Transportation electrification is reshaping the way vehicles are designed, manufactured, and experienced by customers. Among all powertrain alternatives, electric cars have drawn worldwide attention because they provide zero tailpipe emissions, high energy efficiency, and strong low-end torque. However, the shift from internal combustion engines to electric propulsion systems has introduced a remarkable change in vehicle dynamic characteristics. The drive motor of an electric car responds to torque commands almost instantly, and the powertrain structure often lacks large damping components such as torque converters and clutches. As a result, the coupled effect between fast motor torque dynamics and the underdamped driveline can lead to noticeable longitudinal oscillations during acceleration. These oscillations degrade the dynamic response quality and are directly perceived by drivers and passengers as uncomfortable jerks and vibrations.
In my opinion, the evaluation of dynamic response quality is therefore a critical step in the development of modern electric cars. Traditionally, the evaluation of vehicle dynamic performance relies on two separate approaches. Subjective evaluation methods ask trained raters to assess a vehicle through real driving experience, while objective evaluation methods use measurable dynamical quantities to characterize vehicle behavior. Both methods have their own limitations. Subjective evaluation can accurately reflect human perception but suffers from poor repeatability, high cost, and inconsistency between raters. Objective evaluation is repeatable and quantitative, but it does not fully capture how humans experience the motion. Therefore, I have focused my research on combining subjective and objective information in a unified evaluation framework, which has become an important trend in vehicle ride comfort research.
The acceleration process of electric cars deserves special attention because the human body is very sensitive to sudden changes in longitudinal acceleration. During tip-in maneuvers, the electric motor produces a rapid torque increase, which excites torsional vibration in the driveline. Because of the underdamped nature of the system, this torsional oscillation propagates through gears and half-shafts to the vehicle body, generating longitudinal jerk and acceleration oscillation. These undesired motions have a direct influence on ride comfort and can seriously affect the perceived quality of an electric car. Therefore, I have selected the acceleration process as the target scenario in this study, and I have built a research framework that covers dynamic modeling, subjective and objective evaluation experiments, physiological signal analysis, machine learning based model construction, and model predictive control optimization.
The first objective of my research was to establish a reliable electromechanical simulation platform for electric cars. The powertrain architecture considered in this study includes a permanent magnet synchronous motor (PMSM) driven by a voltage source inverter, a two-stage gear reducer, half-shafts, wheels, and the equivalent vehicle body inertia. I modeled the electric machine in the rotor-oriented d-q reference frame, because vector control based on field orientation provides excellent dynamic torque control for electric cars. The stator voltage equations can be written as:
$$u_{d}=R_{s}i_{d}+L_{d}\frac{di_{d}}{dt}-\omega_{e}L_{q}i_{q}$$
$$u_{q}=R_{s}i_{q}+L_{q}\frac{di_{q}}{dt}+\omega_{e}\left(L_{d}i_{d}+\psi_{f}\right)$$
where $u_d$ and $u_q$ are the d-axis and q-axis stator voltages, $i_d$ and $i_q$ are the d-axis and q-axis currents, $R_s$ is the stator resistance, $L_d$ and $L_q$ are the d-axis and q-axis inductances, $\omega_e$ is the electrical angular speed, and $\psi_f$ is the permanent magnet flux linkage. In this study I used a surface-mounted PMSM, for which $L_d$ and $L_q$ are almost equal. Therefore, the electromagnetic torque can be simplified as:
$$T_{e}=\frac{3}{2}p_{n}\psi_{f}i_{q}$$
where $p_n$ is the number of pole pairs. The above equation shows that torque is proportional to the q-axis current when the d-axis current is controlled to zero. I employed a standard $i_d=0$ field-oriented control scheme in the simulation environment. The outer torque command is converted into a q-axis current reference, while two proportional-integral controllers regulate the d-q currents. A space vector pulse width modulation module generates the switching signals for the three-phase inverter. The current loop parameters were tuned according to the system bandwidth. Table 1 summarizes the key parameters of the electric machine and control system used in my simulation model.
| Symbol | Parameter | Value |
|---|---|---|
| $P_n$ | Rated power | 120 kW |
| $T_\text{max}$ | Peak torque | 250 Nm |
| $p_n$ | Number of pole pairs | 4 |
| $R_s$ | Stator resistance | 0.02 Ω |
| $L_d$ | d-axis inductance | 0.15 mH |
| $L_q$ | q-axis inductance | 0.15 mH |
| $\psi_f$ | Flux linkage | 0.08 Wb |
| $U_{dc}$ | DC bus voltage | 380 V |
The driveline model was constructed using the lumped-mass method. I divided the entire powertrain into several rotational inertia elements connected by massless elastic shafts. The model contains one inertia for the motor rotor, four gear inertias for the two-stage reduction gear pair, wheel inertias, and one equivalent inertia representing the vehicle body. Gear meshing was represented by a linear spring-damper element, while the tire was also modeled as a torsional spring-damper element between the wheel and the vehicle body. The equations of motion for the driveline can be expressed in a compact matrix form as:
$$\mathbf{J}\ddot{\boldsymbol{\theta}}+\mathbf{C}\dot{\boldsymbol{\theta}}+\mathbf{K}\boldsymbol{\theta}=\mathbf{T}$$
where $\mathbf{J}$ is the inertia matrix, $\mathbf{K}$ is the stiffness matrix, $\mathbf{C}$ is the damping matrix, $\boldsymbol{\theta}$ is the vector of angular displacements, and $\mathbf{T}$ is the torque vector containing the motor torque and external resistance torque. The external resistance torque comes from aerodynamic drag, rolling resistance, and acceleration resistance. The dynamic behavior of this driveline system shows a low-frequency torsional mode, which is weakly damped because of the absence of traditional damping elements. This mode is quite important for evaluating the ride comfort of electric cars.
I integrated the PMSM vector control model with the driveline dynamic model in MATLAB/Simulink to form an electromechanical coupled simulation platform. The motor torque output serves as the input excitation to the mechanical system, while the mechanical speed at the motor shaft is fed back to the motor model. This coupled model effectively reproduces the transient torque oscillation that occurs during acceleration of electric cars. In the simulation of a full acceleration process, I set the target motor torque to 100 Nm and applied a finite torque rise time. The resulting driveline responses show that the half-shaft torque exhibits a damped oscillation at a frequency of approximately 10–15 Hz. This oscillation can be transmitted to the vehicle body and appears as a longitudinal acceleration oscillation. The peak value of the longitudinal acceleration is significantly higher than the steady-state value, and the acceleration signal oscillates for a noticeable period before reaching a stable value.
I also studied the influence of several structural and control parameters on the dynamic response of electric cars. The torque rise time has an especially strong effect. In the simulation, I compared torque rise times of 0.10 s, 0.14 s, and 0.18 s. When the torque rise time is shorter, the initial excitation contains more high-frequency energy, and the resulting longitudinal acceleration oscillation becomes stronger. This observation is closely related to driver behavior, because a fast accelerator pedal tip-in leads to a short torque rise time and consequently poor dynamic response quality. I also investigated the effect of half-shaft stiffness. Although half-shaft stiffness has some influence on the oscillation amplitude and frequency, its effect is weaker than that of the torque rise time. This conclusion further highlights the importance of controlling the motor torque profile in electric cars.
Having established the simulation platform, I next designed a subjective-objective evaluation experiment. For subjective evaluation, I used the well-known SAE J1441 ten-point scale because it provides a convenient and widely accepted standard for rating vehicle dynamic behavior. The subjective evaluation in this study focuses only on the dynamic comfort perceived during acceleration. I identified several important evaluation points from the dynamic response analysis, including the presence of abnormal shocks, the existence of longitudinal oscillation, the acceptability of initial tip-in shock, overall smoothness of acceleration, and the sensation of unexpected power interruption. Table 2 presents the subjective scoring criteria used in the experiment.
| Score | Description | Comfort level |
|---|---|---|
| 1–4 | Continuous and strong vibration, severe shock, obvious discomfort | Unacceptable |
| 5–6 | Occasional slight shock or vibration, no fatal problem | Acceptable |
| 7–8 | Smooth operation, rare abnormal vibration, good comfort | Good |
| 8–10 | Almost no perceptible shock, excellent experience | Very good |
Objective evaluation was based on measurable vehicle dynamic signals. I collected the longitudinal acceleration, velocity, and motor-related signals during each test run using a high-precision GPS and inertial measurement system. To convert the raw acceleration signals into meaningful objective evaluation indices, I defined a set of acceleration-domain and jerk-domain features. Jerk is the derivative of longitudinal acceleration with respect to time:
$$j(t)=\frac{da(t)}{dt}$$
where $a(t)$ is the longitudinal acceleration. The jerk peak value is defined as:
$$j_{\max}=\max\left(\left|\max_{t}j(t)\right|,\left|\min_{t}j(t)\right|\right)$$
This feature represents the instantaneous impact intensity. In addition, I introduced the shock decay time, which measures how quickly the jerk peak decays. Let $t_{p}$ denote the time at which the jerk reaches its peak value, and let $t_{d}$ denote the time at which the jerk first falls below a prescribed threshold. The shock decay time is then:
$$t_{j}=t_{d}-t_{p}$$
This parameter reflects the energy dissipation speed of the driveline system. A longer shock decay time means that passengers are exposed to impact-related excitation for a longer duration.
The root mean square acceleration is widely used in ride comfort assessment. I calculated the RMS value of the longitudinal acceleration over the test time window $T$:
$$a_{\text{RMS}}=\sqrt{\frac{1}{T}\int_{0}^{T}a^{2}(t)dt}$$
The RMS value provides a time-averaged measure of vibration energy, but it is not sufficiently sensitive to short-duration peaks. Therefore, I also computed the vibration dose value (VDV), which emphasizes the fourth power of the acceleration signal:
$$\text{VDV}=\left[\int_{0}^{T}a^{4}(t)dt\right]^{1/4}$$
Compared with the RMS value, the VDV amplifies transient shock events and better reflects the discomfort caused by non-stationary vibration.
In addition to amplitude-based features, I extracted several frequency-related features. The acceleration oscillation frequency $f_a$ is calculated as:
$$f_{a}=\frac{N_{a}}{T}$$
where $N_a$ is the number of extrema in the longitudinal acceleration signal within the time interval $T$. Similarly, I defined the jerk oscillation frequency:
$$f_{j}=\frac{N_{j}}{T}$$
where $N_j$ is the number of extrema in the jerk signal. A high value of $f_j$ indicates that the driver can feel repeated small changes in acceleration, which may be described as roughness or unsteady acceleration. I also extracted the dominant frequency of the acceleration signal using power spectral density analysis. The dominant frequency is relevant because human beings are particularly sensitive to vibration in certain frequency ranges. Finally, I used the ratio of positive and negative jerk peaks and the average time interval between adjacent acceleration extrema as additional descriptors of waveform symmetry and periodicity. Table 3 lists all objective evaluation indices initially considered in my study.
| Symbol | Description | Formula |
|---|---|---|
| $j_{\max}$ | Peak jerk value | $j_{\max}=\max(|j|)$ |
| $t_j$ | Shock decay time | $t_j=t_d-t_p$ |
| $a_{\text{RMS}}$ | Root mean square acceleration | $a_{\text{RMS}}=\sqrt{(1/T)\int a^2 dt}$ |
| $\text{VDV}$ | Vibration dose value | $\text{VDV}=[\int a^4 dt]^{1/4}$ |
| $f_a$ | Acceleration oscillation frequency | $f_a=N_a/T$ |
| $f_j$ | Jerk oscillation frequency | $f_j=N_j/T$ |
| $j_{\max,3}$ | Third jerk peak following the first peak | extracted from jerk curve |
| $f_{a,d}$ | Dominant acceleration frequency | from PSD |
| $R_j$ | Positive-to-negative jerk peak ratio | $R_j=N_j^+/N_j^-$ |
| $T_{a,\text{mean}}$ | Mean interval between acceleration extrema | $T_{a,\text{mean}}=\frac{1}{N-1}\sum(t_{i+1}-t_i)$ |
Because the final goal of the evaluation model is to reflect human subjective perception, I incorporated physiological signals as an additional bridge between objective dynamics and subjective feelings. Previous research has shown that electrocardiogram signals, especially heart rate variability (HRV), can reflect psychological and physiological responses to external stimuli. Therefore, I decided to include HRV features in the evaluation experiment. During each driving test, I recorded ECG signals from a passenger seated in the rear of the vehicle. The passenger was instructed to remain calm and avoid large body movements. After the experiment, I detected the R-wave peaks and computed the NN interval series, where an NN interval is the time between two consecutive normal heartbeats.
In the time domain, I computed the standard deviation of NN intervals and the root mean square of successive NN interval differences. The standard deviation of NN intervals is given by:
$$\text{SDNN}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(NN_i-\overline{NN}\right)^2}$$
and the root mean square of successive differences is given by:
$$\text{RMSSD}=\sqrt{\frac{1}{N-1}\sum_{i=2}^{N}\left(NN_i-NN_{i-1}\right)^2}$$
In the frequency domain, I used the power spectral density of the NN interval series and extracted the low-frequency power (LF), high-frequency power (HF), and the LF/HF ratio. The time-domain features mainly reflect overall autonomic activity, while the frequency-domain features can give additional information about sympathetic and parasympathetic balance. Table 4 shows the HRV features selected in this study.
| Domain | Feature | Meaning |
|---|---|---|
| Time | SDNN | Standard deviation of NN intervals |
| Time | RMSSD | Root mean square of successive differences |
| Time | NNmin | Minimum NN interval |
| Time | NNmax | Maximum NN interval |
| Frequency | LF | Low-frequency power |
| Frequency | HF | High-frequency power |
| Frequency | LF/HF | Ratio of low to high frequency power |
I conducted the subjective-objective evaluation experiment on a flat and dry asphalt road to eliminate unnecessary road excitation. Three acceleration conditions were designed to represent different intensities of acceleration in electric cars. The first condition was a normal acceleration maneuver. The second condition was a rapid acceleration to about 60 km/h. The third condition was a rapid acceleration to about 100 km/h. Before each experiment, the passenger rested for several minutes to ensure that the heart rate reached a stable baseline level. The vehicle dynamics data and ECG data were recorded synchronously during each test. Several trained participants with experience in vehicle ride comfort evaluation took part in the experiments. All experimental procedures followed ethical requirements, and each participant signed an informed consent form before the experiments.
I processed the measured data through several steps. First, the abnormal RR intervals caused by motion artifacts or ectopic beats were removed from the ECG signal. Second, the longitudinal acceleration signal was low-pass filtered with a cutoff frequency of 10 Hz, because this frequency range is most relevant to human whole-body vibration perception. Third, I calculated all the objective evaluation indices defined in Table 3 for every test run. Fourth, I computed the HRV features from the corrected NN interval series. Fifth, every test run was associated with the subjective score provided by the corresponding participant. Finally, a comprehensive sample database was established. This database contains the subjective score, several objective dynamic indicators, and several physiological features. The database serves as the foundation for developing the integrated evaluation model.
Before building the model, I performed a statistical analysis to determine which HRV features differed significantly across the three acceleration conditions. Since the same participants underwent multiple conditions, I used repeated measures analysis of variance. This method appropriately accounts for the correlation between repeated observations from the same participant. In addition, a Bonferroni post-hoc test was used to identify pairwise differences between acceleration conditions. Table 5 reports the statistical results.
| HRV feature | Condition 1 | Condition 2 | Condition 3 | F-value | p-value |
|---|---|---|---|---|---|
| SDNN | 56.77 (10.75) | 37.46 (8.67) | 28.84 (9.32) | 185.45 | <0.001 |
| RMSSD | 38.96 (11.04) | 32.53 (10.97) | 23.85 (9.25) | 21.98 | <0.001 |
| NNmin | 698.96 (82.25) | 702.81 (110.44) | 663.57 (119.92) | 1.78 | NS |
| NNmax | 966.77 (100.53) | 794.00 (165.90) | 881.58 (135.19) | / | <0.01 |
| LF | 37.00 (17.47) | 34.80 (22.95) | 33.46 (23.36) | / | NS |
| HF | 18.92 (11.73) | 32.77 (27.14) | 44.84 (33.28) | 6.87 | <0.05 |
| LF/HF | 3.00 (3.00) | 6.66 (20.48) | 5.25 (16.10) | / | NS |
The results show that SDNN and RMSSD decreased significantly as the acceleration intensity increased. SDNN is generally considered to reflect overall heart rate variability, while RMSSD is often used as an index of parasympathetic activity. A reduction in SDNN and RMSSD indicates a stress-related response of the passenger. In contrast, HF increased with stronger acceleration, which may reflect changes in breathing patterns or psychological tension. Among all the HRV features examined, SDNN and RMSSD showed significant differences across all three acceleration conditions. Therefore, I used SDNN and RMSSD as physiological indicators for the next step of objective indicator selection.
The next important problem was to determine which of the ten objective indices had the strongest relationship with the physiological responses of the passengers. Since the physiological indicators form one set of variables and the objective dynamic indicators form another set, I needed a method that can examine the correlation between two sets of variables. Canonical correlation analysis (CCA) is an appropriate method for this purpose. Given two variable sets $\mathbf{X}$ and $\mathbf{Y}$, CCA finds linear combinations $U_1=\mathbf{a}^T\mathbf{X}$ and $V_1=\mathbf{b}^T\mathbf{Y}$ that maximize the correlation between $U_1$ and $V_1$:
$$\rho=\frac{\mathbf{a}^T\mathbf{\Sigma}_{XY}\mathbf{b}}{\sqrt{\mathbf{a}^T\mathbf{\Sigma}_{XX}\mathbf{a}}\sqrt{\mathbf{b}^T\mathbf{\Sigma}_{YY}\mathbf{b}}}$$
where $\mathbf{\Sigma}_{XX}$ and $\mathbf{\Sigma}_{YY}$ are the covariance matrices of the two sets, and $\mathbf{\Sigma}_{XY}$ is the cross-covariance matrix. I applied CCA between the HRV features SDNN and RMSSD on one side, and the ten initially selected objective indicators on the other side. The first pair of canonical variables had a canonical correlation of 0.912 and a p-value below 0.05, indicating a statistically significant association. The second pair was not statistically significant and was therefore omitted from further analysis.
The cross-loadings from the first canonical pair reflect the correlation between each original objective indicator and the HRV canonical variate. Table 6 presents these cross-loadings.
| Objective indicator | $j_{\max}$ | $t_j$ | $a_{\text{RMS}}$ | VDV | $f_a$ | $f_j$ | $j_{\max,3}$ | $f_{a,d}$ | $R_j$ | $T_{a,\text{mean}}$ |
|---|---|---|---|---|---|---|---|---|---|---|
| Cross-loading | 0.713 | -0.685 | 0.663 | 0.825 | 0.283 | 0.109 | 0.771 | 0.177 | -0.151 | -0.290 |
The indicators with high absolute cross-loadings were $j_{\max}$, $t_j$, $a_{\text{RMS}}$, VDV, and $j_{\max,3}$. Therefore, I selected these five indicators as the final objective evaluation indices for the acceleration process. The selected indices cover different aspects of dynamic quality: $j_{\max}$ reflects the initial impact intensity, $t_j$ reflects the duration of the impact decay, $a_{\text{RMS}}$ reflects the overall vibration energy, VDV reflects the cumulative effect of transient shocks, and $j_{\max,3}$ reflects the persistence of successive shock events. By reducing the number of objective indicators, I simplified the model input space and removed redundancy that would otherwise reduce the generalization ability of the prediction model.
After the objective evaluation system was determined, my next goal was to establish a nonlinear mapping between the objective indicators and the subjective score. I chose machine learning regression methods because the relationship between objective vehicle dynamics and subjective comfort is not necessarily linear. The first model I developed was a support vector machine for regression. In support vector regression, the goal is to find a function $f(\mathbf{x})=\mathbf{w}^T\phi(\mathbf{x})+b$ that deviates from the actual target values by a value no greater than $\epsilon$ in the insensitive region. The optimization problem can be stated as:
$$\min_{\mathbf{w},b,\xi_i,\xi_i^*}\frac{1}{2}\|\mathbf{w}\|^2+C\sum_{i=1}^{n}\left(\xi_i+\xi_i^*\right)$$
subject to:
$$y_i-f(\mathbf{x}_i)\leq \epsilon+\xi_i$$
$$f(\mathbf{x}_i)-y_i \leq \epsilon+\xi_i^*$$
$$\xi_i,\xi_i^* \geq 0$$
where $C$ is the penalty factor and $\epsilon$ defines the error tolerance tube. I chose the Gaussian radial basis function as the kernel function because it can describe complex nonlinear relationships:
$$K\left(\mathbf{x}_i,\mathbf{x}_j\right)=\exp\left(-\gamma\|\mathbf{x}_i-\mathbf{x}_j\|^2\right)$$
where $\gamma$ is the kernel width parameter. I used a grid search method to determine the optimal combination of $C$ and $\gamma$ for the support vector machine model.
The second model I constructed was a random forest regressor. Random forest is an ensemble learning method that constructs many decision trees and averages their outputs. For regression, the predicted value $\hat{y}$ of the random forest is the mean of the predictions from all individual trees $h_t(\mathbf{x})$:
$$\hat{y}=\frac{1}{M}\sum_{t=1}^{M}h_t(\mathbf{x})$$
where $M$ is the number of trees. The randomness in random forest comes from two sources: bootstrap sampling of the training data and random feature selection at each node split. These two mechanisms reduce variance and help avoid overfitting. Unlike support vector machines, random forest does not require feature normalization, which simplifies the model building process.
I trained both models using the sample database, which contains the five selected objective indicators as inputs and the subjective score as the output. To evaluate the prediction accuracy, I used the mean absolute error, the mean relative error, and the root mean square error. These error metrics are defined as follows:
$$\text{MAE}=\frac{1}{N}\sum_{i=1}^{N}\left|y_{\text{real},i}-y_{\text{predict},i}\right|$$
$$\text{MRE}=\frac{1}{N}\sum_{i=1}^{N}\frac{\left|y_{\text{real},i}-y_{\text{predict},i}\right|}{y_{\text{real},i}}$$
$$\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(y_{\text{real},i}-y_{\text{predict},i}\right)^2}$$
Table 7 compares the prediction errors of the support vector regression and random forest models.
| Model | Mean relative error | Mean absolute error | Root mean square error |
|---|---|---|---|
| Support vector regression | 9.55% | 0.59 | 0.67 |
| Random forest regression | 3.54% | 0.24 | 0.28 |
The random forest model produced lower errors than the support vector regression model. Therefore, I selected the random forest model as the final integrated evaluation model for the acceleration process. This model allows me to predict the subjective score of a given dynamic response without performing an actual subjective evaluation test. Because the subjective score is a direct reflection of the passenger experience, the prediction result is highly valuable for optimizing controller parameters and vehicle calibration parameters.
After I completed the integrated subjective-objective evaluation model, I turned to the task of improving the dynamic response quality of electric cars. The main problem is longitudinal vibration caused by fast motor torque changes. To suppress this vibration, I designed a model predictive controller (MPC) based on a reduced-order driveline model. The simplified model represents the driveline as two equivalent inertias connected by a torsional spring and damper. The motor-side equivalent inertia includes the motor rotor, gear wheels, and part of the shafts, while the vehicle-side equivalent inertia includes the wheels and the vehicle body. The state equation of this reduced-order model can be written as:
$$\dot{\mathbf{x}}=\mathbf{A}_c\mathbf{x}+\mathbf{B}_c u+\mathbf{H}_c d$$
where the state vector is chosen as $\mathbf{x}=[\theta\ \ \omega_m\ \ \omega_v]^T$, $\theta$ is the torsional angle difference between the motor-side and vehicle-side equivalent inertias, $\omega_m$ is the motor angular speed, $\omega_v$ is the vehicle-side angular speed, $u$ is the motor torque, and $d$ is the load torque. The state matrices are given by:
$$\mathbf{A}_c=\begin{bmatrix}0 & \dfrac{1}{i} & -1\\ -\dfrac{k_e}{iJ_{me}} & -\dfrac{c_e}{i^2J_{me}} & \dfrac{c_e}{iJ_{me}}\\ \dfrac{k_e}{J_{ve}} & \dfrac{c_e}{iJ_{ve}} & -\dfrac{c_e}{J_{ve}}\end{bmatrix}$$
$$\mathbf{B}_c=\begin{bmatrix}0 & \dfrac{1}{J_{me}} & 0\end{bmatrix}^T,\ \ \ \mathbf{H}_c=\begin{bmatrix}0 & 0 & -\dfrac{1}{J_{ve}}\end{bmatrix}^T$$
where $i$ is the total reduction ratio, $J_{me}$ and $J_{ve}$ are the equivalent inertias referred to the motor side and vehicle side, $k_e$ is the equivalent torsional stiffness, and $c_e$ is the equivalent damping coefficient. The controlled output is the torsional speed difference $\omega_m/i-\omega_v$, which is closely related to driveline oscillation. I discretized the continuous state-space model using a fixed sampling time.
The model predictive controller was designed to achieve two goals simultaneously. First, it should reduce the torsional speed difference so that the longitudinal oscillation of electric cars is suppressed. Second, it should still track the driver’s desired torque command so that the vehicle retains good acceleration response. To combine these two goals, I defined the following objective function:
$$J=\left\|\mathbf{W}_y\mathbf{Y}(k+1|k)\right\|^2+\left\|\mathbf{W}_u\left(\mathbf{U}(k)-\mathbf{R}(k)\right)\right\|^2$$
where $\mathbf{Y}(k+1|k)$ is the predicted future output sequence, $\mathbf{U}(k)$ is the control sequence, $\mathbf{R}(k)$ is the desired torque sequence, and $\mathbf{W}_y$ and $\mathbf{W}_u$ are diagonal weighting matrices. The weighting factors $w_y$ and $w_u$ directly balance the vibration suppression performance and the torque tracking performance. In every sampling instant, the MPC controller solves a constrained quadratic programming problem to find the optimal control sequence. The torque output is constrained by the motor’s minimum and maximum torque limits at the current speed. Only the first element of the optimal control sequence is applied to the system. At the next sampling instant, the optimization is repeated with newly measured state information. This rolling horizon mechanism gives the controller robustness against model mismatch and external disturbances.
I verified the effectiveness of the MPC controller through co-simulation with the full electromechanical powertrain model. Table 8 compares the simulation results with and without MPC.
| Response metric | Without control | With MPC |
|---|---|---|
| Peak longitudinal acceleration | 2.16 m/s² | 1.13 m/s² |
| Time to peak acceleration | 0.06 s | 0.64 s |
| Sustained oscillation | Evident until about 2.5 s | Almost eliminated |
The comparison shows that the MPC controller significantly improves the damping characteristic of the driveline. Without MPC, the longitudinal acceleration has a large overshoot and oscillates for a long time. With MPC, the acceleration reaches the steady-state value smoothly without visible oscillations. The torque tracking ability is maintained because the output torque quickly returns to the desired torque value after the initial transient.
In practical application, the weighting factors of the MPC controller need to be tuned carefully. If $w_u$ is too large, the controller prioritizes torque tracking and may not adequately suppress longitudinal vibration. If $w_y$ is too large, the controller may reduce vibration too aggressively, but the torque response becomes slow and the driver may feel a noticeable delay. To find a reasonable balance, I carried out a series of simulations with different values of $w_y$. I kept $w_u=1$ and varied $w_y$ from 1 to 100. For each simulation, I computed the selected objective indicators and used the trained random forest model to predict the subjective score. Table 9 shows the objective indicators and predicted subjective scores for representative weight values.
| $w_y$ | $a_{\text{RMS}}$ (m/s²) | VDV (m/s^{1.75}) | $j_{\max}$ (m/s³) | Predicted subjective score |
|---|---|---|---|---|
| 1 | 0.58 | 2.10 | 14.5 | 5.2 |
| 30 | 0.46 | 1.55 | 8.2 | 7.4 |
| 60 | 0.42 | 1.31 | 5.1 | 8.1 |
| 90 | 0.41 | 1.26 | 4.3 | 8.3 |
The predicted subjective score increases with $w_y$, but the rate of improvement slows down when $w_y$ exceeds 60. At the same time, increasing $w_y$ further lengthens the response time. Therefore, I chose $w_y=60$ as the optimal weighting factor for the controller. This choice achieves a good balance between comfort and dynamic response. The predicted subjective score at this setting is 8.1, which is in the “good” to “very good” range according to the subjective evaluation criteria.
To highlight the advantage of the proposed integrated evaluation model, I compared the parameter selection result with two conventional objective evaluation approaches. The first conventional approach uses only the RMS acceleration as the criterion. The second conventional approach uses a weighted multi-objective function that combines VDV, RMS acceleration, and the torque response time:
$$J_c=w_1\text{VDV}+w_2 a_{\text{RMS}}+w_3 t_r$$
where $t_r$ is the time necessary for the vehicle acceleration to reach 90% of its steady-state value, and $w_1$, $w_2$, and $w_3$ are scalar weights. The selected controller parameters under each method were different. Table 10 reports the final selected weights, predicted subjective scores, and response times collected under the two conventional methods and my integrated evaluation method.
| Evaluation method | Selected $w_y$ | Predicted subjective score | Response time (s) |
|---|---|---|---|
| Proposed integrated subjective-objective model | 60 | 8.1 | 0.12 |
| RMS acceleration criterion | 80 | 8.3 | 0.26 |
| Weighted multi-objective function | 20 | 6.9 | 0.08 |
| Without control baseline | / | 4.2 | 0.045 |
The comparison shows that optimizing according to the RMS acceleration criterion leads to a slightly higher subjective score but doubles the response time compared with my integrated method. Conversely, optimizing according to the conventional multi-objective function produces a very short response time but the predicted subjective score is only 6.9, which is close to the lower boundary of acceptable comfort. My proposed method, which uses the random forest integrated evaluation model as the objective function, can simultaneously maintain a high subjective score and an acceptable dynamic response time. This result proves that the subjective-objective integrated evaluation model has valuable engineering significance for electric cars.
Another advantage of my approach is that the evaluation model can be reused for many different controller configurations without requiring a new subjective experiment. Once the mapping from objective indicators to subjective score has been trained, designers can predict the subjective effect of any proposed control strategy through numerical simulation. This capability greatly accelerates the calibration process. Instead of running time-consuming road tests for every candidate parameter set, an engineer can first screen the parameter space in simulation and then verify only the most promising candidates on a real vehicle. This is especially beneficial during the development of electric cars because these vehicles exhibit wide variations in powertrain configuration and motor control software.
In the course of my research, I also realized that the quality of the objective indicators directly affects the generalization ability of the main model. Although the ten initially selected indicators successfully described many aspects of the acceleration response, several indicators turned out to be weakly related to the physiological responses of the passengers. The canonical correlation analysis provided an effective way to remove those less relevant indicators. One potential improvement for future work is to explore non-linear correlation analysis methods. The relationship between vehicle dynamics and human physiological response is complex, and kernel-based canonical correlation analysis may help identify even more powerful objective indicators.
In addition, the current evaluation model is based on longitudinal acceleration only. Passenger comfort in real electric cars, however, is also influenced by lateral and vertical motions. In future work, I intend to expand the evaluation model to include multi-axis acceleration data, yaw rate, pitch rate, and roll rate. Such an expanded model will be able to evaluate more complex driving scenarios such as cornering acceleration or acceleration on undulating roads. Another promising direction is the combination of physiological signals measured from drivers and passengers simultaneously. Driver comfort is closely related to workload and control confidence, while passenger comfort is more related to motion sickness and perceived safety. The differences between these two groups have not yet been fully explored.
Another issue that deserves further attention is the design of the model predictive controller. The linear MPC used in this study is effective for small deviations around the nominal operating point. However, the driveline parameters of electric cars can vary with temperature, tooth wear, and road conditions. Nonlinear tire behavior and gear backlash may also influence the control performance under severe maneuvers. In future work, I plan to investigate adaptive MPC or nonlinear MPC based on online parameter estimation. The subjective-objective evaluation model can also be used as a performance monitor in real time, allowing the control system to adapt its weighting factors according to the predicted comfort score.
Looking back at this research, I have developed a complete framework from simulation modeling to real experimental evaluation and then to controller optimization. The key novelty of my study is the introduction of ECG-derived HRV features into the evaluation process. This addition allowed me to select objective evaluation indicators that are truly relevant to the human perception of dynamic response quality. The framework has been successfully applied to the acceleration process of electric cars, and it has demonstrated clear advantages over traditional objective evaluation approaches.
In summary, my research has drawn the following important conclusions. First, the electromechanical coupled simulation model composed of a PMSM vector control model and a multi-degree-of-freedom driveline dynamic model reproduces the transient longitudinal vibration phenomenon that commonly occurs during acceleration of electric cars. Second, the torque rise time has a significant influence on the longitudinal acceleration response, while the half-shaft stiffness affects the response to a smaller extent. Third, the heart rate variability features SDNN and RMSSD change significantly with the intensity of acceleration, which verifies the feasibility of using physiological signals to indicate subjective discomfort. Fourth, the canonical correlation analysis between HRV indicators and candidate objective indicators successfully reduced the number of objective evaluation indicators and produced a more compact and effective objective evaluation system. Fifth, the random forest regression model predicted the subjective score with much lower error than support vector regression. Sixth, the model predictive controller designed on the basis of a two-inertia driveline model can effectively suppress longitudinal vibration while maintaining good torque tracking. Finally, the choice of MPC weighting factors has a considerable impact on both comfort and responsiveness, and the proposed integrated evaluation model provides a reliable way to determine the optimal weights without extensive subjective testing.
The work presented here has several limitations. The experiments were conducted on a limited number of participants and vehicles. A larger sample pool would make the statistical analysis more robust. Moreover, the random forest model is a black-box model, which means that it is difficult to interpret the physical meaning of the mapping between objective indicators and subjective scores. More interpretable models or feature contribution analysis could be added in future work to improve the transparency of the evaluation model. Despite these limitations, the results confirm that combining physiological signals with machine learning is a promising path toward objective quantification of subjective ride comfort in electric cars.
From an application perspective, this research is not limited to the acceleration process. With some modifications, the same methodology can be extended to other driving conditions such as deceleration, lane change, highway driving, and off-road driving. The idea of using physiological signals to filter objective indicators is especially valuable when the target vehicle exhibits complex dynamic responses that are difficult to assess by a simple physical measure. For future electric cars with increasing levels of automation, the ride comfort evaluation system will be even more important because drivers will become passengers and will no longer be occupied by vehicle control tasks. In such automated driving scenarios, the dynamic response quality of electric cars will directly determine user acceptance and satisfaction.
Therefore, the development of a multi-dimensional and physiologically meaningful evaluation model is a crucial engineering task. I believe that the integrated method proposed in this thesis provides a solid foundation for this task and can be continuously improved by incorporating more physiological channels, more advanced machine learning algorithms, and richer vehicle dynamics data. I hope that the theoretical analysis and experimental framework developed in this study can support the development of high-performance electric cars with excellent dynamic quality.