Figure 8(a), theacceleration pedal interpretation (α) is nonlinear (less sensitive at high pedal inputs) and varieswith the engine speed (less torque is delivered for high engine speeds) whereas the brakingpedal interpretation (β) is linear. Moreover, the selection of ωice = 4000 rpm indicates lessavailability of the regenerative braking torque when it is limited to the ISG torque capacityparticularly at a high engine speed.

A conventional driver model employing a PI speed control law [16] has very limited abilityof tackling the above-mentioned discontinuity in torque at wheels. Unfavourable oscillationsusually occur when the hybrid electric vehicle is simulated over e.g. NEDC cycles. The cause ofthe oscillations is explained inAppendix 1 in details and the simulation results, obtained on theHiL system under study and shown in Figure 9, provide numerical evidences. The time–speedprofile used in the simulation is only part of the NEDC cycle, in which the vehicle performsacceleration, braking, cruising and idling during a 100-s period. The simulation results ofa conventional vehicle with no regenerative braking are given in Figure 10 for comparison.Although the driver model implemented has the full capability of regulating the vehicle tofollow desired speed commands, it is proved to be useful only when there is no regenerative

Figure 8. Pedal map and discontinuous torque at wheels.

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314 S. Shen et al.

Figure 9. HiL simulation results: Conventional driver model with regenerative braking.

Figure 10. HiL simulation results: Conventional driver model without regenerative braking.

braking. The oscillations in the pedal input and powertrain torque during braking (at aroundt = 25 and 92 s) clearly indicate its intrinsic deficiency in handling the discontinuity. Hence,in order to evaluate an HCU for an ISG HEV, it is vital to establish a driver model that cancope with this problem.

4. A rule-based driver model incorporating look-ahead information

The key function of a driver model is to generate the appropriate inputs of gas and brakepedals so that the vehicle could follow the given speed-time profile of the pre-defined drivingcycles. It has been shown in the last section that the driver model employing a PID speedcontrol technique leads to oscillations and is not appropriate for ISG HEVs. A two-degree-of-freedom driver model incorporating look-ahead information with a hybrid control structureis introduced to solve the problem.

4.1. A two-degree-of-freedom driver model

Assume that the following conditions hold:

• There is no energy loss at the transmission and at the clutch.• The dynamics of both the ICE and ISG comparing to that of the vehicle are much quicker

and thus can be neglected.• The tyres work entirely at the linear region, implying that they can provide driving and

braking forces if required.• The vehicle is on a flat road.• The clutch engagement process and shifting process are instantaneous and thus can be

neglected.

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Vehicle System Dynamics 315

Then the motion of the vehicle is governed by

m�vv = Pctr − Pres + d, (14)

where Pctr, Pres and d represent the control input power, the power of the rolling and airresistance and the un-modelled error and/or disturbance, respectively, and m� is the equivalenttotal mass of the entire vehicle. The power introduced by the disturbance is less significant inthis application because the testing environment has to be set up according to the regulationwhen performing the NEDC driving cycle. Thus, un-modelled errors (e.g. shift control, clutchcontrol and idle speed control) are the main concerns. The equivalent total mass m� and thepower consumed owing to the resistance Pres are further expressed as

m� = m + (Jice + Jisg)r2r2

f + Jwh

R2. (15)

Pres = (Fair + Froll)v,

= 12ρCdAv3 + frollmgv, (16)

where air resistance Fair and rolling resistance Froll are given explicitly but the knowledge aboutthe vehicle aerodynamic coefficient Cd and the rolling friction coefficient froll is assumed. Sincethe rolling resistance varies with the vehicle speed v, it can be concluded that the resistancepower depends largely on the vehicle speed. Without causing much of the confusion, Pres isreplaced with Pres(v) hereinafter. Furthermore, the impact of the variation of the tyre pressure,tyre temperature and any other factors on the air and rolling resistance is treated as un-modellederror, which are left to be compensated by the driver model.

Since the vehicle desired speed vd and the desired acceleration vd are known, the drivermodel (to generate the control input power Pctr) can be designed as the following two-degree-of-freedom controller

Pctr = m�vdvd + Pres(vd) + PI (ev)vd, (17)

where the speed error ev is defined as

ev = vd − v. (18)

The first two terms on the RHS of Equation (17) is the feed-forward part of the controller,which is based on the model of the powertrain. The term m�vdvd represents the power requiredto produce the desired acceleration/deceleration and the term Pres(vd) represents the powerrequired to overcome the resistances. The last term PI(ev) on the RHS of Equation (17) is thefeedback part of the controller. It uses the speed error ev to compensate any deviation of v

away from vd caused by any un-modelled error. PI(ev) is further expressed as

PI(ev) = Pev + I

∫ev dt, (19)

where P and I are proportional and integral gains, respectively. Comparing to the conventionalPI controller, the P and I gains are significantly reduced. What is important is that the risk of theclosed-loop system to become unstable is reduced. Similar to the conventional PI controller,the I-term is essential for the system to adapt to the unknown un-modelled errors. However,the purpose of having the P-term is different. The P-gain of the conventional PI controllerdetermines both the closed-loop system dynamics in response to the varying reference inputand its sensitivity to the unknown un-modelled error. Only the later is the case for the above

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316 S. Shen et al.

PI controller and the dynamics of the close-loop system depends largely on the feed-forwardpart.

Combining Equations (14) and (17), one obtains

m�vv = m�vdvd + PI (ev)vd + [Pres(vd) − Pres(v) + d]. (20)

This is analysed in details in the following two cases.Case 1: I = 0, focusing on the impact of the P-gain. Equation (20) is rewritten as

m�vev + (m�vd + Pvd)ev + [Pres(vd) − Pres(v) + d] = 0. (21)

Comparing it with the system controlled only by a proportional controller, which is given by

m�vev + Pvdev + [−m�vvd − Pres(v) + d] = 0, (22)

it can be concluded that:

• The steady-state speed errors essv are, respectively, (Pres(v) − d)/P and − d/P. Although

both the feed-forward and feedback term can reduce the steady-state speed error, theirmechanisms are different. The feed-forward term improves the un-modelled error whereasthe feedback term makes the system less sensitive to it.

• The closed-loop dynamics is not affected by the feed-forward term normally. This is trueif the feed-forward term in Equation (17) is defined as m�vvd + Pres(vd). However, owingto the particular form of the proposed model-based controller, it has an impact on theclosed loop system. The presence of the un-modelled error m�vvd causes a latency ofthe PI control in response to a constant acceleration of the reference input. Furthermore,linearising Equation (21) at a given vehicle speed yields

m�vev +(

m�vd + 3

2ρCdAv2

d + frollmg + Pvd

)ev + d = 0. (23)

This clearly indicates that the bandwidth of the closed-loop system is increased. Conse-quently, the earlier-mentioned controller brings a faster response, particularly during thevehicle acceleration.

• Because the feed-forward term can reduce the modelling error and enhance the systemdynamic performance without the risk of destabilising the system, the contribution from thefeedback term could be minimised. A small P-gain may be sufficient to meet design criteria.

Case 2: I �= 0, highlighting the contribution of the I-gain. In this case, Equation (20)represents a second order system, being rewritten and linearised as

[ev

U

]=

[− vd

vm�

Peqv − vd

vm�

I 0

] [ev

U

]+

⎡⎢⎣

1

m�

0

⎤⎥⎦ �P, (24)

where U represents the contribution of the integral term in the control action,

Peqv =⎧⎨⎩

P conventional,

P + m�vd + frollmg

vd+ 3

2ρCdAvd model-based,

(25)

and

�P ={

m�vvd + Pres(v) − d conventional,

−d model-based.(26)

Investigation of Equation (24) leads to the following observations:

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Vehicle System Dynamics 317

• The closed-loop system with I �= 0 has the capability of learning and adapting to bothdisturbances and modelling errors. Subsequently, at steady-state, ev = 0 and U = v/vd�P.

• Eigenvalues of the system are

λ1,2 = −vd/(vm�)Peqv ∓ √(vd/(vm�)Peqv)2 − 4vd/vm�I

2, (27)

where λ1 corresponds to the mode in response to the reference input and λ2 to the mode ofadapting to the disturbance and modelling errors.

• The dynamics in response to the reference input is faster than the dynamics of convergingto modelling errors when I < Icri with

Icri = vd

4vm�

P 2eqv. (28)

The dynamics in response to the reference input is the same as the dynamics of convergingto model errors when I > Icri. The system exhibits oscillations if the system damping ratio,defined as

ξ = 1

2

√vd

vm�IPeqv, (29)

is sufficiently low.

In contrast to the conventional PI, the above PI has a high equivalent P-gain, as indicated inEquation (25). This eventually makes the closed-loop system faster in response to the referenceinput and less sensitive to disturbance and modelling errors. Also, the high equivalent P-gainallows a significant improvement in adaption to disturbances and model errors without riskingthe system stability, which is the result that a high I-gain can be accommodated by the increasedcritical level Icri in Equation (28) and the reduced damping ratio in Equation (29).

More generalised comparisons of the model-based feed-forward and feedback control canbe conducted in the frequency domain using the block diagram shown in Figure 11, wherer(s) refers to the reference input, y(s) the output, e(s) the error between the reference and theoutput, and d(s) a disturbance and/or modelling error. Gf (s), Gb(s) and Gp(s) are the respectivetransfer functions of a model based feed-forward control, a feedback control, and a plant underthe control. The output y(s) and error e(s) relate to the reference r(s) and the disturbance d(s)by the following equations:

y(s) = [Gf (s) + Gb(s)]Gp(s)

1 + Gb(s)Gp(s)r(s) + 1

1 + Gb(s)Gp(s)d(s), (30)

e(s) = 1 − Gf (s)Gp(s)

1 + Gb(s)Gp(s)r(s) − 1

1 + Gb(s)Gp(s)d(s). (31)

Figure 11. Model-based control.

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318 S. Shen et al.

The model-based feed-forward approach is to approximate the inverse of the plant modelwith sufficient accuracy. Therefore, it is reasonable to assume

Gf (s)Gp(s) = 1 + δ.

The norm of δ, denoted by ‖δ‖, is small. As a consequence, Equations (30) and (31) reduce to

y(s) = r(s) + δ

1 + Gb(s)Gp(s)r(s) + 1

1 + Gb(s)Gp(s)d(s), (32)

e(s) = δ

1 + Gb(s)Gp(s)r(s) − 1

1 + Gb(s)Gp(s)d(s). (33)

This two-degree-of-freedom control design involves the design of both the feed-forward andfeedback control channels. This is in contrast to the so-called one-degree-of-freedom controldesign, in which only the feedback control channel needs to be designed. Setting Gf (s) = 0in Equations (30) and (31) yields the expression of y(s) and e(s). The comparison betweenthe feed-forward and feedback channel clearly indicates the benefits of the introduction of theformer: (1) both the close-loop dynamics which is largely dominated by the denominator terms1 +Gb(s)Gp(s) and the sensitivity of the closed-loop system to the disturbance inputs are notaffected by the design of the feed-forward channel, but the system error e(s) and the systemability to trace the reference input r(s) are greatly impacted; (2) by inversely approximating theplant model (when possible), the feed-forward channel makes the closed-loop system followthe reference input with sufficient accuracy and without the need of high feedback gain.

4.2. Incorporating look-ahead information

The use of Equation (17) to represent a driver does not include the preview function of humanbeing. A driver normally controls the vehicle according to the traffic and road conditions ahead[23]. Therefore, it is more appropriate to include the human being preview function by usinglook-ahead information.

Let tad be the look-ahead time, then look-ahead acceleration/deceleration vad is defined as

vad = vd(t + tad) − vd(t)

tad. (34)

In this study, tad = 1 s is adopted as the human response time. Replacing vd with vad in Equation(17) results in

Pctr = m�vdvad + Pres(vd) + PI (ev)vd. (35)

The first and second terms on the RHS of the equation are treated as feed-forward terms inSection 3.1. Actually, vad defined in Equation (34) is a difference term with a time step of tad.It is therefore more appropriate to treat it as a D-term. Introducing u to represent P, I and Dterms, we can rewrite Equation (35) as

Pctr = uvd + Pres(vd). (36)

Three different forms of the u-control laws are proposed as follows:

u =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

PI (ev) + Dvd(t + T ) − vd(t)

Tspeed look-ahead control,

PI (ev) + De(t + T ) − e(t)

Terror look-ahead control,

PI (ev) + De(t) − e(t − T )

TPID control,

(37)

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Vehicle System Dynamics 319

where D and T are the generalised D-gain and look-ahead time step. First, replacing the differ-ential term with a difference term eliminates the injection of a potentially significant amount ofenergy into the system caused by the rapid change of the reference input. Next, speed and errorlook-ahead control use future information, making them non-causal. Particularly, since futureerror information is not available, certain estimation is therefore required. Finally, assumingaccelerations of both the reference and actual signals are constant during a sufficient smalltime step T, the differences in Equation (37) can be replaced with the average accelerationsduring the same period.

u =

⎧⎪⎨⎪⎩

PI (ev) + DAd(t, t + T ) speed look-ahead control,

PI (ev) + D[Ad(t, t + T ) − A(t − T , t)] error look-ahead control,

P I (ev) + D[Ad(t − T , t) − A(t − T , t)] PID control,

(38)

where Ad(t, t + T ) and A(t, t + T ) are, respectively, the average reference and the actualvehicle acceleration during the period from t to t +T, similarly for the period from t −T tot. Since the actual acceleration from t to t +T is not available, A(t − T , t) is used in errorlook-ahead control. Although the above expressions are still not causal, it shall cause noimplementation problems provided that the future reference information is known.

It is worth noting that the D-terms of all the three control laws play no part at the steadystate. By contrast, during transient when the vehicle acceleration is varying or during quasitransient when the vehicle is undergoing a constant acceleration/deceleration, the D-terms ofthe three control laws lead to very different effects. The D term of speed look-ahead controlreduces un-modelled error and increase the closed-loop system frequency. This is true becausethat replacing vd, and m� with Ad(t, t + T ) and D in Equation (20) results in a close-looprepresentation of the speed look-ahead control system. The analysis of the last subsection stillholds. However, this is not true for the conventional PI controller. To investigate the impact ofthe D term, the following is obtained by combining Equations (14), (35) and (38):

(m�v + Dvd)ev = −(P ev + U)vd + [m�vvd + Pres(v) − d] + D[ev − ev(t − T , t)],U = Iev, (39)

where ev(t − T , t) = Ad(t − T , t) − A(t − T , t). As a consequence, the closed-loopsystem frequency ωn and damping ratio ξ are

ωn =√

I

m�v/vd + D,

ξ = P

2

√1

I (m�v/vd + D).

(40)

Clearly, it indicates that the introduction of D term in PI control slows down the systemfrequency response and reduces the damping ratio. Thus, it is not a surprise to see in Figure12(a) and 12(b) that the overshoot of the step responses is increased with non-zero D-gain sincethe system is the less damped. However, the overshoot of ramp responses is rather reduced asshown in Figure 12(c) and Figure 12(d) when increasing the D-gain because the augmentedlatency as indicated by Figure 13 allows more time of the system to adapt to the input variation.

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320 S. Shen et al.

Figure 12. PID control results.

Figure 13. Overshoot vs D-gain.

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Vehicle System Dynamics 321

Rewriting the second term of Equation (38) for error look-ahead control as

Ad(t, t + T ) − A(t − T , t) = [Ad(t, t + T ) − Ad(t − T , t)] + [Ad(t − T , t) − A(t − T , t)](41)

allows the investigation of the difference between PI control and error look-ahead control. Likethe PI control, the term Ad(t − T , t) − A(t − T , t) deteriorates both the system dampingratio and frequency response. However, unlike the PI control, Ad(t, t + T ) − Ad(t − T , t)

introduces an extra phase lead w.r.t. the reference input.Figure 14 visualises the phase lead introduced by Ad(t, t + T ) − Ad(t − T , t). Ad(t −

T , t) represents the measure of the reference acceleration over the interval [t −T, t] whileAd(t, t + T ) is over the interval [t, t +T ]. Apparently, Ad(t, t + T ) leads Ad(t − T , t) by aquantity of T. When computing the difference between Ad(t, t + T ) and Ad(t − T , t), somefavourable actions appear, for instance, as indicated by the shading area of Figure 14. Theterm Ad(t, t + T ) − Ad(t − T , t) acts as a ‘boost’ when acceleration starts, resulting in aquick and early vehicle response. Conversely, it demands a ‘brake’ of the vehicle at the endof the acceleration phase, preventing unfavourable overshoot. Comparing Figure 14(a) with(b), these effects remain the same for ramp inputs. Thus, the extra phase lead counteracts theeffect of the conventional D-term. Consequently, the overshoot in response to the step inputis nearly irrelevant to the change of the D-gain whereas the improvement in overshoot w.r.t.ramp input almost linearly depends on the D-gain. Both are clearly evidenced in Figure 13.Also shown in the figure are: (1) the closed-loop performances, overshoot in particular, oflook-ahead control are very similar though the mechanisms of improving them are different;(2) reproduction of the results in Figure 12(a) and (c) in light colour highlights the differencebetween look-ahead and PI control. The superiority of look-ahead control over PI control isclearly indicated, particularly when the comparison is focusing on the overshoot of the stepresponse. The trade off between the ramp and step inputs of the conventional PI control showsthe system has a limitation on input frequency. Further comparisons are given in Figure 13 withD = 0 for conventional PI and D =m� for look-ahead and preview control, and observationsare given as follows:

• For a step input, speed look-ahead, error look-ahead and PI control all produce very similarovershoot as shown in Figure 15(b), but look-ahead controls have higher bandwidths.

• As evidenced in Figure 15(d), speed and error look-ahead control make much less overshootthan that of PI control for ramp inputs. In the meanwhile, look-ahead controls still lead PIcontrol in response to transient inputs.

Figure 14. The impact of the phase lead.

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322 S. Shen et al.

Figure 15. Vehicle responses of different control laws (P = 500 and I = 30).

• The look-ahead controls have much better performance than PI control during transientsince they respond to input fast while the systems are well damped.

4.3. Rule-based architecture for the driver model

Although regulating the vehicle speed via the power governed by the Equation (35) is dis-cussed in the above subsections, the driver uses the gas and brake pedals and selects thegear ratio to control the vehicle speed in reality. Bearing in mind that the gear ratio isgiven in NEDC drive cycle, converting the power input into pedal inputs is described inthis subsection. To avoid the oscillation discussed in Section 3, a rule-based architectureis proposed for the driver model. The look-ahead acceleration information vad is used forselecting different conversion algorithms and it has the following three states, as illustratedin Figure 16:

H1 There is an acceleration pedal input α, i.e. α > 0, but no brake pedal input β, i.e. β = 0.The vehicle may accelerate and decelerate, depending on how much power the engine andISG provide to the powertrain.

H2 There is no acceleration or brake pedal input, i.e. α = 0, β = 0. The vehicle decelerates.The deceleration depends on the vehicle resistance power, the engine and ISG brakingpower.

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Vehicle System Dynamics 323

Figure 16. Rule-based architecture for the driver model.

H3 There is no acceleration pedal input but a brake pedal input, i.e. α = 0, β > 0. The vehicledecelerates. The deceleration depends on the vehicle resistance power, the engine and theISG braking power, and the vehicle brake power.

At each state, the driver model is the same, as given in Equation (35). However, thealgorithms of converting the power input Pctr into pedal inputs α and β at each state aredifferent. The transition out of one state to another depends on the value of the look-aheadacceleration/deceleration vad.

Let u = [α, β] be control inputs, x a set of state variables necessary for determining theu, �(hi, hj), {i = 1, 2, 3; j = 1, 2, 3} an transition event from Hi to Hj, � the threshold set fordetermining the transition, then a hybrid controller can be expressed as

u = fhi(Pctr, x), hi = {h1, h2, h3}, (42)

�n+1(hi, hj ) = g[vad, �, �n(hi, hj )]. (43)

Equation (42) is a set of continuous functions fh1, fh2 and fh3, representing the determinationof the u in state H1, H2, and H3, respectively. On the other hand, Equation (43) describes adiscrete-event function g. The (n + 1)th transition of �n+1(hi, hj) is determined by the look-ahead acceleration/deceleration vad, the threshold set � and the nth transition �n(hi, hj). Thetransitions are graphically illustrated in Figure 16, where � = {vpass, v′

pass, vreg, v′reg}. vpass and

v′pass determine the transition between H1 and H2. vpass �= v′

pass (actually v′pass = vpass + Cpass

in this application, and Cpass is a positive constant) is to prevent frequent switch betweenH1 and H2. Similarly, vreg and v′

reg (v′reg = vreg + Creg and Creg is a positive constant) are a

pair of thresholds for the transition between H2 and H3, and vreg and v′pass for the transition

between H1 and H3.Conversions of the power control input to pedal input in each state are presented in the fol-

lowing section. However, all these conversions require the knowledge of the engine maximumtorque T max

ice , engine minimum torque T minice , ISG maximum torque T max

isg , regenerative brakingtorques in Phase 1 T 1

reg and Phase 2 T 2reg, as shown in Figure 17. Also shown in the figure is

the ICE and ISG combined working region.

Conversion of the power control input to pedal input in state H1. As β = 0, only accelerationpedal input α need to be resolved, which is given implicitly as

Pctr = αωice[T maxisg (ωice) + T max

ice (ωice)], (44)

where a linear pedal interpretation is assumed. However, to interpret the acceleration pedalas a torque is somewhat nonlinear in practise, and the interpretation also depends on enginespeed and vehicle speed. The pedal input α derived from Equation (44) is

α = m�vdvad + Pres(vd)

[T maxisg (ωice) + Ticemax(ωice)]ωice

+ PI (ev)vd

[T maxisg (ωice) + T max

ice (ωice)]ωice, (45)

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324 S. Shen et al.

Figure 17. Torque speed characteristics of the ISG hybrid powertrain under study.

where the first term on the RHS represents the feed-forward part while the second term is thefeedback part. To further reduce the complexity of dynamics of the closed loop system, rvvd isintroduced to replace ωice, where rv = rfr/R is the ratio between the engine and vehicle speed.

Conversion of the power control input to pedal input in state H2. There is no brake pedalinput, but the determination of the acceleration pedal input is not straightforward. Let thethreshold parameter vpass and vreg be derived at first. vpass refers to the vehicle decelerationproduced only by the vehicle resistance, i.e.

vpass = −Fres(v)

m�

,

= −Fair + Froll

m�

when α = 0+, (46)

where α = 0+ implies that there is no acceleration pedal input but it approaches zero fromthe positive direction. Thus, no regenerative braking shall be triggered and the engine shallmanage to deliver a torque only sufficient to overcome its internal losses. vreg refers to thevehicle deceleration when both the engine brake and the regenerative braking Phase 2 areapplied but no mechanical brake applies:

vreg = −Fres(v)

m�

+ T 2reg(ωice) + T min

ice (ωice)

m�

ωice

vwhen β = 0+, (47)

where β = 0+ implies that the brake pedal approaches zero from the positive direction. Thus,both regenerative braking Phase 2 and the engine brake are triggered. Also, v1

reg is introducedto represent the vehicle deceleration when both the engine brake and the regenerative brakingPhase 1 are applied:

v1reg = −Fres(v)

m�

+ T 1reg(ωice) + T min

ice (ωice)

m�

ωice

vwhen α = 0−, (48)

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Vehicle System Dynamics 325

where α = 0− implies that the acceleration pedal approaches zero from the negative direction.Consequently, both regenerative braking Phase 1 and the engine brake are triggered but nomechanical brake shall apply.

Equations (46)–(48) indicate that vpass, vreg and v1reg depend only on the vehicle speed v and

the engine speed wice. The vehicle acceleration/deceleration can be represented by a vectorat the space spanned by the vehicle speed v and the time t as shown in Figure 18, where theacceleration/deceleration of the vehicle is represented by the direction of the speed variation inv − t space, thus it is a direction vector and only the normalised direction vector is interested.Clearly, the subspace spanned by the vector vpass and vreg defined as

v = k1vpass + k2vreg, k1 > 0, k2 > 0 (49)

represents the vehicle deceleration that is unreachable at instantaneous moment. It is onlyreachable in a sense of the average deceleration over a period of T since

T v = T1vpass + T2vreg, T1 > 0, T2 > 0, (50)

where v represents the average vehicle deceleration during period T, T 1 is the period that thevehicle decelerates at the rate of vpass and T 2 is the period that the vehicle decelerates at therate of vreg, and T 1 +T 2 =T. Rewriting Equation (50) as follows

v = T1

Tvpass + T2

Tvreg, T1 > 0, T2 > 0. (51)

Its equivalence to Equation (49) suggests that this subspace is reachable over the period T.Furthermore, if

v ∈ {v|v = k1vpass + k2v1reg k1 > 0, k2 > 0},

then v can be reached via a switch between vpass and v1reg over a period T, i.e. a switch between

releasing and applying the acceleration pedal. Similarly for

v ∈ {v|v = k1v1reg + k2vreg, k1 > 0, k2 > 0},

i.e. a switch between releasing and applying the brake pedal.High frequency switch between applying and releasing the acceleration pedal, and between

applying and releasing the brake pedal is essential to trace the desired vehicle deceleration,but this is neither manageable nor comfortable for human beings. Moreover, applying andreleasing a torque in a high frequency manner may damage the ISG machine and the engine.Therefore, the vehicle deceleration defined in Equation (49) is not controllable for humandrivers. In practice, the driver often uses the look-ahead information to decelerate the vehicleearly but with less magnitude.

Figure 18. Unreachable deceleration.

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326 S. Shen et al.

In State H2, α = 0− and α = 0+ are different as engine idle speed control8 is active whenα = 0+. As a consequence, the engine controller shall manage to overcome the engine internalloss and to maintain the engine speed around the reference idle speed. By contrast, the fuelcut is activated when α = 0−. The engine internal loss become a resistance to slow the vehicledown as a result. Meanwhile, regenerative braking Phase 1 is also triggered. Figure 19 showsthe strategy of determining the acceleration pedal position based on the look-ahead accel-eration/deceleration vad, and the threshold parameter v1

reg. State H2,1 corresponds to α = 0+while State H2,2 corresponds to α = 0−.

As a means of simplifying the closed loop system dynamics, the engine speed ωice is replacedwith rvvd and the vehicle speed v is replaced with desired speed vd in Equations (46), (47)and (48).

Conversion of the power control input to pedal input in state H3. As α = 0, only the brakepedal input β requires determination. Similar to the approach described for that in state H1,the following is given

Pctr = [T 2reg(ωice) + T min

ice (ωice)]ωice − F maxbrk βv, (52)

where a linear pedal interpretation is assumed. The pedal input β is then derived as

β = −[T 2reg(ωice) + T min

ice (ωice)]ωice

F maxbrk v

− m�vdvad + Pre(vd)

F maxbrk v

− PI (ev)vd

F maxbrk v

. (53)

For the same reason as stated in determining α in H1, ωice is replaced with rvvd and v isreplaced with vd.

5. Validation of the proposed driver model

This section presents numerical results of the proposed driver model when following NEDCdrive cycles on the HiL system described in Section 2.2. The NEDC drive cycle is composedof four ECE driving cycles simulating city driving and one EUDC simulating highway drivingconditions. Its characteristics are summarised in Table 2. Effectiveness of the proposed drivermodel is first investigated, followed by some performance of the ISG hybrid vehicle resultedfrom several HiL simulations.

Figure 19. Hybrid speed control architecture in state H2.

Table 2. Characteristics of NEDC drive cycle.

Characteristics Unit ECE EUDC

Distance km 4 × 1.013 = 4.052 6.955Duration s 4 × 195 = 780 400Average speed km/h 18.7 (with idling) 62.6Maximum speed km/h 50 120

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Vehicle System Dynamics 327

5.1. Under-shoot problem of the driver model

Figure 20(a) shows the desired and resulted vehicle speed during a particular part of the NEDCcycle in which the vehicle acceleration and deceleration are both covered. When the vehicle isdecelerating faster than the required, a significant undershoot occurs. This is caused by onlythe look-ahead vehicle acceleration/deceleration being used for determining the transitionbetween states. If the look-ahead vehicle speed could also be used in such a way that theswitch to State H1 is taking place at t2 rather than at t1 in Figure 20(a), the undershoot isavoidable. Figure 20(b) shows the results of the improved driver model. Nevertheless, thetwin pair of the so-called undershoot problem, namely significant overshoot at the end phaseof the acceleration, will not arise. This is because the vehicle acceleration is manageable evenby the conventional driver model.

5.2. Effectiveness of the proposed driver model

Figure 21 presents some simulation results of the proposed driver model. For comparison, thetarget vehicle speeds and the duration of the simulation are identical to those presented in Figure9. The following observations are made by investigating and comparing the simulation results.

Observation 1. The proposed driver model can eliminate the oscillations during decelerationphase caused by the discontinuity. This is clearly evidenced when comparing the resultspresented in Figure 21(b) and Figure 21(c) with those in Figure 9(b) and Figure 9(c).

Figure 20. Under-shoot problem with the speed control.

Figure 21. HiL simulation results: Improved driver model with regenerative braking.

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328 S. Shen et al.

Observation 2. The proposed driver model produces no worse performance than that of theconventional driver model in tracing the target vehicle speed during acceleration or cruisingspeed phase. Actually, the overshoot when transition from acceleration to cruise speed isimproved. The comparison of results presented in Figure 21(a) with those in Figure 9(a) andFigure 10(a) provides an evidence.

Figure 22 provides the simulation results for the whole NEDC drive cycle. The actualsimulation on the HiL system is conducted over five NEDC cycles in series, allowing theconvergence of the SoC control so that the SoC at end of the cycle is equal (or almost equal) tothat at the start of the cycle. This is essential for fuel economy analysis. The simulation resultsgiven in Figure 22 are those of the last drive cycle. The resulted SoC in Figure 22 indicatesthat the energy moving to and from the battery is well balanced and that the battery convergesto a SoC with which it starts.

The following could be concluded by investigating the results presented in Figure 22: (1) Theability of the proposed driver model is further established as there is no oscillation throughoutthe entire NEDC drive cycle simulation; (2) The control performance of tracing the pre-definedreference speed is further proved under the absence of the significant error between the actual(thin-solid-line in Figure 22) and the reference (thick-dash-line) speed. Also proved is thevehicle’s acceleration performance, which is more important during extra urban driving cycle;(3) Different from presenting the sum of the engine and ISG torque in Figures 9, 10 and21, notated as the torque inputs to the driveline, the engine and ISG torques are presentedseparately and independently in Figure 22. This allows the discovery of the function of thehybridised powertrain. Clearly, the ISG acts as a torque booster during acceleration and asan energy absorber during deceleration; (4) The resulted engine speed illustrates the engine

Figure 22. Numerical experiment results based on NEDC drive cycle.

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Vehicle System Dynamics 329

IdleStop when the engine is not necessary for propelling the vehicle and when it is shutdownand disconnected from the rest of the powertrain. The engine IdleStop could save extra fuelas it will be consumed otherwise by the unnecessary engine idling.

5.3. Some facts of the simple hybrid powertrain control strategy

Although the assessment of the HCU is not the objective of this study, some primary resultsof the hybrid vehicle simulation are presented. As shown in Table 3, a ‘1.6L base vehicle’refers to a conventional competitor which has an gasoline engine of 1.6L, while a ‘1.3L basevehicle’ refers to an ISG hybrid vehicle but all hybrid functions are turned off. The enginesize is reduced from 1.6L to 1.3L because of hybridisation. The ‘Stop Start’ denotes the‘1.3L base vehicle’ with stop start function while the ‘Torque split’ vehicle has fully hybridfunctions. Clearly, when comparing with the ‘1.6L base vehicle’, 6.4% fuel saving is obtainedthrough reducing the engine size, 2.4% fuel saving by stop start function, and a further 4.1%achieved when the driver demanded torque is appropriately distributed between the engineand the ISG.

To visualise how the hybrid functions improve the fuel economy, Figure 23 compares theengine working points over NEDC cycles with and without hybrid functions. Apparently, thediagram of Figure 23(a) indicates that the engine has more time working at very high torqueregion and therefore is less efficient. Owing to the hybrid function, the time of the engineworking at high torque region is significantly reduced.

Table 3. Fuel Consumptions on NEDC cycle.

Fuel consumed Improvement IncrementalFunctions added (L/100 km) (%) (%)

1.6L base vehicle 7.34 – –1.3L base vehicle 6.87 6.4 6.4Stop Start 6.68 9.0 2.6Torque split 6.38 13.1 4.1

Figure 23. Scattered engine working points over NEDC cycles.

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330 S. Shen et al.

6. Conclusion

When the travel of the brake pedal is not involved in determining the amount of regenerativebraking, stringent braking safety regulations are not applicable, saving development timesignificantly. However, if the regenerative braking is triggered by the release of accelerationpedal or application of brake pedal, and if the amount of the braking is dominated by theengine speed, a discontinuity of torque delivered to powertrain at the emergence or vanish ofregenerative braking arises. This inevitably leads to oscillations, making the assessment of theISG control strategy unnecessarily difficult. The method to improve the conventional drivermodel by means of introducing the look-ahead information is proposed and investigated. Theuse of the look-ahead information is twofold: (1) To determine when the target deceleration isnot admissible thus the trace of it is not essential; (2) To improve the speed control performance,particularly during transient. Simulation results conducted on a HiL system indicate the successof proposed driver model in eliminating the oscillation problem, which paves the way forevaluating the ISG control system systematically. Some primary evaluation shows that theISG strategy has the potential of cutting the fuel consumption by up to 13%.

Since the focus of this paper is on a practical solution to resolve the oscillation issues, thecomprehensive study of the use of the look-ahead information is not conducted. Especially,the following two questions need further attention: (1) The optimal look-ahead time step tad inEquation (34); (2) Only one step look-ahead is proposed but multi-step look-ahead may havebetter performance. All of these are left for future studies.

Although only HiL simulation results are presented in the paper, the proposed driver modelhas been used in commercial hybrid vehicle development as a tool to evaluation of torquemanagement strategy and the vehicle performances.

Acknowledgements

The collaboration of S. Shen and Q.-C. Zhong on this work is an outcome of the Network for New Academics inControl Engineering (New-ACE, www.newace.org.uk) funded by EPSRC, UK under grant No. EP/E055877/1.

The authors thank the reviewers for their very helpful and valuable comments, which have improved the qualityof this article.

Acronyms

BMS battery management system

BSFC brake specific fuel consumption

DoD depth of discharge

ECE European city environment driving cycle

ECU electronic control unit

EMS engine management system

EUDC extra urban driving cycle

HCU hybrid control unit

HEV hybrid electric vehicle

HiL hardware-in-the-loop

ICE internal combustion engine

ISG integrated starter generator

MCU motor control unit

MiL model-in-the-loop

NEDC new European driving cycle

NVH noise, vibration and harshness

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Vehicle System Dynamics 331

PID proportional-integral-derivative (control)

PiL processor-in-the-loop

PWM pulse-width modulation

REG regenerative braking

SiL software-in-the-loop

SoC state of charge

SoH state of health

SoL state of life

TCU transmission control unit

WOT widely opened throttle

Nomenclature

d disturbances and unmodelled errors (–)

ev speed error (–)

froll rolling resistant coefficient (–)

fs static friction coefficient (–)

g the strength of gravity (=9.8 m/s2)

khm, klm SoC control gains (–)

m vehicle mass (=1564 kg)

mf engine fuel rate (g/s)

mv, m� equivalent and total equivalent vehicle mass (kg)

r,rf gear ratios, final ratio (rf = 3.912; –)

rv ratio between engine and vehicle speed (rad/m)

tad look-ahead time (=1 s)

v, vd vehicle velocity, demanding vehicle velocity (m/s)

v vehicle acceleration (m/s2)

vad look-ahead vehicle acceleration (m/s2)

vpass,v′pass,vreg,v′

reg,v1reg,v1 ′

reg hybrid speed control parameters (m/s2)

y output (–)

A vehicle front area (=2.03 m2)

A, Ad average and average demanding acceleration (m/s2)

Cd air resistant coefficient (0.31; –)

Cbatt capacity of the battery (=6.5 A h)

Eair, Eroll energy used for air, rolling resistance (kJ)

Ebrk mechanical brake energy (kJ)

Ekn vehicle kinetic energy (kJ)

Elosseng energy lost due to engine brake torque (kJ)

Fair, Froll air, rolling resistance force (N)

Fbrk brake force (N)

F load resistance forces (N)

F maxbrk maximum break force (N)

Gb, Gf , Gp transfer function (–)

Icri critical integral gain (–)

J ice, J isg inertia of the engine and ISG (=0.1, 0.05 kg m2)

Je combined inertia of the engine and ISG (=0.15 kg m2)

Jwh inertia of the wheel (kg m2)

P, I, D proportional, integral and derivative gains (–)

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332 S. Shen et al.

Peqv equivalent proportional gain (–)

Pbatt battery charging/discharging power (watt)

Pctr, Pres control input and vehicle resistance power (watt)

R radius of the tyre (=0.3 m)

SoC battery state of charge (–)

SoChm, SoClm SoC high and low limit (–)

SoChmhl , SoClm

hl SoC high and low hard limit (–)

T, T1, T2 time (s)

T clt, T c, T stick clutch torque, clutch demanding torque, clutch stick torque (N m)

T ice, T isg engine torque, ISG torque (N m)

T drvdem driver demanding torque (N m)

T maxice , T min

ice , T elice maximum, minimum and E-line engine torque (N m)

T maxisg T max

isg maximum ISG torques (N m)

T 1reg, T 2

reg regenerative phase 1 and phase 2 torque (N m)

U integral contribution (–)

ωice engine angular speed (rad/s)

ωidle engine idle speed (=81.7 rad/s)

ωn closed-loop system frequency (Hz)

H1, H2, H3 hybrid states (–)

x, u states, control input (–)

α, β gas pedal, brake pedal (–)

ρ air density (=1.20 kg/m3)

ηchg,ηdis charging and discharging path efficiency (–)

λ eigenvalue (–)

ξ damping ratio (–)

�P unmodelled errors and disturbances (–)

�ω clutch slipping speed (rad/s)

Notes

1. It is also referred to as integrated starter alternator in some references [13].2. Hybrid vehicle model is referred to as a model that can present all the components that the hybrid powertrain

is made up of but the model does exclude the torque management system. A typical ISG vehicle plant modelis composed of a gasoline engine, a transmission with a clutch, a gear box and a final drive, a high-voltagebattery, a ISG machine, a vehicle and a driver.

3. The clutch and gearbox might be controlled by a computer system, e.g. in automated manual transmission,auto-gear shift, dual clutch transmission.

4. Interpreting the driver pedal input as demanding torque and splitting the demanding torque between theICE and the ISG is very common for HEVs with finite and discrete transmission ratios but it is less com-mon for those with infinite and continuous variable transmission ratios, where interpreting the driver pedalinput as demanding power and splitting the demanding power between the ICE and the ISG is widelyadopted [21,24].

5. The optimisation of the engine operation in this way is only static but not dynamic. This is because the E-lineis calculated based on static BSFC maps.

6. The use of the vehicle speed to determine the regenerative braking torque is another option [25].7. Category M1 refers to any passenger carrying vehicles that can seat up to eight passengers excluding the driver.8. The ISG hybrid engine idle speed control can be different from the conventional engine idle speed control. The

idle speed control law will be presented in an another article.9. k = 0 and k = ± 1 are bifurcation points since when k cross these points, the system behaves qualitatively

very different [26], or topological structure of the system dynamics represented by the phase portrait changesdramatically.

10. The method used converting the differential equation to difference equation does not make any real differencefor this particular problem, but the step-size of calculation does.

11. Using interval [ − 1 1] indicates sgn(x) can take all value at this interval.

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Vehicle System Dynamics 333

References

[1] J. Fredriksson, Improved driveability of a hybrid electric vehicle using powertrain control, Int. J. AlternativePropulsion 1(1) (2006), pp. 97–111.

[2] S. Delprat, J. Lauber, T.M. Guerra, and J. Rimaux, Control of a parallel hybrid powertrain: optimal control,IEEE Trans. Vehicular Technol. 53(3) (2004), pp. 872–881.

[3] X. Wei, L. Guzzella, V.I. Utkin, and G. Rizzoni, Model-based fuel optimal control of hybrid electrical vehicleusing variable structure control system, J. Dynam. Syst., Meas. Control 129 (2007), pp. 13–19.

[4] P. Pisu and G. Rizzoni, A comparative study of supervisory control strategies of hybrid electric vehicles, IEEETrans. Control Syst. Technol. 15(3) (2007), pp. 506–518.

[5] A. Sciarretta and L. Guzzella, Control of hybrid electric vehicles, IEEE Control Syst. Mag.April 2007, pp. 60–70.[6] P. Rodatz, G. Paganelli, A. Sciarretta, and L. Guzzella, Optimal power management of an experimental fuel

cell/supercapacitor-power hybrid vehicle, Control Engrg. Practice 13 (2005), pp. 41–53.[7] Y. Gao and M. Ehsani, A torque and speed coupling hybrid drivetrain-Architecture, control and simulation,

IEEE Trans. Power Electron. 21(3) (2006), pp. 741–748.[8] S. Sasaki, Toyota’s newly developed hybrid powertrain, 1998, pp. 17–22, Kyoto, Japan.[9] C. Gühmann, Model-based testing of automotive electronic control units, 2005.

[10] J. Gerhardt, H. Hnninger, and H. Bischof, A new approach to functional and software structure for enginemanagement systems – BOSCH ME7, SAE Paper 98P-178 (1998).

[11] L. Guzzella and C.H. Onder, Introduction to modelling and control of internal combustion engine systems,Springer, Berlin, 2004.

[12] J.R. Wagner, D.M. Dawson, and Z. Liu, Nonlinear air-to-fuel and engine speed control for hybrid vehicles,IEEE Trans. Vehicular Technol. 52(1) (2003), pp. 184–195.

[13] R.I. Davis and R.D. Lorenz, Engine torque ripple cancellation with an integrated starter alternator in a hybridelectric vehicle: Implementation and control, IEEE Trans. Ind. Appl. 39(6) (2003), pp. 1765–1774.

[14] J. Malan and M.J. Kamper, Performance of a hybrid electric vehicle using reluctance synchronous machinetechnology, IEEE Trans. Ind. Appl. 37(5) (2001), pp. 1319–1324.

[15] B.S. Bhangu, P. Bentley, D.A. Stone, and C.M. Bingham, Nonlinear observers for predicting state-of-chargeand state-of-health of lead-acid batteries for hybrid-electric vehicles, IEEE Trans. Vehicular Technol. 54(3)(2005), pp. 783–794.

[16] B.K. Powell, K.E. Bailey, and S.R. Cikanek, Dynamic modelling and control of hybrid electric vehicle powertrainsystems, IEEE Control System Mag. October 1998, pp. 17–33.

[17] D. Ramaswamy, R. McGee, S. Sivashankar, A. Deshpande, J. Allen, K. Rzemien, and W. Stuart, A case study inhardware-in-the-loop testing: Development of an ECU for a hybrid electric vehicle, SAE Paper 2004-01-0303(2004).

[18] D. Karnopp, Computer simulation of stick-slip friction in mechanical dynamics systems, J. Dynam. Syst. Meas.Control 107(1) (1985), pp. 100–103.

[19] S. Shen, B. Vroeman, and F. Veldpaus, IdleStop and Go: a way to improve fuel economy, Vehicle Syst. Dynam.44(6) (2006), pp. 449–476.

[20] J. Mierlo, P. Bossche, and G. Maggetto, Models for energy sources for EV and EHV: fuel cells, batteries,ultracapacitors, flywheels and engine-generators, J. Power Source 128 (2004), pp. 76–89.

[21] S. Shen and F.E. Veldpuas, Analysis and control of a flywheel hybrid vehicular powertrain, IEEE Trans. ControlSyst. Technol. 12(5) (2004), pp. 645–660.

[22] UNECE13H, Uniform provisions concerning the approval of passenger cars with regards to braking, UnitedNations Economic Commission for Europe.

[23] R.S. Sharp, D. Casanova, and P. Symonds, A mathematical model for driver steering control, with design, tuningand performance results, Vehicle Syst. Dynam. 33(5) (2000), pp. 289–326.

[24] H. Lee and H. Kim, Improvement in fuel economy for a parallel hybrid electric vehicle by continuous variabletransmission ratio control, 2005, 219, pp. 43–51.

[25] H. Yeo, S. Hwang, and H. Kim, Regenerative braking algorithm for a hybrid electric vehicle with CVT ratiocontrol, 2006, 220, pp. 1589–1599.

[26] D.K. Arrowsmith and C.M. Place, An introduction to dynamical systems, Cambridge University Press,Cambridge, UK, 1991.

[27] A.F. Filippov, Differential equation with discontinuous right-hand side, Amer. Math. Soc. Transl. 42(2) (1964),pp. 199–231.

[28] J.J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ, 1991.

Appendix 1. Dynamics of a discontinuous system: An simplified case study

The oscillation problem with a discontinuity is illustrated in this appendix using a simple model. Assuming x is theplant state, and xd is the reference state. The derivative of the state x depends only on the sign of the control input u, i.e.

x = sgn(u). (A1)

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334 S. Shen et al.

The control input u is designed to force the state x follow a reference xd. As such, a pure control gain p can beintroduced to control the plant and to avoid add any unnecessary complicity

u = p(xd − x). (A2)

Setting p = 1 further simplifies the problem. The reference state is governed by the equation

xd = −k, (A3)

where k is a positive constant. By introducing x = xd − x, we can combine Equations (A1)–(A3) into

˙x + sgn(x) + k = 0. (A4)

This is a simple ordinary differential equation with a discontinuity at x = 0 but to derive an analytic solution is indeeddifficult if it is not at all impossible. Nevertheless, the qualitative behaviour for the system governed by the aboveequation is still predictable.

k > 1 sgn(x) + k = 0 has no solution but ˙x < 0 and thus x grows unbounded in negative direction.k = 1 Any x satisfying x < 0 will be the solution of sgn(x) + k = 0, which is also a fixed-point

(or steady-state) solution of the system. However, since there are infinite fixed-point solutionscompeting for stability, any disturbance will drive the system moving from one fixed pointsolution to another.

0 < k < 1 Since sgn(x) + k = 0 is never true, there is no fixed point solution. Also, the system is verysensitive around x = 0.

k = 0 x = 0 is a stable fixed-point solution of the system but as the system has an infinite gain aroundx = 0.

− 1 < k < 0 The system behaviour is similar to the case of 0 < k < 1.k = − 1 The system behaviour is similar to the case of k = 1 but x > 0 is the fixed-point solutions of

the system.k < − 1 The system behaviour is similar to the case of k > 1 but x grows unbounded in positive direction

as ˙x > 0.

Only 0 < k < 1 is interesting in this study because: (1) To maintain parameter k has the values of k = 0 andk = ± 19 is almost impossible in physical world; (2) Qualitative behaviour of the system when k > 1 and k < − 1is crystal clear; and (3) Qualitative behaviour of the system when − 1 < k < 0 is similar to that of when 0 < k < 1.Furthermore, as the system dynamic behaviour resembles across the region of 0 < k < 1, taking k = 0.5 will not losemuch of the generality. Hereinafter, only the dynamics of the system with k = 0.5 is discussed, and Equation (A4)can be rewritten as

˙x = −sgn(x) − 0.5,

=

⎧⎪⎨⎪⎩

−1.5, x > 0,

−0.5, x = 0,

0.5, x < 0.

(A5)

For a given initial condition x �= 0, the follow can be obtained

x ={

C1 − 1.5t for x > 0,

C2 + 0.5t for x < 0,(A6)

with C1 and C2 being determined by the initial condition. Each solution reaches x = 0 in finite time. At first look,it seems that x = 0 is stable because ˙x > 0 for x < 0, and ˙x < 0 for x > 0. However, x = 0 is not a solution andthe system cannot stay at x = 0 as ˙x = −0.5. So the system must bounce back and forward in the neighbourhoodof x = 0. An evidence of such behaviour can be provided when converting the differential equation into differenceequation10. Defining

˙x = xn+1 − xn

�t, (A7)

in which �t is sufficiently small and n = 0, 1, 2, · · · . Equation (A5) is represented in a discrete from as

xn+1 ={

xn − 1.5�t, for xn > 0,

xn + 0.5�t, for xn < 0.(A8)

The simulation results based on the above difference equation are given in Figure A1. The useful observations aresummarised as follows:

(1) Periodic-4 solutions (i.e. it takes four steps come back to where it starts) do exist regardless of the step-size. Toprove this conclusion, we assumed that

xn ∈ {x | ‖ x ‖< �}, (A9)

in which � � �t is a small positive constant. i.e. xn is in the neighbourhood of x = 0. For xn not starting fromthis neighbourhood of x = 0, it will reach the neighbourhood according to Equation (A6). Without the loss of

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Vehicle System Dynamics 335

Figure A1. Period solutions with difference equation for different step-sizes.

generality, let

xn = δ > 0, (A10)

in which δ < �. Applying Equation (A8), the following hold:

xn+1 = δ − 1.5 �t < 0,

xn+2 = δ − 1.5 �t + 0.5�t,

= δ − 1.0�t < 0,

xn+3 = δ − 1.0 �t + 0.5�t,

= δ − 0.5�t < 0,

xn+4 = δ − 0.5�t + 0.5�t,

= δ,

(A11)

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336 S. Shen et al.

namely,

xn+4 = xn. (A12)

This is still true for xn = −δ < 0.(2) For a fixed simulation step �t, all the periodic solution have the same frequency and same amplitude, which is

clearly indicated in Figure A1(a).(3) The frequency and amplitude vary with the calculation steps. The frequency fx is 1/4�t while the amplitude Ax

is 3/4�t. The proof of this conclusion is straightforward thus it is not presented.(4) As �t → 0, Ax → 0 but fx → ∞. Thus, depending on the step-size of the simulation, oscillation can be very

quick though the amplitude become small.(5) Due to the existence of the periodic solution, the mean value of the speed of x should be zero, i.e.

mean( ˙x) = limT →∞

1

T

∫ T

0

˙x dt = 0, (A13)

otherwise, system should move to the direction defined by mean( ˙x) gradually, which contradicts to that the systemhas periodic solution. As a consequence, mean value of sgn(x) must counterbalance k = 0.5, i.e.

mean (sgn(x)) = −0.5. (A14)

The mechanism behind the mean (sgn(x)) = −0.5 is similar to that of pulse-width modulation (PWM). As shownin Figure A1, the sign function takes position value only during one step of calculation, it takes negative value forthe rest three steps in one period. The effect of this mechanism is even clear when investigating Equations (A1) and(A3). Since the sign function cannot offer a constant value by which the state x shall follow the reference xd, it has tovary its value constantly. The implication of mean(sgn(x)) = −0.5 is that at this mean value the state x will followthe reference xd. The quicker sign function alters its value, the better the x follows the xd.

Combining Observations 4 and 5, the system would have a stable fixed-point solution of x = 0 if sgn(x) is definedas

sgn(x) =

⎧⎪⎨⎪⎩

1.0, x > 0,

−0.5, x = 0,

−1.0, x < 0,

(A15)

or more generally defined as11

sgn(x) =

⎧⎪⎨⎪⎩

1, x > 0,

[−1, 1], x = 0,

−1, x < 0.

(A16)

Figure A2. Simulation results.

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Vehicle System Dynamics 337

This idea of cancelling the oscillation was discovered by Filippo [26,27], this is why the discontinuity introducedby sign function is often known as Filippo type of discontinuity. The oscillation introduced by discontinuity is alsointensively studied in the area of the sliding mode control. The discontinuous control input will occur at a slidingsurface where control law is changed from one mode to another. This oscillation is referred to as chattering[28], andelimination of the oscillation involves replacing the sign function with a saturation function. The simulation resultsgiven in Figure A2 use the same principle.

The simulation results as given in Figure A2 are presented to conclude this Appendix 1. It is clearly shown thatwith the sign function the state x is oscillating around the reference xd, and control input u alters its sign in a PWMmanner, which makes the x follow the xd. On the other hand, with saturation function, there is no oscillation, andcontrol input converges to u = − 0.5 quickly to counterbalance the effect of k.

Methods presented to cancel the oscillation in the Appendix are to replace the discontinuous function with asomehow continuous function artificially. However, if the discontinuity is introduced by the physical world, thejust-mentioned replacement cannot be implemented. Therefore, special approaches have to be investigated.

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