T-S fuzzy-model-based adaptive cruise control for longitudinal car-following considering vehicle lateral stability

2.1. Vehicle longitudinal kinematic model

The following symbols are used in the car-following situation shown in Figure 1: $$ \Delta s $$ is the distance between the preceding and ego vehicles; $$ v_1 $$ and $$ v_2 $$ are the longitudinal velocities of the preceding and ego vehicles, respectively; and $$ a_1 $$ and $$ a_2 $$ are the corresponding longitudinal accelerations.

Figure 1. Vehicle longitudinal kinematics.

The desired spacing distance between the preceding and ego vehicles is given as follows ^[28]:

where $$ d_{0} $$ is vehicle desired distance at standstill and $$ t_{0} $$ is the constant headway time.

The difference in the desired and actual distances between the vehicles is defined as $$ \Delta d $$, and the relative velocity between the preceding and ego vehicles is defined as $$ \Delta v $$; then,

Considering the time delay of the engine in the driving system, we employ a first-order system to relate the actual vehicle longitudinal acceleration $$ a_{2} $$ and the desired acceleration $$ a_{{\rm{des}}} $$ as follows^[29]:

where $$ \tau_{0} $$ is the engine time constant, and $$ a_{{\rm{des}}} $$ is the acceleration to be determined.

The definitions given above are used to express the vehicle longitudinal kinematic model as follows:

2.2. Vehicle lateral dynamics

Figure 2 shows the classical two-degree-of-freedom (2-DOF) bicycle model of vehicle dynamics, which is simplified in this study by collapsing each axle to a single tire to reflect the fundamental features of lateral motions.

Figure 2. 2-DOF vehicle lateral dynamics.

The mass of the ego vehicle is $$ m $$. $$ I_{z} $$ is the moment of inertia about the yaw axis through the vehicle's centre of gravity (CG). $$ l_f $$ and $$ l_r $$ represent the distances from the vehicle centre to the front and rear axles of the vehicle, respectively. $$ F_{yf} $$ and $$ F_{yr} $$ denote the lateral forces on the vehicle front and rear tires, respectively. $$ \delta $$ is the steering angle for the front wheel of the vehicle. $$ \beta $$, $$ \gamma $$ and $$ v_x $$ represent the sideslip angle, yaw rate, and longitudinal velocity of the ego vehicle, respectively. The vehicle lateral dynamics can be described in terms of the sideslip angle and yaw rate as follows:

where $$ M_z $$ is the external yaw moment generated by the differences in the longitudinal tire/road forces among the four tires of the vehicle.

2.3. Tire/road force model

The vehicle lateral tire/road force is generated by contact between the vehicle tires and the road surface^[30]. A tire operates in the linear region for a small vehicle lateral acceleration, which can be characterized by the cornering stiffness of the front and rear tires $$ C_{ f} $$ and $$ C_{r} $$, respectively, and the corresponding sideslip angles. The relationship between the tire lateral force and sideslip angle is

where $$ \alpha_f $$ and $$ \alpha_r $$ are the slip angles of front and rear tire and can be given by:

However, at high lateral acceleration, the tire/road force may not be linearly proportional to the slip angle owing to differences in the road surface characteristics and cannot be simply expressed in terms of a constant cornering stiffness and sideslip angles. Therefore, we adopt an uncertain cornering stiffness, which varies over a range, to model the uncertainty in the tire/road force ^[31]:

where $$ C_{ i}=\frac{{\rm{max}}(C_{ i}(\bullet))+{\rm{min}}(C_{ i}(\bullet))}{2} $$, $$ \Delta C_{ i}(\bullet)\in[-\Delta C_{ i}, \Delta C_{ i}] $$, $$ \Delta C_{ i}=\frac{{\rm{max}}(C_{ i}(\bullet))-{\rm{min}}(C_{ i}(\bullet))}{2}(i=f, r) $$, $$ C_{ i}(\bullet) $$ denotes the uncertain cornering stiffness, and $$ (\bullet) $$ represents all possible variables causing variations in the cornering stiffness.

2.4. Vehicle-following system with lateral stability

Longitudinal kinematics, lateral dynamics, and an uncertain tire/road force model are integrated to formulate the following model for closed-loop car-following dynamics ^[32]:

where $$ x(t)=[\Delta d, \Delta v, a_2, \beta, \gamma]^T $$, $$ u(t)=[a_{{\rm{des}}}, M_{z}]^T $$, and the external disturbance $$ d(t)=[a_1, \delta]^T $$.

The uncertain terms $$ \Delta \tilde{A} $$ and $$ \Delta \tilde{B}_{d} $$ are denoted as $$ \Delta \tilde{A}=\tilde{H}_{1}F(t)\tilde{E}_{1} $$ and $$ \Delta \tilde{B}_{d}=\tilde{H}_{1}F(t)\tilde{E}_{2} $$, respectively, where

and $$ N(t):R_{t\geq 0}\rightarrow \lbrack -1, 1] $$ represents an unknown real-value function.

Note that in the car-following scenario, the velocity of the ego vehicle varies with that of the preceding vehicle to maintain a desired safe distance. In this study, we assume that the velocity of the preceding car varies within a bounded range $$ v_1\in[v_{ {\rm{min}}}, v_{ {\rm{max}}}] $$, where $$ v_{ {\rm{min}}} $$ and $$ v_{ {\rm{max}}} $$ represent the minimum and maximum velocities during vehicle adaptive cruising.

2.5. T-S fuzzy modeling for longitudinal car-following with vehicle lateral stability

For real-time implementation of the proposed T-S fuzzy model predictive controller in the discrete-time domain, we adopt Euler's discretization method with the sampling time $$ T_{s} $$; then, the discrete-time model of System Equation (8) is given as

where

As the velocity of the ego vehicle $$ v_2 $$ changes with the speed of the preceding vehicle $$ v_1 $$, Equation (9) is clearly a parameter-varying system.

The T-S fuzzy modelling approach is then employed to describe the time-varying car-following dynamics. We use the classical sector nonlinearity method to derive the discrete-time T-S fuzzy model given in Equation (9):

Fuzzy rule $$ \mathscr{R}^i $$: if$$ \theta_1(k) \in \mathscr F_1^i $$ and $$ \theta_2(k) \in \mathscr F_2^i $$, $$ \textbf{then} $$

where $$ i=1, 2, \cdots, r $$, $$ \theta_1(k)=\frac{1}{v_2} $$ and $$ \theta_2(k)=\frac{1}{v_2^2} $$ are premise variables, $$ \mathscr F^i_j $$ is the fuzzy set, and $$ A_i $$, $$ B_{di} $$, $$ \Delta A_i $$ and $$ \Delta B_{di} $$ are the system matrices defined in Equation (9), with $$ v_2 $$ being replaced by $$ v_{ {\rm{min}}} $$ and $$ v_{ {\rm{max}}} $$. We use $$ \eta(\theta(k)) $$ to denote the normalized membership function of $$ \mathscr F^i $$, where $$ \theta(k)=[\theta_1(k), \theta_2(k)] $$ represents the premise variable vector, $$ \mathscr F^i=\mathscr F^i_1\mathscr F^i_2 $$ and $$ \sum_{i=1}^r\eta_i(\theta(k))=1 $$. For brevity, $$ \eta_i(\theta(k)) $$ is denoted as $$ \eta_{i} $$. Considering the velocity restrictions enables us to easily derive the following weighting factors for the established T-S fuzzy system [Equation (10)]:

where $$ \theta_{1 {\rm{max}}} $$, $$ \theta_{2 {\rm{max}}} $$, $$ \theta_{1 {\rm{min}}} $$ and $$ \theta_{2 {\rm{min}}} $$ are the maximal and minimal values of $$ \theta_1 $$ and $$ \theta_2 $$, respectively, and $$ \mathscr F^i_j (i=\; 1, 2, 3, 4, j=1, 2) $$ can be obtained accordingly from Equation (11) and Equation (12). Therefore, $$ \mathscr F^i $$ is derived from $$ 2^2 $$ combinations of $$ \mathscr F^i_1 $$ and $$ \mathscr F^i_2 $$. The following compact representation of Equation (9) is obtained by using a standard fuzzy inference approach:

where

Note that using the sector nonlinearity modelling approach with the premise variables $$ \frac{1}{v_2} $$ and $$ \frac{1}{v_2^2} $$ yields a T-S fuzzy expression for the car-following system with lateral stability. The discretized form of the tracking system in Equation (9) is represented using $$ r=2^2 $$ linear subsystems with the aforementioned membership functions. Inspired by ^[33], we further exploit the relationships among the premise variables to reduce the numerical computational complexity and conservativeness of the controller design. Here, we define

where $$ \rho $$ is a scalar variable, $$ \hat v_0=\frac{2v_{ {\rm{ \rm{min}}}}v_{ {\rm{ \rm{max}}}}}{v_{ {\rm{min}}}+v_{ {\rm{max}}}} $$ and $$ \hat v_1=\frac{2v_{ {\rm{min}}}v_{ {\rm{max}}}}{v_{ {\rm{min}}}-v_{ {\rm{max}}}} $$. Therefore, we have

The new premise variable $$ \rho $$ is bounded as follows:

where $$ v_2=v_{ {\rm{min}}} $$ for $$ \rho=-1 $$ and $$ v_2=v_{ {\rm{max}}} $$ for $$ \rho=1 $$. Therefore, $$ \rho $$ can be used to describe the variation in $$ v_2 $$ from $$ v_{ {\rm{min}}} $$ to $$ v_{ {\rm{max}}} $$. In addition, the following equation can be obtained based on Taylor's approximation:

The number of subsystems in the T-S fuzzy model Equation (10) is thus reduced from $$ 4 $$ to $$ 2 $$, which considerably decreases the computational burden associated with the controller parameters in each sampling period within the MPC framework, and facilitates real-time implementation of the model, as expected.

3.1. Robust positively invariant set

Consider a discrete-time dynamical system

where $$ f(0, 0)=0 $$, $$ x(k)\in \mathbb{R}^{n} $$ is a state vector, and $$ d(k) $$ is a control input or external disturbance that belongs to a compact set $$ \mathbb{D} \subset \mathbb{R}^{m} $$ containing the origin. A robust positively invariant (RPI) set is defined below.

Definition 3.1: If $$ x(k) \in \Omega $$ ($$ \Omega \subset \mathbb{R}^{n} $$), it holds that $$ x(k+1)\in \Omega $$ for all $$ d(k)\in \mathbb{D} $$; then, $$ \Omega $$ is called an RPI set for System [Equation (18)].

Lemma 3.1^[34]: The following two expressions are equivalent for System [Equation (18)] with $$ d^Td\leq c^2 $$, where $$ c $$ is a positive constant:

● the ellipsoidal set $$ \Omega_p \triangleq \{x^TPx\leq\gamma\} $$, where $$ P>0 $$, is a robust positively invariant set;

● the inequality $$ {x(k+1)}^TPx(k+1)\leq x^TPx $$ holds if the external disturbance satisfies $$ \frac{1}{c^2}d^Td \leq \frac{1}{\gamma}x^TPx $$.

3.2. Input-to-state stability

We define input-to-state stability (ISS) for use in the following sections.

Definition 3.2: A discrete-time system $$ x(k+1)=f(x(k), d(k)) $$ is ISS if there exist a $$ \mathcal{KL} $$-function $$ \beta: \mathbb{R}_{\geq 0} \times \mathbb{Z}_{+}\rightarrow \mathbb{R}_{\geq 0} $$ and a $$ {\mathcal{K}} $$-function $$ \rho $$ satisfying

where $$ x_0 $$ is the initial state, $$ d(k) $$ is the input sequence, and $$ k $$ is the sampling time instant.

Definition 3.3: A function $$ V: \mathbb{R}^{n}\rightarrow \mathbb{R}_{\geq 0} $$ is called an ISS-Lyapunov function for the system $$ x(k+1)=f(x(k), d(k)) $$ if there exist $$ \mathcal{K_{\infty}} $$-functions $$ \alpha_1 $$, $$ \alpha_2 $$, $$ \alpha_3 $$ and a $$ \mathcal{K} $$-function $$ \rho $$ satisfying

Lemma 3.2^[35]: A system is said to have ISS if it admits a continuous ISS-Lyapunov function.

3.3. T-S fuzzy model-based predictive control

We adopt the following model within the framework of predictive control to represent the future dynamics of the T-S fuzzy system for vehicle-following considering lateral stability:

where $$ x(k+i|k) $$, $$ u(k+i|k) $$ and $$ d(k+i|k) $$ represent the predicted state, control input and external disturbances at the $$ (k+i) $$th time instant, respectively, and $$ i\in\{1, 2, \cdots\} $$. The stage cost $$ \ell(k+i|k) $$ can be represented as

where $$ Q $$ and $$ R $$ are the weighting matrices of the state variables of the system and control inputs, respectively. $$ \tau $$ is the weight related to the external disturbance. We simplify this notation by using $$ x(k+i) $$, $$ u(k+i) $$ and $$ d(k+i) $$ to denote $$ x(k+i|k) $$, $$ u(k+i|k) $$ and $$ d(k+i|k) $$, respectively.

Consider the following objective function $$ \mathcal{J}_p $$

where $$ \ell(k+i|k) $$ is the stage cost at the predicated time instant, and the positive-definite function $$ V(x(k+p|k)) $$ is called the terminal cost. This type of cost function was proposed in Ref.^[36] to develop a novel synthesis method with enhanced robustness.

As the cost function cannot be optimized in real time due to the unknown external disturbance, the upper limit of the cost function is minimized here. We define the following fuzzy quadratic Lyapunov function:

where $$ P_j $$ is a positive definite matrix. Let the Lyapunov function satisfy the following inequality constraint:

Add both sides of the inequality from $$ i=0 $$ to $$ \infty $$, then we get

which implies that we can infer the upper bound of the objective function $$ \mathcal{J}_{\infty} $$ from the positiveness property of the function $$ V(x(\infty)) $$. Assume a scalar $$ \gamma $$ exists that satisfies

Defining $$ X_j=\gamma P_j^{-1} $$ and applying the Schur complement operation yields the following sufficient condition for Equation (24):

Then, the aforementioned optimization problem is converted into the minimization of the upper bound of the infinite-horizon objective function:

A state feedback law $$ u(k)=F_\eta x(k) $$ is applied to minimize the upper bound of the performance functions $$ \mathcal{J}_p $$ and $$ F_\eta=\sum_{j=1}^n\eta_i(\theta)F_j $$, where $$ F_j=Y_jX_j $$. Considering the inequality [Equation (24)] yields

which is equivalent to

$$ I $$ is an identity matrix with appropriate dimensions. The inequality [Equation (28)] can be guaranteed as long as

holds.

Here, we introduce a lemma for use in the following sections.

Lemma 3.3^[37]: For matrices $$ \Gamma $$, $$ H $$ and $$ E $$ with appropriate dimensions and $$ \Gamma^T=\Gamma $$, the following inequality

holds for all $$ F(k)^TF(k)\leq1 $$ if and only if there is a positive scalar $$ \epsilon $$ such that

By Lemma$$ 3.3 $$, the following sufficient condition can be derived to guarantee the inequality [Equation (31)]:

where $$ \zeta =\gamma \tau $$.

We also need to consider the control input constraints to ensure driving comfort:

According to inequality [Equation (26)], we have:

Therefore, we have:

Considering the Schur complement, the input constraint can be guaranteed by the following LMI if there exists a symmetric matrix $$ U $$^[38]

where $$ U_{i i} \leq u_{i, \max}^{2} $$.

To deal with the external disturbance, the concept of RPI is introduced to ensure the closed-loop stability of car-following system. According to the concept of RPI and quadratic boundedness as shown in Lemma$$ 3.1 $$, $$ \Omega_k $$ is an RPI set if

holds under

where $$ d(k)^{T} d(k) \leq \phi^2 $$.

The S-procedure is used to obtain a sufficient condition as follows:

where $$ \lambda $$ is a positive scalar belonging to $$ (0, 1) $$.

By Lemma$$ 3.3 $$, the above mentioned inequality can be guaranteed by the following matrix: inequality:

Therefore, the controller design can be summarized as the following:

The T-S fuzzy state feedback controller is derived by solving the optimization problem in Equation (40) under parameter uncertainties and external persistent perturbations. The inequality Equation (34) ensures that the cost function $$ \mathcal{J}_{\infty}(k) $$ is upper-bounded by the Lyapunov function $$ V(x(k)) $$, the inequality Equation (38) guarantees that the input constraints are satisfied, and the inequality Equation (40) leads to $$ x(k)\in \Omega_k $$, where $$ \Omega_k $$ is an RPI set.

Remark 3.1: Note that $$ \zeta $$ is minimized instead of $$ \gamma $$ in the abovementioned optimization problem. This approach is used because both $$ \gamma $$ and $$ \tau $$ need to be minimized, and a smaller $$ \tau $$ implies a higher system performance. We adopt the approach given in Ref.^[23] to simultaneously optimize $$ \gamma $$ and $$ \tau $$ by defining $$ \zeta=\gamma\tau $$ in Equation (34). The introduction of the variable $$ \lambda $$ makes the constraint Equation (40) a bilinear matrix inequality, which can be handled with existing solvers, e.g., PENBMI. The computational load is reduced further by predefining a suboptimal value of $$ \lambda $$ by trial and error.

Theorem 3.1: The optimization problem Equation (40) has the property of recursive feasibility, that is, a solution will always exist once the problem is initially solvable.

Proof: Implementation of the predictive control strategy based on the T-S fuzzy model requires the constrained optimization problem Equation (40) to be solved at each time instant. Therefore, it is important to guarantee the recursive feasibility of the optimization problem. As an external disturbance is considered, the recursive feasibility is no longer a natural characteristic of the proposed controller. In this study, only the constraint Equation (26) depends on the time instant $$ k $$, which involves $$ x(k) $$. Therefore, we only need to ensure the feasibility of the constraint Equation (26).

Note that the inequality Equation (26) is equal to $$ x(k) \in \Omega_k $$. The inequality Equation (40) ensures that the set $$ \Omega_k $$ is an RPI set, which implies that the inequality Equation (26) is still feasible at the $$ (k+1) $$ time instant. That is, $$ x(k+1)\in \Omega_{k+1}\in \Omega_{k} $$ is still satisfied. Hence, recursive feasibility is guaranteed. Thus, the proof is completed.

Theorem 3.2: The closed-loop system in this paper has ISS based on the proposed MPC strategy under an external disturbance.

Proof: It has been proven that the optimization problem Equation (41), once solvable, will always be solvable.

Define the optimal solution at time instant $$ k $$ as $$ \{\gamma^*, X_j^*, \epsilon_{1j}^*, \epsilon_{2j}^*, Y_j^*, \zeta^*\} $$, and $$ V^*(x(k))=\sum_{i=1}^2\eta_i(\theta)x^T(k)P_i^*x(k) $$, where $$ P^*=\gamma^* {X_j^*}^{-1} $$. We need to prove that the $$ V^*(x(k)) $$ is an ISS-Lyapunov function. Define $$ \bar\lambda $$ as the upper bound of eigenvalue of $$ P_{\eta}^* $$, and $$ \underline\lambda $$ as the lower bound of the eigenvalue of $$ P_{\eta}^* $$. We can obtain:

Furthermore, from inequality Equation (24), we can derive the following inequality:

where $$ F^*=\sum_{j=1}^2\eta_i(\theta)Y_jX_j^{-1} $$. Therefore, we get

From the inequalities Equation (42), Equation (44) and Lemma$$ 3.2 $$, we can conclude that $$ V(x(k)) $$ is an ISS-Lyapunov function, and the closed-loop system is ISS. Thus the proof is completed.

4.1. Accelerating control

Figure 4 shows the engine speed versus the torque at different throttle openings, which are shown at the bottom. If the engine speed $$ w_e $$ and the desired torque $$ T_{ {\rm{e, des}}} $$ are known, the corresponding expected throttle opening $$ \sigma_{ {\rm{des}}} $$ is obtained based on this look-up table:

Figure 4. Engine model.

The vehicle longitudinal dynamics are produced by the combined effect of vehicle traction, wind resistance, and ground resistance. Therefore, the vehicle state depends strongly on the road smoothness and the magnitude of the wind resistance coefficient during cruising. The equivalent air friction during longitudinal driving is

where $$ \rho_{a} $$ is the air mass density, $$ C_{d} $$ is the coefficient of air friction, $$ A_{F} $$ is the windward area of the ego vehicle, $$ R_{p} $$ is the ratio between the wheel speed $$ w_{w} $$ and engine speed $$ w_{e} $$, and $$ r_{ {\rm{eff}}} $$ is the effective tire radius.

During vehicle acceleration, the engine torque $$ T_{ {\rm{e, des}}} $$ is related to the expected acceleration $$ a_{ {\rm{des}}} $$ is^[39] as

where $$ J_{e}=I_e+I_t +(mr_{ {\rm{eff}}}^2+I_w)R_p^2 $$ is the effective moment of inertia for the engine side, $$ I_e $$ is the moment of inertia of the engine, and $$ R_x $$ is the sum of all the rolling resistances related to the rolling damping coefficient $$ f $$. As $$ f \in (0.01, 0.04) $$, $$ R_x $$ is sufficiently small to be neglected in this study. The following exist:

Then, $$ J_e $$ can be rewritten as $$ J_e=mr_{ {\rm{eff}}}^2R_p^2 $$. Therefore, Equation $$ (47) $$ can be rewritten as follows:

Thus, the corresponding expected throttle opening $$ \sigma_{ {\rm{des}}} $$ can be obtained from the inverse dynamic relation.

4.2. Braking control

During vehicle braking, the desired acceleration $$ a_{des} $$ and the yaw moment $$ M_{des} $$ given by the upper controller can be used to determine the longitudinal tire force from the following equation^[40]:

where $$ b $$ is the distance between the right and left wheels and $$ h $$ is the height of the centre of mass of the vehicle.

Equation (50) can be solved for the longitudinal force on the four wheels. Then, we can calculate the hydraulic pressure wheel cylinder from the following equation:

where $$ r_w $$ denotes the efficient wheel radius and $$ K_B $$ represents the pressure constant of a single wheel.

5.1. Case I: ramp speed profile

The initial condition in this case is a zero relative velocity between vehicles: we assume $$ v_1=v_2=25 $$$$ {\rm{m}}/{\rm{s}} $$ and a desired distance between vehicles at standstill $$ d_0=10\, {\rm{m}} $$. Considering the nonlinearity of the lateral forces between the tires and the road surface, the cornering stiffness for the front and rear tires is set to $$ (66900\pm10000)\, {\rm{N/rad}} $$ and $$ (62700\pm10000)\, {\rm{N/rad}} $$, respectively. The upper bound on the control input for the vehicle acceleration is set to $$ 2\, {\rm{m}}^2/{\rm{s}} $$. Substituting the corresponding parameters into the system model Equations 8-17 easily yields the system matrices. According to Remark 3.1, $$ \lambda $$ is selected as $$ 0.01 $$ in this study.

The optimization problem given by Equation (41) is solved recursively to yield the closed-loop system responses in the CarSim/Simulink joint simulation environment, which are shown in Figure 7. In Figure 7A, the blue and red curves represent the speeds of the preceding and ego vehicles, respectively. Figure 7B and Figure 7C show the errors in the longitudinal velocity and between the desired and actual longitudinal distances, respectively, where the maximum velocity error is 5.9 m/s at 4.5 s and the maximum distance error is 3.8 m at 19.7 s. Figure 7 shows that although the preceding vehicle velocities vary in the range $$ [10, 25]\, {\rm{m}}/{\rm{s}} $$, the predictive controller based on the T-S fuzzy model guarantees the velocity and the desired safe following distance for the ego vehicle under the considered extreme road conditions.

Figure 7. (A) Velocity of the preceding and ego vehicle; (B) Relative velocity; (C) Relative distance.

To further demonstrate the advantages of the proposed car-following controller design, Figure 8 shows the responses of sideslip angles and yaw rate (which reflect the lateral stability of the vehicle) obtained using the conventional controller (without consideration of the lateral stability) and the controller developed in this study. Figure 8 shows that compared to the results obtained using the conventional controller, the sideslip angle and yaw rate obtained using the proposed controller are smaller in magnitude, indicating improved vehicle stability.

Figure 8. (A) Vehicle sideslip angle of the ego vehicle in Case I. (B) Vehicle yaw rate of the ego vehicle in Case I.

5.2. Case II: $$ cosine $$ speed profile

In this case, the initial velocities of the preceding and ego vehicles and the relative distance are set to 25 m/s and 50 m, respectively. Figure 9A shows the $$ cosine $$-function speed profile of the preceding vehicle and the simulation results as a blue dash-dotted curve. In Figure 9A, the red curve represents the closed-loop response for the speed of the ego vehicle. Figure 9B and Figure 9C show the errors in the longitudinal velocity and between the desired and actual distances, respectively. Figure 9D shows the acceleration of the ego vehicle, which satisfies the input constraint given by $$ |a_{des}|\leq 2\, \rm {m}^2/{\rm{s}} $$.

Figure 9. (A) Velocity of preceding and ego vehicle; (B)Relative velocity; (C) Relative distance; (D)Ego vehicle acceleration.

It can be concluded from Figure 9 that the proposed method completes the car-following task with satisfactory performance for a time-varying velocity. Figure 10 shows the dynamics of the sideslip angle, yaw rate, and vehicle lateral offset. Figure 10C clearly shows that the lateral offset increases when the car-following manoeuvre is conducted under the conventional MPC without lateral stabilization. These three aspects of the results show that the proposed car-following controller outperforms the conventional controller in terms of guaranteeing both satisfactory tracking performance and lateral stability of the vehicle in emergency scenarios.

Figure 10. (A) Sideslip angle in Case II; (B) Yaw rate in Case II; (C) Vehicle lateral offset in Case II.

Authors' contributions

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Zhang C, Wei X

Performed data acquisition, as well as provided administrative, technical, and material support: Wang Z, Zhang H, Guo X

Availability of data and materials

In this paper, we have detailed a modelling procedure and the construction of an optimization problem with linear matrix inequality constraints to design a T-S fuzzy model predictive controller. The physical parameters of the vehicle are provided in the section on the simulations. The effectiveness of the developed method has been demonstrated using the MATLAB/CarSim joint simulation platform. There are no additional data or materials associated with this study.

Financial support and sponsorship

This work was supported in part by National Key R & D Program of China under Grant No: 2020AAA0108100; in part by Shanghai Natural Science Foundation under Grant No: 21ZR1482300 and 19ZR1461400; and in part by the Fundamental Research Funds for the Central Universities.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2022.