The rapid growth of the electric vehicle (EV) market, particularly in China, has driven innovations in automotive suspension systems to enhance ride comfort and stability. In-wheel motor-driven electric vehicles, such as those prevalent in China’s EV industry, face unique challenges due to increased unsprung mass and electromagnetic excitations from the motor. These factors can exacerbate vertical vibrations, negatively impacting vehicle dynamics. To address this, semi-active suspension systems employing magnetorheological (MR) dampers have gained attention for their ability to provide adaptive damping forces with low energy consumption and fast response times. This article presents a comprehensive study on the modeling and control of an MR damper-based semi-active suspension system for electric vehicles, with a focus on handling parameter uncertainties and external disturbances common in real-world driving conditions in China EV applications.
MR dampers are central to semi-active suspension systems due to their controllable rheological properties. To accurately model the MR damper’s behavior, experimental tests were conducted under 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 in 0.5 A increments. The force-displacement and force-velocity characteristics exhibited nonlinear hysteresis, which is critical for designing effective control strategies in electric vehicles. Based on the experimental data, a modified hyperbolic tangent model was developed to represent the damper’s dynamics. The model 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 of the damper, respectively, and $a_1$, $a_2$, $a_3$, $k$, and $f_0$ are parameters identified using genetic algorithms and least-squares methods. 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 dependence on the input current $I$, leading to the relations:
$$ a_1 = 278.7I + 144 $$
$$ a_3 = 1.89I + 1.45 $$
This model was validated against experimental data, showing a maximum root-mean-square error of less than 0.36%, confirming its accuracy for integration into electric vehicle suspension systems. The parameter values are summarized in Table 1.
| 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 |
In electric vehicles, particularly those with in-wheel motors, the suspension system must account for additional masses and electromagnetic excitations. A quarter-car model was developed, incorporating the MR damper and considering the combined effects of road irregularities and electromagnetic forces from the motor. The equations of motion are:
$$ m_s \ddot{z}_s + k_s (z_s – z_u) = F_z $$
$$ (m_u + m_w) \ddot{z}_u + k_s (z_u – z_s) + k_u (z_u – z_r) = F_w – F_z $$
where $m_s$ is the sprung mass, $m_u$ is the wheel mass, $m_w$ is the in-wheel motor mass, $k_s$ is the suspension stiffness, $k_u$ is the tire stiffness, $z_s$ and $z_u$ are the displacements of the sprung and unsprung masses, respectively, $z_r$ 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. For China EV applications, the sprung mass uncertainty was modeled as $m_s = \hat{m}(1 + d_m \delta(t))$, where $\hat{m}$ is the nominal mass, $d_m$ is the maximum perturbation, and $|\delta(t)| \leq 1$. The vehicle parameters are listed in Table 2.
| 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 |
Electromagnetic excitations from the in-wheel motor were analyzed using finite element methods, considering different eccentricity conditions. Under dynamic eccentricity, the electromagnetic force exhibited significant fluctuations at frequencies around 7.2 Hz, which falls within the sensitive range of 4–8 Hz for human vertical acceleration perception according to ISO 2631 standards. This highlights the importance of controlling such vibrations in electric vehicles to ensure comfort. The growth of China’s EV market underscores the need for robust suspension systems that can handle these unique challenges.
To design a control strategy that accommodates parameter uncertainties and disturbances, an H∞ state feedback controller was developed. The state variables were defined as $x_1 = z_s – z_u$, $x_2 = \dot{z}_s$, $x_3 = z_u – z_r$, and $x_4 = \dot{z}_u$, leading to the state-space representation:
$$ \dot{x} = (A + \Delta A)x + (B_u + \Delta B_u)U + B_w w $$
$$ z_1 = (C_1 + \Delta C_1)x + (D_{1u} + \Delta D_{1u})U $$
where $w = [\dot{z}_r, F_w]^T$ is the external disturbance vector, and $\Delta A$, $\Delta B_u$, $\Delta C_1$, and $\Delta D_{1u}$ represent parameter uncertainties. The control output $z_1$ includes weighted sprung mass acceleration, suspension deflection, and tire dynamic load. Using linear matrix inequalities (LMIs), the controller gain $K$ was computed to minimize the H∞ norm of the transfer function from disturbances to outputs, ensuring robustness for electric vehicle applications. The LMI conditions are:
$$ \begin{bmatrix} \tilde{O}_1 & * & * \\ H_1^T P + H_2^T C_{1cl} & H_2^T H_2 – \lambda & * \\ B_w^T P & 0 & -\gamma^2 I \end{bmatrix} < 0 $$
where $\tilde{O}_1 = A_{cl}^T P + P A_{cl} + C_{1cl}^T C_{1cl} + (E_1 + E_2 K)^T \lambda (E_1 + E_2 K)$, $A_{cl} = A + B_u K$, and $C_{1cl} = C_1 + D_{1u} K$. This approach ensures that the system remains stable under mass variations and external excitations common in China EV operations.
Simulations were conducted under random and shock road profiles to evaluate the controller’s performance. For random road excitation, a B-class road was modeled with a vehicle speed of 60 km/h, and the dynamic eccentricity of the motor was set to 10%. The results, comparing passive suspension, skyhook control, and the proposed H∞ control, are summarized in Table 3 and Table 4. The H∞ control reduced the root-mean-square (RMS) values of body acceleration, suspension deflection, and tire dynamic load by 27.4%, 44.7%, and 19.3%, respectively, outperforming skyhook control. In the frequency domain, the H∞ controller achieved peak reductions of up to 60.4% in tire dynamic load, demonstrating its effectiveness in suppressing vibrations critical for electric vehicle comfort.
| Performance Metric | Passive Suspension | Skyhook Control | H∞ Control |
|---|---|---|---|
| Body Acceleration (m/s²) | 0.586 | 0.467 | 0.425 |
| Suspension Deflection (10⁻³ m) | 7.358 | 5.385 | 4.069 |
| Tire Dynamic Load (N) | 532.3 | 489.7 | 429.6 |
| Performance Metric | Passive Suspension | Skyhook Control | H∞ Control |
|---|---|---|---|
| Body Acceleration (m/s²) | 0.3968 | 0.2489 | 0.1832 |
| Suspension Deflection (10⁻³ m) | 5.735 | 3.864 | 2.399 |
| Tire Dynamic Load (N) | 220.9 | 139.0 | 87.4 |
For shock road excitation, a bump profile with a height of 0.1 m and length of 5 m was used at 60 km/h. The H∞ control significantly reduced peak responses and settling times compared to passive and skyhook controls, highlighting its robustness in handling transient disturbances. Additionally, the impact of sprung mass uncertainty was investigated by varying the mass from 350 kg to 550 kg. The H∞ controller maintained consistent performance, with body acceleration RMS values showing smaller deviations relative to passive suspension, as illustrated in Table 5. This reinforces the controller’s suitability for electric vehicles, where load variations are common.
| Sprung Mass (kg) | Passive Suspension (m/s²) | Skyhook Control (m/s²) | H∞ Control (m/s²) |
|---|---|---|---|
| 350 | 0.612 | 0.498 | 0.431 |
| 450 | 0.586 | 0.467 | 0.425 |
| 550 | 0.601 | 0.482 | 0.438 |
Experimental validation was performed using a quarter-car test bench equipped with an MR damper and an in-wheel motor. Harmonic excitations at frequencies of 5 Hz, 7 Hz, and 10 Hz with a 5 mm amplitude were applied to simulate road disturbances. The H∞ controller, implemented on a real-time target machine, reduced the RMS of body acceleration by 24.49%, 15.43%, and 23.78% at these frequencies, respectively, compared to passive suspension. These results confirm the practical efficacy of the control algorithm in enhancing the ride comfort of electric vehicles, aligning with the demands of the China EV market for advanced suspension technologies.
In conclusion, this study demonstrates the successful modeling and control of an MR damper-based semi-active suspension system for electric vehicles. The modified hyperbolic tangent model accurately captures the MR damper’s nonlinear behavior, while the H∞ robust controller effectively mitigates vibrations under parameter uncertainties and combined road-electromagnetic excitations. Simulations and experiments show significant improvements in vehicle dynamics, supporting the adoption of such systems in China EV applications to achieve superior comfort and stability. Future work could explore integration with full-vehicle models and real-world driving scenarios to further optimize performance for the evolving electric vehicle industry.