Design Optimization of the Control System for the Powertrain of an Electric Vehicle a Cyber-Physical System Approach

Design Optimization of the Control System for the Powertrain of an Electric Vehicle: A Cyber-Physical

System Approach

Chen Lv and Junzhi Zhang Pierluigi Nuzzo and

Alberto Sangiovanni-Vincentelli Yutong Li and Ye Yuan

State Key Laboratory of Automotive Safety and Energy

Department of Electrical Engineering and Computer Sciences

State Key Laboratory of Automotive Safety and Energy

Tsinghua University University of California, Berkeley Tsinghua University Beijing 100084, P.R. China Berkeley, California 94720, USA Beijing 100084, P.R. China

[email protected]; [email protected] {nuzzo, alberto}@eecs.berkeley.edu [email protected];

[email protected]

Abstract— By leveraging the interaction between the physical

and the computation worlds, cyber-physical systems provide the capability of augmenting the available design space in several application domains, possibly improving the quality of the final design. In this paper, we propose a new, optimization-based methodology for the co-design of the gear ratio and the active damping controller of the powertrain system in an electric vehicle. Our goal is to explore the trade-off between vehicle acceleration performance and drivability. Using a platform-based approach, we first define the system architecture, the requirements, and quality metrics of interest. Then, we formulate the design problem for the powertrain control system as an optimization problem, and propose a procedure to derive an optimal system sizing, by relying on the simulation of the vehicle performance for a set of driving scenarios. Optimization results show that the driveline control performance can be substantially improved with respect to conventional solutions, using the proposed methodology. This further highlights the relevance and effectiveness of a cyber-physical system approach to system design across the boundary between plant architecture and control law.

Index Terms- Design optimization, powertrain control system, electric vehicle, cyber-physical system, platform-based design.

I. INTRODUCTION The ever-heavier burden on the environment and energy use requires automobiles to be cleaner and more efficient. Technologies such as powertrain electrification and alternative fuel have been researched and developed. Among these solutions, various types of electrified vehicles show promises, due to their high-efficiency powertrain and the decreased or even zero emission [1,2].

Compared to a conventional ICE vehicle, an electric vehicle can achieve better acceleration performance, thanks to the fast response of motor torque. But the fast transient of the motor torque also causes powertrain vibration due to powertrain flexibility and backlash nonlinearity, resulting in unexpected jerk at the vehicle level [3]. The conventional approach to

solve this issue is to implement a powertrain active damping controller. Focusing on the control part, engineers designed different control algorithms and selected suitable variables for the powertrain system to coordinate vehicle acceleration and drivability [4].

However, as a complex cyber-physical system, an automotive electric powertrain involves two subsystems, i.e. the physical plant and the controller [5, 6]. These two parts, which are highly coupled, affect vehicle’s behavior and performance. By using the conventional design method, which deals with the two parts independently, even if the controller is very well designed, the improvement of vehicle performance could be limited, since the physical architecture and parameters are not optimized in synch with the controller, and the system potential capability is not fully explored.

The cyber-physical system approach, which is a newly emerging concept featuring the interaction between the physical world and controller, is of great significance for electro-mechanical system design and optimization, and worthwhile researching [7]. Co-design of the physical architecture and controller parameters provides the ability to extend system design space and improve further overall system performances.

In this paper, we present a methodology for the co-design of architecture parameters, i.e. the gear ratio, and the active damping controller ones for the powertrain system of an electric vehicle using a cyber-physical system approach, targeting the “trade-off” between vehicle’s acceleration and drivability. The system structure, requirements, and optimization goals are described first in Section 2 and 3. The methodology is proposed in Section 4. Section 5 presents the formulation of the proposed powertrain co-design problem. Section 6 reports the simulation-based design optimization results, and is followed by the conclusion in Section 7.

II. SYSTEM DESCRIPTION AND REQUIREMENTS

A. System Description Figure 1 shows the overall structure of the powertrain control system of an electric passenger car that considered in this study. A central electric motor is installed at the front axle of the vehicle. The battery, which is electrically connected to the motor through the DC bus, can be discharged or charged for motoring or absorb the regenerative power during the braking process respectively.

Fig. 1. Structure of the powertrain control system of an electric car.

B. Requirements The requirements for vehicle design are generally expressed in terms of safety [8], dynamical performance [9], energy efficiency [10], and drivability (comfort) [11]. In this study, we focus on some of the performances that are relevant to the co-design problem of the electrified powertrain system:

• Vehicle Performance. Typically, vehicle performance, including maximum speed, acceleration, deceleration, and grade ability, is closely related to the powertrain parameters. In this case, we set the requirement of maximum vehicle speed over 100 km h and below 120 km h [12]. Besides, we set the acceleration performance, i.e. the 0-50 km/h acceleration time, as one of the optimization goals.

• Vehicle Drivability. The drivability of a vehicle reflects the comfort level. It can be evaluated by the value of vehicle’s longitudinal acceleration. During a torque fast transient process, oscillation may occur in the half-shafts, deteriorating vehicle drivability. Therefore, implementation of an active damping controller is required.

There usually exists a conflict between vehicle performance and drivability. The main drawback of current implementations is the lack of optimality in powertrain architecture parameter design and controller parameter selection. In the next section, with the aims of coordinating the vehicle performance and drivability, we propose a co-design method to optimize gear ratio and parameters of the active damping controller.

III. DESIGN OPTIMIZATION GOALS FOR THE POWERTRAIN SYSTEM

A. Acceleration Performance The acceleration time is a key parameter for evaluating the vehicle’s dynamical performance. When accelerating, the accelerator pedal is fully depressed, i.e. the pedal is at its 100% position. Here, we adopt the 0-50 km/h acceleration time as the parameter for evaluating the acceleration performance.

For a given electric motor, the vehicle’s acceleration performance is mainly decided by the value of gear ratio, which is the key architecture parameter of powertrain. Hence, we need to optimize the value of the gear ratio to achieve the goal.

B. Driveline Control Performance Beyond the drivability requirement proposed in Section 2,

we expect to further improve the control performance of the driveline. Hence, our second optimization goal is to improve the control accuracy of the half-shaft torque by implementing an active damping controller. However, the control performance of the half-shaft torque is not only related to the control algorithm, but also greatly affected by the gear ratio. Therefore, we need to co-design the architecture parameter and control variables to reach the goal.

IV. HIERARCHICAL DESIGN OPTIMIZATION METHODOLOGY In this study, the methodology that we adopt is Platform-Based Design (PBD) [7], a paradigm that allows reasoning about design in a structured way.

Fig. 2. Methodology for the design optimization of a powertrain system.

As figure 2 shows, PBD is a meet-in-the-middle approach. The top layer is the high-level requirements, described in detailed in Section 2. The bottom layer is defined by a design platform, which is a library of models that represent behaviors and performance of the components. High-level requirements are mapped onto bottom-up abstractions and characterizations

of the physical plant, controller, and environment. Design space exploration and performance optimization are performed at the middle layer.

In this way, co-design of the architecture parameter and controller variables for the powertrain system of the electric vehicle can be achieved.

V. PROBLEM FORMULATION

A. System Dynamics

Fig. 3. Simplified two-inetia architecture model of powertrain system.

Figure 3 shows a two-inertia powertrain model, a simplification of the architecture presented in figure 1. One inertia corresponds to the electric motor, and the other to the contribution of the wheel.

Electric Motor Dynamics: The motor torque mT is modelled as a first-order reaction with a small time constant [12], as shown in equation (1).

11m

m

Gτ

=+

(1)

Gearbox: The gearbox connects the output shaft of the electric motor and the half shafts. The transmitted torque via gearbox can be shown as:

2 /m m m hs gJ T T iθ = − (2)

where mJ is the motor inertia, gi is the gearbox ratio, and hsT

is the half-shaft torque.

Half-shaft Torque: A flexible half shaft connects the gearbox and the wheel inertia. The model for the half-shaft torque can be given by [3]:

hs hs d hs dT k cθ θ= + (3)

1 2dθ θ θ= − (4)

where dθ is the half-shaft angular position. 1θ , and 2θ are the

angles at the indicated positions on the shaft. mθ and lθ are

the angular speeds of electric motor and load, respectively.

1 /m giθ θ= , and 2 lθ θ= . hsk and

hsc are the stiffness and damping coefficients of the half shaft, respectively.

Vehicle Dynamics: The model for the longitudinal vehicle dynamics is as follows:

2l l hs lJ T Tθ = − (5)

where lJ is the equivalent inertia of vehicle body, powertrain

system, and wheels. And lT is the resistance torque contributed by road, l wT fmgR= .

B. Active Damping Controller Active damping of powertrain can be seen as a torque tracking problem. The control objective is to track the target torque ,m tgtT with the half-shaft torque /hs gT i . A combined feed-forward and feed-back control structure is adopted [11]:

,m ref ff fbT T T= + (6)

where ffT is the feed-forward input term required for tracking

and fbT is the feed-back component designed to reduce the control error.

Based on the control objective, the feed-forward term can be determined by:

,ff m tgtT T= (7)

, ,limm tgt acc mT P T= ⋅ (8)

where ,m tgtT is the target value of the motor torque, which is determined by accP the position of driver’s accelerator pedal. And ,limmT is the torque limit of the electric motor, as equation 5 shows.

,lim

145 0 2650409550 2650

m

mm

m

nT

nn

≤ ≤⎧⎪= ⎨ ⋅ >⎪⎩

(9)

The error term between the target and the real value of half-shaft torque can be represented by:

, 2 /m tgt hs ge T T i= − (10)

For the feedback term, a linear PID control law is adopted:

fb P I DdT K e K edt K edt

= + +∫ (11)

Fig. 4. Overall structure of the powertrain active damping controller.

A block diagram of the active damping controller for powertrain system is shown in figure 4.

C. Environment Model The environment model describes the driver’s operating behavior, i.e. the accelerator pedal position accP

(0 100%)accP≤ ≤ in this study. It represents the torque demand requested by the driver.

Scenario 1: 0-50km/h Acceleration: In this scenario, the driver’s accelerator accP is fully pressed, i.e. 100%accP = . The car is fully accelerated from the speed of 0 to 50 km/h. The acceleration performance requirement proposed in section 2 can be checked in this scenario.

Scenario 2: Positive Torque Fast Transient: In this scenario, the accelerator pedal is taken as a ramp input stabilizing at 70% of its full stroke. The output torque of the electric motor experiences a fast transient starting from zero, which may lead to driveline oscillations. Hence, the drivability requirement can be verified in this scenario.

D. Requirement Formulation Maximum Vehicle Speed:

max100 120km h v km h≤ ≤ (12)

Tracking Error of Vehicle Acceleration: During the defined acceleration process (Scenario 2), the average tracking error between the expected and real values of the longitudinal acceleration aσ is required to be within 0.03m/s2.

E. Formulation of Optimization Goals 0-50km/h Acceleration Time 0 50t − : 0 50t − is the time taken in

scenario 1.

Tracking Error of Half-shaft Torque: Corresponding to the second optimization goal, the average tracking error of the half-shaft is minimized in Scenario 2.

VI. PERFORMANCE OPTIMIZATION AND RESULTS

Fig. 5. Implementation of the design in Matlab/Simulink.

Based on the above sections, models of physical plant, controller, and environment, and the high-level requirements formulated above are implemented in Matlab/Simulink, as shown in figure 5. Then, the design optimization of the powertrain control system is carried out.

A. Architecture Design Exploration According to our experiment results, the vehicle’s 0-50 km/h

acceleration performance is majorly related to the gear ratio gi .

Based on the upper and lower limits of the maximum vehicle speed required, the reasonable range for gear ratio

gi optimization is bonded in [7.78 9.33] , as figure 6 shows.

Fig. 6. Simulation results for the acceleration performance and architecture space exploration.

B. Controller Parameter Exploration Driveability of a vehicle is related to gear ratio, but it is also correlated closely with the controller parameters. In this study, the active damping controller is a PID controller.

Based on the experiments, the fixed value of D at 0.1 is acceptable for all the situations. Therefore, we only optimize the values of P and I. Here we give an example whereas the gear ratio gi is at 8.0.

Figure 7 shows the results of drivability variation within the explored area of control variables. Only the area where the value is within 0.03 m/s2 can satisfy the second requirement.

Within the valid space explored above, further optimization of control performance of the half-shaft torque is finished.

As shown in figure 8, when gear ratio is set at 8.0, the driveline control performance becomes better when values of control variables P and I get closer to 2 and 4.5, respectively.

For each value of the gear ratio in the valid range, the above processes for design optimization exploration of control variables need to be repeated.

Fig. 7. Results for drivability within the explored area of control variables.

Fig. 8. Results for the half-shaft torque control performance exploration.

Finally, under both requirements, based on simulation, the correlation between gear ratio and the optimized control performance of half-shaft torque is figured out, as the blue line shows in figure 9. Moreover, the valid range for gear ratio design optimization is further narrowed.

C. Performance Optimization and Results Based on the above exploration, the overall performance optimization is carried out. A trade-off problem between acceleration and drivability is set as follows.

,1 0 50 2

,

min ( ) hs hs

hs

T T baselinebaseline

T baseline

e et t

eα α−

−⋅ − + ⋅ (13)

Fig. 9. Relationship between gear ratio and optimized half-shaft torque control performance.

For the optimization, we target two different driving modes, namely the sport mode and comfort mode, as listed in table 1. The design optimization results of architecture and controller parameters obtained under different driving modes are shown in table 1.

TABLE I. PARAMETER SELECTION FOR DIFFERENT MODES

Mode Weights Values

1α 2α Gear Ratio P I

Baseline - - 7.9 2 4.5 Sport 0.75 0.25 9.0 1.2 3.5

Comfort 0.25 0.75 8.2 2.5 4.5

As figure 10 shows, both the sport and comfort set-up are better when compared to the baseline in terms of the 0-50 acceleration time. Moreover, in the sport mode, the acceleration time is 7.708 s, which is 0.4s less than that of the comfort mode.

Fig. 10. Acceleration performance in different scenarios.

Fig. 11. Driveline control performance in different scenarios.

Fig. 12. Value of longitudinal acceleration in different modes.

As shown in figure 11 and figure 12, the results of control performances of half-shaft torque and vehicle drivability validate our different optimization goals. Figure 11 shows that the average tracking error of the half-shaft torque is 2.74 Nm, improving the drivability performance of baseline by over 20%. However, since the sport mode targets the acceleration performance, it deteriorates the drivability performance compared to the baseline.

The above results validate our optimization goals under different system set-up, and also demonstrate the significance and effectiveness of system co-design.

VII. CONCLUSION In this paper, a co-design optimization methodology for the

powertrain system of an electric vehicle is proposed to harmonize vehicle’s acceleration and drivability. Using a platform-based approach, the system architecture, the requirements, and quality metrics of interest are firstly defined. Then, the design problem for the powertrain control system is formulated as an optimization problem. A procedure to derive an optimal system sizing is proposed by relying on the

simulation of the vehicle performance for a set of driving scenarios implemented in Matlab/Simulink. Optimization results show that the driveline control performance can be substantially improved with respect to conventional solutions, using the proposed methodology. This further highlights the relevance and effectiveness of a cyber-physical system approach to system design across the boundary between plant architecture and control law.

ACKNOWLEDGMENT The authors would like to thank the Natural Science

Foundation of China [Project no. 51475253], the China Scholarship Council [File no. 201306210121], and the TerraSwarm Research Center, one of six centers supported by the STARnet phase of the Focus Center Research Program (FCRP) a Semiconductor Research Corporation program sponsored by MARCO and DARPA, for partially funding this work.

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