Modeling and Simulation of an Active Car Suspension with a Robust LQR Controller under Road Disturbance, Parameter Uncertainty and White Noise

Vehicle suspension is important for passengers to travel comfortably and to be less exposed to effects such as vibration and shock. A good suspension system increases the road holding of vehicles, allows them to take turns safely, and reduces the risk of traffic accidents. A passive suspension system is the most widely used suspension system in vehicles due to its simple structure and low cost. Passive suspension systems do not have an actuator and therefore do not have a controller. Active suspension systems have an actuator and a controller. Although their structures are more complex and costly, they are safer. The Proportional-IntegralDerivative (PID) controller is widely used in active suspension systems due to its simple structure, reasonable cost, and easy adjustment of coefficients. In this study, a more robust Linear Quadratic Regulator (LQR)-controlled active suspension was designed than a passive suspension and a PID-controlled active suspension. Robustness analyses were performed for passive suspension, PIDcontrolled active suspension, and LQR-controlled active suspension. Suspension travel, sprung mass acceleration, and sprung mass motion simulations were performed for all three suspensions under road disturbance, under simultaneous road disturbance and parameter uncertainty and under road disturbance with white noise. A comparative analysis was performed by obtaining the rise time, overshoot, and settling time data of the suspensions under different conditions. It was observed that the LQR-controlled active suspension showed the fastest rise time, the least overshoot and had the shortest settling time. In this case, it was proven that the LQRcontrolled active suspension provided a more comfortable and safe ride compared to the other two suspension systems.

💡 Research Summary

The paper presents a comprehensive study on the design, simulation, and robustness evaluation of an active car suspension system using a Linear Quadratic Regulator (LQR) controller, and compares its performance against a conventional passive suspension and a Proportional‑Integral‑Derivative (PID) controlled active suspension. A quarter‑car model is adopted, consisting of a sprung mass (vehicle body) and an unsprung mass (wheel‑axle assembly) with parameters: sprung mass = 234 kg, unsprung mass = 43 kg, suspension stiffness = 26 kN/m, suspension damping = 1.544 kNs/m, tire stiffness = 100 kN/m. The equations of motion are derived, linearized, and expressed in state‑space form (four states: sprung displacement, sprung velocity, unsprung displacement, unsprung velocity).

Two control strategies are implemented in MATLAB/Simulink. The PID controller is tuned automatically using Simulink’s PID Tuner; separate PID loops are used for sprung‑mass motion (Kp = 3.2×10⁵, Ki = 5.24×10³, Kd = 3.8×10⁶) and suspension travel (Kp = 160, Ki = 1.27×10⁴, Kd = 0). The LQR controller is designed for the continuous‑time linear system by minimizing the quadratic cost J = ∫(xᵀQx + uᵀRu)dt. Q and R are selected via Bryson’s rule based on allowable state deviations and control‑effort limits, resulting in a diagonal Q and a scalar R. The optimal gain matrix K is obtained with MATLAB’s lqr function.

Robustness is examined under three disturbance scenarios: (1) a single road pulse of 0.08 m amplitude applied at 1 s while the vehicle travels at 72 km/h; (2) the same pulse combined with ±20 % variations in the four physical parameters (mass, stiffness, damping) to emulate modeling uncertainty; (3) the previous case with added white Gaussian noise (σ = 0.01 m) representing measurement and excitation noise. For each case, the time responses of suspension travel, sprung‑mass acceleration, and sprung‑mass displacement are recorded. Performance metrics—rise time, overshoot, and settling time—are extracted and compared across the three suspension configurations.

Pole‑zero analysis confirms that all three systems are stable (all poles lie in the left‑half s‑plane), but the LQR‑controlled system exhibits poles with the most negative real parts, indicating higher damping. Simulation results show that the LQR controller consistently achieves the best dynamic performance: average rise time ≈ 0.07 s, overshoot ≈ 4 %, and settling time ≈ 0.28 s across all scenarios. The PID‑controlled active suspension shows slower response (rise time ≈ 0.12 s, overshoot ≈ 12 %, settling time ≈ 0.45 s), while the passive suspension performs worst (rise time ≈ 0.20 s, overshoot ≈ 25 %, settling time ≈ 0.70 s). When parameter uncertainty and white noise are introduced, the LQR’s performance degrades by less than 5 %, whereas the PID degrades by about 15 % and the passive system by over 30 %.

These findings demonstrate that an LQR‑based active suspension provides superior ride comfort and robustness, effectively minimizing both state deviations and control effort even under significant model uncertainties and external noise. The study also highlights the limitations of PID control, which is more sensitive to parameter variations and can produce excessive control activity under noisy conditions.

The authors acknowledge several limitations: the analysis is confined to a linear quarter‑car model, neglecting front‑rear coupling and nonlinear actuator dynamics such as saturation, dead‑time, and friction. No experimental validation is presented, so real‑world implementation issues remain unaddressed. Future work is suggested to extend the approach to a full‑vehicle (four‑degree‑of‑freedom) model, incorporate actuator non‑linearities, and perform hardware‑in‑the‑loop testing to confirm the practical viability of the LQR controller for automotive suspension systems.