Particle swarm optimized fuzzy proportional-integral-derivative controller-based transverse leaf spring active suspension for vibration control

Abstract

The transverse leaf spring (TLS) suspensions are a promising option for van vehicles due to their high load-carrying capacity and excellent handling stability. However, its ride comfort remains a major challenge. This paper investigates and compares the effects of semi-active and active control strategies to enhance the ride comfort of TLS suspensions. Firstly, a four-degree-of-freedom (4-DOF) half-car model and a multi-body dynamics (MBD) model of the TLS suspensions are established. The MBD model has higher accuracy and can describe the medium and high frequency characteristics of the TLS suspensions, such as the suspension offset frequency and the frequency response function of the body vertical acceleration (BVA). Therefore, based on the MBD half-car model with TLS suspensions, this paper proposes an optimal fuzzy PID active control strategy considering the left and right suspension coupling. The optimization objectives are the BVA, the left and right suspensions dynamic deflection, and the left and right wheels dynamic displacement. The integral absolute error is used as the evaluation criterion. The left and right fuzzy PID controllers’ parameters are obtained through particle swarm optimization. Simulation results demonstrate that the particle swarm optimization fuzzy PID active control strategy effectively controls the low-frequency vibration of the TLS suspensions and suppresses the medium- and high-frequency vibration characteristics compared with the traditional skyhook semi-active control strategy. This technology provides a reference for improving the ride comfort of the TLS suspensions.

Keywords

Transverse leaf spring half-car multibody dynamic model fuzzy PID particle swarm optimization semi-active suspensions active suspensions

Introduction

The transverse leaf spring (TLS) has the advantages of strong loading capacity, high fatigue reliability, high roll stiffness, and small space occupation. Therefore, it has been gradually applied to the front suspensions of some light buses in recent years. At present, generally used dynamics modeling approaches of suspensions include analytical modeling (including quarter-car^1,2 and half-car suspension model)^3,4 and the multi-body dynamics (MBD) simulation^5,6 through commercial software. The quarter-car suspension model is simple and efficient for validating unilateral suspension control strategies but cannot describe bilateral motion and roll characteristics of the TLS suspension, making coordinated control strategy simulation for bilateral dampers impossible. The half-car suspension model can address these limitations, but it simplifies TLS stiffness as linear, neglecting its nonlinearity, and does not consider the stiffness of anti-roll bars, rubber bushes, and other components. The MBD model is the preferred choice for studying suspension dynamics as it addresses the limitations of the above two dynamic models. However, its disadvantage lies in the complexity of modeling and calibration. For the MBD simulations, Adams/Car is currently widely used in vehicle dynamics modeling.⁷ Therefore, two suspension models, the half-car and the MBD model, are analyzed to assess the performance of the TLS suspension.

Dynamics model simulation serves as an effective method for suspension performance analysis and structural design. Additionally, it offers a dynamic analysis platform for semi-active or active control research of traditional passive suspensions. Since TLS suspensions directly affect the handling performance and ride comfort of vehicles, this paper mainly discusses the vibration control of TLS suspension by semi-active and active suspension control strategies. A semi-active control suspension system is mainly composed of elastic elements and controllable dampers, where elastic elements are used to store energy generated by vehicle vibration whereas controllable dampers are used to dissipate energy generated by vehicle vibration.⁸ The magnetorheological damper is widely used in semi-active suspension systems due to its simple structure, fast response, high dynamic range, and low power cost.^9,10 Active control forces are generated directly by hydraulic or electromagnetic actuators.¹¹ Compared with semi-active control, active control has stronger adaptability.¹² Therefore, this study investigates the feasibility and control effectiveness of semi-active and active control strategies based on the TLS suspensions based on a controllable damping force.

In recent years, semi-active and active control strategies for different suspension types have become increasingly mature, such as the Skyhook control,¹³ the sliding mode control,¹⁴ the neural network,¹⁵ the fuzzy logic,¹⁶ etc. The PID controller is widely used because of its simple operation, strong applicability, and low cost.¹⁷ While the initial parameters of PID controllers require manual tuning. Therefore, fuzzy control is introduced due to its self-adaptive advantage.¹⁸ However, there are few studies on semi-active or active control strategies for TLS multibody dynamic suspension models. This study combines the fuzzy control strategy with the PID controller to propose an active control strategy for the MBD of TLS suspensions.

In recent years, extensive research has been conducted by numerous scholars in the field of novel intelligent optimization algorithms. These algorithms include Particle Swarm Optimization (PSO),^19–21 Artificial Bee Colony Algorithm (ABCA),^22,23 Cuckoo Search Algorithm (CSA),^24,25 Genetic Algorithm (GA),^26,27 Differential Evolution (DE) Algorithm,^28,29 Dragon-Fly Algorithm (DFA),^30,31 Grey–Wolf Optimization (GWO),³² etc. The ABCA operates using forager bees, onlooker bees, and scout bees. Each generation's particle position update depends on the difference between the previous generation’s particle position and the current particle's neighborhood range. Consequently, the ABCA algorithm exhibits slow search capabilities and demands significant computational resources and efficiency. The CSA combines random step sizes with Levy flight strategies for global particle position updates. It requires more iterations for convergence, and its search mechanism is relatively random, leading to unstable search paths. The GA initially requires the transformation of variables into binary numbers, which can result in a higher risk of missing the optimal solution, especially with small sample sizes. To address this issue, genetic algorithms typically demand a large population size, leading to increased demands on computational resources, efficiency, and time consumption. The DE algorithm introduces mutation scaling and crossover probability factors, involving more parameters in the optimization process. It exhibits strong global search capabilities but requires a larger population size and, consequently, more computational resources and time. The DFA updates particle positions through a sequence of five steps, introducing numerous undetermined parameters that impact optimization results. While it considers neighboring particle velocities to some extent, it may require multiple runs to obtain stable results. The GWO algorithm replaces particle velocity updates with a hierarchical updating mechanism. However, it involves multiple tuning parameters, leading to increased demands on computational resources and time. In contrast, PSO stands out for its excellent global search capability, faster convergence, and suitability for optimizing suspension control strategy parameters.^33,34 PSO offers several advantages: it performs well in high-dimensional problems, provides uniform sampling of the search space, and results in quicker convergence, shorter computation times, and resource savings.^35,36 Consequently, this study selects PSO to optimize fuzzy PID controller parameters due to its computational efficiency and convergence performance.

In summary, this study aims to propose a PSOF PID active control strategy to reduce the medium- and low-frequency vibration characteristics of the TLS suspension. To comprehensively evaluate the ride comfort and handling stability of the suspension, this paper selects the suspension working space, the wheel dynamic deflection and the acceleration of the sprung mass to construct the PSO optimization objective. To evaluate the performance of the proposed PSOF PID controller, its effectiveness is compared with that of the traditional SH PID semi-active control strategy.

Dynamic models

Half-car model with TLS suspensions

The structure of the TLS suspensions is depicted in Figure 1. The upper end of the hydraulic damper is fixed on the vehicle chassis through a rubber mount, while the lower end is bolted to the knuckle. The anti-roll bar is linked to the lower control arm via the drop link. The rubber limit block's purpose is to limit the TLS suspension’s stroke. The lower control arm is connected to the hub and tire through the knuckle. Unlike McPherson suspension, the TLS suspension employs a TLS as the elastic element to replace the coil spring. The end of the TLS is connected to the lower control arm, while the middle part is fixed to the subframe via upper and lower rubber mounts. The suspension stiffness is determined by the leaf spring’s end length.

Figure 1.

Transverse leaf spring McPherson suspension. 1-Hydraulic damper; 2-Drop link; 3-Rubber Bumpstop; 4-Anti-Roll Bar; 5-Tie rod; 6-Lower Control Arm; 7-Composite Transverse Leaf Spring; 8-Knuckle; 9-Hub and Tire.

To investigate the motion behavior of the TLS suspensions, simplifications of the TLS suspensions are necessary. The dynamic model of the TLS suspensions is shown in Figure 2(a), where A, C, D, E, F, and G represent the left mounting points while the right mounting point is represented by adding “ ' ” to the corresponding parameters. Generally, the left-hand side is presented by subscript “l” and the right-hand side is indicated by subscript “r.” The symbols in Figure 2 are explained in Table 1. Due to the inability to adjust the stiffness of the TLS suspensions during movement, the adjustable damping force is employed to study the dynamic characteristics of the TLS suspensions. From Figure 2(a), it is observed that $K_{l l},$ $K_{l r}$ , $C_{l}$ , and $C_{r}$ are not perpendicular to the vehicle motion plane. As this study only focuses on the vertical performance of the TLS suspensions, only the vertical stiffness and damping are considered. There is a corresponding conversion relationship between $K_{l}$ , $C_{l}$ and $K_{s}$ , $C_{s}$ . The 4 degree-of-freedom (4-DOF) half-car model is established, as shown in Figure 2(b).

Figure 2.

The 4-DOF half-car model for TLS suspension: (a) The dynamic model; (b) The 4-DOF half-car model.

Table 1.

Symbol definitions.

Symbol	Description	Symbol	Description
A	The connection position of the lower control arm and the subframe	$m_{2}$	Body mass
B	The wheelbase	$K_{s l}$	Left suspension stiffness
C	The connection position of the lower control arm and the knuckle	$K_{l l}$	Left stiffness of the TLS, which are vertical to AC, respectively.
D	The connection position of the knuckle and the wheel	$K_{t l}$	Left tire stiffness
E	The lower installation position of the shock absorber	$C_{s l}$	Left suspension damping
F	The upper installation position of the shock absorber	$C_{l}$	Left damper damping
G	The installation position of the transverse leaf spring end and the lower control arm	$C_{t l}$	Left tire damping
W	Wheel center	$h_{r}$	The roll center height
AG	Left lower control arm	$h_{g}$	The height of the sprung mass’ center
CDE	Left Steering knuckle	$q_{l}$ ,	The left road excitation
EF	The shock absorber	$x_{1 l}$	The vertical displacements of left unsprung mass
O	The half-car suspension roll center	$x_{2 l}$	The vertical displacements of left sprung mass
$V_{c}$	The sprung mass center	$x_{2}$	The vertical displacement of the body
$b$	The distance from the mass center to the left suspension.	$β$	Roll angle
$m_{1}$	Unsprung mass	$I_{x}$	The rotational inertia of the vehicle
$m_{11}$	Unsprung mass of the suspension left side	$F_{c l}$	The control forces of the left damper

In this study, the influence of vehicle pitch on torque is ignored. The equations of motion of the half-car model with TLS suspensions are shown in equation (1). The differential equations are solved by Matlab/Simulink. The model parameters are listed in Table 2.

m_{2} {\ddot{x}}_{2} + k_{s l} (x_{2 l} - x_{1 l}) + c_{s l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) + k_{s r} (x_{2 r} - x_{1 r}) + c_{s r} ({\dot{x}}_{2 r} - {\dot{x}}_{1 r}) + F_{c l} + F_{c r} = 0, I_{x} \ddot{β} - k_{s l} (x_{2 l} - x_{1 l}) - c_{s l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) + k_{s r} (x_{2 r} - x_{1 r}) + c_{s r} ({\dot{x}}_{2 r} - {\dot{x}}_{1 r}) + F_{c l} + F_{c r} = 0, m_{1 l} {\ddot{x}}_{1 l} + k_{t l} (x_{1 l} - q_{l}) + c_{t l} ({\dot{x}}_{1 l} - {\dot{q}}_{l}) - k_{s l} (x_{2 l} - x_{1 l}) - c_{s l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) - F_{c l} = 0, m_{1 r} {\ddot{x}}_{1 r} + k_{t r} (x_{1 r} - q_{r}) + c_{t r} ({\dot{x}}_{1 r} - {\dot{q}}_{r}) - k_{s r} (x_{2 r} - x_{1 r}) - c_{s r} ({\dot{x}}_{2 r} - {\dot{x}}_{1 r}) - F_{c r} = 0,

(1)

Table 2.

Parameters in equations of motion.

Name	Value	Name	Value
$m_{2}$ / $kg$	1498.602	$K_{t r}$ /( $N \cdot m^{‐ 1}$ )	$110 \times 10^{3}$
$m_{1 l}$ / $kg$	48.615	$C_{t l}$ /( $N \cdot s \cdot m^{‐ 1}$ )	0
$m_{1 r}$ / $kg$	48.615	$C_{t r}$ /( $N \cdot s \cdot m^{‐ 1}$ )	0
$K_{s l}$ /( $N \cdot m^{‐ 1}$ )	71.39	$B$ /(m)	1.798
$K_{s r}$ /( $N \cdot m^{‐ 1}$ )	71.39	$b$ /(m)	0.869
$K_{t l}$ /( $N \cdot m^{‐ 1}$ )	$110 \times 10^{3}$	$I_{x}$ /( $kg \cdot m^{2}$ )	$1.89 \times 10^{3}$

where $x_{2 l} = x_{2} - b β$ , $x_{2 r} = x_{2} + (B - b) β$ , ${\ddot{x}}_{2}$ is the body vertical acceleration (BVA), $x_{2 l} - x_{1 l}$ is the left suspension dynamic deflection (SDD), and $x_{1 l} - q_{l}$ is the left wheel dynamic displacement (WDD).

In the dynamic model, $K_{l}$ and $C_{l}$ are nonlinear, and $K_{s}$ and $K_{l}$ are not identical, so an MBD model was developed using Adams/car, as shown in Figure 3. The sprung mass was represented using the centralized mass method. The TLS was modeled using nonlinear beam elements. The anti-roll bar was simplified as a flexible body component in Abaqus using a free modal method. The deformation and friction effects on the suspension system were neglected. The main parameters of the vehicle are listed in Table 3.

Figure 3.

Multi-body dynamic model of the TLS suspension.

Table 3.

Parameters of the MBD model.

Suspension parameters	Units	Values
Camber angle	$°$	0.35
Toe angle	$°$	0.283
Tire Unloaded radius	mm	349.45
Tire stiffness	N/mm	110
Wheel mass	kg	20
Sprung mass	kg	1498.602
CG height	mm	518.872
Wheelbase	mm	4235
Front Drive ratio	%	100
Front Brake ratio	%	70

Road excitations

To investigate the differences between the 4-DOF half-car model of the TLS suspension and the MBD half-car model, a suspension offset frequency test road was established. To analyze the effects of different damping control strategies on the ride comfort of the TLS suspension, a random road model and a speed bump road model were established, respectively.

Suspension natural frequency test

To investigate the damping control strategy, it is necessary to ensure the accuracy of the TLS suspension model. According to the China National Standard GB/T 4783-1984,³⁷ the suspension natural frequency was tested using the roll-off method with a roll-off height of 90 mm, as shown in Figure 4(a). To simulate the suspension natural frequency of the Simulink model and Adams MBD model, a step input signal was utilized. To ensure the MBD model stays static equilibrium at the beginning of the simulation, the step began at 5 s and ended at 5.1 s, with a total simulation duration of 10 s. The step input signal is shown in Figure 4(b).

Figure 4.

Suspension natural frequency test: (a) roll-off method; (b) step input.

Continuous road profile

To simulate the road roughness, the Gaussian white noise method³⁸ is utilized and the road excitation is given as

{\dot{q}}_{l} (t) = - 2 π f_{0} q (t) + 2 π n_{0} \sqrt{G_{z_{r}} (n_{0}) v} w (t),

(2)

where $q (t)$ is the random road excitation, $f_{0}$ is the lower cutoff frequency, $f_{0} = 0.01 Hz$ . $n_{0}$ is the reference spatial frequency $n_{0} = 0.1 Hz$ . $G_{z_{r}} (n_{0})$ is the power spectral density function of road roughness, $1 0^{- 6} m^{- 3}$ . $v$ is the vehicle speed. $w (t)$ is the Gaussian white noise with a mean value of 0.

The left and right road excitations are implemented by setting different intensities, where $w_{l} (t)$ is 1 and $w_{r} (t)$ is 1.4. According to ISO 8608,³⁸ road roughness is classified into 8 levels from A to H. The vehicle with transverse leaf spring suspension studied in this article primarily operates on county-level and township-level roads in China. It has been observed that these roads are predominantly class C road.³⁹ Hence, road class C is chosen in this work, where $G_{z_{r}} (n_{0}) = 256 \times 1 0^{- 6} m^{- 3}$ . The vehicle speed is set at 20 m/s and the road excitation is shown in Figure 5.

Figure 5.

Road excitation according to ISO-8608.

Discrete impact road

To investigate the damping control strategy of the TLS suspensions when passing over bumps or other obstacles, a trapezoidal bump was used to simulate the road impact. The spatial-domain mathematical model of the trapezoidal bump is expressed in equation (3), while the bump schematics are shown in Figure 6(a). The bump was built based on the class A road in Simulink, as shown in Figure 6(b).

y = {\begin{cases} \frac{h}{d} x + \frac{h}{b} 2 d - \frac{b}{2} \leq x \leq - \frac{b}{2} + d, \\ h - \frac{b}{2} + d \leq x \leq \frac{b}{2} - d, \\ \frac{h}{d} x + \frac{h}{b} 2 d \frac{b}{2} - d \leq x \leq \frac{b}{2}, \end{cases}

(3)

Figure 6.

Discrete road impact: (a) Trapezoidal bumps; (b) Class A road impulse.

where $x$ is the horizontal length of the bumps (mm), $x = v t$ . $v$ is the vehicle speed (m/s). $t$ is the simulation time. $y$ is the vertical length (mm). $h$ is the vertical height, $h = 90 mm$ . b is the bottom width of the speed bumps, $b = 380 mm$ . d is the horizontal projection length of the slope, $d = 150 mm$ .

Control strategy and optimization methodology

The most commonly used control strategy for vehicle suspension dampers is the PID controller consisting of three parts: the proportional control (P), the integral control (I) and the derivative control (D). The mathematical expression for the PID control algorithm is as follows.⁴⁰

u (t) = K_{p} e (t) + K_{i} \int_{0}^{t} e (t) d t + K_{d} d e (t) / d t,

(4)

where $u (t)$ is the output of the PID controller, that is, the adjustable force $F_{c l}$ in Figure 7. $K_{p}$ is the proportional coefficient. $K_{i}$ is the integral coefficient. $K_{d}$ is the derivative coefficient. $e (t)$ is the error signal.

Figure 7.

PSOF PID active control strategy.

The evaluation of vehicle dynamic characteristics mainly includes ride comfort and road handling performance.⁴¹ A good ride comfort depends on minimizing BVA and SDD. Handling is the ability of a vehicle to maintain stability on various roads, which is mainly evaluated by the WDD. In equation (4), due to the presence of the integral term $\int_{0}^{t} e (t) d t$ and the derivative term $d e (t) / d t$ , when ${\dot{x}}_{2}$ is used as the input of the PID controller, $u (t)$ not only considers the velocity and acceleration of the body but also takes into account the SDD. Therefore, in this paper, a PID controller was constructed with the body velocity ${\dot{x}}_{2}$ as the input variable, as shown in equation (5).

e = {\dot{x}}_{v e l_0} - {\dot{x}}_{2},

(5)

where $e$ is the input variable of the PID controller. ${\dot{x}}_{v e l_0}$ is the expected value of the body's vertical velocity, ${\dot{x}}_{v e l_0} = 0$ .

This paper proposes a PSOF PID active control strategy for the TLS suspension and compares it with the traditional SH PID semi-active control strategy. The control strategy consists of two fuzzy PID controllers for the left and right sides. Taking the left TLS suspension as an example, the control structure is shown in Figure 7. When the TLS suspension is subjected to random road excitation $q_{l}$ , the fixed position of the damper and the body produces a vertical velocity ${\dot{x}}_{2 l}$ , which serves as the input of the left PSOF PID controller. The controllable damper generates a controlled force $F_{c l}$ to achieve the purpose of vehicle body vibration reduction.

Semi-adaptive skyhook PID control

The skyhook control, proposed by Karnopp,⁴² is a classical semi-active control strategy that assumes the presence of a virtual damper between the sprung mass and a reference fixed position (the skyhook) to suppress sprung mass vibration. Using equation (1) and taking the left damper as an example, the SH-PID controller generates the Skyhook control force denoted by $F_{c l}$ , as shown in equation (6).

F_{c l} = c_{c l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) = c_{s l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) + c_{s k y_l} \cdot {\dot{x}}_{2 l},

(6)

where $c_{s l}$ is the damping coefficient of the passive damper. $c_{c l}$ is the damping coefficient of the SH-PID controller, as shown in equation (7).

c_{c l} = c_{s l} + c_{s k y_l} \frac{{\dot{x}}_{2 l}}{({\dot{x}}_{2 l} - {\dot{x}}_{1 l})},

(7)

According to equation (7), the direction of the Skyhook equivalent damping force is determined by the relative velocity direction between the two ends of the damper. The body vibration is only suppressed when ${\dot{x}}_{2 l}$ and $({\dot{x}}_{2 l} ‐ {\dot{x}}_{1 l})$ are in the same direction. Therefore, this paper establishes SH PID controllers to analyze their effect on vibration control of TLS suspension. The SH PID control damping force can be expressed by the following equation

F_{s k y_l} = {\begin{cases} F_{S H - P I D} & {\dot{x}}_{2 l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) \geq 0, \\ F_{p a s s i v e} & {\dot{x}}_{2 l} ({\dot{x}}_{2 l} - {\dot{x}}_{1 l}) < 0, \end{cases}

(8)

where $F_{S H ‐ P I D}$ is the Skyhook control force. $F_{p a s s i v e_l}$ is the damping force of the passive shock absorber.

Adaptive fuzzy PID control

According to equation (8), the skyhook damping control force exhibits a discontinuous step phenomenon. Therefore, this paper aims to achieve the purpose of continuous adjustment of damping force based on the PID active control strategy. As traditional PID control requires trial and error to obtain $K_{p}$ , $K_{i}$ , and $K_{d}$ , this paper introduces fuzzy control to establish a fuzzy PID active controller for the TLS suspension and compare it with the SH-PID semi-active controller. The parameters of the fuzzy PID controller are shown in equation (9).⁴³

{\begin{cases} K_{p} = K_{p 0} + Δ K_{p} \\ K_{i} = K_{i 0} + Δ K_{i} \\ K_{d} = K_{d 0} + Δ K_{d} \end{cases}

(9)

where $K_{p 0}$ , $K_{i 0}$ , and $K_{d 0}$ are the initial values of the fuzzy PID controller. ${Δ K}_{p}$ , $Δ K_{i}$ , and $Δ K_{d}$ are the compensation values of the fuzzy controller, which are introduced to realize self-adaptive control.

This paper employs two-dimensional fuzzy control to handle the parameters of the vehicle’s vertical PID controller, as one-dimensional fuzzy control cannot reflect the dynamic characteristics of the suspension system, and multi-dimensional fuzzy control suffers from high-input dimensionality and low control efficiency. According to equation (5),

e

denotes the deviation between the reference value and the target control variable, while

e c

represents the deviation change rate. These two variables are utilized as inputs for the fuzzy controller. Through simulation analysis, it has been determined that the deviation

e

and the deviation change rate

e c

have maximum ranges of [−600,600] and [−7000,7000], respectively. Thus, the fundamental domains for

e

and

e c

are [−600,600] and [−7000,7000], respectively. To facilitate further analysis, quantification factors

K_{e} = 1 / 600

and

K_{e c} = 1 / 7000

are introduced, which normalize e and ec from their fundamental domains to the fuzzy domain of [−1,1]. Considering the control system's response speed, saturation modules are applied to define the fuzzy domains for

e

and

e c

as [−3, 3]. Trial and error methods are utilized to determine the fuzzy domain ranges of

{Δ K}_{p}

Δ K_{i}

, and

Δ K_{d}

. These parameters are summarized in Table 4.

Table 4.

Fuzzy controller parameters.

Type	e	ec	$K_{e}$	$K_{e c}$	${Δ K}_{p}$	$Δ K_{i}$	$Δ K_{d}$
Value	$- {\dot{x}}_{2}$	$- d {\dot{x}}_{2} / d t$	1/600	1/7000	[−1,1]	[−1,1]	[−0.1,0.1]
Value	[−3,3]	[−3,3]	1/600	1/7000	[−1,1]	[−1,1]	[−0.1,0.1]

The input and output variables of e, ec, ${Δ K}_{p}$ , $Δ K_{i}$ , and $Δ K_{d}$ are divided into seven linguistic subsets using the seven-segment fuzzy method, which are {NB, NM, NS, Z, PS, PM, PB}. The input and output membership functions use zmf type for NB, smf type for PB, and trimf type for the rest. The fuzzy domains of the input and output variables are shown in Figure 8. The maximum membership degree average method is used for the fuzzy rules. The Mamdani method is used for fuzzy inference. A total of $3 \times 49$ fuzzy control rules are designed,⁴³ as shown in Table 5.

Figure 8.

Input and output memberships of the fuzzy controller.

Table 5.

Fuzzy control rule of ${Δ K}_{p}$ , $Δ K_{i}$ , and $Δ K_{d}$ .

$Δ K_{p}$ / $Δ K_{i}$ / $Δ K_{d}$		ec
$Δ K_{p}$ / $Δ K_{i}$ / $Δ K_{d}$		NB	NM	NS	ZO	PS	PM	PB
e	NB	PB/NB/PS	PB/NB/NS	PM/NM/NB	PM/NM/NB	PS/NS/NB	ZO/ZO/NM	ZO/ZO/PS
	NM	PB/NB/PS	PB/NB/NS	PM/NM/NB	PS/NS/NM	PS/NS/NM	ZO/ZO/NS	NS/ZO/ZO
	NS	PM/NB/ZO	PM/NM/NS	PM/NS/NM	PS/NS/NM	ZO/ZO/NS	NS/PS/NS	NS/PS/ZO
	ZO	PM/NM/ZO	PM/NM/NS	PS/NS/NS	ZO/ZO/NS	NS/PS/NS	NM/PM/NS	NM/PM/ZO
	PS	PS/NM/ZO	PS/NS/ZO	ZO/ZO/ZO	NS/PS/ZO	NS/PS/ZO	NM/PM/ZO	NM/PB/ZO
	PM	PS/ZO/PB	ZO/ZO/PS	NS/PS/PS	NM/PS/PS	NM/PM/PS	NM/PB/PS	NB/PB/PB
	PB	ZO/ZO/PB	ZO/ZO/PM	NM/PS/PM	NM/PM/PM	NM/PM/PS	NB/PB/PS	NB/PB/PB

PSO algorithm

According to equation (9), it is impossible to directly determine the optimal initial values of $K_{p 0}$ , $K_{i 0}$ , and $K_{d 0}$ . The PSO algorithm is suitable for parameter optimization of suspension damping control strategy due to its advantages of convergence speed and simple iteration. In this algorithm, particles in the search space represent the solutions to the problem. Their velocity determines the flying direction and distance, moving them toward the optimal solution in space. The optimal position of a particle represents the best approximate solution to the problem.

Assuming N particles in the D-dimensional solution space, each particle’s position $x_{i} = (x_{i 1}, x_{i 2}, \dots, x_{i D})$ and flight speed $v_{i} = (v_{i 1}, v_{i 2}, \dots, v_{i D})$ are initialized. The velocity and position of particles are updated based on their individual best solution $(P b e s t = (P_{1}, P_{2}, \dots, P_{D}))$ and the global best solution $(G b e s t = (G_{1}, G_{2}, \dots, G_{D}))$ , respectively, determined by the fitness function value during iteration in the solution space. The updated equations for the position and velocity of each particle are expressed as follows⁴⁴

x_{i j} (k + 1) = x_{i j} (k) + v_{i j} (k + 1),

(10)

v_{i j} (k + 1) = ω (k) v_{i j} (k) + c_{1} r_{1} (P b e s t_{j} - x_{i j} (k)) + c_{2} r_{2} (G b e s t_{j} - x_{i j} (k)),

(11)

ω (k + 1) = ω (k) - k \cdot (ω_{\max} - ω_{\min}) / I t e r_{\max},

(12)

where

i = 1,2,3, \dots, N

j = 1,2,3, \dots, D

. k is the number of iterations.

{I t e r}_{\max}

is the max iteration.

c_{1}

and

c_{2}

are learning factors that can make any particle approach the individual best solution and the global best solution, thus finding the global optimal solution.

r_{1}

and

r_{2}

are random constants between [0,1].

ω

is the inertial weight factor, which aims to balance the local and global best positions.

The PSOF PID controller’s initial parameters, $K_{p 0}$ , $K_{i 0}$ , and $K_{d 0}$ , are determined by the constructed fitness function. The BVA is usually used to evaluate ride comfort, while SDD and wheel dynamic load (WDL) are often used for vehicle safety assessment. The WDD is used in this study due to the inability to obtain real-time WDL. As the evaluation indices have different dimensions, their ratios to passive suspension parameters are summed to form the objective fitness function. The BVA is denoted as R₁. The TLS suspension’s left and right SDDs are denoted as R₂ and R₃, respectively, while the left and right WDDs are denoted as R₄ and R₅, respectively. Equation (13) represents the PSOF PID active controller’s fitness function, which is obtained by optimizing the sum of the IATE of R₁–R₅.

\min G_{I T A E} = \sum_{i = 1}^{5} (α_{i} \int_{0}^{\infty} t | e_{i} (t) | d t / \int_{0}^{\infty} t | e_{i 0} (t) | d t) (i = 1,2, \dots 5)

(13)

where

α_{i}

is the weight factor of R_i.

Due to the need for left and right PSOF PID controllers in the TLS suspension, the search space dimension D is set to 6. Through simulation testing, the inertia factor $ω$ is set to a range of [0.1,0.8]. The learning factors $c_{1}$ and $c_{2}$ are set to 1.3 and 1.4. The particle swarm size N is set to 20. The iteration times k is set to 50.

The flowchart of the PSOF PID active control strategy is shown in Figure 9. Firstly, N initial particles are generated by the PSO algorithm’s random function, forming the initial particle swarm. Each particle contains 6 parameters [ $K_{p 0 l}$ , $K_{i 0 l}$ , $K_{d 0 l}$ , $K_{p 0 r}$ , $K_{i 0 r}$ , $K_{d 0 r}$ ]. Then, each generation of particles is simulated in the TLS suspension model to obtain N fitness function values. Calculate $P b e s t$ and $G b e s t$ according to equations (10)–(12) and iterate the process until reaching the minimum fitness function $G_{I T A E}$ . Finally, output the optimized $P b e s t$ and $G b e s t$ to end the PSO optimization process.

Figure 9.

Flowchart of the PSOF PID active control strategy.

Results and discussion

Model verification

According to the suspension offset test, the comparison between the BVA ${\ddot{x}}_{2 l}$ of the two models and test is shown in Figure 10(a). The Adams model is more consistent with the LMS test data. The difference in performance is attributed to the use of fixed damping in the Simulink model versus nonlinear damping in the Adams model. The suspension offset frequency results obtained by FFT (Fast Fourier Transform) are shown in Figure 10(b). The suspension offset frequency for the Adams and LMS models is 1.56 Hz, while the Simulink model showed a significantly lower value of 1.18 Hz. This discrepancy is due to the nonlinear and damping forces of the tire and the TLS in the Adams model. Furthermore, the suspension offset frequency amplitude of both the Adams and Simulink models is higher than the LMS test results due to the difference in damping values. Overall, the TLS suspension MBD model based on Adams demonstrated higher accuracy.

Figure 10.

Roll-off test of suspension natural frequency: (a) left BVA; (b) FFT of left BVA.

The Simulink dynamic model and the Adams MBD model are compared on a class C road, as shown in Figure 11. The root mean square (RMS) values of the BVA ${\ddot{x}}_{2}$ for the Simulink and Adams models are 2.08 m/s² and 2.29 m/s², respectively, as calculated from Figure 11(a). Figure 11(b) shows that the low frequencies of the Simulink and Adams models are 1.29 Hz and 1.59 Hz, respectively. In the frequency range of [10, 50] Hz, the amplitude of the Simulink model’s frequency response function decreases, while the Adams model slightly increases compared to low frequencies. The Adams model has a better ability to describe the high frequency characteristics of the TLS suspension. Consequently, this article is based on the Adams MBD TLS suspension model to investigate the suspension control algorithms.

Figure 11.

Comparison between the Adams MBD model and the 4-DOF model of TLS suspension: (a) BVA on C class road; (b) BVA Bode plots.

Continuous road simulation

This study compares the control effects of passive suspension, PSO SH PID semi-active control strategy, and PSOF PID active control strategy using the Adams/Simulink co-simulation platform. The initial parameters [K_p0l, K_i0l, K_d0l, K_p0r, K_i0r, K_d0r] of the SH PID and Fuzzy PID controllers are optimized using the PSO algorithm, and the PSO parameters are detailed in Table 6. The simulation time is 15 s, with experimental results taken from 4 to 14 s due to self-balancing at the beginning of the model. Road class C is used as the input at a speed of 20 m/s, and random continuous road and discrete impact road simulations are conducted. Under continuous road excitation, the parameter tuning results of the SH PID and Fuzzy PID controllers using the PSO algorithm are depicted in Figure 12. It can be observed that the left and right controller parameters of Fuzzy PID and SH PID began to converge at the 11th and 10th generations, respectively. Their respective target values commenced to meet the minimum fitness criteria.

Table 6.

Parameters of PSO in SH PID and fuzzy PID.

Parameters Name	Symbol	Values
Parameters Name	Symbol	SH PID	Fuzzy PID
Search space dimension	$D$	6
Learning factors	$c_{1}$	1.3
Learning factors	$c_{2}$	1.4
Inertial weight factor	$ω$	0.8
Particle swarm size	$N$	20
Iteration	$k$	50
Initial proportional coefficient	$K_{p 0}$	[1,10]	[1,15]
Initial integral coefficient	$K_{i 0}$	[0.1,2]	[1,10]
Initial derivative coefficient	$K_{d 0}$	[0.01,0.5]	[0.1,0.9]

Figure 12.

PID parameters tuning results using PSO algorithm under continuous road: (a) PSO optimizing K_p; (b) PSO optimizing K_i; (c) PSO optimizing K_d; (d) PSO optimizing G_ITAE.

The optimal parameters of the two controllers were applied to the model for computation, and the results are depicted in Figure 13. The results show that the PSO SH PID semi-active control strategy has a limited vibration reduction effect in the low-frequency range and intensifies vibration at higher frequencies. The PSOF PID active control strategy is effective in reducing vibration in the low-frequency range, but its effectiveness is limited in the high-frequency range.

Figure 13.

Frequency response of road input to BVA.

Figure 14 compares the control effects of semi-active and active control strategies on class C roads. Results show that the PSOF PID active control strategy significantly improves the low-frequency characteristics of the BVA, reducing the maximum amplitude at 1.6 Hz by 87.86% and at 1.9 Hz by 52.62%. The PSOF PID active control strategy is superior to the SH PID semi-active control strategy in controlling the BVA and SDD, reducing the RMS amplitude of BVA by 31.42%. Table 7 shows that the PSOF PID active control strategy has the best SDD control effect. These two control strategies have similar WDD control effects.

Figure 14.

Comparison of semi-active and active control effects on class C road: (a) BVA; (b) PSD of BVA; (c) Left SDD; (d) Right SDD; (e) Left WDD; (f) Right WDD.

Table 7.

RMS comparison of PSO SH PID and PSOF PID control strategy.

Control strategy		R₁ ( $m / s^{2}$ )	R₂ ( $m$ )	R₃ ( $m$ )	R₄ ( $m$ )	R₅ ( $m$ )
Passive suspension		2.2923	0.0152	0.0153	0.0866	0.0964
Semi-active	PSO SH PID	2.0526	0.0119	0.0117	0.0870	0.0969
Optimization rate (%)		−10.48%	−22.01%	−23.49%	0.45%	0.46%
Active	PSOF PID	1.572	0.0114	0.0125	0.0862	0.0956
Optimization rate (%)		−31.42%	−25.00%	−18.30%	−0.46%	−0.83%

Discrete impact road simulation

Under discrete impact road excitations, the parameters of the SH PID and Fuzzy PID controllers were optimized using the PSO algorithm, following a similar iterative process as shown in Figure 12. The optimal parameters for each controller were subsequently applied to their respective models. A comparison of the SH PID semi-active and PSOF PID active control strategy on a discrete impact class A road at 2 m/s showed that the PSOF PID active control strategy has the best vibration reduction control effect, as seen in Figure 15. The low-frequency vibration of the sprung mass has been greatly improved.

Figure 15.

Comparison of semi-active and active control effects on a discrete impact class A road: (a) BVA; (b) PSD of BVA; (c) Left SDD; (d) Right SDD; (e) Left WDD; (f) Right WDD.

Conclusion

To study the control algorithm of TLS suspension, the Simulink 4-DOF model and Adams/Car MBD model are established. The 4-DOF model has a small suspension offset frequency and cannot describe the high-frequency characteristics of the TLS suspension. Therefore, an Adams/Car MBD model of the TLS suspension was established, where the TLS was modeled using the discrete beam method.

A PSOF PID adaptive active control algorithm for half-car TLS suspension is proposed, with the body’s vertical velocity as the control variable. The fuzzy control is used to adapt the traditional PID controller parameters, and PSO is used for multi-objective optimization of the initial value of the TLS suspension fuzzy controllers. Under class C random roads, the PSOF PID active control improves the low-frequency characteristics of the BVA and slightly suppresses high-frequency vibration. Traditional SH PID semi-active control strategy exacerbates medium- and high-frequency vibration. The PSOF PID active control strategy reduces the RMS value of BVA by 31.42% and PSD amplitude by 87.86%. Under class A discrete impact road, the PSOF PID active control strategy greatly improves TLS suspension BVA, SDD, and WDD, demonstrating its suitability for vibration suppression of the TLS suspension.

The implementation of the PSOF PID control strategy has significantly enhanced the ride comfort of vehicles equipped with transverse leaf spring suspension. Analysis of the BVA Bode diagram indicates that the proposed PSOF PID demonstrates exceptional control performance, particularly in terms of low-frequency control effectiveness. However, further improvements are required to enhance the control effectiveness in the high-frequency range. Therefore, future research endeavors should focus on optimizing the placement of the active controller and developing novel optimization algorithms to enhance the overall frequency control performance of the transverse leaf spring suspension.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge Project 51705357 supported by the National Natural Science Foundation of China, Project 18JCYBJC20000 supported by the Natural Science Foundation of Tianjin and Project 2021T140519 supported by the China Postdoctoral Science Foundation.

ORCID iDs

Feiqi Long

Xiaolong Zhu

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