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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Dynamic coordinated control during mode transition process for a compound power-split hybrid electric vehicle Yanzhao Su a,b, Minghui Hu a,b,⇑, Ling Su b,c, Datong Qin a,b, Tong Zhang d, Chunyun Fu a,b a

State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing 400044, China School of Automotive Engineering, Chongqing University, Chongqing 400044, China c Changan New Energy Automobile Research Institute, Chongqing 401120, China d Corun CHS Technology Co., Ltd., Shanghai 201501, China b

a r t i c l e

i n f o

Article history: Received 24 July 2017 Received in revised form 12 November 2017 Accepted 15 January 2018 Available online 2 February 2018 Keywords: Hybrid electric vehicles (HEVs) Compound power-split transmission Torque estimation Mode transition Dynamic coordinated control strategy

a b s t r a c t The fuel economy of the hybrid electric vehicles (HEVs) can be effectively improved by the mode transition (MT). However, for a power-split powertrain whose power-split transmission is directly connected to the engine, the engine ripple torque (ERT), inconsistent dynamic characteristics (IDC) of engine and motors, model estimation inaccuracies (MEI), system parameter uncertainties (SPU) can cause jerk and vibration of transmission system during the MT process, which will reduce the driving comfort and the life of the drive parts. To tackle these problems, a dynamic coordinated control strategy (DCCS), including a staged engine torque feedforward and feedback estimation (ETFBC) and an active damping feedback compensation (ADBC) based on drive shaft torque estimation (DSTE), is proposed. And the effectiveness of this strategy is verified using a plant model. Firstly, the powertrain plant model is established, and the MT process and problems are analyzed. Secondly, considering the characteristics of the engine torque estimation (ETE) model before and after engine ignition, a motor torque compensation control based on the staged ERT estimation is developed. Then, considering the MEI, SPU and the load change, an ADBC based on a real-time nonlinear reduced-order robust observer of the DSTE is designed. Finally, the simulation results show that the proposed DCCS can effectively improve the driving comfort. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction As the energy crisis and environmental pollution problems become increasingly serious, the development of electric vehicles has become a focus of the automotive industry. The power-split hybrid electric vehicle (PS-HEV) has become one of the most promising schemes among various powertrain system configurations, and there have been a variety of such HEVs [1,2]. Examples of PS-HEV models include Toyota Prius, GM Cadillac and Geely Dili. The PS-HEVs control the mode transition (MT) to reduce fuel consumption by means of its energy management control strategy. However, since the engine is directly connected to a power-split transmission by a torsional damper spring (TDS), during the MT process (especially with engine start/stop) between electric vehicle (EV) mode and HEV mode, the engine ripple torque (ERT) [3–6], inconsistent dynamic characteristics (IDC) of engine and motors [7,8] can directly cause jerk and

⇑ Corresponding author. E-mail address: [email protected] (M. Hu). https://doi.org/10.1016/j.ymssp.2018.01.023 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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vibration to the powertrain. Besides, the model estimation inaccuracies (MEI) (e.g. engine torque estimation (ETE) inaccuracies) and system parameter uncertainties (SPU) (e.g. stiffness and damping uncertainties of the TDS and drive shaft) can also lead to the jerk and vibration of the powertrain [9]. Therefore, in order to improve the driving comfort, research efforts have been put into analyzing system dynamic characteristics (SDC) using a powertrain dynamic simulation model (plant model) and studying dynamic coordinated control strategy (DCCS) during the MT process. For the PS-HEVs, measures have been taken to suppress the jerk and vibration of the powertrain system. Such measures include establishing a simplified engine torque physical model (ETPM) [10,11], creating an ERT approximation function [12], and using an ETE method based on motor torque feedback values (MTBV) [13,14]. Then based on these measures, motor torque compensation control is used to improve the driving comfort. Most of these methods took into account the ERT, but only feedforward or feedback single ETE method was used and the characteristics of the ETE model before and after engine ignition, and the influence of MEI and SPU on the powertrain system were neglected. Zeng [15] limited the power source torque change rate to improve the driving comfort, but ignored the ERT. Zhuang [16] used the dynamic programming algorithm to optimize the appropriate torque and brake pressure, but this method is mainly adopted to provide guidance for the formulation of control strategy and can not be used for real-time control. Besides, to suppress the ripple torque that passes to the wheels, a motor active damping and weighted feedback filtering control method [17] and an active damping control method based on PID control [12] have been proposed. However, the MEI and SPU were neglected by these methods, so their robustness in practice is not satisfactory. Several control methods are proposed to cope with the jerk and vibration caused by the SPU, such as the method based on online parameter identification [9], the fuzzy adaptive sliding mode control [18], the H1 robust coordinated control method [19] and the ml integrated robust control method [20,21]. However, these methods not only neglected the ERT but also mainly solve the coordinated control problem of clutch engagement process for a parallel HEV. In addition, these works did not establish a detailed plant model to analyze the SDC of a powertrain, and thus could hardly provide guidance for the development of DCCS. For the PS-HEV considered in this paper, a simplified ETPM method can be used before engine ignition due to its simple structure and suitability for feedforward control. After ignition the engine speed becomes higher (greater than 800 rpm) [14] and the ERT has little effect on system jerk and vibration, the use of an ETE method based on the MTBV is simpler than the ETPM method, avoiding the effect of ignition and injection on the estimated model. Therefore, the staged ETE compensation control may become a choice. However, factors such as the MEI and SPU make it difficult to completely compensate the fluctuation torque, if only the ETE compensation control method is used. As a result, the uncompensated fluctuation torque will be transmitted to the wheels by the drive shaft and in turn jeopardize the driving comfort. To solve this problem, this paper designs a drive shaft torque (DST) observer with robust and real-time performance, and employs the motors to perform feedback compensation control for the drive shaft torque estimation (DSTE) (i.e. active damping feedback compensation control (ADBC)). Therefore, different from the existing solutions, we aim to develop a DCCS including ‘‘a staged ETFBC and a DSTE-based ADBC” during the MT process, in order to suppress the jerk and vibration of the powertrain. This study shows the following three major contributions: (1) A detailed plant model, including the ERT model and the planetary dynamic model etc., is established to reflect the SDC with higher accuracy. (2) The characteristics of the ETE model before and after the engine ignition are taken into account, and the motor torque compensation control based on the ERT model feedforward estimation before engine ignition and the ERT estimation with motor feedback values after ignition is developed. (3) The MEI, SPU and load change are considered, and a DSTE-based ADBC is designed for all work modes, which presents good robustness and real-time performance. The rest of this paper is organized as follows: In Section 2, the plant model is established, and then the MT process and problems are analyzed. In Section 3, the detailed DCCS is developed. In Section 4, the proposed DCCS is validated using the plant model. In Section 5, some concluding remarks are provided.

2. Powertrain dynamic modeling and analysis In this paper, a compound PS-HEV based on the Ravigneaux planetary gear set (shown in Fig. 1) is investigated. This is a typical power-split HEV configuration, which will been applied on a Changan HEV. S1 and Pf are the small sun gear and the front planetary gear, respectively. S2 and Pr are the big sun gear and the rear planetary gear, respectively. C represents the sharing planetary carrier. R denotes the sharing ring gear, whose output torque drives the vehicle. The big motor 2 (MG2) is connected to the S2 shaft. The small motor 1 (MG1) is connected to the S1 shaft and it can be locked by the brake B2. The engine is connected to the carrier shaft by the TDS and can be locked by the brake B1. 2.1. Dynamic modeling To ensure that the plant model is accurate enough to reflect the SDC and to verify the effectiveness of the strategy, a detailed plant model including engine, battery, motor, compound power-split transmission and brake, is established in the MATLAB/Simulink environment.

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223

Fig. 1. The diagram of the compound PS-HEV.

2.1.1. Engine modeling and verification The engine model is established based on theoretical mathematic derivations, as well as some parameter and coefficient maps obtained by some experimental data tables from the manufacturers. (1) Dynamic model of intake manifold The intake manifold pressure (MAP) model [16,22] is expressed as:

_ ¼ MAP

RT atm _ im _ oÞ ðm Vm

ð1Þ

_ i denotes the intake air flow rate, m _ o denotes the exhaust air flow rate, R denotes the ideal gas constant, T atm repwhere m resents the atmosphere temperature, V m is the manifold volume.

Pcyl ,i Pamb

l θ Eng

rc

θ Eng

Fig. 2. Schematic of an operating engine cylinder.

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(2) In-cylinder thermodynamic model For a thermodynamic system shown in Fig. 2, the pressure in the cylinder [23,24] is obtained:

dPcyl;i dP R dm dT ¼ ¼ dhEng dhEng dhdV dhEng dhEng

ð2Þ

Eng

where the cylinder working volume is defined as:

VðhEng Þ ¼

Vs 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 2 1 1 k2s sin hEng þ 1 cos hEng þ ks ec 1

ð3Þ

where hEng is the engine crank angle, V s is the geometric working volume, P (Pcyl;i ), V and T are the gas pressure in the i-th cylinder, volume and temperature, respectively, m denotes the gas quality, R is the gas constant, ks is the crank connection ratio, and ec is the geometric compression ratio. (3) Engine ripple torque model In general, ERT includes the engine cylinder pressure ripple torque TP, piston and connecting rod reciprocating inertia torque TI, and piston and air valve friction torque TF [14]. The ERT in the i-th cylinder is as follows:

T cyl;i ¼ T P þ T I þ T F

ð4Þ

The engine cylinder pressure ripple torque [10,11,14,22] is given by:

T P ¼ AP ðPcyl;i Pamb Þrc f ðhEng Þ

ð5Þ

rc sin hEng cos hEng l ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f ðhEng Þ ¼ sin hEng þ q 2 1 ðrlc sin hEng Þ

ð6Þ

where

where AP ¼ pr2cyl , rcyl is the cylinder radius, P amb is the environment pressure, r c is the crankshaft radius, l is the connection rod length. The piston and connecting rod reciprocating inertia torque [10,11,14] is written as:

T I ¼ mp rc sin hEng

where lv ¼

! r 2c sin 2hEng € r 2c cos 2hEng r 4c cos2 2hEng _ 2 hEng hEng þ rc cos hEng þ þ þ 3 2lv lv 4lv

ð7Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 l ðr c sin hEng Þ , mp is the equivalent piston mass.

The piston and air valve friction torque [10,11] is as follows:

T F ¼ k0 þ k1 gh_ Eng þ k2 Pcyl gjf ðhEng Þj

ð8Þ

where g is the kinematic viscosity, kj is the adjustment factor of friction torque, j = 0, 1, 2, and these parameters are selected by the look-up tables (e.g. temperature-g and temperature -kj ), which are from experimental data tested by a manufacturer. The total output torque of the four-cylinder engine is equal to the superposition of the torque generated by every cylinder with the phase shifted by 180° for a four-stroke engine.

T Eng ¼

X T cyl;i

ð9Þ

i

(4) Verification of engine dynamic model Because the engine dynamic model is directly related to the SDC of the MT process [14,25], the verification of the engine model is important. Note: because the engine dynamic characteristics are quite complicated, this article assumes that the engine performance is good in the modeling process. In this study, parameters such as the crank angle, cylinder pressure, engine speed and torque, etc., are collected by the engine dynamic test bench (shown in Fig. 3) and the Kibox combustion analyzer (shown in Fig. 4). Then, the output torque of the dynamic engine model is compared with the experimental result. At the different speeds, the simulation results are close to the corresponding experimental values both before ignition (shown in Fig. 5) and after ignition (shown in Fig. 6).

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225

Fig. 3. Engine dynamic test bench.

Considering the page limits and the similar follow-up performance at different speeds, only the ERT results under some certain speeds are demonstrated in this paper. This implies that the established engine model can generally reflect the ERT characteristics. 2.1.2. Battery-motor dynamic model and verification The battery, DC/DC converter and motor models are built using the basic modules of Battery, SPSdrives Converter and PM Synchronous Motor Drive in the MATLAB/Simulink/Simscape Power Systems [26]. In order to investigate the torque-following characteristics of the battery-motor model, a permanent magnet synchronous motor with a maximum power of 62 kW is tested, as shown in Fig. 7. The motor torque curves obtained from simulation and bench test at 2000 rpm are plotted in Fig. 8. The section between 0.68 s and 0.7 s and that between 4.64 s and 4.67 s are zoomed in, in order to better demonstrate the increasing and decreasing characteristics of the motor torque. The results show that the simulation and experimental data are in good agreement generally, which verifies the validity of the battery-motor dynamic model. 2.1.3. Torsional dynamic model of dynamic coupling mechanism (1) Dynamic model description of compound planetary gears In order to make the torsional dynamic characteristics of the power-split transmission more close to the actual situation, as shown in Fig. 9, a planetary pure-torsional dynamic model at variable speed is established using the lumped-parameter method [27,28]. There are three coordinate systems, which are the stationary coordinate system OXY, the kinetic coordinate system oxy, and the moving coordinate system onnngn (n = 1 . . . N). Besides, h, k, c, and e represent the corresponding angular displacement, time-varying mesh stiffness, mesh damping, and meshing errors, respectively, and a and b denote the rear and front planetary gear, respectively. (2) Equations of motion for the dynamic model

Fig. 4. Kibox combustion analyzer.

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Fig. 5. ERT for motoring stage before ignition.

The Newton’s law for non-inertial system [29] is applied to derive the motion equations. Clockwise direction is defined as the positive direction, while the counter-clockwise direction is defined as the negative direction. These motion equations are as follows:

Is1 €hs1 ¼ T s1 þ

N X F ybn s1 rs1

ð10Þ

n¼1

Is2 €hs2 ¼ T s2 þ

N X F yan s2 rs2

ð11Þ

n¼1

Ice €hc ¼ T c þ

N N N X X X F ys2 an cos as2 rc þ F ys1 bn cos as1 rc F yrbn cos ar rc n¼1

Ir €hr ¼

n¼1

ð12Þ

n¼1

N X F ybn r rr T L

ð13Þ

n¼1

Ia €han c ¼ F ybn an r a F ys2 an r a

ð14Þ

Ib €hbn c ¼ F yan bn r b F yrbn rb F ys1 bn r b

ð15Þ

with the dynamic engagement force given by [27,28]:

Fy ¼

ceilð ec Þ X

kdi cos bb þ cd_ i cos bb

ð16Þ

i¼1

where ec , d, and bb are the helical coincidence, elastic deformation and base circle helix angle, respectively, I, r, a, N, € h, and T denote the subscript corresponding inertia, the base circle radius, engagement angle, planet number, angular acceleration, and external torque, respectively. Besides, the equivalent carrier inertia is Ice ¼ Ic þ NIa þ NIb .

Fig. 6. ERT for firing stage after ignition.

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227

Fig. 7. Motor dynamic characteristics test.

2.1.4. Brake model The braking operation consists of three states: disengagement, sliding and engagement. In order to reflect its dynamic characteristics, a brake dynamic model is established [18]:

TB ¼

8 0; > > " > > < > > > > :

disengagement

lsl þ ðlst lsl Þe

½lst r B PB AB NB

jh_ c g jGB lst lsl

# r B PB AB NB sgnðh_ c g Þ; h_ c

g

– 0;

sliding

ð17Þ

lst rB PB AB NB ; h_ c g ¼ 0; engagement

where T B is the transmitted torque, h_ c g denotes the carrier rotating speed, lsl and lst represent the sliding and static friction coefficient of the friction plate, respectively, GB is the friction gradient, r B denotes the effective friction radius, P B represents the brake oil pressure, AB is the equivalent friction area, and N B is the number of friction surfaces. 2.1.5. Vehicle longitudinal dynamic model The vehicle longitudinal dynamic model is composed of the Tire module and the Longitudinal Vehicle Dynamics module in MATLAB/Simulink/Simscape Power Systems. The involved equations are as follows:

mV_ x ¼ F x F d mgf cos b mg sin b

ð18Þ

F d ¼ 0:5C d qAV 2x sgnðV x Þ

ð19Þ

where m is the vehicle mass, V x denotes the longitudinal speed, F x represents the tire driving force, F d stands for the air resistance, g is the gravitational acceleration, f is the tire rolling resistance coefficient, b denotes the ramp angle, C d represents the air resistance coefficient, q is the air density, and A is the vehicle frontal area.

Fig. 8. Motor torque following characteristics.

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Fig. 9. Planetary pure-torsional dynamic model at variable speed.

Table 1 The main parameters of the powertrain. Parameters

Values

Parameters

Values

Maximum engine torque (Nm/rpm) Battery voltage (V) MG1/MG2 maximum torque (Nm) Transmission ratio of front row i1 Transmission ratio of rear row i2 Final reduction gear ratio i0 Carrier inertia Ic (kg m2) Ring gear inertia IR (kg m2) Inertia of MG1 and S1 ðIMG1 þ Is1 Þ (kg m2)

170/4200 288 96/246 3.174 2.355 3.529 0.001 0.002 0.041

Inertia of MG2 and S2 ðIMG2 þ Is2 Þ (kg m2) Engine inertia IEng (kg m2) Equivalent TDS torsional stiffness (N m rad1) Equivalent TDS torsional damping (N m s rad1) Tire inertia (kg m2) Vehicle mass (kg) Wheel radius (m) Equivalent torsional stiffness of tire and half shaft (N m rad1) Equivalent torsional damping of tire and half shaft (N m s rad1)

0.072 0.18 618 10 1.83 1530 0.31 2864 15

Many parameters and coefficients used for modeling are provided by the manufacturers, and the some main parameters of the powertrain are shown in Table 1 and literature [28]. 2.2. Mode transition process and problem description 2.2.1. Mode transition process analysis The HEV starts with an EV stage with B1 locked. When the driver’s demand torque is greater than the maximum torque threshold, or the battery state-of-charge (SOC) is less than the minimum threshold, or the vehicle speed is greater than the maximum vehicle speed threshold, the MT process from EV-to-HEV mode is initiated. As illustrated in Fig. 10, the MT process includes four stages: (a) the EV stage with B1 locked, (b) the EV stage with B1 opened, (c) the motoring stage before engine ignition, and (d) the HEV stage after engine ignition. The speed and torque relationships derived for all four modes can be obtained in the literature [13]. (a) In the first stage, MG2 provides the driving torque T MG2 , the brake B1 is locked and provides the balance torque T B1 , and MG1 is rotating freely. (b) In the second stage, the brake B1 is opened soon. The carrier is mainly subject to the brake torque T B1 and the engine static friction torque. To prevent engine reversal, the torque at the planet carrier T C g is equal toT B1 . (c) In the third stage, after B1 is fully opened, MG1 superimposes an additional torque T0 MG1 on the current torque and drags the engine to the ignition preset speed (800 rpm). At the same time, MG2 superimposes a compensation torque T’MG2 on the current torque to eliminate the influence of the MG1 drag torque on the output of the ring gear. (d) In the final stage, when the engine speed reaches the preset speed, the engine starts firing, and the engine torque T Eng and torques of MG1 and MG2 will overcome the load torque T R together. The engine speed and torque are obtained from the engine optimal speed and torque maps. 2.2.2. Problem description for mode transition process As seen from Fig. 11a, in the absence of DCCS, stage 3 is entered at t = 3.106 s. The engine reaches the preset speed and begins to enter stage 4 at t = 3.446 s. It is shown that there exists a noticeable discrepancy in the engine dynamics (see

Y. Su et al. / Mechanical Systems and Signal Processing 107 (2018) 221–240

Fig. 10. Lever diagram of the MT process from EV mode to HEV mode.

Fig. 11. Problem description for the MT process.

229

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Fig. 11c) and the motors dynamics (see Fig. 8). There exits significant oscillation in the engine output torque, and the engine torque response is delayed. At the same time, Fig. 11b shows that the ERT also causes a significant low frequency fluctuation in the output speed of stage 3. This means that the dynamic characteristics during the MT process obtained from the engine steady-state model can not fully reflect the actual situation. Fig. 11d shows that different TI shaft dampings (i.e. 15 Nm/(rad/ s) and 20 Nm/(rad/s)) can produce different vehicle jerks. Note that the TI shaft is the equivalent elastic shaft of the tire and half shaft in Fig. 13. Moreover, as shown in Fig. 11d and f, the maximum jerk caused by the ERT reaches about 80 m/s3. Then, the absolute value of jerk gradually reduces to less than 11 m/s3 as the torque stabilizes. These indicate that the ERT is the main cause of vehicle jerk and vibration during the motoring stage and the initial combustion process in stage 4. In Fig. 11e and f, different ring gear ripple torques are observed under different TDS damping (i.e. 5 Nm/(rad/s) and 10 Nm/(rad/s)), which causes a change in the jerk. Thus, these indicate that the SPU can lead to jerk and vibration. Also, the factors that cause the torque ripple at the ring gear output, such as MEI of the engine torque, can result in jerk and vibration of the transmission system. The jerk not only reduces the driving comfort, but also affects vehicle safety (especially in crowded traffic conditions). Therefore, it is necessary to carry out a DCCS in the MT process. 3. Dynamic coordinated control strategy for the MT 3.1. Control objectives Since the MT time is quite short (e.g. less than 0.5 s in Fig. 11), the fuel consumption and emissions are negligible [30]. However, we know from the problem description in Section 2.2.2 that significant jerk and vibration are present in MT process. Thus, minimizing the jerk and vibration has become the main goal of the DCCS. As the jerk is synchronous with the human feeling and not affected by the road bumps and driver operations, it can better reflect the nature of vehicle dynamics [31]. Besides, the acceleration root mean square (ARMS), which reflects longitudinal speed change rate, is employed to directly affect the running smooth [32]. Therefore, this study uses the jerk and ARMS as the evaluation indexes for the driving comfort in the MT process. Their expressions are as follows:

Jerk : j ¼ daðiÞ=dt sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z tf 1 ARMS : ARMS ¼ a2 ðiÞdt t f t0 t0

ð20Þ ð21Þ

where aðiÞ is the vehicle instantaneous acceleration, t0 denotes the starting time of each step, and t f represents the end time of each step. 3.2. Block diagram of the dynamic coordinated control To minimize the jerk and vibration during the MT process, a DCCS including ‘‘a staged ETFBC and a DSTE-based ADBC” is developed. As shown in Fig. 12, the DCCS mainly includes three parts: engine torque estimation, motor torque distribution and active damping feedback compensation control. (1) Engine torque estimation includes the ERT model feedforward estimation before engine ignition and the ERT estimation with motor feedback values after ignition, considering the characteristics of the ETE model before and after the engine ignition. (2) Motor torque distribution is composed of the motor target torques obtained by planetary dynamics at each stage based on the ETE. The purpose of motor torque distribution is to suppress the ripple torque transmitted to the ring gear (point A), caused by the ERT, IDC of engine and motors. (3) Active damping feedback compensation control consists of the drive shaft torque estimation, the carrier torque estimation and the active damping feedback compensation torque. This control suppresses the fluctuation torques transmitted to the wheel (point B) via the drive shaft, caused by the engine MEI and SPU of stiffness and damping of the TDS and drive shaft. The proposed control presents good robustness and real-time performance, and can be applied to all work modes. The strategies described above will be explained in detail in the following text. 3.3. Engine torque estimation and motor compensation control 3.3.1. Feedforward compensation control before engine ignition For an ETE method based on MTBV, the ERT has been generated and applied to the transmission system, and the actual feedback control must have a time delay to cause torque estimation errors. In order to effectively eliminate the ERT in stage 3, a feedforward control method can be applied. (1) ERT estimation during the motoring stage before engine ignition To estimate the cylinder pressure trajectory during the motoring stage, the following assumptions are made based on the established thermodynamic model: (a) When the engine is being started by the motor, the intake and exhaust valves are

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Fig. 12. Block diagram of the DCCS for the MT process.

always closed during the compression and expansion strokes of each working cycle (720°); (b) The loss of heat exchange is neglected as this process is short (approximately 4 working cycles [10]); (c) Fuel injection has not started and the amount of fuel consumed is 0. A single-zone engine thermodynamic model [10], expressed by Eq. (22), is used to calculate the pressure in the cylinder:

dPcyl;i Pcyl;i dV c 1 dQ B ¼ c þ dhEng VðhEng Þ dhEng VðhEng Þ dhEng

ð22Þ

where c is the in-cylinder gas specific heat ratio, and Q B is the gross heat release. The engine cylinder pressure in this stage can be obtained by simplifying Eq. (22):

dPcyl;i cranking Pcyl;i cranking dV ¼ c dhEng VðhEng Þ dhEng

ð23Þ

where P cyl;i cranking is the gas pressure in the i-th cylinder during the motoring stage. Substituting the estimated cylinder pressure in Eq. (9) yields the estimated ERT T Eng

me

in motoring stage.

(2) Feedforward motor compensation torque According to the lever diagram in Fig. 10c, the feed-forward compensation torques T MG1 the two motors can be derived:

T MG1

cranking

¼

T MG2

cranking

¼

ði2 1ÞT WH

Lim =i0

þ i2 f c ðPcyl;i i1 i2

cranking Þ

ði1 1ÞT WH

Lim =i0

cranking Þ

þ i1 f c ðPcyl;i i2 i1

cranking

and T MG2

cranking

allocated to

þ ðIs1 þ IMG1 Þði1 i0 €hWH

req

þ ð1 i1 Þ€hEng

req Þ

ð24Þ

þ ðIs2 þ IMG2 Þði2 i0 €hWH

req

þ ð1 i2 Þ€hEng

req Þ

ð25Þ

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with

f c ðPcyl;i

cranking Þ

¼ T Eng

me

Ic ði2 €hMG1 i1 €hMG2 Þ IEng €hEng i2 i1

ð26Þ

is the demand torque at the wheels based on the motor external characteristic [14], € hWH req is the wheel demand angular acceleration, € hEng req is the engine target angular acceleration, € hMG1 , € hMG2 and € hEng are the angular accelerations of MG1, MG2 and engine, respectively, i1 and i2 are the transmission ratios of front and rear row, respectively, i0 is the final ratio, Is1 , Is2 , IMG1 , IMG2 , IEng and Ic are the inertias of S1 gear, S2 gears, MG1, MG2, engine and planet carrier, respectively. where T WH

Lim

3.3.2. Feedback compensation control of engine torque after engine ignition Since the IDC of engine and motors can cause great shock [7,8,13], it is necessary to estimate the actual engine torque, and offset the torque ripple by MG1 and MG2. However, the ignition and injection process makes the feedforward ETE model quite complex. And the ETE method based on MTBV has been able to greatly reduce the jerk and vibration [14]. Therefore, a feedback control method can be applied in stage 4. Based on the lever diagram in Fig. 10d, the estimated ERT T Eng fe in HEV firing stage and the feedback torques (T MG1 fe and €Eng req is obtained T MG2 fe ) compensated by the two motors are derived. In this stage, the engine target angular acceleration h by a PID controller whose input is the difference between the actual engine speed and the optimal target speed, and the P, I, D parameters are 4, 1 and 0.5 respectively. T MG1 and T MG2 are the actual feedback torques from MG1 and MG2, respectively.

T Eng

fe

¼

Ic ði2 €hMG1 i1 €hMG2 Þ þ IEng €hEng þ ½T MG1 ðIs1 þ IMG1 Þ€hMG1 ði1 1Þ þ ½T MG2 ðIs2 þ IMG2 Þ€hMG2 ði2 1Þ i2 i1

T MG1

fe

¼

T MG2

fe

¼

ði2 1ÞT WH

Lim =i0

þ i2 ðT Eng fe ðIc þ IEng Þ€hEng i1 i2

req Þ

ði1 1ÞT WH

Lim =i0

þ i1 ðT Eng fe ðIc þ IEng Þ€hEng i2 i1

req Þ

ð27Þ

þ ðIs1 þ IMG1 Þði1 i0 €hWH

req

þ ð1 i1 Þ€hEng

req Þ

ð28Þ

þ ðIs2 þ IMG2 Þði2 i0 €hWH

req

þ ð1 i2 Þ€hEng

req Þ

ð29Þ

3.4. Active damping feedback compensation control Even with the ERT feedforward and feedback compensation controls on-board, the ETE method can not completely compensate for the torque fluctuation in the presence of the ETE errors and SPU. In order to effectively cancel out the ripple torque before it is transmitted to the wheels, it is necessary to estimate and cancel out the torque variations at the drive shaft in real time, with sufficient robustness against external disturbances. To this end, a nonlinear reduced-order robust observer for DSTE is designed and employed for the ADBC during the MT process. 3.4.1. Drive shaft torque estimation Although the precise knowledge of the DST will facilitate the design of the ADBC, the DST is rarely measured in production vehicles due to cost, reliability and installation considerations [33,34]. Therefore, it is a good choice to design an on-line torque observer that does not require additional sensors. Considering only the vehicle longitudinal dynamics and powertrain pure-torsional characteristics, and based on the simplified spring-mass model from the ring shaft to wheels using the lumped-parameter method (shown in Fig. 13), a nonlinear reduced-order observer for all operating modes is designed to estimate the DST, using the method of correcting angular

Fig. 13. Simplified spring-mass model of the powertrain.

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233

acceleration errors. The observer gain is solved by using the Linear Matrix Inequalities (LMIs) to make the observer good robustness. (1) Problem description of drive shaft torque estimation Based on the above simplified spring-mass model, the following relationships are derived:

T R g T TI ¼ IR €hR

ð30Þ

T TI T L ¼ IL €hL

ð31Þ

T_ TI ¼ kTI ðh_ R h_ L Þ þ C TI ð€hR €hL Þ

ð32Þ

where T L ¼ T WH Lim =i0 , T R g is the torque at gear ring, the subscript L indicates the TI shaft position, and T TI denotes the actual torque of drive shaft, h_ R and h_ L are the speed at gear ring and TI shaft, respectively, € hR and € hL are the angular acceleration at gear ring and TI shaft, respectively, IR and IL are the inertias of gear ring and TI shaft, respectively, kTI and C TI are the equivalent stiffness and damping at TI shaft, respectively. The following equations are obtained by selecting x1 ¼ h_ R , x2 ¼ h_ L , x3 ¼ T TI =T TI as the state variables and T TI as the nominal torque of ring gear output shaft:

T TI x3 þ f 1 IR

ð33Þ

T TI x3 þ f 2 ðx2 Þ IL

ð34Þ

x_ 1 ¼ x_ 2 ¼

kTI C TI x_ 3 ¼ ðx1 x2 Þ þ ðx_ 1 x_ 2 Þ T TI T TI €

where f 1 ¼ I1R T R

ð35Þ

€

g est ðT MG1 ; T MG2 ; hMG1 ; hMG2 Þ,

f 2 ðx2 Þ ¼ I1L T L ðx2 Þ and T R g est is the estimated ring torque. ^ The estimated drive shaft torque T S can be obtained by means of the following equation:

T^ S ¼ T^ TI i0

ð36Þ

where T^ TI represents the estimated torque at TI shaft. (2) Nonlinear reduced-order observer design The speed at the ring gear can be estimated using the planetary row kinematics:

ð1 i1 Þh_ MG2 ð1 i2 Þh_ MG1 h_ R ¼ i2 i1

ð37Þ

where h_ MG1 and h_ MG2 are the speed of MG1 and MG2, respectively. The estimated ring speed x1 and the speed x2 of the TI shaft are combined as follows:

y ¼ ½ x1

x2 T

ð38Þ

The estimated state is ^z (^z ¼ ^ x3 ), and we have:

(

y_ ¼ Fðx2 Þ þ G^z þ Hxðx2 ; ^zÞ ^z_ ¼ Ay þ By_

ð39Þ

where x represents the model uncertainties, H is the constant matrix obtained by x, and

Fðx2 Þ ¼

f1 f 2 ðx2 Þ

;

2

G¼4

TIRTI T TI IL

3

5;

A¼

kTI kTI ; ; T TI T TI

B¼

C TI C TI ; T TI T TI

Since the torque directly affects the angular acceleration of the rotating components [33], a nonlinear reduced- order observer can be designed using the angular acceleration error as the correction term:

^z_ ¼ Ay þ By_ þ Lðy_ Fðx2 Þ G^zÞ where L 2 R12 is a time-invariant observer gain to be determined.

ð40Þ

234

Y. Su et al. / Mechanical Systems and Signal Processing 107 (2018) 221–240

In order to attenuate the influence of the measurement noise and avoid direct differentiation on y, Eq. (40) is rearranged as follows:

g_ ¼ Ay LGðg þ By þ LyÞ LFðx2 Þ

ð41Þ

where g ¼ ^z By Ly Thus, Eq. (41) constitutes a nonlinear reduced-order observer for estimating the DST. The estimated value ^z is obtained by substituting gainL in Eq. (41). (3) Determination of the observer gain To achieve good robustness, LMIs is used to solve Eq. (42) to get the suitable observer gain L with the given static coefficient k1 ðk1 > 0Þ and error attenuation coefficient k2 ðk2 > 0Þ.

8 k1 k2 P 0 < LG LG k1 k2 LH : P0 T T H L I

ð42Þ

Since the modes of HEV cannot be switched very frequently, the model uncertainty functions x1 and x2 are assumed to be step signals when calculating the static error [33,35]. The upper bound elim ð1Þ of the static estimation error can be calculated using the obtained gain L. Then, by judging the upper bound of the static error, we can know whether the accuracy range of the DST estimation meets the requirements of control accuracy. The upper bound of the static error is given by:

a 1 a2 elim ð1Þ ¼ þ LG LG

ð43Þ

where aj is thej-th element of LHi . 3.4.2. Wheel speed estimation We see from Eq. (41) that the estimation accuracy and latency of the DST are directly dependent on the estimation accuracy and latency of the ring gear speed and the wheel speed. For this HEV studied in this article, as the resolution of wheel speed obtained from the anti-lock brake system (ABS) via the controller area network (CAN) bus is low and some latency is present, the torque accuracy and real-time requirements can not be met [14]. Therefore, the wheel speed requires to be estimated online. Based on the estimated ring gear torque T R g est , the state equation from gear ring output shaft to wheels (see Appendix) is discretized to estimate the wheel speed. The discretized Eq. (45) is as follows:

TR

g est

¼ i1 T MG1 i2 T MG2 þ i1 ðIs1 þ IMG1 Þ€hMG1 þ i2 ðIs2 þ IMG2 Þ€hMG2

ð44Þ

8 1 C K ^ M 0 1 N > < Xðk ^ þ 1Þ ¼ M 0 XðkÞ þ UðkÞ I 0 0 I 0 0 I > : ^ ^ W ob ðkÞ ¼ Ið1; 1ÞXðkÞ=i 0

ð45Þ

^ represents the estimated state variable, and W ^ ob is the estimated wheel speed. where k denotes the sample number, X 3.4.3. Active damping feedback compensation torque Employing the estimated DST and the estimated carrier torque T c g est , the real-time feedback compensation torque of the two motors (T MG1 Damp and T MG2 Damp ) can be obtained by means of the lever diagram in Fig. 10, which will compensate for the torque fluctuation transmitted to the drive shaft.

Tc

g est

T MG1

T MG2

¼ ði1 1ÞT MG1 þ ði2 1ÞT MG2 ði1 1ÞðIs1 þ IMG1 Þ€hMG1 ði2 1ÞðIs2 þ IMG2 Þ€hMG2

Damp

¼

i2 1 ^ ð1 i1 Þ€hMG2 ð1 i2 Þ€hMG1 ðT TI T L Þ þ IR i1 i2 i2 i1

Damp

¼

i1 1 ^ ð1 i1 Þ€hMG2 ð1 i2 Þ€hMG1 ðT TI T L Þ þ IR i2 i1 i2 i1

!

ð46Þ

þ

i2 Tc i1 i2

g est

þ ðIs1 þ IMG1 Þ€hMG1

ð47Þ

þ

i1 Tc i2 i1

g est

þ ðIs2 þ IMG2 Þ€hMG2

ð48Þ

!

4. Simulation and analysis 4.1. Simulation scenarios In order to verify the effectiveness of the proposed DCCS, the MT process from EV mode to HEV mode is selected as the simulation scenario using the established plant model. In the simulation, the vehicle accelerates and reaches a speed of 30

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235

km/h at t = 2.5 s in EV mode, and then assume that when the battery SOC is less than the minimum threshold at t = 3 s, a MT command is received and this MT process is commenced. In the course of simulation, three control methods are employed for comparative purposes. These methods include: (a) Baseline Control (BC) without any coordination; (b) Engine torque feedback compensation control (ETBC) only; (c) Proposed engine torque feedforward and feedback compensation control + Active damping feedback compensation control (ETFBC + ADBC). The main parameters of the powertrain used in the simulation are shown in Table 1. 4.2. Speed and torque estimation results 4.2.1. Engine torque estimation It can be seen in Fig. 14 that the estimated engine torque using ETFBC well follows the engine output torque obtained from the plant model, however the estimated value by means of ETBC fails to do so. The reason for this failure is that the TDS is ignored and the engine output shaft is only considered as a rigid body. 4.2.2. Ring gear and wheel speed estimation It is seen that the estimated ring gear speed in Fig. 15 and the estimated wheel speed in Fig. 16 well follow the ring gear speed and wheel speed obtained from the plant model. The maximum estimation errors of these two speeds are less than 3% and 1.5%, respectively, which provides lays a good foundation for accurate the DSTE. 4.2.3. Torque estimation of ring gear and planetary carrier As shown in Figs. 17 and 18, the estimated ring gear torque using Eq. (44) and the estimated carrier torque using Eq. (46) can well follow the ring torque and carrier torque obtained from the plant model. The maximum estimation errors of these two torques are less than 2% and 3.5% respectively, which ensures the accuracy of the ADBC. 4.2.4. Drive shaft torque estimation Set the nominal torque of the ring gear output shaft to T TI = 500 Nm and select k2 to meet the required error decay rate. The estimation error is required to converge in the 10 ms receive/send time interval of the CAN bus. Since the first-order system to establish the time is 4 times the time constant [35] (namely 4=k2 =0.01 s), one can readily achieve that k2 = 400. Assume that the ring gear torque estimation error is 3% of the actual value, the wheel speed estimation error is 2% of the actual value, the vehicle mass is increased from the curb mass of 1530 kg to the full load mass of 2000 kg, 2560 0 . and the road slope increases from 0 to 5° at the speed of 80 km/h [36]. Then, we obtain H ¼ 0 19:44 The value of k1 is incremented from 1 to 200 using the ‘for loop’ and the LMIs tool is applied to Eq. (42) to achieve the smallest possible gain L to meet the design requirements of the observer. The calculation result is as follows: when k1 = 26 there exits an appropriate gain L ¼ ð 0:0024 0:007 Þ and the upper bound of the static error for all modes is jeð1Þj 6 elim ð1Þ ¼ 00:0105. The maximum error value of the DST is 00:0105 T TI i0 ¼ 18:53N:m and the maximum static error is less than 3.03% of the maximum DST, which has met the requirements on control precision. It is shown in Fig. 19 that the estimated DSTs (m = 1530 kg, b = 0° or m = 2000 kg, b = 5°) are consistent with the torques obtained from the plant model. And the section between 3.2 s and 3.3 s is zoomed into better demonstrate the details in Fig. 19. These indicate that the proposed nonlinear reduced-order observer possesses good robustness. Besides, since the control system is simple and the control gain L is time-invariant, the proposed estimation can satisfy the real-time requirement of vehicle control. 4.3. Dynamic coordinated control results After the estimated speed and torque are verified, the EV-to-HEV MT process with the engine start-up process is selected as the simulation scenario using the plant model, in order to verify the effectiveness of the proposed DCCS.

Fig. 14. Engine torque estimation results.

Y. Su et al. / Mechanical Systems and Signal Processing 107 (2018) 221–240

Speed of ring gear(rad/s)

236

97 Speed from plant model Estimated speed

96.5 96 95.5 95 94.5 2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Time (s) Fig. 15. Estimation results of the ring gear speed.

Fig. 16. Estimation results of the wheel speed.

Fig. 17. Torque estimation results for ring gear.

Fig. 18. Torque estimation results for planetary carrier.

Fig. 20 demonstrates the motor target torque commands for MG1 and MG2 obtained by the three control methods. Based on the ERT feedforward and feedback control, the two motor target torques resulting from the ETFBC-ADBC method are obtained using real-time estimation and compensation feedback for the ripple torque, which has been transmitted to the drive shaft and is caused by the MEI and SPU. Therefore, the two motor target torques are able to compensate for the ERT, and the IDC of engine and motors can be eliminated. Due to the maximum torque limit of MG1, the ERT cannot be completely compensated. This issue can be addressed by changing the MG1 motor characteristic curve or optimizing the engine

237

Y. Su et al. / Mechanical Systems and Signal Processing 107 (2018) 221–240

Fig. 19. Drive shaft torque estimation results: (a) m =1530 kg, b = 0°; (b) m = 2000 kg, b = 5°.

Fig. 20. MG1 and MG2 target torque commands.

angular acceleration during the motoring stage. This paper does not look into this issue in detail. Fig. 21 and Table 3 demonstrate the comparitive simulation results between the three control methods. We see that compared with the BC method, the Ja-max resulting from the ETBC method is reduced from 74.64 m/s3 to 23.86 m/s3, with an improvement of 68.03%. However, the Ja-max produced by the ETFBC-ADBC method is further decreased from 74.64 m/s3 to 10.4 m/s3, providing a even better improvement of 86.55%. Besides, compared with the ETBC method, the proposed method reduces the vehicle jerk by 58%.

Fig. 21. Vehicle jerk and ARMS.

Table 2 Cases for ETFBC-ADBC robustness verification during the MT process. Cases

TDS equivalent damping (Nm/(rad/s))

TI shaft equivalent damping (Nm/(rad/s))

ETE error

Case 1 Case 2 Case 3

5 10 10

15 15 20

3% 3% 3%

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Y. Su et al. / Mechanical Systems and Signal Processing 107 (2018) 221–240

Table 3 The results of the DCCS using the three control methods during the MT process. Control methods

Ja-max (m/s3)

Improvement of Ja-max (%)

ARMS-max (m/s2)

Improvement of ARMS-max (%)

BC ETBC ETFBC-ADBC

74.64 23.86 10.04

— 68.03 86.55

0.92 0.31 0.15

— 66.3 92

Note: Ja-max is the maximum of absolute value of jerk. ARMS-max is the maximum of ARMS.

Fig. 22. ETFBC-ADBC robustness verification for jerk.

Fig. 23. ETFBC-ADBC robustness verification for ARMS.

Moreover, the ARMS resulting from the ETFBC-ADBC method is also significantly smaller than the other two methods’. To verify the robustness of the proposed control strategy against disturbances, the following three sets of simulation studies have been conducted Table 2. The resulting simulation outcomes (plotted in Figs. 22 and 23) show that the vehicle jerk and the ARMS can be controlled within 10.5 m/s3 and 0.15 m/s2, respectively, when subject to ETE inaccuracies, TDS and TI shaft damping parameter uncertainties. Therefore, the proposed method is proven to be effective in reducing the jerk and vibration of the powertrain and improving the driving comfort.

5. Conclusions To improve the driving comfort of a compound PS-HEV during the MT process, this study proposed a DCCS including ‘‘a staged ETFBC and a DSTE-based ADBC”. A powertrain plant model was established for control strategy verification. The EV-to-HEV MT process was analyzed, and the jerk and vibration problem existing in the transmission system (caused by the ERT, IDC of engine and motors, MEI and SPU) during the MT process was investigated using the plant model. Then, the jerk and ARMS were used as the evaluation indicators for the driving comfort. The BC, ETBC and the proposed ETFBCADBC method were employed for comparative analysis. Finally, the results showed that compared with the BC method, the jerk produced by the proposed method is decreased from 74.64 m/s3 to 10.4 m/s3, leading to a significant improvement of 86.55%. Besides, the ARMS resulting from the ETFBC-ADBC method is also significantly smaller than the other two methods’. It was proven that the proposed method could effectively reduce the jerk and vibration of the powertrain and improve the vehicle driving comfort. The proposed DCCS was not only applicable to the MT process from EV mode to HEV mode, but also to unsteady operating processes such as engine shut-down and engine torque mutation. In the next step, experimental verification will be conducted on prototype vehicles, and the effect of other facts (e.g. the power source torque change rate limit, the engine angular acceleration trajectory, the engine start/stop operating point, and the engine ECU injection/ignition timing control) on the jerk and vibration of the powertrain will be investigated, in order to further improve the driving comfort of the PS-HEV.

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Acknowledgments This work is financially supported by the National Natural Science Foundation of the People’s Republic of China [No. 51675062], the National Key Research and Development Project [No. 2016YFD0701100] and the Major Program of Chongqing Municipality [No. cstc2015yykfC60003]. Appendix A Select X and U as the state variable and control variable, respectively.

X ¼ h_ L

h_ R

hL

U ¼ ½ TL

TR

g est

hR

T

ðA:1Þ

T

ðA:2Þ

Based on the simplified spring-mass model in Fig. 13, the state equation (describing the dynamics of the components from ring gear output shaft to wheels) is obtained as follows:

M 0 1 C K M 0 1 N X_ ¼ Xþ U 0 I I 0 0 I 0

ðA:3Þ

Y ¼ IX

ðA:4Þ

with,

M¼

IL

0

0

IR

;

N¼

1 0 0

1

;

C¼

C TI

C TI

C TI

C TI

;

K¼

kTI

kTI

kTI

kTI

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