In recent years, the rapid development of electric vehicles, particularly in the context of China EV markets, has highlighted the importance of advanced suspension systems to enhance ride comfort and stability. As a researcher in automotive dynamics, I have focused on addressing the challenges posed by in-wheel motor-driven electric cars, where increased unsprung mass and electromagnetic excitations can negatively impact vehicle performance. This study investigates the modeling and control of a magnetorheological (MR) damper-based semi-active suspension system tailored for electric cars. By integrating experimental data, finite element analysis, and robust control strategies, I aim to improve the vertical vibration characteristics of these vehicles under various road conditions.
The core of this work involves developing an accurate model for the MR damper, which is crucial for effective semi-active control. I conducted extensive mechanical property tests on an MR damper using a hydraulic actuator system, applying harmonic excitations with amplitudes of 5 mm and frequencies of 0.5 Hz and 1 Hz, while varying the input current from 0 to 2.5 A. The force-displacement and force-velocity characteristics exhibited nonlinear hysteresis, which I captured using a modified hyperbolic tangent model. The model equation is expressed as:
$$ F = a_1 \tanh(a_2(\dot{x} + kx)) + a_3(\dot{x} + kx) + f_0 $$
where $F$ is the damping force, $x$ and $\dot{x}$ are the relative displacement and velocity, respectively, and $a_1$, $a_2$, $a_3$, $k$, and $f_0$ are parameters identified through genetic algorithms and least-squares fitting. The parameters $a_2$, $k$, and $f_0$ were found to be relatively constant with current variations, while $a_1$ and $a_3$ showed linear dependencies on the current $I$, leading to the relationships:
$$ a_1 = 278.7I + 144 $$
$$ a_3 = 1.89I + 1.45 $$
This model was validated against experimental data, showing a maximum root mean square error of less than 0.36%, confirming its accuracy for subsequent control applications. The following table summarizes the identified parameters for different currents:
| Current (A) | $a_1$ | $a_2$ | $k$ | $a_3$ | $f_0$ |
|---|---|---|---|---|---|
| 0 | 149.14 | 0.22 | 0.39 | 1.90 | 44.35 |
| 0.5 | 256.57 | 0.31 | 0.34 | 2.07 | 45.19 |
| 1.0 | 434.21 | 0.36 | 0.36 | 3.06 | 34.52 |
| 1.5 | 580.35 | 0.37 | 0.38 | 4.13 | 53.99 |
| 2.0 | 713.77 | 0.36 | 0.42 | 5.35 | 47.19 |
| 2.5 | 820.67 | 0.31 | 0.44 | 6.36 | 34.44 |
Building on this, I developed a quarter-car model for the electric car semi-active suspension system, incorporating the MR damper and accounting for the in-wheel motor’s effects. The system dynamics are described by the following equations:
$$ m_s \ddot{z}_s + k_s (z_s – z_u) = F_z $$
$$ (m_u + m_w) \ddot{z}_u + k_s (z_u – z_s) + k_u (z_u – z_r) = F_w – F_z $$
where $m_s$ is the sprung mass, $m_u$ is the wheel mass, $m_w$ is the in-wheel motor mass, $k_s$ and $k_u$ are the suspension and tire stiffness coefficients, $z_s$ and $z_u$ are the displacements of the sprung and unsprung masses, $z_r$ is the road excitation, $F_z$ is the controllable damping force from the MR damper, and $F_w$ is the electromagnetic force from the motor. To address practical uncertainties, I considered variations in the sprung mass, modeled as $m_s = \hat{m}(1 + d_m \delta(t))$, where $\hat{m}$ is the nominal mass, $d_m$ is the perturbation range, and $|\delta(t)| \leq 1$. The vehicle parameters used in this study are listed below:
| Parameter | Value |
|---|---|
| Nominal Sprung Mass $\hat{m}$ (kg) | 450 |
| Wheel Mass $m_u$ (kg) | 21 |
| Motor Mass $m_w$ (kg) | 51.9 |
| Suspension Stiffness $k_s$ (N/m) | 35,714 |
| Tire Stiffness $k_u$ (N/m) | 200,330 |
The electromagnetic excitations from the in-wheel motor were analyzed using finite element methods, considering different eccentricity conditions. For a dynamic eccentricity of 10%, the electromagnetic force exhibited harmonic-like fluctuations with a dominant frequency of 7.2 Hz, which falls within the critical 4–8 Hz range for human sensitivity to vertical vibrations. This emphasizes the need for effective control strategies in electric cars to mitigate such disturbances. The integration of these elements into the suspension system is visually represented below, highlighting the components of an electric car suspension setup.
To handle parameter uncertainties and external disturbances, I designed an H∞ robust state feedback controller. The state-space representation of the system is given by:
$$ \dot{x} = (A + \Delta A)x + (B_u + \Delta B_u)U + B_w w $$
$$ z_1 = (C_1 + \Delta C_1)x + (D_{1u} + \Delta D_{1u})U $$
where $x = [z_s – z_u, \dot{z}_s, z_u – z_r, \dot{z}_u]^T$ is the state vector, $w = [\dot{z}_r, F_w]^T$ is the disturbance input, $U$ is the control input, and $\Delta A$, $\Delta B_u$, $\Delta C_1$, $\Delta D_{1u}$ represent parameter uncertainties. Using linear matrix inequalities (LMIs), I derived the control law $U = Kx$, ensuring system stability and performance optimization. The LMI conditions were solved via MATLAB’s LMI toolbox, resulting in a controller that minimizes the H∞ norm of the transfer function from disturbances to controlled outputs, such as body acceleration and suspension deflection.
In simulation analyses, I evaluated the system under random and impact road excitations. For random road conditions, I used a B-class road profile with a vehicle speed of 60 km/h, generated by the filter:
$$ \dot{q} = -0.111[vq(t) + 40 \sqrt{G_q(n_0)} v w_0(t)] $$
where $v$ is the speed, $G_q(n_0)$ is the road roughness coefficient, and $w_0(t)$ is white noise. The performance of the H∞ controller was compared against passive and skyhook-controlled semi-active suspensions. The results demonstrated significant improvements: the H∞ control reduced the root mean square (RMS) of body acceleration by 27.4%, suspension deflection by 44.7%, and tire dynamic load by 19.3%, outperforming skyhook control. In frequency domain analyses, the H∞ controller achieved peak reductions of up to 60.4% in key metrics, highlighting its effectiveness in suppressing resonances around the body and wheel natural frequencies.
For impact road conditions, I simulated a bump excitation described by:
$$ z_r(t) = \begin{cases}
\frac{A_m}{2} \left(1 – \cos \frac{2\pi v}{L} t\right) & 0 \leq t \leq \frac{L}{v} \\
0 & \text{otherwise}
\end{cases} $$
with $A_m = 0.1$ m and $L = 5$ m. The H∞ controller showed faster settling times and lower peak responses compared to other strategies, reducing the risk of hitting suspension limits and enhancing ride comfort. Additionally, I investigated the impact of sprung mass variations (±20%) on control performance. The H∞ controller maintained robust performance across different masses, whereas skyhook control exhibited greater sensitivity, underscoring the advantage of incorporating uncertainty in the design for electric cars.
To validate the simulations, I developed a quarter-car test bench for the electric wheel suspension system, incorporating an MR damper, an out-runner BLDC motor, and sensors for acceleration measurement. The experimental setup used a Speedgoat real-time target machine for control implementation. Harmonic excitations at frequencies of 5 Hz, 7 Hz, and 10 Hz were applied, and the H∞ controller consistently reduced the RMS of body acceleration by 15.43% to 24.49% compared to passive suspension, confirming the practical efficacy of the approach for China EV applications.
In conclusion, this study presents a comprehensive framework for modeling and controlling MR semi-active suspensions in electric cars. The modified hyperbolic tangent model accurately captures MR damper dynamics, while the H∞ robust controller effectively handles parameter uncertainties and external disturbances. Simulations and experiments under diverse conditions validate the superiority of this approach in improving ride comfort and stability. Future work could explore integration with full-vehicle models and adaptive control strategies to further enhance performance in the rapidly evolving electric car industry, particularly for China EV markets where innovation in suspension systems is critical.