1. Introduction
With the emergence of modern technology in the automobile sector, improving the dynamic riding performance of suspension systems has become a critical design feature. In the literature, various control strategies, like the PID controller, model predictive controller (MPC), fuzzy logic network (FLC), and artificial neural network (ANN), have been reported to address the trade-off between driving comfort and handling capability [
1]. The primary role of the vehicle’s suspension is to efficiently shield the chassis from external roadside disturbances and shocks, thereby providing comfort, stability, and improved handling [
2,
3]. This isolation helps to enhance ride comfort by reducing the transmission of road shocks and vibration to the occupants while maintaining a firm grip of the tire on the road surface [
4]. In some cases, inherent stability challenges in traditional suspension systems can be non-trivial, even at low speeds. Isolation of these oscillations and overcoming the conflicting objectives are essential considerations in the design of dynamic control for vehicle suspension.
Based on the energy dissipation mode, the basic three types of suspension systems are the passive suspension system (PSS), active suspension system (ASS), and semi-active suspension system (SASS) [
5,
6]. A PSS consists of a metal spring, a standard damper, and a wishbone, which require no external energy for their operation. In contrast, an ASS requires external power for actuator operation to tackle exogenous disturbances and uncertainties encountered during vehicle forward motion and taking on various road maneuvers [
7,
8]. However, a SASS requires less energy than the active suspension system [
9], as only a limited amount of actuator power is needed to change the orifice of the variable damper.
There are several types of variable dampers, including magnetorheological dampers, electrorheological dampers, electrohydraulic dampers, and electromagnetic dampers, which require limited power for their operation and only a few parameters [
10]. However, with the ability to dissipate energy without the expenditure of external power, the main problem associated with semi-active suspension system is the passivity constraints, which limit their ability to minimize the vertical acceleration and mitigate undesired forces during the height and attitude motions of the vehicle [
11]. An integration-based, linear parameter-varying controller (LPV) was employed to track road velocity under certain irregularities to enhance a semi-active suspension system’s driving comfort and system stability performance. However, improvements in suspension deflection were not considerable compared to the ride comfort owing to the conflicting nature of the target parameters [
12]. Based on preview information, a vibration isolation platform was designed, and an MPC algorithm was applied to enhance the suspension acceleration across different road surfaces. The proposed study utilized a vehicle-mounted vibration mitigation controller by employing a damping control damper model [
13]. A PID controller was optimized using particle swarm optimization (PSO) to improve the passenger comfort and road holding of a quarter-car SASS. The controller was verified by a hardware-in-the-loop approach, while the MR damper was modeled using an ANN [
14]. However, the main focus was improving ride comfort performance compared to a passive suspension system.
To achieve a better compromise between passenger comfort and road-holding goals, the authors in [
15] employed several controllers in an in-wheel electric vehicle to reduce the magnitude of oscillations by considering random road profiles. The authors considered the unbalanced vertical forces arising from a switched reluctance motor. However, each controller could provide the best improvement depending on the particular type of vehicle and the direction in which the destabilizing forces acted on the vehicle. To reduce the variation in the magnitude of the passenger compartment acceleration, the authors in [
16] demonstrated an optimal preview control approach to eliminate the jerk in car body acceleration, thereby improving the comfort level of a quarter-car SASS. The results showed reduced amplitude vibrations compared to disregarding the preview controller. However, the jerk in the dynamic response of the ride comfort-oriented index was flattened at the cost of a
reduction in the road-holding performance.
The authors in [
17] investigated using an active aerodynamic surface to track roll motion and minimize the tracking error to address the conflicting parameters in a half-car model. The optimal controller effectively reduced the magnitude of transverse forces acting on the car body, enabling the tracking of ideal roll motion during circular and different lane maneuvers. To attenuate an anticipated road disturbances and oscillations affecting vehicle attitude motion, the MPC strategy was demonstrated to ensure the desired maneuvering of the four-DOF half-car ASS [
18]. A detailed frequency-domain analysis was conducted on a sports car using an AAS on different road surfaces, and an improvement of 30% in ride comfort was achieved but at the expense of power requirements and associated complex circuitry [
19].
Classical on–off controllers like sky-hook and ground-hook control techniques have been experimentally validated extensively in the literature. However, they are not well suited for a multi-objective function. Similarly, a mix-one-sensor controller offers an exclusive control strategy that focuses on the conflicting objectives but not both simultaneously [
20]. To enhance sports cars’ suspension travel and safety in a cornering maneuver, actively controlled spoilers were employed using numerical techniques. The normal loads in the front and rear axles were controlled by changing the angle of attack in the opposite direction of the vehicle. However, the study focused only on the critical stability of the vehicle for passive and ASS [
21]. A review article published by [
22] presented different models of the movable aerodynamic elements for the solution of conflicting parameters, safety, and low fuel consumption of various modern cars. Therefore, the major factor in enhancing ride comfort and maintaining a firm grip along the road surface lay in the effective consideration of the ASS force. By optimizing the application of this force alongside the control force of a SASS, the vehicle could achieve improved stability and enhanced handling capability. Based on the above findings, this paper presents an effective control scheme to enhance a quarter-car semi-active suspension system’s ride comfort and road-holding capability in conjunction with an active airfoil. The classical linear quadratic regulator (LQR), which is well suited for a multi-objective cost function, is employed to meet the conflicting objectives and minimize the objective performance index of the system [
23].
The objective of this study can be outlined as follows:
In the first place, the effect of the downward active aerodynamic force on a quarter-car model with a SASS is analyzed. The performance measure is optimized to enhance driving comfort and road-holding capabilities simultaneously.
Then, the simulation result of an active suspension system that requires external power for the actuator operation and incorporates an active aerodynamic surface is presented as a benchmark case.
Finally, a comprehensive comparative study is conducted with the benchmark case and other suspension systems. This evaluation is based on both time- and frequency-domain analyses to assess the suspension performance.
The remainder of the paper is organized in the following manner.
Section 2 presents the modeling of the SASS supplemented by an active airfoil. The problem formulation is illustrated in
Section 3.
Section 4 outlines the optimal control design strategy.
Section 5 discusses the results and their implication. Finally,
Section 6 concludes the paper.
4. Optimal Controller Design
The design of a 1/4 car SASS system involves adjusting the target parameters to improve the dynamic response against road unevenness and irregularities. This work aimed to design the optimal controller that could attenuate the exogenous disturbances induced into the car body during the vehicle’s forward motion. For driving comfort, deviation in the car body acceleration was minimized without losing tire grip on the road surface. All these objectives were obtained, keeping the rattle space and the passivity constraints of the variable damper of the SASS. The target parameters were optimized by suitably tuning the weighting constant parameters. The objective cost function of the proposed model is depicted in Equation (
10).
The optimal damping coefficient
minimizes the performance objective function subject to the passivity constraints, as described in Equations (
11) and (
14). Similarly, the bilinear time-varying state equation of the SASS supplemented by an active dynamic surface is described by Equation (
11).
where
represents the initial condition,
w shows the input velocity of the kinematic disturbance, and
and
are the input constant matrices. The variable damper coefficient
is approximated in Equation (
12) by:
Similarly, Equation (
13) represents the aerodynamic force generated by the AAS in the downward direction.
where the variable damper satisfies the following passivity constraints:
Alternatively, the time-varying state equation based on the variable damper coefficient can be described by Equation (
15).
The constant matrices, which represent fixed parameters in the system and remain unchanged throughout the analysis, are depicted as follows:
where
is the output matrix,
is a positive definite weighting matrix, and
and
are positive semi-definite symmetric matrices [
38]. The dynamic model of the SASS is nonlinear, and a limited amount of control force is required to adjust the orifice of the variable damper. Therefore, in the first place, the unconstrained optimization problem for the active suspension system incorporated by an AAS is analytically solved. From Equation (
12), the active force from the ASS with an AAS is compared with the control force of the SASS. Following this, the updated value of the variable damper
for the SASS is calculated by selecting either the minimum, maximum, or no change in the control force. For the ASS collaborated by an AAS, the control law calculates the optimal feedback gains
and
, which minimizes the total performance criterion and is described by Equation (
16).
The target performance indices includes the vertical car body acceleration
, suspension deflection
, and dynamic tire load
. These indices are associated with passenger comfort, handling, and vehicle dynamic stability. The weighting factors
in the synthesized model are selected based on ride comfort and road holding preferences. In the conventional design of the LQR controller, we can penalize the passenger compartment acceleration (ride comfort) by
at the expense of degradation in the road holding. Similarly,
prioritizes the road holding index,
is selected to minimize the control effort, and
is related to the airfoil force. These were the optimized values tuned based on trial and error method and were related to our previous work [
16]. The weighting factors were chosen according to specific road and speed conditions. The resultant control force that minimized the above performance criterion is illustrated in Equation (
17) as:
is the solution of the algebraic Riccati equation and is positive definite [
39,
40]. The Riccati equation for the active suspension system, incorporated by an AAS is given by Equation (
18):
where the matrices
and
are given by:
is symmetric and non-negative matrix. The nonlinear state equation of the SASS Equations is described by Equation (
21), which involves a control switching mechanism of the control variable in the presence of a disturbance signal.
Equation (
22) depicts the closed-loop state-space matrix of the system under the feedback control law:
Since the SASS does not provide any active supply to the suspension to counter the external vibratory forces, the system is always stable. Only a limited amount of power is required for the modulation mechanism of the variable damper [
41]. From Equation (
12), the control force from the variable damper of the SASS equipped with an AAS is similar to the ASS without considering the passivity constraints of Equations (
11) and (
14).
The variable damper force
is determined by using the control algorithm in Equation (
23). The resultant value is subsequently substituted into the system matrix
, which becomes piecewise linear. This integrated approach of utilizing the control force of the resultant SASS and aerodynamic force is used to achieve the desired performance metrics.
From Equation (
12), the objective of the controller is the minimization of the least squares difference
between the active and semi-active control forces. By using Equation (
23) and the well-established rule of calculating the variable damper coefficient
, the subsequent values of the variable damping coefficient are replaced with
and
in the time-varying state Equation (
21). When encountering a singularity at the point
, the optimal controller retains the most recent value of
in the simulation.
6. Conclusions
In this research work, the simulation of a SASS in the presence of an AAS was successfully modeled and implemented. By optimizing the two sets of weighting factors, the optimal controller managed to provide a smoother ride comfort while maintaining a solid tire grip on the road, addressing the inherent trade-off between the two target parameters. The frequency-domain analysis confirmed that the added effect of the AAS successfully mitigated the deviation in the first hop frequency (body hop) and second hop frequency (tire hop) without negatively affecting the road-holding capability. As a benchmark, the active suspension with an active airfoil could effectively reduce the effect of disturbances at the resonant peaks in the car body’s acceleration. However, at the same time, the proposed suspension system had a better low frequency response compared to the PSS, and the passenger comfort was greatly improved at the tire resonant frequency compared to all other systems. From the time-domain results, it was evident that the proposed system significantly reduced the amplitude of vibration under extreme conditions like road bumps, where the total performance was enhanced by 49.17% and 44.88% compared to that of the PSS and ASS for the first weighting set. At the same time, ride comfort saw an improvement of about 76.37% at the expense of about −35% in road holding. However, all target parameters significantly improved compared to those of the ASS. Emphasizing road holding, the indices of the target parameters were considerably enhanced. The ride comfort, road holding, and suspension deflection improved by 59.58%, 42.56%, and 34.07%, respectively, compared to the PSS. When compared to the ASS, these indices showed enhancements of 66%, 18.43%, and 13.25%, respectively. A similar trend was observed for the asphalt road with a stationary stochastic white Gaussian signal. From an overall performance perspective, the proposed model successfully balanced the conflicting goals of passenger comfort and road-holding capability. This model can be extended to the half-car model and full-car models, where a detailed aerodynamic analysis can further substantiate these findings.