# A Novel Torque Coordination Control Strategy of a Single-Shaft Parallel Hybrid Electric Vehicle Based on Model Predictive Control

Authors: Jing Sun; Guojing Xing; Xudong Liu; Xiaoling Fu; Chenghui Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The torque coordination control during mode transition is a very important task for hybrid electric vehicle (HEV) with a clutch serving as the key enabling actuator element. Poor coordination will deteriorate the drivability of the driver and lead to excessive wearing to the clutch friction plates. In this paper, a novel torque coordination control strategy for a single-shaft parallel hybrid electric vehicle is presented to coordinate the motor torque, engine torque, and clutch torque so that the seamless mode switching can be achieved. Different to the existing model predictive control (MPC) methods, only one model predictive controller is needed and the clutch torque is taken as an optimized variable rather than a known parameter. Furthermore, the successful idea of model reference control (MRC) is also used for reference to generate the set-point signal required by MPC. The parameter sensitivity is studied for better performance of the proposed model predictive controller. The simulation results validate that the proposed novel torque coordination control strategy has less vehicle jerk, less torque interruption, and smaller clutch frictional losses, compared with the baseline method. In addition, the sensitivity and adaptiveness of the proposed novel torque coordination control strategy are evaluated.


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## Body

## 1. Introduction

HEV represents an effective solution to significantly reduce the consumption of fossil fuels and carbon emissions [1], whose control problem is generally regarded as an energy management optimization problem to minimize the fuel consumption and the exhaust emissions over an arbitrary driving cycle. Many researches have focused on the energy management strategy (EMS) which aims to improve fuel economies and reduce emissions [1–8]. However, to our knowledge, the drivability and smoothness are often disregarded in energy management strategies. In fact, HEV should be regarded not only for its fuel saving potential but also for its additional driving comfort and add on performance. To improve the HEV’s fuel economies, the optimized vehicle operations require frequent mode switching which will cause impact and jitter to the powertrain. For the clutch-enabled HEV, one special challenge in managing mode transition is the switch from pure electric running to parallel hybrid operation, as the clutch friction torque introduces nonlinear dynamics to the powertrain. Sharp torque variation may occur at this time due to the abrupt introduction of the engine torque into the driveline by a controlled clutch torque. Obviously, the driving comfort of the driver will be deteriorated and the clutch lifespan will be shortened unless the transition process is carefully controlled. The core problem of the mode transition process is torque coordination control which is a critical control task for HEV drivability and clutch lifespan.As far as we know, Tong defined the control issue of how to coordinate engine and motor under mode transition as a coordinated control issue for the first time and proposed engine torque open-loop control with dynamical engine torque estimation and motor torque compensation [9]. Davis and Lorenz built engine state observer and cancelled the engine torque ripple using integrated starter alternator [10]. Wang et al. limited the changing rate of the load signal of engine and motor to decrease the torque fluctuation of mode transition process [11]. Li et al. proposed a model matching control method for the dynamic coordinated control of a parallel hybrid electric vehicle [12]. In the aforementioned works, the difference of dynamic response characteristic between engine and electric motor was considered, and the fast response characteristic of motor was used to compensate the engine. Nevertheless, the clutch dynamic characteristic which is a crucial factor to the mode transition performance is neglected in the above works.Taking the clutch dynamic characteristic into account, many research works handled the torque coordination problem from the perspective of state equations under different running modes. To achieve a seamless transition from pure electric running to parallel hybrid operation of HEV, Wang et al. applied a fuzzy adaptive sliding mode approach to the mode transition control for a series-parallel hybrid electric bus [13]. Koprubasi et al. applied switched hybrid theory to the control of a simplified model of a parallel hybrid electric vehicle drivetrain [14]. Minh and Rashid built up a typical model of a parallel hybrid electric vehicle and developed two model predictive controllers for this model to control the speeds and torques for fast clutch engagement [15]. This paper finished the preparation phase for the clutch slipping friction phase but did not involve the control during the clutch slipping friction phase. Beck et al. also applied two model predictive controllers to regulate HEV mode transitions [16]. Nevertheless, in the aforementioned works, the clutch normal pressure in its slipping friction phase, which is proportional to clutch torque and plays a crucial role for the improvement of transition performance, was viewed as a known parameter and its optimizing function was neglected. In fact, the clutch torque in the slipping friction phase can be controlled independently by its actuator [17] and thus can be used as an optimized control variable. To coordinate the motor torque, engine torque, and clutch torque to manage transitions, Chen et al. proposed a MRC law [18] which is a very valuable idea.Obviously, as an important and gordian knot, torque coordination control has already caused the extensive concerns of the academia and business circles.MPC, which can be regarded as a kind of optimal control methods, is one of the most practical advanced control techniques in industrial applications [19]. At each sampling time, it employs a dynamic model of plant to forecast the future behavior of states and determines the future control action according to optimization of a certain performance target function or an operating cost function. MPC possesses the advantages including robustness, simplicity of modeling, and good capability of handling constraints, so it can ensure a satisfying system performance [20]. MPC has been successfully applied in many engineering problems, such as the electrical motor drive system [21], the power plant coordinate control [22], and the vehicle active steering system [23]. Several automotive companies have contributed and supported MPC research, including Ford, BMW, Honda, Honeywell, PSA, and Toyota [24].This paper proposes a novel torque coordination control strategy for the typical switching from pure electric running to parallel hybrid operation for a single-shaft parallel HEV based on MPC. The design objective is to coordinate the engine torque, the motor torque, and the clutch torque to achieve smooth transition with reduced driveline jerk and clutch frictional losses. The innovations of this paper are outlined as follows. Firstly, only one model predictive controller is needed compared with the existing MPC methods about torque coordination control in [15, 16], which is helpful to reduce the computation burden. Secondly, as a pivotal factor, the clutch torque is used as an optimized variable rather than a known parameter in MPC, which will increase the design degree of freedom. Lastly, the successful experience of MRC in [18] is introduced into the proposed novel torque coordination control strategy, and the transient output of the reference model is used as the set-point signal of MPC algorithm. The simulation results show that the proposed novel torque coordination control strategy can achieve smoother transition with less driveline torque interruption and smaller clutch frictional losses compared with the outcomes of the conventional method.The remainder of this paper is organized as follows. Section2 establishes the HEV dynamic models during different running modes. Section 3 introduces the block diagram and detailed implementation of the proposed novel torque coordination control strategy for the mode transition from pure electric running to parallel hybrid operation. In Section 4, the results of the proposed torque coordination control strategy are compared with those of the conventional method, and the sensitivity and adaptiveness of the proposed novel strategy are also evaluated. The paper is concluded in Section 5.


## 2. HEV Modeling

It is well known that the well-built dynamic model is the precondition of the analysis and control to HEV powertrain. To implement the proposed novel torque coordination control strategy, a control-oriented HEV model capturing the mode transition dynamics will be built in this section.Figure1 shows the configuration of the single-shaft parallel hybrid powertrain to be studied in this paper. It consists of a conventional engine, an integrated starter and generator (ISG), and a traction motor. A clutch separates the powertrain into two parts. The first part includes the engine and the ISG, and the second part consists of the traction motor and the rest of the transmission. The driven wheels are connected with a standard automated transferred gearbox and a differential gearbox.Figure 1
Configuration of a single-shaft parallel hybrid powertrain.When HEV runs at low speeds (below 30 km/h), the powertrain runs in pure electric mode. At this time, the clutch is open and the traction motor drives the vehicle using solely electric energy from the battery. The transition from pure electric running to parallel hybrid operation takes place at high speeds (above 30 km/h) by closing the clutch when the traction motor is no longer sufficient to propel the vehicle. In the hybrid driving mode, the clutch is locked and the traction motor drives the vehicle together with the engine.


### 2.1. Vehicle Resistance Model

The vehicle resistance model is composed of three separate parts: the rolling resistance, the air drag, and the combined mechanical losses in gearbox, differential, and shaft bearings due to friction. The third part can be approximated from measured characteristic curves as a second order polynomial. The resulting resistance torqueTwh can be expressed as follows: (1)Twh=mgsinα+frmgcosα+ρ2cDAVωwhR2R+a0+a1ωwh+a2ωwh2,where m is the vehicle mass, g is the gravitational acceleration constant, α is the road inclination angle, fr is the tire rolling resistance coefficient, ρ is the air density, cD is the aerodynamic drag coefficient, AV is the vehicle frontal area, ωwh is the vehicle speed, R is the tire dynamic rolling radius, and a0, a1, a2 are the transmission friction curve fitting factors. The linearization Twh=Twh0+a1ωwh is reasonable without significant loss of accuracy since the developed powertrain model is used for low vehicle speeds and the terms proportional to ωwh2 have less contributions to Twh [16]. The vehicle mass m, the road inclination angle α, and the tire rolling resistance coefficient fr can be considered to be constant during the short mode transiton process, so Twh0=(mgsinα+frmgcosα)R+a0 can be seen as the constant part of vehicle resistance. Since the actual mode transition may occur under different road inclination angel, vehicle mass, or tire rolling resistance coefficient, the adaptiveness of the proposed torque coordination control strategy to these factors will be discussed in Section 4.


### 2.2. Clutch Model

The clutch engagement process can be divided into three phases, namely, the open phase, the slipping friction phase, and the locked phase.When the clutch is open, the two sides of clutch are not contacted. Thus the open phase is not important to the vehicle dynamics and the clutch torqueTc=0 at this circumstance.When the clutch is engaged and the clutch primary speed is not equal to the clutch secondary speed, the clutch is in the slipping friction phase, during which the vehicle longitudinal dynamics are sensitive to the clutch torque profile. In this case, the clutch torqueTc can be expressed as follows [25]: (2)Tc=μs·Fc·Rc·signωp-ωs,where μs is the slipping friction coefficient of clutch, Fc is the clutch normal pressure, Rc is the equivalent acting radius of the friction torque on clutch plate, ωp is the clutch primary speed, ωs is the clutch secondary speed, and sign(·) is a signum function.When the clutch primary speed is equal to the clutch secondary speed, the clutch is locked, and the clutch torque no longer affects the vehicle dynamics. Moreover,Tc can be derived from the dynamics of the powertrain and is subject to Tc∈[-μl·Fc·Rc,μl·Fc·Rc], where μl is the static friction coefficient of the clutch.


### 2.3. HEV Dynamic Models during Different Running Modes

The transition from pure electric running to parallel hybrid operation is classified as the following four modes.Mode 1. When the vehicle speed is lower than 30 km/h, the clutch is open. The vehicle is solely propelled by the traction motor and the engine is unstarted.Neglecting the torsional displacement of the half shafts and assuming that the differential gear ratio is 1, the state-space equation during this phase can be denoted by(3)Jm′ω˙m=Tm-bmωm-1iTwh0+a1ωmi,ωe=0,where Jm′=Jm+Jc+Jgb+(mR2/i2) is the combined inertia of the motor, the clutch, the gearbox, and the transmission at the motor shaft (Jm is the inertia of the motor, Jc is the inertia of the clutch, and Jgb is the inertia of the gearbox), ωm is the traction motor speed, Tm is the traction motor torque, bm is the traction motor friction coefficient, i is the gear ratio, and ωe is the engine speed.Mode 2. When the vehicle speed exceeds 30 km/h, the clutch remains open. The vehicle is still propelled by the traction motor separately and the engine is started with the help of ISG to its idle speed which is generally 800 r/min.The state-space equation during this phase can be denoted by(4)Jm′ω˙m=Tm-bmωm-1iTwh0+a1ωmi,Je′ω˙e=Te-beωe,where Je′ is the total inertia moment of engine and ISG, be is the engine friction coefficient, and Te denotes the total torque of engine and ISG (the internal dynamics of engine and ISG are ignored here for simplicity).Mode 3. When the speed difference between two sides of the clutch is less than a given threshold value, the clutch enters into the slipping friction phase. As a transitional stage from pure electric running to parallel hybrid operation, this is the most important phase to coordinate the motor torque, the clutch torque, and the engine torque for improved drivability and reduced clutch frictional losses.The state-space equation during this phase can be denoted by(5)Jm′ω˙m=Tm+Tc-bmωm-1iTwh0+a1ωmi,Je′ω˙e=Te-beωe-Tc.As is shown in (5), the clutch frictional torque Tc begins to affect the dynamic characteristics of the powertrain. It is notable that Tc in the slipping friction phase was viewed as a known parameter in the existing MPC methods about torque coordination control. In fact, it can be controlled independently by its actuator in the slipping friction phase. Tc will be used as an optimized control variable in the proposed novel torque coordination control strategy based on MPC, which will increase the control flexibility.Mode 4. When the speed difference between two sides of the clutch equals zero (i.e. ωm=ωe), the cultch is locked, and the clutch enters into the static friction phase.The state-space equation in this phase can be denoted by(6)Je′+Jm′ω˙m=Te+Tm-be+bmωm-1iTwh0+a1ωmi.


## 2.1. Vehicle Resistance Model

The vehicle resistance model is composed of three separate parts: the rolling resistance, the air drag, and the combined mechanical losses in gearbox, differential, and shaft bearings due to friction. The third part can be approximated from measured characteristic curves as a second order polynomial. The resulting resistance torqueTwh can be expressed as follows: (1)Twh=mgsinα+frmgcosα+ρ2cDAVωwhR2R+a0+a1ωwh+a2ωwh2,where m is the vehicle mass, g is the gravitational acceleration constant, α is the road inclination angle, fr is the tire rolling resistance coefficient, ρ is the air density, cD is the aerodynamic drag coefficient, AV is the vehicle frontal area, ωwh is the vehicle speed, R is the tire dynamic rolling radius, and a0, a1, a2 are the transmission friction curve fitting factors. The linearization Twh=Twh0+a1ωwh is reasonable without significant loss of accuracy since the developed powertrain model is used for low vehicle speeds and the terms proportional to ωwh2 have less contributions to Twh [16]. The vehicle mass m, the road inclination angle α, and the tire rolling resistance coefficient fr can be considered to be constant during the short mode transiton process, so Twh0=(mgsinα+frmgcosα)R+a0 can be seen as the constant part of vehicle resistance. Since the actual mode transition may occur under different road inclination angel, vehicle mass, or tire rolling resistance coefficient, the adaptiveness of the proposed torque coordination control strategy to these factors will be discussed in Section 4.


## 2.2. Clutch Model

The clutch engagement process can be divided into three phases, namely, the open phase, the slipping friction phase, and the locked phase.When the clutch is open, the two sides of clutch are not contacted. Thus the open phase is not important to the vehicle dynamics and the clutch torqueTc=0 at this circumstance.When the clutch is engaged and the clutch primary speed is not equal to the clutch secondary speed, the clutch is in the slipping friction phase, during which the vehicle longitudinal dynamics are sensitive to the clutch torque profile. In this case, the clutch torqueTc can be expressed as follows [25]: (2)Tc=μs·Fc·Rc·signωp-ωs,where μs is the slipping friction coefficient of clutch, Fc is the clutch normal pressure, Rc is the equivalent acting radius of the friction torque on clutch plate, ωp is the clutch primary speed, ωs is the clutch secondary speed, and sign(·) is a signum function.When the clutch primary speed is equal to the clutch secondary speed, the clutch is locked, and the clutch torque no longer affects the vehicle dynamics. Moreover,Tc can be derived from the dynamics of the powertrain and is subject to Tc∈[-μl·Fc·Rc,μl·Fc·Rc], where μl is the static friction coefficient of the clutch.


## 2.3. HEV Dynamic Models during Different Running Modes

The transition from pure electric running to parallel hybrid operation is classified as the following four modes.Mode 1. When the vehicle speed is lower than 30 km/h, the clutch is open. The vehicle is solely propelled by the traction motor and the engine is unstarted.Neglecting the torsional displacement of the half shafts and assuming that the differential gear ratio is 1, the state-space equation during this phase can be denoted by(3)Jm′ω˙m=Tm-bmωm-1iTwh0+a1ωmi,ωe=0,where Jm′=Jm+Jc+Jgb+(mR2/i2) is the combined inertia of the motor, the clutch, the gearbox, and the transmission at the motor shaft (Jm is the inertia of the motor, Jc is the inertia of the clutch, and Jgb is the inertia of the gearbox), ωm is the traction motor speed, Tm is the traction motor torque, bm is the traction motor friction coefficient, i is the gear ratio, and ωe is the engine speed.Mode 2. When the vehicle speed exceeds 30 km/h, the clutch remains open. The vehicle is still propelled by the traction motor separately and the engine is started with the help of ISG to its idle speed which is generally 800 r/min.The state-space equation during this phase can be denoted by(4)Jm′ω˙m=Tm-bmωm-1iTwh0+a1ωmi,Je′ω˙e=Te-beωe,where Je′ is the total inertia moment of engine and ISG, be is the engine friction coefficient, and Te denotes the total torque of engine and ISG (the internal dynamics of engine and ISG are ignored here for simplicity).Mode 3. When the speed difference between two sides of the clutch is less than a given threshold value, the clutch enters into the slipping friction phase. As a transitional stage from pure electric running to parallel hybrid operation, this is the most important phase to coordinate the motor torque, the clutch torque, and the engine torque for improved drivability and reduced clutch frictional losses.The state-space equation during this phase can be denoted by(5)Jm′ω˙m=Tm+Tc-bmωm-1iTwh0+a1ωmi,Je′ω˙e=Te-beωe-Tc.As is shown in (5), the clutch frictional torque Tc begins to affect the dynamic characteristics of the powertrain. It is notable that Tc in the slipping friction phase was viewed as a known parameter in the existing MPC methods about torque coordination control. In fact, it can be controlled independently by its actuator in the slipping friction phase. Tc will be used as an optimized control variable in the proposed novel torque coordination control strategy based on MPC, which will increase the control flexibility.Mode 4. When the speed difference between two sides of the clutch equals zero (i.e. ωm=ωe), the cultch is locked, and the clutch enters into the static friction phase.The state-space equation in this phase can be denoted by(6)Je′+Jm′ω˙m=Te+Tm-be+bmωm-1iTwh0+a1ωmi.


## 3. The Model Predictive Controller Design

A block diagram of the proposed novel torque coordination control strategy is shown in Figure2. Different to the existing MPC methods about torque coordination control in [15, 16], only one model predictive controller is needed and the clutch torque Tc is used as an optimized variable rather than a known parameter to improve the performance of MPC. The idea of MRC in [18] is used for reference in the proposed novel strategy. The reference model is built to capture the desired powertrain dynamics of the pure electric mode. The input of reference model is the equivalent driver’s demand torque determined by the acceleration pedal. The transient output of the reference model is used as the target output of the plant (i.e., the set-point signal of MPC algorithm). The control goal is that both the clutch primary speed (i.e., ωe) and the clutch secondary speed (i.e., ωm) can track the output of the reference model. This will not only assure that the clutch primary speed is always equal to the clutch secondary speed to reduce clutch wear during the slipping friction phase, but also assure that the vehicle runs as if it were still in pure electric mode to reduce the longitudinal jerk and assure driving comfort.Figure 2
The structure of the proposed novel torque coordination control strategy.Letε represent the speed difference of the two sides of the clutch; that is, ε=ωm-ωe; the proposed control strategy can be described as follows.Whenε is larger than a predefined threshold, the vehicle runs in mode 2. The traction motor propels the vehicle alone.Whenε is smaller than this threshold, the vehicle runs in mode 3. The engine torque, the clutch torque, and the traction motor torque are all determined by the model predictive controller.Whenε is equal to zero, the vehicle runs in mode 4. The engine torque and the traction motor torque are determined by EMS and the clutch torque is no longer controllable.The discrete-time model predictive control algorithm is used here, since it is convenient to be implemented in the current digitalized vehicle control units. LetTs denote the sampling time interval; then the discrete form of state-space equation (5) in mode 3 is (7)ωmk+1=1-bmTsJm′-a1TsJm′i2ωmk+TsJm′Tmk+TsJm′Tck-Twh0TsJm′i,ωek+1=1-beTsJe′ωek-TsJe′Tck+TsJe′Tek.System (7) can be rewritten in the following matrix form:(8)
ω
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.It can be seen that this plant has 2 states and 3 inputs. The introduced input variableTc(k) is to be optimized that is significantly different to the existing MPC methods about torque coordination control.Let(9)xdk=ωmkωek;then (8) can be rewritten as the following form: (10)xdk+1=Adxdk+Bduk+Bξξ,where(11)
A
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.Applying the differential operator to both sides of (10) yields [26, 27] (12)xdk+1-xdk=Adxdk-xdk-1+Bduk-uk-1.The state increment is defined as(13)Δxdk=xdk-xdk-1,and the control increment is defined as (14)Δuk=uk-uk-1.Thus (12) can be denoted by (15)Δxdk+1=AdΔxdk+BdΔuk.Let the following new state vector be(16)xk=Δxd(k)Txd(k)TT,where the superscript T indicates matrix transpose. The following augmented model can be obtained:(17)Δxdk+1xdk+1=Ad0dTAdIΔxdkxdk+BdBdΔuk,yk=0dIΔxdkxdk.The output dimension of (17) is 2. The state dimension of (17) is 4. I is the identity matrix with dimensions 2×2 and 0d is a 2×2 zero matrix.Let(18)A=Ad0dTAdI,B=BdBd,C=0dI;then (17) can be rewritten in the following form: (19)xk+1=Axk+BΔuk,yk=Cxk,where(20)
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.N
p and Nc are used to represent the prediction horizon and the control horizon, respectively. When Nc is less than Np, at the sample time k, u(k+i)(i=0,1,…,Np-1) keeps invariant after i=Nc-1 [28]. The well-known disadvantage of MPC method is that the calculation of control action requires a cumbersome computation load since it requires the solution of an optimization problem at each sampling time. Np and Nc determine the computation load to a great extent. To reduce the computation burden, Np is chosen to be 3, and Nc is chosen to be 1; thus u(k+2)=u(k+1)=u(k). The future state variables can be sequentially calculated based on the state-space model (A,B,C): (21)xk+1∣k=Axk+BΔuk,xk+2∣k=A2xk+ABΔuk+BΔuk+1=A2xk+ABΔuk,xk+3∣k=A3xk+A2BΔuk+ABΔuk+1+BΔuk+2=A3xk+A2BΔuk.On the basis of the predicted state variables, the predicted output variables can be indicated by(22)yk+1k=CAxk+CBΔuk,yk+2k=CA2xk+CABΔuk,yk+3k=CA3xk+CA2BΔuk.Defining vectors(23)Y=yk+1∣kTyk+2∣kTyk+3∣kTT=ωmk+1∣k,ωek+1∣k,ωmk+2∣k,mmωek+2∣k,ωmk+3∣k,ωek+3∣kT,ΔU=Δuk=ΔTmkΔTckΔTekT,then (22) can be expressed as the following compact matrix form: (24)Y=Fxk+ϕΔU,where (25)F=CACA2CA3,ϕ=CBCABCA2B.At the sample timek, the objective of MPC is to bring the predicted outputs ωm(k+i∣k)(i=1,2,3) and ωe(k+i∣k)(i=1,2,3) as close as possible to the set-point signal r(k). To generate an ideal r(k) required by MPC, the idea of MRC in [18] is introduced. To achieve a smooth mode transition from pure electric running to parallel hybrid operation, the vehicle is expected to run as if it were still in the pure electric running. This desired performance can be expressed by a reference model that will be mimicked by the plant through MPC. Because the clutch is indeed locked when the engagement is completed, the clutch is assumed to be locked in the reference model; thus, the desired clutch primary speed is equal to the clutch secondary speed. The dynamic equation of the reference model is (26)Je′+Jm′ω˙ref=Tdem-be+bmωref-1iTwh0+a1ωrefi,where Tdem is the equivalent driver’s demand torque with respect to gearbox input side and ωref is the output of the reference model.The discrete form of this reference model is(27)ωrefk+1=-Tsbe+bmJe′+Jm′+Tsa1Je′+Jm′i2-1ωrefk+TsTdemJe′+Jm′-TsTwh0Je′+Jm′i.Thus, the reference speedωref(k) can be obtained from (27). ωref(k) is just the value of ωref at the sample time k in Figure 2, and r(k)=ωrefkωrefkT will be used as the reference (i.e., the set-point signal) of the motor speed ωm(k) and the engine speed ωe(k) in the model predictive controller.The objective is then changed to find the best control vectorΔU to minimize the error function between the set-point and the predicted output. The cost function J that reflects the control objective is defined as(28)J=Rs-YTRs-Y+ΔUTR¯ΔU=∑i=13‍ωrefk-ωmk+i∣k2+ωrefk-ωek+i∣k2+rw1ΔTmk2+rw2ΔTck2+rw3ΔTek2.In the above equation,Rs is a data vector that contains the set-point information, and the weight matrix R¯ is a diagonal matrix whose elements are used as the tuning parameters for the desired closed-loop performance: (29)Rs=ωrefkωrefkωrefkωrefkωrefkωrefkT,R¯=rw1000rw2000rw3,where positive constants rw1, rw2, and rw3 are used to restrict the drastic changes of Tm(k), Tc(k), and Te(k), respectively.The necessary condition of the minimumJ is obtained as ∂J/∂ΔU=0, from which the optimal control increment within one optimization window is shown as (30)ΔU=ϕTϕ+R¯-1ϕTR¯srk-ϕTFxk,where ϕTR¯s is equal to the last two columns of ϕTF.From (30), it can be seen that the performance of the designed model predictive controller is affected by the choice of tuning parameters rw1, rw2, and rw3. For example, the clutch torque change rate and the clutch frictional losses caused by (30) may be affected by rw2 to a large extent, which will be analyzed in the sensitivity analysis in Section 4.


## 4. Simulation Results

This section consists of three parts:(1) the simulation results to compare the proposed novel torque coordination control strategy with the conventional method; (2) sensitivity analysis of the selection of parameters on the mode transition performance in the proposed strategy; and (3) adaptiveness analysis of the proposed strategy.Vehicle jerk, driveline torque interruption, and clutch frictional losses are crucial indices for mode transition performance. Vehicle jerk indicates smoothness, which is measured by the derivative of the vehicle acceleration. Vehicle acceleration quantitatively describes the driveline torque interruption. Vehicle jerk and driveline torque interruption represent the longitudinal dynamics of a HEV. Clutch frictional losses can negatively affect the service life of clutch in a HEV. Letj be the absolute value of vehicle jerk. The recommended value of j is 10 m/s3 in Germany and 17.64 m/s3 in China [29]. Clutch frictional losses can be calculated by using the following formula [30]: (31)Wsl=∫t0tf‍Tcωe-ωmdt,where Wsl is the clutch friction work, ωe-ωm stands for the speed difference of the two sides of clutch, and t0 and tf are the initial time and the terminal time of the slipping friction phase, respectively.In order to verify the proposed novel torque coordination control strategy, the numerical simulations are carried out within MATLAB. The following simulations are conducted by using the nominal parameters of a hybrid electric car [14]: m=2300 kg, α=0, fr=0.02 (ordinary concrete pavement), R=0.308 m, Jm=0.22 kg·m2, Jc=0.05 kg·m2, Jgb=0.1 kg·m2, Je′=0.4 kg·m2, a1=0.4, i=5, bm=0.005 N·m·s/rad, be=0.1 N·m·s/rad, Ts=0.01 s, rw1=1, rw2=4, rw3=3.


### 4.1. Comparison with the Conventional Method

HEV drivers have found that the transition time and power interruption can be reduced by pressing the accelerator pedal gradually and then releasing the clutch pedal quickly during the mode transition from pure electric running to parallel hybrid operation [18]. To evaluate the performance of the proposed novel torque coordination control strategy, this conventional method will be used as the baseline. The algorithm of the baseline operation is given by (32)Tm=Tdem,Tc=-300t,Te=Te0+30t.The equivalent driver’s demand torque with respect to gearbox input sideTdem is shown in Figure 3. Besides, Figures 4~10 show the results of the proposed novel torque coordination control strategy and the baseline operation.Figure 3
Equivalent driver’s demand torque with respect to gearbox input side.Figure 4
Angular speed of motor and engine.As shown in Figure4, the mode transition duration of the novel strategy is 0.59 s longer than that of the baseline, but the slipping friction duration of clutch in the novel strategy is 0.09 s shorter than that of the baseline (the specific index is shown in Table 1), which is helpful to reduce the clutch frictional losses.Table 1
The specific indices of the proposed novel strategy and the baseline.
The novel strategy
The baseline
The mode transition duration
0.91 s
0.32 s
The slipping friction duration of clutch
0.23 s (from 0.68 to 0.91 s)
0.32 s (from 0 to 0.32 s)
The absolute value of jerk amplitude
1.6 m/s3
56 m/s3
The fluctuation range of vehicle acceleration
0.14~0.26 m/s2
−0.46~0.12 m/s2
The clutch frictional losses
7.47 J
397.96 JThe absolute value of jerk amplitude in the novel strategy is far less than that of the baseline as shown in Figure5 (the specific index is shown in Table 1). The reason for the smaller jerk of the novel strategy at 0.68 s is that the speed difference of the two sides of clutch has reached a small threshold value. For the novel strategy, the clutch synchronization point occurs at 0.91 s. Before this point, the set-point signal of the novel strategy comes from the reference model which happens to be the state-space equation behind the point, so very small jerk appears at the synchronization point.Figure 5
Vehicle jerk.As shown in Figure6, vehicle acceleration of the novel strategy maintains a more stable fluctuation range than that of the baseline (the specific index is shown in Table 1).Figure 6
Vehicle acceleration.Since vehicle jerk and driveline torque interruption are the indices of longitudinal dynamics of vehicles, both Figures5 and 6 can prove that the proposed novel torque coordination control strategy is able to significantly improve HEV’s smoothness and dynamic performance.Figure7 further indicates that the vehicle speed of the novel strategy changes more smoothly than that of the baseline, and this will improve HEV’s driving comfort greatly.Figure 7
Vehicle speed.Figure8 shows that the clutch frictional losses resulting from the novel strategy are much less than those of the baseline operation (the specific index is shown in Table 1). All three factors in formula (31) are much less in the novel strategy than in the baseline; therefore, the clutch frictional losses are greatly reduced so that the proposed novel strategy is helpful to prolong the service life of clutch.Figure 8
Clutch torqueTc.Figures9 and 10 show that the proposed novel strategy increases the motor torque and the engine torque to compensate the negative clutch torque so that the smooth mode transition can be achieved.Figure 9
Motor torqueTm.Figure 10
Engine torqueTe.


### 4.2. Sensitivity Analysis of the Proposed Novel Torque Coordination Control Strategy

Several key parameters that may influence the mode transition performance are to be studied in this section for better performance of the proposed novel strategy.


#### 4.2.1. Effect of the Tuning Parameterrw2 on Clutch Frictional Losses

Figure11 shows the punishment role of different tuning parameters rw2 on clutch torque Tc. Here, rw2 is chosen to be 1, 4, and 10, respectively, and the other parameters remain the same. The resulting clutch frictional losses of the novel strategy which are calculated according to formula (31) are 20.56 J, 7.47 J, and 3.33 J, separately. Besides, the slipping friction duration is 0.18 s, 0.23 s, and 0.24 s, respectively. It can be seen that the bigger the tuning parameter rw2 is, the slower the clutch torque change is, the smaller the clutch torque is, and the smaller the clutch frictional losses are. But the slipping friction duration becomes longer as rw2 increases. Therefore, the proposed novel torque coordination strategy is sensitive to rw2, so rw2 should be carefully chosen. Considering these factors comprehensively, rw2=4 is assumed.Figure 11
Clutch torqueTc under different tuning parameters rw2.


#### 4.2.2. Effect of the Switch Triggerε on the Vehicle Jerk and Clutch Frictional Losses

The control role of different switch triggerε on the vehicle jerk and clutch frictional losses is shown in Figures 12 and 13, respectively. It can be seen that the proposed novel strategy is sensitive to different ε. Figure 12 shows that the bigger the switch trigger is, the larger the vehicle jerk is.Figure 12
Vehicle jerk under different switch triggerε.Figure 13
Clutch torqueTc and corresponding clutch frictional losses under different switch trigger ε.As shown in Figure13, small switch trigger implies less usage of clutch torque. The choice of small switch trigger is helpful for reducing slipping friction duration and frictional losses. However, the smaller the switch trigger is, the longer the mode transition time is. Taking sensor inaccuracy and actuator delay into account, it is impractical to choose ε too small.


### 4.3. Adaptiveness Analysis of the Proposed Novel Torque Coordination Control Strategy

For a running HEV, the mode transition from pure electric running to parallel hybrid operation may occur under different inclination angles, tire rolling resistance coefficients, vehicle masses, driving styles, and so on. As examples, the adaptiveness of the proposed novel strategy to different tire rolling resistance coefficients and driving styles is to be studied in this section.


#### 4.3.1. The Adaptiveness of the Proposed Novel Strategy to Different Tire Rolling Resistance Coefficients

The tire rolling resistance coefficientfr is equal to 0.02 for ordinary concrete pavement, whereas fr becomes 0.04 for snow road. Figures 14 and 15 show that the absolute values of vehicle jerk and the clutch frictional losses are all very small in the case of either fr=0.01, fr=0.02, or fr=0.04. This indicates that the proposed novel strategy is adaptive to different tire rolling resistance coefficients.Figure 14
Vehicle jerk under different tire rolling resistance coefficientfr.Figure 15
Clutch torqueTc and corresponding clutch frictional losses under different tire rolling resistance coefficient fr.


#### 4.3.2. The Adaptiveness of the Proposed Novel Strategy to Different Driving Styles

Vehicle jerk and clutch torque under different driving styles are shown in Figures16 and 17, respectively. The change rates of acceleration pedal 30 N·m/s, 50 N·m/s, and 70 N·m/s mimic a mild driver, a normal driver, and an aggressive driver, respectively. Simulation results show that the vehicle jerk and the clutch frictional losses are all very small which demonstrate that the proposed novel strategy is adaptive to different driving styles.Figure 16
Vehicle jerk under different driving styles.Figure 17
Clutch torqueTc and corresponding clutch frictional losses under different driving styles.


## 4.1. Comparison with the Conventional Method

HEV drivers have found that the transition time and power interruption can be reduced by pressing the accelerator pedal gradually and then releasing the clutch pedal quickly during the mode transition from pure electric running to parallel hybrid operation [18]. To evaluate the performance of the proposed novel torque coordination control strategy, this conventional method will be used as the baseline. The algorithm of the baseline operation is given by (32)Tm=Tdem,Tc=-300t,Te=Te0+30t.The equivalent driver’s demand torque with respect to gearbox input sideTdem is shown in Figure 3. Besides, Figures 4~10 show the results of the proposed novel torque coordination control strategy and the baseline operation.Figure 3
Equivalent driver’s demand torque with respect to gearbox input side.Figure 4
Angular speed of motor and engine.As shown in Figure4, the mode transition duration of the novel strategy is 0.59 s longer than that of the baseline, but the slipping friction duration of clutch in the novel strategy is 0.09 s shorter than that of the baseline (the specific index is shown in Table 1), which is helpful to reduce the clutch frictional losses.Table 1
The specific indices of the proposed novel strategy and the baseline.
The novel strategy
The baseline
The mode transition duration
0.91 s
0.32 s
The slipping friction duration of clutch
0.23 s (from 0.68 to 0.91 s)
0.32 s (from 0 to 0.32 s)
The absolute value of jerk amplitude
1.6 m/s3
56 m/s3
The fluctuation range of vehicle acceleration
0.14~0.26 m/s2
−0.46~0.12 m/s2
The clutch frictional losses
7.47 J
397.96 JThe absolute value of jerk amplitude in the novel strategy is far less than that of the baseline as shown in Figure5 (the specific index is shown in Table 1). The reason for the smaller jerk of the novel strategy at 0.68 s is that the speed difference of the two sides of clutch has reached a small threshold value. For the novel strategy, the clutch synchronization point occurs at 0.91 s. Before this point, the set-point signal of the novel strategy comes from the reference model which happens to be the state-space equation behind the point, so very small jerk appears at the synchronization point.Figure 5
Vehicle jerk.As shown in Figure6, vehicle acceleration of the novel strategy maintains a more stable fluctuation range than that of the baseline (the specific index is shown in Table 1).Figure 6
Vehicle acceleration.Since vehicle jerk and driveline torque interruption are the indices of longitudinal dynamics of vehicles, both Figures5 and 6 can prove that the proposed novel torque coordination control strategy is able to significantly improve HEV’s smoothness and dynamic performance.Figure7 further indicates that the vehicle speed of the novel strategy changes more smoothly than that of the baseline, and this will improve HEV’s driving comfort greatly.Figure 7
Vehicle speed.Figure8 shows that the clutch frictional losses resulting from the novel strategy are much less than those of the baseline operation (the specific index is shown in Table 1). All three factors in formula (31) are much less in the novel strategy than in the baseline; therefore, the clutch frictional losses are greatly reduced so that the proposed novel strategy is helpful to prolong the service life of clutch.Figure 8
Clutch torqueTc.Figures9 and 10 show that the proposed novel strategy increases the motor torque and the engine torque to compensate the negative clutch torque so that the smooth mode transition can be achieved.Figure 9
Motor torqueTm.Figure 10
Engine torqueTe.


## 4.2. Sensitivity Analysis of the Proposed Novel Torque Coordination Control Strategy

Several key parameters that may influence the mode transition performance are to be studied in this section for better performance of the proposed novel strategy.


### 4.2.1. Effect of the Tuning Parameterrw2 on Clutch Frictional Losses

Figure11 shows the punishment role of different tuning parameters rw2 on clutch torque Tc. Here, rw2 is chosen to be 1, 4, and 10, respectively, and the other parameters remain the same. The resulting clutch frictional losses of the novel strategy which are calculated according to formula (31) are 20.56 J, 7.47 J, and 3.33 J, separately. Besides, the slipping friction duration is 0.18 s, 0.23 s, and 0.24 s, respectively. It can be seen that the bigger the tuning parameter rw2 is, the slower the clutch torque change is, the smaller the clutch torque is, and the smaller the clutch frictional losses are. But the slipping friction duration becomes longer as rw2 increases. Therefore, the proposed novel torque coordination strategy is sensitive to rw2, so rw2 should be carefully chosen. Considering these factors comprehensively, rw2=4 is assumed.Figure 11
Clutch torqueTc under different tuning parameters rw2.


### 4.2.2. Effect of the Switch Triggerε on the Vehicle Jerk and Clutch Frictional Losses

The control role of different switch triggerε on the vehicle jerk and clutch frictional losses is shown in Figures 12 and 13, respectively. It can be seen that the proposed novel strategy is sensitive to different ε. Figure 12 shows that the bigger the switch trigger is, the larger the vehicle jerk is.Figure 12
Vehicle jerk under different switch triggerε.Figure 13
Clutch torqueTc and corresponding clutch frictional losses under different switch trigger ε.As shown in Figure13, small switch trigger implies less usage of clutch torque. The choice of small switch trigger is helpful for reducing slipping friction duration and frictional losses. However, the smaller the switch trigger is, the longer the mode transition time is. Taking sensor inaccuracy and actuator delay into account, it is impractical to choose ε too small.


## 4.2.1. Effect of the Tuning Parameterrw2 on Clutch Frictional Losses

Figure11 shows the punishment role of different tuning parameters rw2 on clutch torque Tc. Here, rw2 is chosen to be 1, 4, and 10, respectively, and the other parameters remain the same. The resulting clutch frictional losses of the novel strategy which are calculated according to formula (31) are 20.56 J, 7.47 J, and 3.33 J, separately. Besides, the slipping friction duration is 0.18 s, 0.23 s, and 0.24 s, respectively. It can be seen that the bigger the tuning parameter rw2 is, the slower the clutch torque change is, the smaller the clutch torque is, and the smaller the clutch frictional losses are. But the slipping friction duration becomes longer as rw2 increases. Therefore, the proposed novel torque coordination strategy is sensitive to rw2, so rw2 should be carefully chosen. Considering these factors comprehensively, rw2=4 is assumed.Figure 11
Clutch torqueTc under different tuning parameters rw2.


## 4.2.2. Effect of the Switch Triggerε on the Vehicle Jerk and Clutch Frictional Losses

The control role of different switch triggerε on the vehicle jerk and clutch frictional losses is shown in Figures 12 and 13, respectively. It can be seen that the proposed novel strategy is sensitive to different ε. Figure 12 shows that the bigger the switch trigger is, the larger the vehicle jerk is.Figure 12
Vehicle jerk under different switch triggerε.Figure 13
Clutch torqueTc and corresponding clutch frictional losses under different switch trigger ε.As shown in Figure13, small switch trigger implies less usage of clutch torque. The choice of small switch trigger is helpful for reducing slipping friction duration and frictional losses. However, the smaller the switch trigger is, the longer the mode transition time is. Taking sensor inaccuracy and actuator delay into account, it is impractical to choose ε too small.


## 4.3. Adaptiveness Analysis of the Proposed Novel Torque Coordination Control Strategy

For a running HEV, the mode transition from pure electric running to parallel hybrid operation may occur under different inclination angles, tire rolling resistance coefficients, vehicle masses, driving styles, and so on. As examples, the adaptiveness of the proposed novel strategy to different tire rolling resistance coefficients and driving styles is to be studied in this section.


### 4.3.1. The Adaptiveness of the Proposed Novel Strategy to Different Tire Rolling Resistance Coefficients

The tire rolling resistance coefficientfr is equal to 0.02 for ordinary concrete pavement, whereas fr becomes 0.04 for snow road. Figures 14 and 15 show that the absolute values of vehicle jerk and the clutch frictional losses are all very small in the case of either fr=0.01, fr=0.02, or fr=0.04. This indicates that the proposed novel strategy is adaptive to different tire rolling resistance coefficients.Figure 14
Vehicle jerk under different tire rolling resistance coefficientfr.Figure 15
Clutch torqueTc and corresponding clutch frictional losses under different tire rolling resistance coefficient fr.


### 4.3.2. The Adaptiveness of the Proposed Novel Strategy to Different Driving Styles

Vehicle jerk and clutch torque under different driving styles are shown in Figures16 and 17, respectively. The change rates of acceleration pedal 30 N·m/s, 50 N·m/s, and 70 N·m/s mimic a mild driver, a normal driver, and an aggressive driver, respectively. Simulation results show that the vehicle jerk and the clutch frictional losses are all very small which demonstrate that the proposed novel strategy is adaptive to different driving styles.Figure 16
Vehicle jerk under different driving styles.Figure 17
Clutch torqueTc and corresponding clutch frictional losses under different driving styles.


## 4.3.1. The Adaptiveness of the Proposed Novel Strategy to Different Tire Rolling Resistance Coefficients

The tire rolling resistance coefficientfr is equal to 0.02 for ordinary concrete pavement, whereas fr becomes 0.04 for snow road. Figures 14 and 15 show that the absolute values of vehicle jerk and the clutch frictional losses are all very small in the case of either fr=0.01, fr=0.02, or fr=0.04. This indicates that the proposed novel strategy is adaptive to different tire rolling resistance coefficients.Figure 14
Vehicle jerk under different tire rolling resistance coefficientfr.Figure 15
Clutch torqueTc and corresponding clutch frictional losses under different tire rolling resistance coefficient fr.


## 4.3.2. The Adaptiveness of the Proposed Novel Strategy to Different Driving Styles

Vehicle jerk and clutch torque under different driving styles are shown in Figures16 and 17, respectively. The change rates of acceleration pedal 30 N·m/s, 50 N·m/s, and 70 N·m/s mimic a mild driver, a normal driver, and an aggressive driver, respectively. Simulation results show that the vehicle jerk and the clutch frictional losses are all very small which demonstrate that the proposed novel strategy is adaptive to different driving styles.Figure 16
Vehicle jerk under different driving styles.Figure 17
Clutch torqueTc and corresponding clutch frictional losses under different driving styles.


## 5. Conclusions

This paper proposes a novel torque coordination control strategy to achieve smooth mode transition from pure electric running to parallel hybrid operation for a single-shaft parallel HEV effectively. Compared with the existing MPC methods about torque coordination control, only one model predictive controller is needed, which reduces the calculation amount of the control strategy and is more suitable for the real-time control application. Since the clutch torque plays a crucial role for the improvement of transition performance, it is used as an optimized variable rather than a known parameter as in the existing MPC methods. Using the precious idea of MRC for reference, the desired performance during the mode transition is expressed by a reference model which directly generates the set-point signal of MPC real-timely. HEV’s dynamic models under different running modes have been developed for the proposed novel strategy.Compared with the baseline operation, the proposed novel torque coordination control strategy can achieve much smoother mode transition with much higher driving comfort, much better dynamic performance, and much lower clutch abrasion. The service life of clutch is greatly prolonged so that the use cost of HEV can be reduced. Several factors that may influence the performance of the proposed novel strategy have been studied through simulations. The proposed novel torque coordination control strategy is found to be sensitive to the tuning parameterrw2 and the threshold value of ε. ε is proved to have significant influence on vehicle jerk and clutch frictional losses. Further simulations have shown that the proposed novel strategy is adaptive to different tire rolling resistance coefficients and driving styles. The proposed novel torque coordination strategy will lay a foundation for the practical application in HEV control units. There are some physical constraints in HEV. For example, the engine torque Te, the clutch friction torque Tc, the motor torque Tm, and their change rates are all bounded. So far, these constraints are not concerned in this paper. Future work will address the model predictive controller with HEV’s realistic constraints.



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*Source: 960678-2015-01-05.xml*