Research on Control Strategy of Semi-Active Suspension System Based on Fuzzy Adaptive PID-MPC

Abstract

To address the dynamic characteristics of vehicle semi-active suspension systems under special operating conditions and multi-source excitations, this paper proposes a fuzzy adaptive proportional–integral–derivative model predictive control (PID-MPC) strategy aimed at enhancing ride comfort during vehicle operation. The proposed approach employs MPC as the primary controller to optimize suspension performance, incorporating a fuzzy adaptive PID compensation mechanism for real-time adjustment of PID parameters, thereby improving control efficacy. A half-car semi-active suspension model was established on the MATLAB/Simulink (2020b) platform, with simulation validation conducted across diverse road profiles, including speed bump road surface, Class B road surface, and Class C road surface. Simulation results demonstrate that the proposed strategy achieves a significant reduction in both vehicle vertical acceleration and vehicle pitch angle acceleration while maintaining appropriate suspension deflection and tire dynamic loads, effectively elevating occupant ride comfort. Research demonstrates that the fuzzy adaptive PID-MPC control strategy exhibits commendable performance under typical road operating conditions, possessing notable potential for practical engineering implementation.

1. Introduction

At present, vehicle suspension systems, particularly damping-adjustable semi-active suspensions, exert a pivotal influence in enhancing ride comfort and ensuring driving safety [1]. The paramount challenge lies in devising efficient control strategies capable of addressing the high dynamic nature of road excitations, pronounced system nonlinearities, and intricate multivariable coupling characteristics, thereby achieving optimal performance in vehicle vertical dynamics. In recent years, the Model Predictive Control (MPC) has emerged as a promising approach within suspension control domains, attributed to its explicit capability to handle constraints, advantages in multivariable coordinated regulation, and predictive optimization features for future system behaviors [2,3].

The fundamental principle of MPC involves rolling optimization of an objective function within a finite temporal horizon to derive the optimal control sequence, with its initial element subsequently applied to the system. Numerous investigations have examined the application of MPC in suspension systems. For instance, Li Lansong et al. proposed a model predictive control-based solution for active suspension systems in hub-motor-driven vehicles, demonstrating efficacy in addressing multifaceted constraints such as electromechanical actuator stroke limitations and tire adhesion capacities [4]. Nevertheless, practical implementation necessitates frequent controller adjustments due to hardware-imposed restrictions on actuator force magnitude and response latency, potentially compromising control performance. Jiang Hong et al. developed a semi-active suspension control methodology grounded in MPC theory, formulating the control problem as an N-step open-loop optimal control task over a finite horizon while optimizing ride comfort and handling stability under constrained conditions [5].

However, contemporary control systems increasingly confront complex challenges encompassing nonlinear characteristics, information scarcity, significant time delays, and strong multivariable coupling effects, rendering it difficult for conventional MPC alone to precisely establish multidimensional dynamic relationships between control variables and system behaviors [6]. Bououden et al. proposed a robust model predictive control (RMPC) control method to improve the performance of active suspension systems with a time-varying shift delay [7]. As for semi-active systems, this strategy may exert an influence on the system. The traditional MPC struggles with real-time implementation due to energy dissipation constraints inherent in semi-active suspensions. To address this limitation, Fredrik introduced a Nonlinear Model Predictive Control (NMPC) approach grounded in a full-vehicle vertical dynamic model, which integrates road preview information to optimize vertical, roll, and pitch accelerations for enhanced ride comfort [8]. While simplified-model NMPC configurations improve computational efficiency and ride quality, they persistently confront trade-offs between model fidelity and multi-constraint coordination. Addressing the substantial computational burden of conventional MPC, Theunissen et al. and Li Lansong et al. developed an active suspension control framework based on Explicit Model Predictive Control (E-MPC) theory. By eliminating iterative online optimization, E-MPC demonstrates markedly higher efficiency compared to implicit MPC formulations [9,10]. Nevertheless, its practical application is constrained by prohibitively complex control law representations, extensive memory requirements, and intensive offline computational demands, limiting scalability for large-scale systems. Chen Xiaokai et al. proposed a Variable Step Length MPC (VSL-MPC) algorithm designed to optimize damping adjustment timing for performance enhancement [11]. Ahmed engineered a semi-active vibration control suspension system leveraging synergistic interaction between MPC and magnetorheological (MR) dampers, achieving improvements in both ride comfort and handling stability through optimized semi-active suspension dynamics [12]. To mitigate displacement delays, Kim et al. implemented an MPC strategy integrating Kalman filtering with shift compensation, effectively compensating for hardware response latency to enhance control efficacy and further elevate vehicle ride comfort [13]. To enhance the controller’s performance, He Guang introduced a hybrid control algorithm combining MPC correction with PID into the altitude and attitude control loops of a tilt-rotor unmanned aerial vehicle. Compared to conventional PID controllers, this approach significantly improves flight control effectiveness during transient modes [14].

Therefore, synthesizing the strengths and limitations of MPC applications in vehicle suspension systems outlined in the aforementioned literature, this study proposes a fuzzy adaptive PID-MPC strategy for semi-active suspension systems. Under the premise of ensuring vehicular driving safety, the objective is to enhance ride comfort. This strategy integrates the maturity, structural simplicity, and rapid responsiveness of PID control with MPC, enabling dynamic adjustment of PID compensation via fuzzy rules to fulfill MPC’s compensatory requirements. To validate its effectiveness, a half-car suspension model and random road profile model were first established. Subsequently, the principles and constraints of the fuzzy adaptive PID-MPC strategy were analyzed. Finally, comprehensive comparative evaluations were conducted against traditional passive suspension, standalone MPC, and PID-MPC strategies across speed bump road surface, Class B road surface, and Class C road surface conditions. Results demonstrate that the proposed fuzzy adaptive PID-MPC strategy significantly improves ride comfort in semi-active suspension systems, offering a valuable reference for future control designs in this domain.

2. Establishment of Semi-Active Suspension System Model

2.1. Half-Car Semi-Active Suspension System Model

In order to improve the ride comfort of the vehicle, this study constructs a state space model of the half-car semi-active suspension system. It is assumed that there is no sliding state between the wheels and the road surface, and the wheels always keep contact with the ground. The vertical vibration characteristics of the wheels are simplified to a spring element without considering the damping effect. The suspension system of the left and right sides of the vehicle is completely symmetrical, and the suspension of the vehicle is regarded as a system composed of the sprung mass, springs, and dampers to carry out the simulation experiments [15]. The dynamic model of the half-car semi-active suspension system is established as shown in Figure 1.

In Figure 1, $m_s$ is the suspension mass; $m_{sf}$ and $m_{sr}$, respectively, denote the mass equivalent to the suspension mass above the front and rear tires; $m_{\omega f}$ and $m_{\omega r}$, respectively, represent the mass equivalent to the unsprung mass above the front and rear tires; $I_{sy}$ is the mass moment of inertia of the suspension mass around the y-axis; $L$ is the wheelbase; $a$ and $b$ are the distances from the center of mass of the body to the front and rear axles; $K_{sf}$ and $K_{sr}$, respectively, denote the stiffnesses of front and rear suspensions; $C_{sf}$ and $C_{sr}$, respectively, represent the damping of the front and rear suspensions; $K_{\omega f}$ and $K_{\omega r}$, respectively, correspond to the front and rear tire stiffnesses; $\varphi$ is the body pitch angle; $q_f$ and $q_r$, respectively, denote the displacements of the front and rear tires for the road surface unevenness; $z_{\omega f}$ and $z_{\omega r}$, respectively, represent the vertical displacements of the unsprung masses of the front and rear axles; $z_{sf}$ and $z_{sr}$, respectively, correspond to the vertical displacements of the suspension masses above the front and rear wheels; $z_s$ is the vertical displacement at the center of mass of the body; and $F_{ucf}$ and $F_{ucr}$, respectively, denote the damping adjusting forces for the front and rear suspensions. The main parameters of the selected semi-active suspension system are shown in

Table 1. The main parameters of the semi-active suspension system.

Parameter	Notation	Value
Sprung mass	ms/kg	1270
Unsprung mass	mω/kg	88.6
Rotational inertia	I/kg·m2	1536.7
Wheelbase	L/m	2.91
Distance from CG to front axle	a/m	1.015
Distance from CG to rear axle	b/m	1.895
Suspension damping	Cs/N·s·m−1	6000
Suspension stiffness	Ks/N·m−1	27,000
Tire stiffness	Kω/N·m−1	268,000

.

According to Newton’s second law, the differential equation for the vertical motion of the center of mass is obtained with the body as the object of study:

$$m_s{\ddot{z}}_{\omega f}=K_{sf}(z_{\omega f}-z_{sf})+C_{sf}({\dot{z}}_{\omega f}-{\dot{z}}_{sf})+K_{sr}(z_{\omega r}-z_{sr})+C_{sr}({\dot{z}}_{\omega r}-{\dot{z}}_{sr})+F_{ucf}+F_{ucr}$$

(1)

The differential equation of motion for the center of mass in the pitch direction is as follows:

$$I_{sy}\ddot{\varphi }=b{K}_{sr}(z_{\omega r}-z_{sr})+b{C}_{sr}({\dot{z}}_{\omega r}-{\dot{z}}_{sr})-a{K}_{sf}(z_{\omega f}-z_{sf})-a{C}_{sf}({\dot{z}}_{\omega f}-{\dot{z}}_{sf})+b{F}_{ucr}-a{F}_{ucf}$$

(2)

Taking the front and rear unsprung masses as the research objects, the differential equation for the unsprung mass of the front suspension is as follows:

$$m_{\omega f}{\ddot{z}}_{\omega f}=K_{\omega f}(q_f-z_{\omega f})-K_{sf}(z_{\omega f}-z_{sf})-C_{sf}({\dot{z}}_{\omega f}-{\dot{z}}_{sf})-F_{ucf}$$

(3)

The differential equation for the unsprung mass of the rear suspension is as follows:

$$m_{\omega r}{\ddot{z}}_{\omega r}=K_{\omega r}(q_r-z_{\omega r})-K_{sr}(z_{\omega r}-z_{sr})-C_{sr}({\dot{z}}_{\omega r}-{\dot{z}}_{sr})-F_{ucr}$$

(4)

The vertical displacement of the vehicle body, pitch angle of the vehicle body, vertical displacements of the unsprung masses at the front and rear axles, road irregularity displacements at the front and rear wheels, vertical velocity of the vehicle body, pitch angle velocity of the vehicle body, and vertical velocities of the unsprung masses at the front and rear axles are selected as the state variables of the system:

$$X={(z_s\hspace{1em}\varphi \hspace{1em}{\mathrm{z}}_{\omega f}\hspace{1em}{z}_{\omega r}\hspace{1em}{q}_f\hspace{1em}{q}_r\hspace{1em}{\dot{z}}_s\hspace{1em}\dot{\varphi }\hspace{1em}{\dot{z}}_{\omega f}\hspace{1em}{\dot{z}}_{\omega r})}^T$$

(5)

Therefore, the state equation of the semi-active suspension system is as follows:

$$\left\{\begin{array}{c}\dot{X}=AX+BU+EW\\ U={(F_{ucf},F_{ucr})}^T\\ W={({\dot{q}}_f,{\dot{q}}_r)}^T\end{array}\right.$$

(6)

In Equation (6), $A$ denotes the 10 × 10 system matrix, $B$ represents the 10 × 2 control matrix, $E$ corresponds to the 10 × 2 disturbance matrix, $U$ is the control vector of the system, and W is the disturbance vector of the system.

Furthermore, the vertical acceleration of the vehicle body, pitch angle acceleration of the vehicle body, dynamic deflections of the front and rear suspensions, and dynamic loads of the front and rear tires are selected as the system’s output variables:

$$X={(z_s\hspace{1em}\varphi \hspace{1em}{\mathrm{z}}_{\omega f}\hspace{1em}{z}_{\omega r}\hspace{1em}{q}_f\hspace{1em}{q}_r\hspace{1em}{\dot{z}}_s\hspace{1em}\dot{\varphi }\hspace{1em}{\dot{z}}_{\omega f}\hspace{1em}{\dot{z}}_{\omega r})}^T$$

(7)

Therefore, the output equation for the semi-car semi-active suspension is as follows:

$$Y=CX+DU$$

(8)

In Equation (8), $C$ is the 6 × 10 output matrix, and $D$ is the 6 × 2 transfer matrix.

2.2. Construction of Random Road Excitation Model

Regarding the measurement indicators for random road surfaces, the concept of road roughness is universally adopted, which reflects the road condition by quantifying the roughness of the road surface. Road roughness is precisely one of the key factors affecting the driving experience of the driver and the ride comfort of passengers [16]. The spatial power spectral density fitting expression is as follows [17]:

$$G_q(n)=G_q(n_0){({\displaystyle \frac{n}{n_0}})}^{-w}$$

(9)

where $n$—the spatial frequency, which is the reciprocal of the wavelength $\lambda$, m⁻¹;

$G_q(n)$—the pavement power spectral density (PSD) at the spatial frequency, m³;

$n_0$—the reference spatial frequency, $n_0=0.1$ m⁻¹;

$G_q(n_0)$—is the pavement power spectral density at the reference spatial frequency, called the pavement unevenness coefficient, m³;

$\omega$—the frequency coefficient, which is the slope of the diagonal line on double logarithmic coordinates that determines the frequency structure of the pavement power spectral density, commonly taken as $\omega =2$.

According to Formula (9), it is concluded that the spatial power spectral density is not only affected by the two factors of road roughness and wavelength, but also by other parameters such as vehicle running time and driving speed. For the convenience of analysis, a concept of time–frequency power spectral density is introduced here, and the relationship between time–frequency power spectral density and spatial frequency power spectral density is as follows:

$$G_q(f)={\displaystyle \frac{G_q(n)}{v}}=G_q(n_0)n_0^2{\displaystyle \frac{v}{f^2}}$$

(10)

where $G_q(f)$ is the time–frequency power spectral density, $f$ denotes the time frequency and $f=vn$, where $v$ represents the traveling speed of the car.

The road excitation input is a stochastic road input, and a random white noise time-domain road surface simulation model is adopted as the road input model for simulation analysis [18]. The filtered white noise time-domain road input model used in this study is as follows:

$$\dot{q}(t)=-2\pi f_{0}q(t)+2\pi \sqrt{v{G}_q(n_0)}\omega (t)$$

(11)

where $q(t)$ denotes road roughness, $f_0$ represents lower temporal cutoff frequency, and $\omega (t)$ corresponds to the randomly distributed Gaussian white noise. The specific value of $G_q(n_0)$ can be found according to the pavement unevenness grading criteria [19], as shown in

Table 2. Classification standard on roughness of road surface.

Road Grade	Lower Limit (×10−6 m3)	Geometric Mean (×10−6 m3)	Upper Limit (×10−6 m3)
A	8	16	32
B	32	64	128
C	128	256	512
D	512	1024	2048
B	2048	4096	8192

.

Due to the wheelbase, there is a time delay in the road input between the front and rear wheels. Therefore, the following corrections are applied to the rear-wheel inputs relative to the front wheels:

$$q_r=q_f\times (1+{\displaystyle \frac{l}{v}})$$

(12)

In the MATLAB/Simulink simulation platform, a random road input simulation model as shown in Figure 2 is established. The two gain modules in the model are $2\pi \sqrt{v{G}_q(n_0)}$ and $-2\pi f_0$, respectively. Here, $out.dot\_Zr$ represents the output of the road’s vertical acceleration, and $out.Zr$ denotes the output of the road’s vertical displacement.

In this study, a simulation analysis was conducted using three different road surfaces: a speed bump road surface, a Class B road surface, and a Class C road surface. To ensure the validity of comparative assessments and reproducibility of results, road excitations were synthesized only once and maintained consistently across all simulations. The excitation for speed bumps was deterministic, whereas those for Class B and Class C roads were generated using fixed random speeds to preserve identical statistical properties. The vehicle traversed the speed bumps at 20 km/h and traveled over both Class B and Class C roads at 40 km/h. According to Table 2, the road power spectral density (PSD) values for Class B and Class C surfaces are $G_q(n_0)=64\times {10}^{-6}$ m³ and $G_q(n_0)=256\times {10}^{-6}$ m³, respectively, with a uniform cutoff frequency of $f_0=0.1$ Hz. The corresponding road excitation simulations are illustrated in Figure 3, Figure 4 and Figure 5.

In order to verify the accuracy of the obtained random pavement excitation, the Welch algorithm was applied to simulate Class B and Class C road surfaces, as shown in Figure 6. The PSD of the simulated measured data were compared with the standard PSD curves, and the results showed that the power spectral densities of the pavements coincided with each other.

3. Fuzzy Adaptive PID-MPC Strategy

3.1. MPC Principles

Model predictive control (MPC) originated in the 1970s and gradually matured in the early 21st century [20,21,22]. MPC is a control strategy that uses the method of the optimal problem to solve the control problem, which includes four major elements, namely, model prediction, rolling optimization, feedback correction, and reference trajectory. The control principles are illustrated in Figure 7.

In Figure 7, $N_c$ represents the control horizon and $N_p$ represents the prediction horizon, which are two unique parameters of MPC. The number of prediction intervals indicates the number of steps predicted, and the control interval represents the change in the number of control steps beyond which the amount of control remains unchanged.

3.2. Predictive Model

The spatial state Equations (6) and (8) discretized using the forward Euler method yield:

$$X(n+1)=\tilde{A}X(n)+\tilde{B}U(n)+\tilde{E}W(n)$$

(13)

$$Y(n)=\tilde{C}X(n)+\tilde{D}U(n)$$

(14)

where $\tilde{A}=(I+TA)$, $\tilde{B}=TB$, $\tilde{E}=TE$, $\tilde{C}=C$, $\tilde{D}=D$, and $T$ denotes discrete-time.

Further, the state variables can be obtained from the state equation, which is derived as follows:

$$\begin{array}{l}{X}_{k+1|k}=\tilde{A}\times X_k+\tilde{B}\times U_{k|k}+\tilde{E}\times W_{k|k}\\ X_{k+2|k}={\tilde{A}}^2\times X_k+\tilde{A}\times \tilde{B}\times U_{k|k}+\tilde{B}\times U_{k+1|k}+\tilde{A}\times \tilde{E}\times W_{k|k}+\tilde{E}\times W_{k+1|k}\\ ⋮\\ X_{k+N_c|k}={\tilde{A}}^{N_c}\times X_k+{\tilde{A}}^{N_c-1}\times \tilde{B}\times U_{k|k}+\dots +\tilde{B}\times U_{k+N_c-1|k}+{\tilde{A}}^{N_c-1}\times \tilde{E}\times W_{k|k}+\dots +\tilde{E}\times W_{k+N_c-1|k}\\ X_{k+N_c+1|k}={\tilde{A}}^{N_c+1}\times X_k+{\tilde{A}}^{N_c}\times \tilde{B}\times U_{k|k}+\dots +\tilde{A}\times \tilde{B}\times U_{k+N_c-1|k}+\tilde{B}\times U_{k+N_c-1|k}+{\tilde{A}}^{N_c}\times \tilde{E}\times W_{k|k}+\dots +\tilde{E}\times W_{k+N_c|k}\\ ⋮\\ X_{k+N_p|k}={\tilde{A}}^{N_p}\times X_k+{\tilde{A}}^{N_p-1}\times \tilde{B}\times U_{k|k}+\dots +{\tilde{A}}^{N_p-N_c}\times \tilde{B}\times U_{k+N_c-1|k}+{\tilde{A}}^{N_p-N_c-1}\times \tilde{B}\times U_{k+N_c-1|k}+\dots +\tilde{B}\times U_{k+N_c-1|k}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\tilde{A}}^{N_p-1}\times \tilde{E}\times W_{k|k}+\dots +\tilde{E}\times W_{k+N_p|k}\end{array}$$

(15)

where $X_k$ represents the actual state of the system at time $k$, $X_{k+i|k}$ denotes the predicted state at time $k+i$ based on the estimation at time $k$, $U_{k+i|k}$ denotes the predicted control input $\theta$ of the system at time $k+i$ based on time $k$, and $W_{k+i|k}$ indicates the predicted external disturbance affecting the system at time $k+i$ based on time $k$.

The state variables, control inputs, and disturbance inputs of the system at time $k$ are combined into a matrix form as:

$${\widehat{X}}_k=\widehat{A}{X}_k+\widehat{B}{\widehat{U}}_k+\widehat{E}{\widehat{W}}_k$$

(16)

Here, the vector representations are given as follows:

$$\begin{gathered} {\widehat{X}}_k={\left[\begin{array}{cccc}{X}_{k+1|k}& X_{k+2|k}& \dots & X_{k+N_p|k}\end{array}\right]}_{N_p\times 1}^T \\ {\widehat{U}}_k={\left[\begin{array}{cccc}{U}_{k|k}& U_{k+1|k}& \dots & U_{k+N_c-1|k}\end{array}\right]}_{N_c\times 1}^T \\ {\widehat{W}}_k={\left[\begin{array}{cccc}{W}_{k|k}& W_{k+1|k}& \dots & W_{k+N_p-1|k}\end{array}\right]}_{N_p\times 1}^T \\ \widehat{A}={\left[\begin{array}{cccc}\tilde{A}& {\tilde{A}}^2& \dots & {\tilde{A}}^{N_p}\end{array}\right]}_{N_p\times 1}^T \\ \tilde{B}={\left[\begin{array}{ccccc}\tilde{B}& 0& \dots & \dots & 0\\ \tilde{A}\tilde{B}& \tilde{B}& 0& \dots & 0\\ ⋮& ⋮& & & ⋮\\ {\tilde{A}}^{N_c-1}\tilde{B}& {\tilde{A}}^{N_c-2}\tilde{B}& \dots & \dots & \tilde{B}\\ ⋮& ⋮& & & ⋮\\ {\tilde{A}}^{N_p-1}\tilde{B}& {\tilde{A}}^{N_p-2}\tilde{B}& \dots & \dots & {\displaystyle \sum _{i=0}^{N_p-N_c}{\tilde{A}}^{i-1}\tilde{B}}\end{array}\right]}_{N_p\times (N_c-1)} \\ \tilde{E}={\left[\begin{array}{cccc}\tilde{E}& 0& \dots & 0\\ \tilde{A}\tilde{E}& \tilde{E}& \dots & 0\\ {\tilde{A}}^2\tilde{E}& \tilde{A}\tilde{E}& \dots & ⋮\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{A}}^{N_p-1}\tilde{E}& {\tilde{A}}^{N_p-2}\tilde{E}& \dots & \tilde{E}\end{array}\right]}_{N_p\times N_p} \end{gathered}$$

From the output equation, the output variables can be derived as follows:

$$\begin{array}{l}{Y}_{k+1|k}=C{X}_{k+1}+D{U}_{k+1|k}=C\tilde{A}{X}_k+C\tilde{B}{U}_{k|k}+C\tilde{E}{W}_{k|k}+D{U}_{k+1|k}\\ Y_{k+2|k}=C{X}_{k+2}+D{U}_{k+2k|k}=C{\tilde{A}}^2{X}_k+C\tilde{A}\tilde{B}{U}_{k|k}+C\tilde{B}{U}_{k+1|k}+C\tilde{A}\tilde{E}{W}_{k|k}+C\tilde{E}{W}_{k+1|k}+D{U}_{k+2|k}\\ ⋮\\ Y_{k+N_c|k}=C{X}_{k+N_c}+D{U}_{k+N_c|k}=C{\tilde{A}}^{N_c}{X}_k+C{\tilde{A}}^{N_c-1}\tilde{B}{U}_{k|k}+\dots +C\tilde{B}{U}_{k+N_c-1|k}+C{\tilde{A}}^{N_c-1}\tilde{E}{W}_{k|k}+\dots +C\tilde{E}{W}_{k+N_c-1|k}+D{U}_{k+N_c|k}\\ ⋮\\ Y_{k+N_p|k}=C{X}_{k+N_p}+D{U}_{k+N_c-1|k}=C{\tilde{A}}^{N_p}{X}_k+C{\tilde{A}}^{N_p-1}\tilde{B}{U}_{k|k}+\dots +C\tilde{B}{U}_{k+N_c-1|k}+C{\tilde{A}}^{N_p-1}\tilde{E}{W}_{k|k}+\dots +C\tilde{E}{W}_{k+N_p-1|k}+D{U}_{k+N_p|k}\end{array}$$

(17)

where $Y_{k+i|k}$ denotes the predicted output of the system at time $k+i$ forecasted at time $k$.

Then, the output equation of the system at moment $k$ is defined as follows:

$${\widehat{Y}}_k=\widehat{C}{X}_k+\widehat{D}{\widehat{U}}_k+\widehat{\Gamma }{\widehat{W}}_k$$

(18)

Here, the vector representations are given as follows:

$$\begin{gathered} {\tilde{Y}}_k={\left[\begin{array}{ccccc}{Y}_{k+1|k}& Y_{k+2|k}& Y_{k+3|k}& \dots & Y_{k+N_p|k}\end{array}\right]}_{N_p\times 1}^T \\ \widehat{C}={\left[\begin{array}{ccccc}C\widehat{A}& C{\widehat{A}}^2& C{\widehat{A}}^3& \dots & C{\widehat{A}}^{N_p}\end{array}\right]}_{N_p\times 1}^T \\ \widehat{D}={\left[\begin{array}{cccc}C\tilde{B}& D& \dots & 0\\ C\tilde{A}\tilde{B}& C\tilde{B}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ C{\tilde{A}}^{N_p-1}\tilde{B}& C{\tilde{A}}^{N_p-2}\tilde{B}& \dots & C{\displaystyle \sum _{i=0}^{N_p-N_c}{\tilde{A}}^{i-1}\tilde{B}+D}\end{array}\right]}_{N_p\times (N_c-1)} \\ \widehat{\Gamma }={\left[\begin{array}{cccc}C\tilde{E}& 0& \dots & 0\\ C\tilde{A}\tilde{E}& C\tilde{E}& \dots & 0\\ C{\tilde{A}}^2\tilde{E}& C\tilde{A}\tilde{E}& \dots & ⋮\\ ⋮& ⋮& \ddots & ⋮\\ C{\tilde{A}}^{N_p-1}\tilde{E}& C{\tilde{A}}^{N_p-2}\tilde{E}& \dots & C\tilde{E}\end{array}\right]}_{N_p\times Np} \end{gathered}$$

Since the output variables $Y$ and control inputs $U$ have different physical units and magnitudes, a weighting design will be applied in the cost function to normalize and balance their contributions.

3.3. Selection of the Cost Function J

For conventional suspension systems, it is generally desired to achieve more precise control of the variable damping amplitude. Simultaneously, it is hoped that the control input can be minimized as much as possible while satisfying the control requirements. Based on these considerations, the following principles are established for selecting the system’s cost function:

$$J={\displaystyle \sum _{i=1}^{N_p}{(Y_{k+i|k}-Y_{ref|k+i|k})}^T}Q(Y_{k+i|k}-Y_{ref|k+i|k})+{\displaystyle \sum _{i=0}^{N_c-1}(U_{k+i|k}^T}R{U}_{k+i|k})$$

(19)

where $Y_{ref|k+i|k}$ represents the predicted output of the system’s output variable at the time $k$, $Q$ denotes the weight matrix for the output error, and $R$ is the weight matrix for the decision variable.

In order to ensure that the output term $Y$ and the control term $U$ contribute proportionally to the objective function, normalization and weighting were applied through the weighting matrices $Q$ and $R$. Specifically, $Q$ is a diagonal matrix with larger weights assigned to the vertical and pitch accelerations to emphasize ride comfort, while smaller weights are given to the suspension deflection and tire dynamic load to maintain safety. The matrix $R$ is set with small but non-zero values to suppress overly frequent or large variations in the control input, thus ensuring actuator feasibility and stability.

Substituting the previously derived Equation (18) into the cost function yields:

$$J=2{(\widehat{C}{X}_k+\widehat{\Gamma }{\widehat{W}}_k-{\widehat{Y}}_{ref|k})}^T\overline{Q}\widehat{D}{\widehat{U}}_k+{\widehat{U}}_k^T(\widehat{D}\overline{Q}\widehat{D}+\overline{R}){\widehat{U}}_k$$

(20)

For a convenient solution using quadratic programming (QP) solvers in MATLAB, the cost function is transformed into quadratic form as follows:

$$J=f^T{\widehat{U}}_k+{\displaystyle \frac{1}{2}}{\widehat{U}}_k^{T}H{\widehat{U}}_k$$

(21)

where $H=\widehat{D}\overline{Q}\widehat{D}+\overline{R}$, $\overline{Q}=Q\times I$, $\overline{R}=R\times I$, $f=\widehat{D}\overline{Q}{(\widehat{C}{X}_k+\widehat{\Gamma }{\widehat{W}}_k-{\widehat{Y}}_{ref|k})}^T$, $Q={\left[\begin{array}{cccc}{q}_1& 0& 0& 0\\ 0& q_2& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& q_n\end{array}\right]}_{n\times n}$, and $R={\left[\begin{array}{cccc}{r}_1& 0& 0& 0\\ 0& r_2& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& r_p\end{array}\right]}_{p\times p}$.

In matrices $\mathrm{Q}$ and $R$, $n$ represents the number of columns in the output matrix $C$, $n=6$, while $p$ denotes the number of columns in the disturbance matrix $E$, $p=2$. The initial values undergo crossover, mutation, and selection operations via the NSGA-II algorithm. To ensure optimal vehicle ride comfort, the optimized parameters obtained through this optimization process are as follows: $q_1=2.5714379531682283E6$, $q_2=4.389836070864168E7$, $q_3=3.8904644479975593E8$, $q_4=6.322802274628704E8$, $q_5=6.640764133865229E8$, $q_2=9.923349517389698E8$, $r_1=3.02711351668428$, $r_2=0.39620336898974906$.

3.4. Constraints for Semi-Active Suspension System Control

In semi-active suspension systems, the dynamic travel of the suspension is subject to physical limitations imposed by the suspension structure, and the damping force output of the variable damping mechanism is also constrained by the inherent structural limits of the dampers. To ensure that the contact force between the wheels and the road surface does not exceed the maximum load capacity designed for the suspension, corresponding constraints on the dynamic tire load are implemented to safeguard both the operational safety of the system and occupant safety.

Therefore, in order to ensure the practical feasibility of the control outputs, in addition to the static constraints on suspension deflection $\left|f_d\right|\le 0.06$ m and the damping force amplitude $\left|F_s\right|\le 5000$ N, the physical characteristics of the actuator are further considered. Specifically, the equivalent damping coefficient is restricted to the range [500, 6000] N·s/m; the rate of change of the damping force satisfies $|\Delta F_s/\Delta t|\le 1000$ N/ms; the rate of change of the equivalent damping coefficient satisfies $|\Delta C_{eq}/\Delta t|\le 200$ N·s/(m·ms); and the actuator exhibits an inherent response delay of approximately 10–20 ms, which can be approximated by a first-order dynamic model:

$$F_a^{act}(k+1)=F_a^{act}(k)+\alpha (F_a^{cmd}(k)-F_a^{act}(k)), \alpha ={\displaystyle \frac{T_s}{{\tau }_a}}$$

(22)

where ${\tau }_a$ denotes the actuator time constant (set to 15 ms), and $T_s$ is the control sampling period. These constraints reflect the physical limitations of the semi-active damper and provide a reference basis for subsequent MPC constraint modeling.

3.5. PID Controller

As a classical control strategy, a PID controller is widely used in various industrial processes. Its basic principle is to adjust the system deviation by proportional control and to combine integral and differential links to further improve the control effect. The conventional PID control principle is illustrated as follows:

$$u_t=K_p\left[e_t+{\displaystyle \frac{1}{T_i}}{\displaystyle {\int }_0^t{e}_{t}dt+T_d\frac{d{e}_t}{dt}}\right]=K_p{e}_t+K_i{\displaystyle {\int }_0^t{e}_{t}dt+K_d\frac{d{e}_t}{dt}}$$

(23)

where $t$ is the sampling time, $K_p$ denotes a proportional coefficient, $K_i$ represents integral coefficients, $K_d$ corresponds to a differential coefficient, $e_t$ refers to error input, $T_i$ denotes an integral time constant, and $T_d$ represents a differential time constant.

The PID control system has a simple structure, good real-time performance, and is easily adjusted [23,24]. Employing it as a compensation component in the MPC strategy not only enhances the system’s tracking performance and disturbance rejection capability with properly tuned parameters, but also optimizes tracking characteristics and stability, rendering the overall control system more precise and flexible. More importantly, by leveraging the PID’s inherent advantages of simplicity and rapid response as a compensatory mechanism, the composite control strategy’s response time is not significantly prolonged. This ensures the system maintains quick adaptability to various dynamic changes.

Research findings regarding vehicle ride comfort indicate that the vehicle’s vertical acceleration and pitch angle acceleration serve as critical influencing factors. Consequently, the following parameters are utilized as inputs to the PID controller: vertical acceleration deviation, rate of change of vertical acceleration deviation, pitch angle acceleration deviation, and rate of change of pitch angle acceleration deviation. Through adjustment of the PID controller parameters using the Ziegler–Nichols method, the initial parameters are determined as $K_{p0}=380$, $K_{i0}=300$, and $K_{d0}=1$.

3.6. Fuzzy Adaptive PID Controller

In the fuzzy adaptive PID controller, a hybrid approach integrating fuzzy control with conventional PID control is adopted. With the objective of optimizing vehicle ride comfort, the fuzzy controller employs vertical acceleration deviation and its rate of change, as well as pitch angular acceleration deviation and its corresponding rate of change, as input variables. The output variables correspond to the corrected incremental gains ($\Delta K_p$, $\Delta K_i$, $\Delta K_d$). These adjustments are then superimposed onto the initial PID parameters ($K_{p0}$, $K_{i0}$, $K_{d0}$), yielding the final parameter configuration for the fuzzy adaptive PID controller [25,26]:

$$\left\{\begin{array}{l}{K}_p=K_{p_0}+\Delta K_p\\ K_i=K_{i0}+\Delta K_i\\ K_d=K_{i0}+\Delta K_d\end{array}\right.$$

(24)

And the controllable damping force constitutes the final output of the fuzzy adaptive PID controller. The architecture of this controller is depicted in Figure 8.

During the fuzzy control design process, the aggregation method adopts the maximum (max) principle, which involves performing a union operation on all fuzzy sets generated by the activated rules. Specifically, the maximum membership degree among these fuzzy sets at each point is selected as the resulting membership degree for that point. For defuzzification, the centroid method is utilized, calculating the center of gravity of the aggregated fuzzy set as the precise output value. This combination effectively integrates the influence of all rules, producing smooth and stable control outputs suitable for continuous damping force regulation in suspension systems. While proportional control within PID systems offers the advantage of rapidly suppressing deviations and effectively reducing their magnitude, excessively large proportional gains may lead to exacerbated system overshoot, thereby adversely impacting dynamic performance. Integral control helps eliminate steady-state errors, but due to its lagging nature, excessive use can easily cause system instability. Derivative control ensures the dynamic performance of the system [27,28]. Consequently, it is crucial to adjust the fuzzy adaptive PID control parameters based on the error and its rate of change. The specific operational methods are as follows:

When the error is significant (e.g., $e>0$), a larger proportional gain $K_p$ should be selected to prevent derivative saturation caused by rapid deviation increases in the short term. However, excessively high $K_p$ may also lead to controller output saturation. Meanwhile, appropriately reducing the integral gain $K_i$ can decrease the overshoot magnitude. To avoid exceeding the system’s control range, a smaller $K_d$ should be adopted.

When the error and its rate of change are moderate (e.g., $\left|e\right|=de$), it is appropriate to select a lower proportional gain $K_p$ and ensure that the differential gain $K_d$ is moderate, so as to ensure the system response speed. Furthermore, to ultimately ensure the final stability of the system, the $K_i$ size can be appropriately increased.

When the error is small (e.g., $e<0$), the proportional gain $K_p$ and the integral gain $K_i$ are appropriately increased to maintain the steady-state performance of the system, improve the anti-interference ability and avoid oscillation.

Based on the aforementioned theoretical analysis, the fuzzy subsets of input and output variables are categorized into seven levels, including negative large (NL), negative medium (NM), negative small (NS), zero (ZO), positive small (PS), positive medium (PM), and positive large (PL). Through iterative experiments, the fundamental domains were determined as follows: the deviation $e$ spans [−10, 10], its rate of change $ec$ ranges within [−3, 3], while the base intervals for $\Delta K_p$, $\Delta K_i$ and $\Delta K_d$ are, respectively, [−5, 5], [−5, 5], and [0, 0.0001]. The membership function curves for the input and output are illustrated in Figure 9.

Control rules for $\Delta K_p$, $\Delta K_i$ and $\Delta K_d$ in the fuzzy controller were formulated via algorithmic synthesis grounded in fuzzy theory, as detailed in

Table 3. The control rule table of $\Delta K_p$ regarding the input variables $e$ and $ec$.

ΔKp		ec
		NL	NM	NS	ZO	PS	PM	PL
$e$	NL	PL	PL	PL	PL	PM	PS	ZO
	NM	PL	PL	PL	PL	PM	ZO	ZO
	NS	PM	PM	PM	PM	ZO	NS	NS
	ZO	PM	PM	PS	ZO	NS	NS	NM
	PS	PS	PS	ZO	NS	NM	NM	NM
	PM	PS	ZO	NS	NM	NM	NM	NL
	PL	ZO	ZO	NM	NM	NM	NL	NL

,

Table 4. The control rule table of $\Delta K_i$ regarding the input variables $e$ and $ec$.

ΔKi		ec
		NL	NM	NS	ZO	PS	PM	PL
$e$	NL	NL	NL	NM	NM	NS	ZO	ZO
	NM	NL	NL	NM	NS	NS	ZO	ZO
	NS	NL	NM	NS	NS	ZO	PS	PS
	ZO	NM	NM	NS	ZO	PS	PM	PM
	PS	NM	NS	ZO	PS	PS	PM	PL
	PM	ZO	ZO	PS	NM	PM	PL	PL
	PL	ZO	ZO	PS	PM	PM	PL	PL

and

Table 5. The control rule table of $\Delta K_d$ regarding the input variables $e$ and $ec$.

ΔKd		ec
		NL	NM	NS	ZO	PS	PM	PL
$e$	NL	PS	NS	NL	NL	NL	NM	PS
	NM	PS	NS	NL	NM	NM	NS	ZO
	NS	ZO	NS	NM	NM	NS	NS	ZO
	ZO	ZO	NS	NS	NS	NS	NS	ZO
	PS	ZO	ZO	ZO	ZO	ZO	ZO	ZO
	PM	PL	PS	PS	PS	PS	PS	PL
	PL	PL	PM	PM	PM	PS	PS	PL

. Fuzzy decision-making was implemented using the centroid method, visualized in Figure 10, Figure 11 and Figure 12.

3.7. Fuzzy Adaptive PID-MPC Controller

The fuzzy adaptive PID-MPC control strategy comprises an integrated framework of three control systems. This architecture employs classical PID compensation theory to enhance the model predictive control (MPC) strategy, while a fuzzy inference mechanism dynamically adjusts the three PID parameters in an adaptive manner. This configuration substantially improves the disturbance rejection capability of the overall controller, thereby mitigating the influence of external perturbations on system performance. Within the control hierarchy, the MPC controller functions as the core component of the inner loop, with both the prediction horizon and control horizon specified as $N_p=N_c=10$ [29]. Concurrently, the fuzzy adaptive PID controller operates in the outer loop, providing compensatory adjustments to refine the performance of the inner MPC module.

It should be noted that the MPC algorithm utilizes a discrete optimization approach for rolling-horizon prediction and control, with a sampling period set to $T_s=0.001$ s. The PID controller is discretized by means of a difference equation and executes at the same sampling interval as the MPC. Simultaneously, the fuzzy inference engine updates its outputs within each sampling period, thereby ensuring temporally synchronized operation among all three components. The fuzzy adaptive PID-MPC controller architecture with compensation mechanism as the underlying logic is shown in Figure 13.

In Figure 13, the negative input terminal of the comparator receives the reference signal, which corresponds to the target values for the vehicle body’s vertical acceleration and pitch angular acceleration. To implement vibration suppression control, these target values are intentionally set to zero. The controller generates a total control force by combining the compensation damping force from the outer-loop fuzzy adaptive PID, which real-time adjusts PID parameters, with the reference damping force output by the inner-loop MPC based on model predictions:

$$U=F_0+\Delta F_1+\Delta F_2$$

(25)

where $U$ is the control vector of the system and $F_0$ serves as the reference component within the system control vector $U={[F_f,F_r]}^T$, generated by the primary Model Predictive Control (MPC) controller. Fundamentally, this constitutes a two-dimensional vector $F_0={[F_{0f},F_{0r}]}^T$, corresponding to the damping force outputs for both the front and rear suspensions. The complete control effort $U$ is formulated as $U=F_0+\Delta F_1+\Delta F_2$, where $\Delta F_1$ and $\Delta F_2$ denote the compensatory outputs from the fuzzy adaptive PID secondary controllers.

4. Simulation Results Analysis

In order to verify the effectiveness and superiority of the proposed fuzzy adaptive PID-MPC control strategy, this study compares traditional passive suspension, uncorrected MPC strategy, and the PID-MPC strategy with the fuzzy adaptive PID-MPC strategy proposed in this study in detail. The simulation model of the semi-car semi-active suspension system based on fuzzy adaptive PID-MPC control shown in Figure 14 is built in the MATLAB/Simulink platform. To further comprehensively capture the ride comfort characteristics of the vehicle during its operational process, the body vertical acceleration, pitch angle acceleration, suspension dynamic deflection and wheel dynamic load are selected as the key evaluation indexes of the suspension system performance. And the root mean square (RMS) is used as the evaluation standard to quantify the influence of each control strategy on the performance of the suspension system.

To further enhance result reproducibility, this work executed repeated simulation validations using identical suspension configurations and controller parameter sets. Notably, the deterministic speed-bump road profile yielded fully consistent simulation outcomes. For stochastic Class B and C road inputs—generated with a fixed random speed to maintain statistical homogeneity—the RMS values exhibited fluctuation margins below 2% across multiple runs, thereby validating the stability and reproducibility of the obtained results.

4.1. Simulation Analysis of Speed Bump Road Surface

For a speed bump road surface, the vehicle vertical acceleration and vehicle pitch angle acceleration of the vehicle body were primarily selected as key performance indicators for evaluating ride comfort. The simulation results are presented in Figure 15 and

Table 6. Simulation results of speed bump road surface.

RMS	Passive Suspension	MPC	PID-MPC	Fuzzy Adaptive PID-MPC
Vehicle vertical acceleration (m·s−2)	0.5853	0.4922	0.4167	0.3740
Vehicle pitch angle acceleration (rad·s−2)	0.1315	0.1123	0.0955	0.0872

.

As can be seen from Table 6, the fuzzy adaptive PID-MPC strategy demonstrates the best control performance in terms of vertical acceleration and pitch angular acceleration of the vehicle body. Compared with the passive suspension, the RMS values of vertical acceleration are reduced by 15.91%, 28.81%, and 36.10% under the MPC, PID-MPC, and fuzzy adaptive PID-MPC strategies, respectively. Similarly, the RMS values of pitch angular acceleration are reduced by 14.60%, 27.38%, and 33.67%, respectively. Figure 15a,b further illustrate that all three active control strategies significantly reduce the peak accelerations and effectively suppress shock responses compared to the passive suspension, with the most notable peak reduction achieved by the fuzzy adaptive PID-MPC strategy. Furthermore, Figure 15c,d present the corresponding front and rear suspension control forces, showing that the fuzzy adaptive PID-MPC strategy produces smoother and more stable damping trajectories compared with MPC and PID-MPC. This avoids abrupt fluctuations while still providing sufficient force to suppress the vibration peaks induced by the speed bump.

Meanwhile, it can be observed that the vibration decay time under speed bump excitation is extended to varying degrees across all strategies, which is related to the adjustment of the system’s damping characteristics. Due to its fixed control parameters, the MPC strategy tends to enter an overdamped state under certain conditions, resulting in slower response times. The PID-MPC strategy partially mitigates this issue, yet its performance remains limited under rapid excitations. In contrast, the fuzzy adaptive PID-MPC strategy dynamically adjusts the PID parameters in real time to maintain the damping ratio within a near-optimal range close to critical damping. As a result, it not only effectively reduces peak vibrations but also alleviates the problem of excessively slow decay, thereby achieving a balance between shock suppression capability and response speed.

4.2. Simulation Analysis of Class B Road Surface

The simulation results under a Class B road surface are shown in Figure 16 and

Table 7. Simulation results of Class B road surface.

RMS	Passive Suspension	MPC	PID-MPC	Fuzzy Adaptive PID-MPC
Vehicle vertical acceleration (m·s−2)	0.5664	0.4580	0.3943	0.3501
Vehicle pitch angle acceleration (rad·s−2)	0.1297	0.1057	0.0883	0.0811
Front suspension dynamic deflection (m)	0.0073	0.0080	0.0081	0.0081
Rear suspension dynamic deflection (m)	0.0034	0.0036	0.0038	0.0039
Front wheel dynamic load (N)	1152.1	1240.9	1294.9	1264.8
Rear wheel dynamic load (N)	635.7	701.5	721.0	735.5

. Compared to the passive suspension, the uncorrected MPC, PID-MPC, and fuzzy adaptive PID-MPC strategies reduced the RMS values of the vehicle body vertical acceleration by 19.14%, 30.38%, and 38.19%, respectively, and those of the pitch angle acceleration by 18.50%, 31.92%, and 37.47%, respectively. This demonstrates that the fuzzy adaptive PID-MPC strategy can more accurately track road irregularities, proactively generate control signals through model prediction, mitigate the impact of road excitations on the vehicle body, effectively suppress pitch vibrations, and significantly improve ride comfort during driving. In addition, Figure 16g,h illustrate the damping force outputs of the front and rear suspensions. Compared with the abrupt variations observed under MPC and the partial improvement under PID-MPC, the fuzzy adaptive PID-MPC strategy yields the most continuous and stable control forces. These smoother outputs better match the dynamic characteristics of magnetorheological (MR) dampers and correspond to the superior reduction in vertical and pitch accelerations shown in Table 7.

However, from the perspective of the root mean square values of the dynamic deflections of the front and rear suspensions and the dynamic loads of the front and rear tires, the uncorrected MPC, PID-MPC, and fuzzy adaptive PID-MPC strategies all exhibit increases compared to the passive suspension. Specifically, the RMS values of front suspension dynamic deflection increased by 0.0007 m, 0.0008 m, and 0.0008 m, respectively, while rear suspension dynamic deflection increased by 0.0002 m, 0.0004 m, and 0.0005 m. The front wheel dynamic load increased by 88.8 N, 142.8 N, and 112.7 N, and the rear wheel dynamic load increased by 65.8 N, 85.3 N, and 99.8 N, respectively. These results reveal the inherent conflict between improving ride comfort and maintaining driving safety, as well as the trade-off between vibration reduction and suspension travel limitations. Reducing vehicle body vibrations necessitates greater suspension deflection and tire dynamic loads. When vehicles traverse highly uneven roads, fluctuating wheel contact forces may lead to wheel liftoff, disrupting the interaction between tires and the road surface. This can compromise the functionality of safety systems such as the antilock braking system (ABS) and traction control system (TCS). To maintain close contact between the wheels and the ground, a stiffer damper is used to reduce wheel hop and improve driving safety. However, excessively stiff dampers transmit road impacts to the vehicle body, degrading comfort. Conversely, a soft damper can effectively filter out road bumps and improve comfort, but it also requires larger suspension travel, which increases the possibility of collision with the limit blocks.

Overall, the research objectives of this study were to enhance handling stability and ride comfort by reducing the vertical acceleration and pitch angle acceleration of the vehicle body in the half-car suspension system. Based on the analysis of simulation data, the fuzzy adaptive PID-MPC strategy dynamically adjusted control parameters through fuzzy rules, achieving improved ride comfort under constrained conditions while effectively maintaining suspension deflection and tire dynamic load variations within reasonable operational limits.

4.3. Simulation Analysis of Class C Road Surface

The simulation results under a Class C road surface are shown in Figure 17 and

Table 8. Simulation results of Class C road surface.

RMS	Passive Suspension	MPC	PID-MPC	Fuzzy Adaptive PID-MPC
Vehicle vertical acceleration (m·s−2)	1.1309	0.9159	0.7886	0.7037
Vehicle pitch angle acceleration (rad·s−2)	0.2538	0.2101	0.1810	0.1642
Front suspension dynamic deflection (m)	0.0145	0.0160	0.0163	0.0162
Rear suspension dynamic deflection (m)	0.0067	0.0072	0.0075	0.0080
Front wheel dynamic load (N)	2304.2	2552.6	2663.7	2601.8
Rear wheel dynamic load (N)	1271.3	1429.0	1468.9	1504.4

. The simulation results on a Class C road surface indicate that, compared with passive suspension, the uncorrected MPC, PID-MPC, and fuzzy adaptive PID-MPC strategies reduce the RMS values of the vehicle vertical acceleration by 19.01%, 30.27%, and 37.78%, and those of its pitch angle acceleration by 17.22%, 28.68%, and 35.30%, respectively. This indicates that the fuzzy adaptive PID-MPC strategy retains superior control performance under adverse road conditions characterized by higher excitation frequencies and larger amplitudes. Moreover, Figure 17g,h demonstrate that under such severe road excitations, this strategy still maintains smoother damping force trajectories than both MPC and PID-MPC, ensuring actuator feasibility while effectively suppressing large-amplitude vibrations.

In terms of suspension dynamic deflection and wheel dynamic load, the fuzzy adaptive PID-MPC strategy demonstrated increases of 11.72% in disturbance of front suspension, 19.40% in disturbance of back suspension, 12.92% in front wheel dynamic load, and 18.34% in rear wheel dynamic load relative to the passive suspension. Despite these increases exceeding those on a Class B road surface, all values remained within the acceptable range defined by control constraints. This implies that under adverse road conditions, heightened control effort is necessary to suppress body vibrations, inherently amplifying suspension deflections and wheel loads. Nevertheless, through optimized control algorithms, the fuzzy PID-MPC strategy achieves an effective compromise between ride comfort and driving safety, thereby minimizing detrimental effects on vehicle stability.

It should be noted that discrepancies exist across studies regarding vehicle parameters and suspension model configurations, rendering the absolute numerical values of simulation results non-comparable in a cross-literature context. However, when examining the trend of relative improvement magnitudes, the proposed fuzzy adaptive PID-MPC approach demonstrates consistently superior performance in ride comfort enhancement compared to existing MPC or PID-MPC methodologies. For instance, while conventional MPC-based control schemes reported in the literature typically achieve approximately 20–30% reduction in RMS (root mean square) metrics under standard operating conditions, our methodology attains 30–38% improvements across a speed bump road surface, Class B road surface, and Class C road surface. This performance advantage signifies enhanced adaptability and responsiveness under representative excitation conditions. Notably, this advancement is accompanied by moderate increases in suspension working space utilization and tire dynamic loads, along with extended vibration damping duration, thereby reflecting the inherent trade-off between ride comfort, response characteristics, and driving safety considerations.

5. Conclusions

To resolve the limitations of uncorrected MPC and PID-MPC controllers in semi-active suspensions, specifically inadequate prediction accuracy and pronounced time lags in vertical and pitch angle acceleration control, this study proposes a novel fuzzy adaptive PID-MPC hybrid strategy. The design objectives are enhanced vehicle stability and ride comfort. In this framework, the primary MPC controller generates preliminary optimal solutions for weighting matrices Q and R, while a fuzzy PID module dynamically adjusts parameters in real time. This synergistic approach significantly improves control performance for both vehicle vertical acceleration and vehicle pitch angle acceleration.

Based on MATLAB/Simulink, a semi-car semi-active suspension simulation model implementing the fuzzy adaptive PID-MPC strategy was developed. Validation was conducted under three typical driving conditions of speed bump road surface, Class B road surface, and Class C road surface. Simulation results demonstrate that under speed bump excitation, although the vibration duration was slightly prolonged after control, the peak values of the vehicle vertical acceleration and vehicle pitch angle acceleration were significantly suppressed, leading to substantial improvements in vehicle handling stability and ride comfort. On a Class B road surface and a Class C road surface, compared to passive suspension, the proposed strategy reduced RMS vehicle vertical acceleration by 38.19% (Class B) and 37.78% (Class C), and vehicle pitch angle acceleration by 37.47% (Class B) and 35.30% (Class C), conclusively validating its effectiveness and superiority.

It should be noted that due to the inherent conflicts between ride comfort and driving safety, as well as between comfort and suspension dynamic travel, a moderate increase in suspension dynamic deflection and tire dynamic load is an expected phenomenon under the control objectives of ensuring ride comfort and safety. The fuzzy adaptive PID-MPC control strategy proposed in this study enables the achievement of enhanced vehicle ride comfort across diverse operating conditions through flexible fuzzy rule design and parameter tuning, thereby providing an effective technical pathway for subsequent optimization of suspension system overall performance.

However, this study primarily considered static constraints such as suspension travel, damping force, and tire dynamic load, without explicitly modeling the dynamic characteristics of actuators, including response speed, delay, and damping variation rate. The analysis of the controller outputs (i.e., the damping forces presented in Figure 15, Figure 16 and Figure 17) indicates that the amplitude and rate of change of the control signals remain within the typical operational range of semi-active actuators such as magnetorheological dampers (damping force around 0–5000 N, response time 10–20 ms), suggesting practical feasibility despite this simplification. Nevertheless, to further enhance the applicability and robustness of the proposed strategy, future work will explicitly integrate actuator physical constraints and dynamic limitations directly into the MPC optimization framework. This will be coupled with parameterized modeling and disturbance analysis to systematically evaluate the controller’s sensitivity, stability, and reliability under different operating conditions. This endeavor will allow a more comprehensive assessment of the control strategy and provide a stronger foundation for its engineering applications.