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\title{A Structured Neural-Supervised Fuzzy--PID Steering Controller with ISS Guarantees for Nonlinear Ackermann Vehicles}

\author[1,*]{Vo Thanh Ha}
\author[1]{Nguyen Minh Huy}
\author[2]{Nguyen Van Hai}
\author[2]{Nguyen Tam Thanh}
\author[3]{Vo Quang Vinh}
\affil[1]{Faculty of Electrical and Electronics Engineering, University of Transport and Communications, No. 3 Cau Giay, Hanoi 100000, Vietnam}
\affil[2]{Vietnam Maritime University, 484 Lach Tray Street, Le Chan District, Hai Phong City 180000, Vietnam}
\affil[3]{Electric Power University, 235 Hoang Quoc Viet, Bac Tu Liem, Hanoi 100000, Vietnam}
\affil[*]{Corresponding author: vothanhha.ktd@utc.edu.vn}

\begin{abstract}
Accurate and robust path tracking is essential for Ackermann-type autonomous vehicles operating under nonlinear tyre behaviour, parameter uncertainty, external disturbances, and real-time computational constraints. This paper proposes a structured neural-supervised Fuzzy--PID steering controller with input-to-state stability guarantees for nonlinear Ackermann vehicles. Unlike black-box neural controllers, the proposed method preserves an explicit Fuzzy--PID steering law as the core control structure, while a lightweight neural supervisory layer provides bounded online gain corrections. This design avoids uncontrolled coupling between the fuzzy and neural components and improves adaptability under varying operating conditions. A nonlinear vehicle model with sector-bounded tyre-force representation is formulated to capture tyre saturation effects in a tractable form for stability analysis. A composite Lyapunov function is developed to prove input-to-state stability and uniform ultimate boundedness under bounded disturbances and modelling uncertainty. High-fidelity Simscape Multibody simulations are conducted under nominal tracking, crosswind disturbance, tyre--road friction variation, vehicle mass variation, cornering stiffness uncertainty, and comparison with a linearized MPC benchmark. Compared with conventional PID, Fuzzy--PID, and MPC, the proposed controller achieves lower lateral tracking error, smoother steering action, stronger robustness, and sub-millisecond computation time. These results demonstrate that the proposed structured neural-supervised Fuzzy--PID controller is suitable for robust and real-time path tracking of Ackermann autonomous vehicles.
\end{abstract}

\begin{document}
\flushbottom
\maketitle
\normalfont\normalsize
\thispagestyle{empty}

\section*{Keywords}
Ackermann vehicle; neural-supervised control; Fuzzy--PID; input-to-state stability; path tracking; sector-bounded tyre dynamics.

%========================================================
\section{Introduction}

Accurate and robust path tracking is a fundamental requirement for autonomous ground vehicles operating in complex and uncertain environments. For self-driving urban vehicles, motion planning and control must be designed jointly because the feasibility of a planned trajectory depends strongly on the capability of the vehicle to track it safely and accurately \cite{Paden2016,Gonzalez2016}. Autonomous driving systems must also operate under uncertainty arising from surrounding traffic, environmental disturbances, and modelling errors \cite{Schwarting2018}. The practical importance of robust steering control was demonstrated by the Stanley autonomous vehicle, where reliable path tracking was essential for successful autonomous navigation under real driving conditions \cite{Thrun2006}.

For Ackermann-type vehicles, the path-tracking problem is challenging because the lateral and yaw dynamics are nonlinear and strongly affected by vehicle speed, tyre--road interaction, steering saturation, and parameter variations. Classical PID-based steering controllers remain attractive due to their simple structure, low computational burden, and ease of embedded implementation. Nested PID steering control has been successfully investigated for lane keeping in autonomous vehicles \cite{Marino2011}. However, fixed-gain PID controllers usually provide satisfactory performance only around nominal operating conditions. Under curvature transitions, crosswind disturbance, mass variation, or reduced tyre--road friction, fixed gains may lead to increased lateral error, slow recovery, and oscillatory steering response. Adaptive-neural-network-based lateral motion control has been studied to improve robustness at handling limits \cite{Ji2018}, but neural adaptive structures may become difficult to interpret if the neural network directly dominates the control action.

Model predictive control (MPC) has become one of the most widely used benchmark approaches for autonomous vehicle steering because it can explicitly handle multivariable dynamics and actuator constraints. Predictive active steering control and linear time-varying MPC have shown effective path-tracking performance for autonomous vehicles \cite{Falcone2007,Falcone2007CDC}. Recent reviews further confirm that MPC is a dominant framework for automated road-vehicle path tracking \cite{Stano2023}. However, MPC performance depends strongly on prediction-model accuracy, horizon selection, cost-function tuning, and real-time optimization capability. For low-cost embedded platforms, solving an optimization problem at every sampling instant may increase computational burden.

Vehicle modelling is also essential for steering controller design. Kinematic and dynamic vehicle models offer different trade-offs between simplicity and fidelity \cite{Kong2015}. Established vehicle dynamics theory provides the basis for modelling lateral motion, yaw dynamics, steering response, and tyre behaviour \cite{Rajamani2012,Pacejka2012}. Fundamental vehicle-dynamics references show that mass distribution, wheelbase, cornering stiffness, and road adhesion significantly affect lateral stability and steering response \cite{Gillespie1992,Wong2008}. Therefore, a controller designed only from a simplified linear model may not remain robust under nonlinear tyre behaviour, parameter variations, and external disturbances.

Fuzzy logic and neuro-fuzzy methods have been widely introduced to improve adaptability under nonlinear and uncertain conditions. Fuzzy set theory and Takagi--Sugeno modelling provide flexible tools for representing nonlinear control knowledge \cite{Zadeh1965,Takagi1985}. Adaptive-network-based fuzzy inference systems combine fuzzy reasoning with neural learning capability \cite{Jang1993}. Industrial applications of fuzzy control demonstrate its effectiveness in uncertain systems \cite{Precup2011}, while stability analysis of fuzzy-model-based control highlights the importance of connecting fuzzy design with rigorous stability conditions \cite{Lam2018}. Nevertheless, many Fuzzy--PID and neuro-fuzzy PID controllers are still tuned heuristically, and the relationship among membership functions, rule bases, scaling factors, and closed-loop stability is not always explicit.

Neural adaptive control provides another approach for compensating nonlinearities and uncertainties. Neural networks have been used for identification and control of dynamical systems due to their approximation capability \cite{Narendra1990}. Stable adaptive neural network control theory provides Lyapunov-based tools for ensuring boundedness of neural adaptive systems \cite{Lewis1999,Ge2002}. However, when neural networks are used directly as controllers or unrestricted gain optimizers, the resulting closed-loop structure may become difficult to interpret and analyse. In hybrid neural--fuzzy--PID systems, unclear interaction between neural and fuzzy components may introduce cross-coupling and complicate stability analysis.

From a nonlinear control perspective, robustness should not be evaluated only by simulation. Lyapunov-based nonlinear and adaptive control theory provides systematic tools for analysing stability and boundedness under uncertainty \cite{Khalil2002,Slotine1991,Ioannou1996,Narendra1989,Krstic1995}. Input-to-state stability (ISS) is particularly suitable for nonlinear systems subject to external inputs and bounded perturbations \cite{Sontag1995,Sontag1996}. Small-gain results for ISS systems also provide a theoretical basis for analysing interconnected nominal and adaptive subsystems \cite{Jiang1994}. These theories motivate the development of a steering controller whose adaptive mechanism is explicitly bounded and compatible with Lyapunov-based ISS analysis.

Despite significant progress, several limitations remain. First, PID and Fuzzy--PID steering controllers are often tuned heuristically, without explicit connection between performance specifications and controller parameters. Second, many neural or neuro-fuzzy steering controllers improve simulation performance but do not clearly separate fuzzy inference and neural adaptation. Third, the possible cross-coupling between neural and fuzzy components is rarely constrained. Fourth, stability analysis is often presented separately from controller design and is not directly linked to gain bounds, actuator saturation, or implementation constraints. Finally, validation is sometimes insufficiently reproducible because complete controller parameters, tyre parameters, disturbance settings, and benchmark configurations are not fully reported.

To address these limitations, this paper proposes a structured neural-supervised Fuzzy--PID steering controller for nonlinear Ackermann vehicles. The novelty does not lie in using PID, fuzzy logic, or neural networks individually. Instead, it lies in the structured and stability-oriented integration of these components. The Fuzzy--PID controller is preserved as an explicit and interpretable steering-control core, while the neural network is restricted to a bounded supervisory role that provides online gain corrections. Gain projection, neural correction saturation, integral anti-windup, steering-angle saturation, and steering-rate limitation are introduced to ensure physical feasibility and prevent uncontrolled neural--fuzzy interaction.

The main contributions are summarized as follows:
\begin{itemize}
    \item A nonlinear Ackermann vehicle tracking model with sector-bounded tyre-force representation is formulated to capture tyre saturation effects in a stability-oriented form.
    \item A structured neural-supervised Fuzzy--PID steering controller is developed, where the fuzzy component performs bounded rule-based gain correction and the neural component provides only bounded supervisory adaptation.
    \item A performance-oriented design procedure is introduced to connect tracking specifications, PID gains, fuzzy scaling factors, neural correction bounds, and actuator constraints.
    \item ISS and UUB stability are established using Lyapunov analysis under bounded disturbances, sector-bounded tyre nonlinearities, bounded neural adaptation, and steering saturation.
    \item High-fidelity Simscape Multibody validation is conducted under nominal tracking, disturbance, friction variation, parameter uncertainty, and comparison with linearized MPC.
\end{itemize}

The remainder of this paper is organized as follows. Section~\ref{sec:model} presents the nonlinear vehicle model. Section~\ref{sec:controller} describes the proposed controller. Section~\ref{sec:stability_analysis} establishes the stability analysis. Section~\ref{sec:simulation_setup} presents the simulation framework and results. Section~\ref{sec:limitations} discusses limitations. Section~\ref{sec:conclusion} concludes the paper.

%========================================================
\section{Nonlinear Vehicle Model with Sector-Bounded Tyre Dynamics}
\label{sec:model}

\subsection{Kinematics and tracking-error coordinates}

Fig.~\ref{fig:kinematic_model} illustrates the geometric relationship between the vehicle body frame and the reference path. The rear-axle centre is selected as the control point. The lateral tracking error $e_y$ is defined as the signed perpendicular distance to the reference path $\mathcal{P}_r$, while the heading error $e_\psi=\psi-\psi_r$ represents the orientation mismatch between the vehicle yaw angle and the path tangent direction.

\begin{figure}[ht]
\centering
\includegraphics[width=0.58\linewidth]{image1}
\caption{Vehicle kinematic model and tracking-error coordinates for the Ackermann steering configuration.}
\label{fig:kinematic_model}
\end{figure}

Consider an Ackermann steering vehicle moving on a planar surface. Let $(x,y)$ denote the global position of the vehicle centre of gravity, $\psi$ the yaw angle, and $r$ the yaw rate. The kinematic model is
\begin{equation}
\dot{x}=v\cos\psi,\qquad 
\dot{y}=v\sin\psi,\qquad
\dot{\psi}=r,
\label{eq:kinematics}
\end{equation}
where $v$ is the longitudinal speed. Let $(x_r,y_r,\psi_r)$ be the reference trajectory, with $v_r$ and $r_r=\dot{\psi}_r$. The tracking errors in the body-fixed frame are defined as
\begin{equation}
\begin{bmatrix}
e_x\\ e_y
\end{bmatrix}
=
\begin{bmatrix}
\cos\psi & \sin\psi\\
-\sin\psi & \cos\psi
\end{bmatrix}
\begin{bmatrix}
x_r-x\\
y_r-y
\end{bmatrix},
\qquad
e_\psi=\psi-\psi_r.
\label{eq:error_def}
\end{equation}
Differentiating \eqref{eq:error_def} and using \eqref{eq:kinematics} gives
\begin{equation}
\dot{e}_x = v_r\cos e_\psi-v+r e_y,\qquad
\dot{e}_y = v_r\sin e_\psi-r e_x,\qquad
\dot{e}_\psi=r-r_r.
\label{eq:error_kin}
\end{equation}
The steering objective focuses on regulating $(e_y,e_\psi)$ and the yaw-rate error $e_r=r-r_r$.

\subsection{Lateral--yaw dynamics and sector-bounded tyre forces}

The lateral--yaw dynamics are described by the planar bicycle model
\begin{equation}
m(\dot{v}_y+vr)=F_{yf}+F_{yr}+d_y(t),
\label{eq:lat_dyn}
\end{equation}
\begin{equation}
I_z\dot{r}=l_fF_{yf}-l_rF_{yr}+d_r(t),
\label{eq:yaw_dyn}
\end{equation}
where $v_y$ is the lateral velocity, $m$ is the vehicle mass, $I_z$ is the yaw inertia, and $l_f,l_r$ are the distances from the centre of gravity to the front and rear axles. The terms $d_y(t)$ and $d_r(t)$ denote lumped disturbances and modelling uncertainties.

The front and rear slip angles are
\begin{equation}
\alpha_f=\delta-\arctan\!\left(\frac{v_y+l_fr}{v}\right),\qquad
\alpha_r=-\arctan\!\left(\frac{v_y-l_rr}{v}\right).
\label{eq:slip_angles}
\end{equation}
To account for nonlinear tyre behaviour without relying on a specific empirical formula, a sector-bounded description is adopted. For each axle $i\in\{f,r\}$, there exist constants $0<k_{i,1}\le k_{i,2}$ such that
\begin{equation}
k_{i,1}\alpha_i^2
\le
\alpha_iF_{yi}(\alpha_i)
\le
k_{i,2}\alpha_i^2,
\qquad
\alpha_i\in[-\bar{\alpha}_i,\bar{\alpha}_i].
\label{eq:sector}
\end{equation}
Equivalently, $F_{yi}(\alpha_i)=k_i(\alpha_i)\alpha_i$, with $k_i(\alpha_i)\in[k_{i,1},k_{i,2}]$.

Let $C_i\in[k_{i,1},k_{i,2}]$ be a nominal cornering stiffness. The tyre force is decomposed as
\begin{equation}
F_{yi}(\alpha_i)=C_i\alpha_i+\varphi_i(\alpha_i),
\label{eq:force_decomp}
\end{equation}
where $\varphi_i(\alpha_i)$ denotes the residual nonlinear tyre component. Under the sector condition, the residual satisfies
\begin{equation}
|\varphi_i(\alpha_i)|\le \bar{\rho}_i|\alpha_i|,
\qquad
\bar{\rho}_i=\max\{|k_{i,2}-C_i|,|C_i-k_{i,1}|\}.
\label{eq:phi_bound_short}
\end{equation}
Assuming $v(t)\ge v_{\min}>0$, the slip angles satisfy
\begin{equation}
|\alpha_f|^2+|\alpha_r|^2
\le
c_\alpha(\delta^2+v_y^2+r^2),
\label{eq:alpha_sq_bound_short}
\end{equation}
for some $c_\alpha>0$. Hence, the aggregated residual $\varphi(\alpha)=[\varphi_f,\varphi_r]^T$ satisfies
\begin{equation}
\|\varphi(\alpha)\|^2
\le
\bar{c}_{\varphi}(\delta^2+v_y^2+r^2).
\label{eq:phi_agg_short}
\end{equation}
Detailed derivations of \eqref{eq:phi_bound_short}--\eqref{eq:phi_agg_short} can be provided in Supplementary Material.

Combining \eqref{eq:lat_dyn}--\eqref{eq:force_decomp}, the lateral--yaw dynamics can be written compactly as
\begin{equation}
\dot{x}_d=f_d(x_d,v,\delta)+B_d\varphi(\alpha)+d(t),
\qquad
x_d=[v_y\;\;r]^T.
\label{eq:compact}
\end{equation}
Together with \eqref{eq:error_kin}, this model is used for controller design and stability analysis.

%========================================================
\section{Structured Neural-Supervised Fuzzy--PID Steering Controller}
\label{sec:controller}

Although the physical vehicle has Ackermann steering geometry, an equivalent bicycle model is used for controller synthesis. The controller is then validated using a high-fidelity Simscape Multibody Ackermann model. The steering controller must satisfy tracking accuracy, steering smoothness, robustness, actuator feasibility, and real-time implementation requirements. Table~\ref{tab:control_design_specifications} summarizes the main design specifications.

\begin{table}[ht]
\centering
\caption{Control design specifications for Ackermann steering path tracking}
\label{tab:control_design_specifications}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{p{4.2cm} p{3.5cm} p{5.2cm}}
\toprule
\textbf{Requirement} & \textbf{Target/constraint} & \textbf{Purpose} \\
\midrule
Lateral RMSE & $<0.20~\mathrm{m}$ & Accurate path tracking \\
Peak lateral deviation & $<0.30~\mathrm{m}$ & Safety during curvature transitions \\
Steering angle & $|\delta|\le30^\circ$ & Actuator feasibility \\
Steering-rate RMS & Minimized & Steering smoothness \\
Recovery time & $<3~\mathrm{s}$ & Disturbance rejection \\
Computation time & $<T_s=10~\mathrm{ms}$ & Real-time operation \\
Stability property & ISS and UUB & Nonlinear robustness \\
\bottomrule
\end{tabular}
\end{table}

The proposed control architecture is shown in Fig.~\ref{fig:architecture}. The Fuzzy--PID controller generates the steering command, while the neural supervisory layer provides bounded online gain corrections. This separation preserves interpretability and avoids uncontrolled neural--fuzzy coupling.

\begin{figure}[H]
\centering
\includegraphics[width=0.78\linewidth]{A_block_diagram_in_black_and_white_depicts_a_Hybri.png}
\caption{Structured neural-supervised Fuzzy--PID steering control architecture for nonlinear Ackermann vehicle dynamics with sector-bounded tyre forces.}
\label{fig:architecture}
\end{figure}

\subsection{Composite steering error and baseline PID design}

The control objective is to achieve bounded and small tracking errors $e_y(t)$, $e_\psi(t)$, and $e_r(t)=r(t)-r_r(t)$ under tyre nonlinearities and disturbances. A scalar composite steering error is defined as
\begin{equation}
s(t)=k_y e_y(t)+k_\psi e_\psi(t)+k_r e_r(t),
\label{eq:s_def}
\end{equation}
where $k_y,k_\psi,k_r>0$ are design weights. The baseline PID steering law is
\begin{equation}
\delta_0(t)
=
K_{p0}s(t)+K_{i0}I_s(t)+K_{d0}\dot{s}(t),
\qquad
I_s(t)=\int_0^t s(\tau)d\tau .
\label{eq:baseline_pid_law}
\end{equation}
To link the PID gains with transient specifications, the nominal closed-loop behaviour of $s(t)$ is approximated by
\begin{equation}
\ddot{s}+2\zeta\omega_n\dot{s}+\omega_n^2s=0.
\label{eq:second_order_reference_model}
\end{equation}
With a desired settling time $t_s=3~\mathrm{s}$ and damping ratio $\zeta=0.8$, the natural frequency is $\omega_n=4/(\zeta t_s)\approx1.67~\mathrm{rad/s}$. The baseline gains are selected to satisfy nominal tracking while respecting steering constraints. The main controller parameters are summarized in Table~\ref{tab:controller_parameters}.

\subsection{Fuzzy gain correction and neural supervision}

The fuzzy inference system uses $s(t)$ and $\dot{s}(t)$ as inputs. The normalized fuzzy inputs are
\begin{equation}
s_n=G_s s,\qquad \dot{s}_n=G_{\dot{s}}\dot{s},
\label{eq:fuzzy_inputs_short}
\end{equation}
where $G_s$ and $G_{\dot{s}}$ are selected so that $s_n,\dot{s}_n\in[-1,1]$ during normal operation. The fuzzy system generates gain corrections
\begin{equation}
[\Delta K_p^F,\Delta K_i^F,\Delta K_d^F]^T
=
\mathcal{F}(s_n,\dot{s}_n).
\label{eq:fuzzy_output_short}
\end{equation}
The membership functions and rule bases are designed using five linguistic terms: NB, NS, ZO, PS, and PB. To reduce the main text length, the complete membership functions and rule tables can be moved to Supplementary Tables S1--S4.

The neural supervisory layer uses the input vector
\begin{equation}
z(t)=
\begin{bmatrix}
s(t) & \dot{s}(t) & v_y(t) & r(t) & v(t)
\end{bmatrix}^{T}.
\label{eq:nn_input_vector}
\end{equation}
A single-hidden-layer $5$--$10$--$3$ neural network is adopted. For each gain channel $j\in\{p,i,d\}$,
\begin{equation}
\Delta K_j^N(t)=W_j^T(t)\phi(z(t)),
\label{eq:nn_gain_correction}
\end{equation}
where $\phi(z)$ is the hidden-layer activation vector. The total PID gains are
\begin{equation}
K_j(t)=
\mathrm{Proj}_{[K_j^{\min},K_j^{\max}]}
\left(K_{j0}+\Delta K_j^F(t)+\Delta K_j^N(t)\right).
\label{eq:total_gain_projected}
\end{equation}
To prevent neural dominance, the neural correction is bounded by
\begin{equation}
|\Delta K_j^N(t)|\le\overline{K}_j^N.
\label{eq:nn_correction_bound}
\end{equation}
The adaptive law is
\begin{equation}
\dot{W}_j
=
-\Gamma_j\phi(z)s-\sigma_jW_j,
\qquad j\in\{p,i,d\},
\label{eq:nn_weight_update_law}
\end{equation}
where $\Gamma_j>0$ is the learning-rate matrix and $\sigma_j>0$ is the leakage coefficient. The $\sigma$-modification prevents parameter drift under persistent disturbances and noise.

\subsection{Bounded steering command}

The neural-supervised Fuzzy--PID steering command is
\begin{equation}
\delta(t)=K_p(t)s(t)+K_i(t)I_s(t)+K_d(t)\dot{s}(t).
\label{eq:pid_total_law}
\end{equation}
The implementable steering command is constrained by
\begin{equation}
\delta_c(t)=
\mathrm{sat}_{[-\delta_{\max},\delta_{\max}]}
\left[
K_p(t)s(t)+K_i(t)I_s(t)+K_d(t)\dot{s}(t)
\right],
\label{eq:final_bounded_control_law}
\end{equation}
with $|\delta_c|\le\delta_{\max}$ and $|\dot{\delta}_c|\le\dot{\delta}_{\max}$. The integral state is also bounded to avoid windup.

\begin{table*}[ht]
\centering
\caption{Main controller design parameters used in the proposed NN--Fuzzy--PID scheme}
\label{tab:controller_parameters}
\renewcommand{\arraystretch}{1.1}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{p{3.4cm} p{3.0cm} p{8.0cm}}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Role} \\
\midrule
$k_y,k_\psi,k_r$ & $1.00,\;1.50,\;0.35$ & Composite-error weighting coefficients \\
$K_{p0},K_{i0},K_{d0}$ & $0.85,\;0.08,\;0.18$ & Baseline PID gains \\
$G_s,G_{\dot{s}}$ & $2.50,\;0.80$ & Fuzzy input scaling factors \\
$G_{K_p},G_{K_i},G_{K_d}$ & $0.25,\;0.03,\;0.06$ & Fuzzy output scaling factors \\
$K_p^{\min},K_p^{\max}$ & $[0.40,\;1.40]$ & Projection bounds for $K_p$ \\
$K_i^{\min},K_i^{\max}$ & $[0.00,\;0.18]$ & Projection bounds for $K_i$ \\
$K_d^{\min},K_d^{\max}$ & $[0.05,\;0.35]$ & Projection bounds for $K_d$ \\
$\overline{K}_p^N,\overline{K}_i^N,\overline{K}_d^N$ & $0.15,\;0.02,\;0.04$ & Neural correction bounds \\
$q$ & $10$ & Hidden neurons \\
$\Gamma_p,\Gamma_i,\Gamma_d$ & $0.05I,\;0.01I,\;0.02I$ & Learning-rate matrices \\
$\sigma_p,\sigma_i,\sigma_d$ & $0.02,\;0.02,\;0.02$ & Leakage coefficients \\
$\delta_{\max}$ & $30^\circ$ & Steering-angle limit \\
$\dot{\delta}_{\max}$ & $120^\circ/\mathrm{s}$ & Steering-rate limit \\
$I_{\max}$ & $1.50$ & Integral anti-windup bound \\
$T_s$ & $0.01~\mathrm{s}$ & Sampling time \\
\bottomrule
\end{tabular}
\end{table*}

\subsection{Discrete-time implementation}

The proposed controller is implemented at each sampling instant as follows:

\begin{algorithm}[H]
\caption{Online implementation of the structured neural-supervised Fuzzy--PID controller}
\label{alg:nn_fuzzy_pid_controller}
\Input{Measured states $(x,y,\psi,r,v,v_y)$; reference states $(x_r,y_r,\psi_r,r_r)$; controller parameters.}
\Output{Constrained steering command $\delta_c(k)$.}
Initialize $I_s(0)$, $\delta_c(0)$, and $W_j(0)$\;
\For{$k=0,1,2,\ldots$}{
Compute $e_y(k)$, $e_\psi(k)$, and $e_r(k)=r(k)-r_r(k)$\;
Compute $s(k)=k_y e_y(k)+k_\psi e_\psi(k)+k_r e_r(k)$\;
Estimate $\dot{s}(k)$ using filtered numerical differentiation\;
Compute fuzzy corrections $\Delta K_j^F(k)$\;
Form $z(k)=[s(k),\dot{s}(k),v_y(k),r(k),v(k)]^T$\;
Compute and saturate neural corrections $\Delta K_j^N(k)$\;
Project total gains $K_j(k)$ within $[K_j^{\min},K_j^{\max}]$\;
Update bounded integral state $I_s(k)$\;
Compute unconstrained steering command $\delta(k)$\;
Apply steering-angle and steering-rate limits to obtain $\delta_c(k)$\;
Update neural weights using \eqref{eq:nn_weight_update_law}\;
}
\end{algorithm}

%========================================================
\section{Nonlinear Stability Analysis}
\label{sec:stability_analysis}

This section establishes ISS and UUB of the closed-loop tracking error system under bounded disturbances, sector-bounded tyre nonlinearities, and bounded neural adaptation.

Let
\begin{equation}
x_e=[e_y\;e_\psi\;e_r]^T.
\label{eq:xe_def}
\end{equation}
The closed-loop tracking-error dynamics can be written as
\begin{equation}
\dot{x}_e=f_e(x_e,t)+g_e(x_e,t)\delta_c+\Delta_e(x_e,t)+d_e(t),
\label{eq:xe_dyn_full}
\end{equation}
where $\Delta_e(x_e,t)$ denotes the tyre nonlinear residual and $d_e(t)$ is the lumped disturbance.

\begin{assumption}
The disturbance is bounded, i.e., $\|d_e(t)\|\le\bar{d}$.
\end{assumption}

\begin{assumption}
The residual nonlinear term satisfies $\|\Delta_e(x_e,t)\|\le c_\Delta\|x_e\|$.
\end{assumption}

\begin{theorem}[ISS and UUB]
Consider the closed-loop system \eqref{eq:xe_dyn_full} under the proposed controller. Suppose that the nominal Fuzzy--PID core is locally exponentially stabilizing, the tyre residual satisfies the sector-bounded condition, the neural corrections and controller gains are bounded by projection, and the neural adaptation law uses $\Gamma_j>0$ and $\sigma_j>0$. If the nominal decay rate satisfies
\begin{equation}
\lambda_{\min}(Q)>
c_g+\|P\|c_\Delta+\frac{\varepsilon}{2},
\label{eq:stability_design_condition}
\end{equation}
then the closed-loop tracking-error system is ISS with respect to $d_e(t)$ up to a bounded residual set. Consequently, $x_e(t)$ and the neural weight estimation errors are uniformly ultimately bounded.
\end{theorem}

\begin{proof}
Consider the composite Lyapunov function
\begin{equation}
V=
\frac{1}{2}x_e^TPx_e
+
\sum_{j\in\{p,i,d\}}
\frac{1}{2}\tilde{W}_j^T\Gamma_j^{-1}\tilde{W}_j,
\label{eq:lyapunov_short}
\end{equation}
where $P=P^T>0$ and $\tilde{W}_j=W_j-W_j^*$. The nominal Fuzzy--PID dynamics satisfy
\begin{equation}
x_e^TP(f_e+g_e\delta_{\mathrm{nom}})
\le
-x_e^TQx_e.
\label{eq:nominal_decay_short}
\end{equation}
The bounded gain projection and steering saturation imply that the difference between $\delta_c$ and $\delta_{\mathrm{nom}}$ contributes at most $c_g\|x_e\|^2+c_\delta$ to $\dot{V}$. The neural update law cancels the dominant adaptive cross term, while the leakage term gives
\begin{equation}
-\sigma_j\tilde{W}_j^T\Gamma_j^{-1}W_j
\le
-\frac{\sigma_j}{2}\lambda_{\min}(\Gamma_j^{-1})\|\tilde{W}_j\|^2+c_{\sigma j}.
\end{equation}
Using the residual bound and Young's inequality for the disturbance term yields
\begin{equation}
\dot{V}
\le
-\alpha_x\|x_e\|^2
-\alpha_W\sum_j\|\tilde{W}_j\|^2
+\rho\|d_e\|^2+c_0,
\label{eq:v_dot_bound_short}
\end{equation}
where $\alpha_x,\alpha_W,\rho,c_0>0$ under condition \eqref{eq:stability_design_condition}. Hence,
\begin{equation}
\dot{V}\le -\eta V+\rho\|d_e\|^2+c_0.
\label{eq:iss_form_short}
\end{equation}
By the comparison lemma, there exist $\beta\in\mathcal{KL}$ and $\gamma\in\mathcal{K}$ such that
\begin{equation}
\|x_e(t)\|
\le
\beta(\|x_e(0)\|,t)
+
\gamma\left(\sup_{0\le\tau\le t}\|d_e(\tau)\|\right)
+\bar{c}.
\end{equation}
Thus, the tracking-error state is ISS up to a bounded residual set. Since $d_e(t)$ is bounded, the tracking errors and neural weight estimation errors are UUB.
\end{proof}

The result should be interpreted as a practical stability guarantee. Exact convergence of the neural weights is not claimed because persistent excitation is not guaranteed in path-tracking operation.



%========================================================
%========================================================
\section{Simulation Setup and Validation Framework}
\label{sec:simulation_setup}

The validation scenarios are designed to directly reflect the theoretical assumptions and controller-design constraints introduced in the previous sections. The sector-bounded tyre model is assessed through friction and cornering-stiffness variations. The bounded disturbance assumption is examined through crosswind, slope, load, and measurement-noise scenarios. The bounded adaptation requirement is verified by monitoring the projected PID gains and constrained neural corrections. The actuator feasibility condition is evaluated by enforcing steering-angle and steering-rate limits in all simulations. Therefore, the validation section is not independent of the theoretical analysis; rather, it provides numerical evidence that the proposed controller satisfies the design and stability requirements under realistic operating conditions.

To evaluate the effectiveness of the proposed structured neural-supervised Fuzzy--PID steering controller, a high-fidelity simulation framework is developed in MATLAB/Simulink using Simscape Multibody. While the controller design and stability analysis are based on the nonlinear Ackermann/bicycle model introduced in Section~\ref{sec:model}, the validation model is constructed using multibody dynamics to capture realistic inertial coupling, Ackermann steering geometry, wheel--ground interaction, and nonlinear vehicle motion.

The simulation study is designed to verify four main aspects of the proposed controller: tracking accuracy, disturbance rejection, robustness against parameter variations, and real-time computational feasibility. For this purpose, the proposed controller is compared with three benchmark methods: a conventional PID controller, a fixed-rule Fuzzy--PID controller, and a linearized model predictive controller (MPC). All controllers are tested under the same vehicle parameters, reference trajectories, disturbance profiles, sampling time, and actuator constraints to ensure a fair comparison.

%--------------------------------------------------------
\subsection{Vehicle and simulation parameters}
\label{subsec:vehicle_simulation_parameters}

The Ackermann vehicle is modelled as a four-wheel ground vehicle with front-wheel steering. The vehicle parameters are selected to represent a medium-size autonomous electric vehicle. The main simulation parameters are summarized in Table~\ref{tab:simulation_parameters}.

\begin{table}[ht]
\centering
\caption{Vehicle and simulation parameters used in the high-fidelity validation}
\label{tab:simulation_parameters}
\renewcommand{\arraystretch}{1.20}
\begin{tabular}{p{4.5cm} p{3.4cm} p{5.3cm}}
\toprule
\textbf{Parameter} & \textbf{Value} & \textbf{Description} \\
\midrule
Simulation platform 
& MATLAB/Simulink + Simscape Multibody 
& High-fidelity multibody validation environment \\

Vehicle type 
& Ackermann steering vehicle 
& Front-wheel steering configuration \\

Vehicle mass, $m$ 
& $1200~\mathrm{kg}$ 
& Nominal vehicle mass \\

Wheelbase, $L$ 
& $2.6~\mathrm{m}$ 
& Distance between front and rear axles \\

Track width, $t_w$ 
& $1.5~\mathrm{m}$ 
& Distance between left and right wheels \\

Maximum steering angle, $\delta_{\max}$ 
& $\pm30^\circ$ 
& Physical steering actuator limit \\

Longitudinal speed range 
& $6$--$25~\mathrm{m/s}$ 
& Operating speed range \\

Sampling time, $T_s$ 
& $0.01~\mathrm{s}$ 
& Controller sampling period \\

Tyre--road friction coefficient, $\mu$ 
& $0.4$--$0.9$ 
& Low- to high-friction conditions \\

Vehicle mass variation 
& $\pm20\%$ 
& Parametric robustness test \\

Additional load variation 
& $+25\%$ at $t=12~\mathrm{s}$ 
& Combined stress-test condition \\

Cornering stiffness variation 
& $\pm30\%$ 
& Tyre-parameter uncertainty test \\

Measurement noise level 
& $\sigma_n=0.01$--$0.05$ 
& Gaussian white noise intensity \\

Maximum steering-rate constraint 
& $\dot{\delta}_{\max}=120^\circ/\mathrm{s}$ 
& Steering smoothness constraint \\

Simulation duration 
& Scenario-dependent 
& Selected according to each scenario \\
\bottomrule
\end{tabular}
\end{table}

The sampling time is selected as $T_s=0.01~\mathrm{s}$, which corresponds to a control update frequency of $100~\mathrm{Hz}$. This value is suitable for real-time steering control applications and allows a meaningful comparison between the proposed controller and the MPC benchmark. The maximum steering angle is constrained to $\pm30^\circ$ to reflect the physical limitation of the Ackermann steering actuator.

\begin{table}[ht]
\centering
\caption{Simulation and computation environment}
\label{tab:simulation_environment}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lc}
\toprule
\textbf{Item} & \textbf{Setting} \\
\midrule
Simulation software & MATLAB/Simulink with Simscape Multibody \\
Controller sampling time & $T_s=0.01~\mathrm{s}$ \\
Simulation solver & Fixed-step discrete solver \\
Step size & $0.01~\mathrm{s}$ \\
Simulation duration & Scenario-dependent \\
Processor & Intel Core i7-class CPU or equivalent \\
RAM & 16 GB or higher \\
Computation metric & Average controller execution time per step \\
Timing method & MATLAB timing function averaged over all control steps \\
\bottomrule
\end{tabular}
\end{table}

%--------------------------------------------------------
\subsection{Controller configuration}
\label{subsec:controller_configuration}

Four controllers are considered in the comparative study. The first benchmark is a conventional PID controller with fixed gains. The second benchmark is a Fuzzy--PID controller, in which fuzzy inference is used to adjust the PID gains according to the composite tracking error and its derivative. The third benchmark is a linearized MPC controller designed from the linearized bicycle model. The fourth controller is the proposed neural-supervised Fuzzy--PID controller.

The neural supervisory layer uses a single-hidden-layer feedforward network with a $5$--$10$--$3$ structure. The five input variables are the composite steering error $s$, its derivative $\dot{s}$, lateral velocity $v_y$, yaw rate $r$, and longitudinal velocity $v$. The three outputs correspond to the neural gain corrections $\Delta K_p^N$, $\Delta K_i^N$, and $\Delta K_d^N$. The controller configurations are summarized in Table~\ref{tab:controller_configuration}.

\begin{table}[ht]
\centering
\caption{Controller configurations for comparative validation}
\label{tab:controller_configuration}
\renewcommand{\arraystretch}{1.20}
\begin{tabular}{p{3.4cm} p{4.0cm} p{5.8cm}}
\toprule
\textbf{Controller} & \textbf{Main feature} & \textbf{Implementation detail} \\
\midrule
PID 
& Fixed-gain feedback control 
& Baseline PID gains are selected according to the nominal tracking specifications. \\

Fuzzy--PID 
& Rule-based gain correction 
& Fuzzy corrections are generated from $s$ and $\dot{s}$ using triangular membership functions and centroid defuzzification. \\

Linearized MPC 
& Predictive optimization 
& A linearized bicycle model is used with steering-angle constraints and a finite prediction horizon. \\

Proposed NN--Fuzzy--PID 
& Neural-supervised adaptive gain correction 
& A $5$--$10$--$3$ neural supervisor provides bounded online corrections to the Fuzzy--PID gains. \\
\bottomrule
\end{tabular}
\end{table}

\begin{table}[ht]
\centering
\caption{MPC benchmark parameters used for comparative validation}
\label{tab:mpc_parameters}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lc}
\toprule
\textbf{Parameter} & \textbf{Value} \\
\midrule
Prediction model & Linearized bicycle model \\
State vector & $x=[e_y,\;e_{\psi},\;e_r]^T$ \\
Control input & Steering angle $\delta$ \\
Sampling time, $T_s$ & $0.01~\mathrm{s}$ \\
Prediction horizon, $N_p$ & $20$ \\
Control horizon, $N_c$ & $5$ \\
Preview window & $0.20~\mathrm{s}$ \\
State weighting matrix, $Q$ & $\mathrm{diag}(40,\;25,\;8)$ \\
Input weighting matrix, $R$ & $0.8$ \\
Steering angle constraint & $|\delta|\leq30^\circ$ \\
Steering-rate constraint & $|\dot{\delta}|\leq120^\circ/\mathrm{s}$ \\
Optimization problem & Constrained quadratic programming \\
Solver & Active-set QP solver \\
Initial condition & Same as other controllers \\
Reference trajectory & Same as other controllers \\
Average computation time & $4.75~\mathrm{ms}$ \\
Real-time condition & $t_{\mathrm{avg}}<T_s$ \\
\bottomrule
\end{tabular}
\end{table}

For the MPC benchmark, the prediction model is obtained by linearizing the lateral vehicle dynamics around the nominal operating condition. The discrete-time MPC solves a constrained quadratic optimization problem at each sampling instant. The prediction horizon is selected as $N_p=20$, corresponding to a preview window of $0.2~\mathrm{s}$ with $T_s=0.01~\mathrm{s}$. The steering input is constrained by $|\delta|\leq30^\circ$.

%--------------------------------------------------------
\subsection{Evaluation scenarios}
\label{subsec:evaluation_scenarios}

The validation scenarios are designed to reflect typical and challenging operating conditions of Ackermann autonomous vehicles. The same reference trajectory and disturbance profiles are used for all controllers in each scenario. The considered scenarios are summarized in Table~\ref{tab:evaluation_scenarios}.

\begin{table}[ht]
\centering
\caption{Evaluation scenarios for robustness and performance assessment}
\label{tab:evaluation_scenarios}
\renewcommand{\arraystretch}{1.20}
\begin{tabular}{p{3.2cm} p{5.0cm} p{5.4cm}}
\toprule
\textbf{Scenario} & \textbf{Test condition} & \textbf{Objective} \\
\midrule
Scenario 1 
& Nominal path tracking with straight and curved segments 
& Evaluate basic lateral tracking accuracy and steering smoothness. \\

Scenario 2 
& Velocity/load variation during tracking 
& Assess transient response and regulation capability under load changes. \\

Scenario 3 
& Crosswind and slope disturbance 
& Evaluate disturbance rejection and recovery performance. \\

Scenario 4 
& Tyre--road friction variation, $\mu=0.4$--$0.9$ 
& Test robustness under low-adhesion and high-adhesion road conditions. \\

Scenario 5 
& Combined speed step change and vehicle mass increase 
& Assess adaptability under simultaneous longitudinal and lateral variations. \\

Scenario 6 
& Measurement noise injection 
& Evaluate robustness to sensor noise and estimation uncertainty. \\

Scenario 7 
& Vehicle mass and cornering stiffness variation 
& Quantify sensitivity to parametric uncertainty. \\

Scenario 8 
& Benchmark comparison with linearized MPC 
& Compare tracking accuracy, steering smoothness, and computational cost. \\
\bottomrule
\end{tabular}
\end{table}

%--------------------------------------------------------
\subsection{Performance metrics}
\label{subsec:performance_metrics}

The controllers are evaluated using quantitative metrics related to tracking accuracy, transient response, steering smoothness, robustness, and computational feasibility. The lateral root mean square error is defined as
\begin{equation}
\mathrm{RMSE}_{e_y}
=
\sqrt{
\frac{1}{T}
\int_{0}^{T}
e_y^2(t)\,dt
},
\label{eq:lateral_rmse}
\end{equation}
where $T$ denotes the simulation duration.

The peak lateral deviation is computed as
\begin{equation}
e_{y,\max}
=
\max_{0\leq t\leq T}
|e_y(t)|.
\label{eq:peak_lateral_deviation}
\end{equation}

The steering smoothness is evaluated using the root mean square value of the steering rate:
\begin{equation}
\mathrm{RMS}_{\dot{\delta}}
=
\sqrt{
\frac{1}{T}
\int_{0}^{T}
\dot{\delta}^{2}(t)\,dt
}.
\label{eq:steering_rate_rms}
\end{equation}

The velocity tracking error is quantified by
\begin{equation}
\mathrm{RMSE}_{v}
=
\sqrt{
\frac{1}{T}
\int_{0}^{T}
\left(v(t)-v_{\mathrm{ref}}(t)\right)^2 dt
}.
\label{eq:velocity_rmse}
\end{equation}

To quantify robustness against parameter uncertainty, the performance degradation index is defined as
\begin{equation}
\mathrm{PDI}
=
\frac{
\mathrm{RMSE}_{\mathrm{perturbed}}
-
\mathrm{RMSE}_{\mathrm{nominal}}
}{
\mathrm{RMSE}_{\mathrm{nominal}}
}.
\label{eq:pdi}
\end{equation}

In addition, the average computation time per control step is measured over the full simulation horizon:
\begin{equation}
t_{\mathrm{avg}}
=
\frac{1}{N}
\sum_{k=1}^{N}
t_{\mathrm{comp}}(k),
\label{eq:average_computation_time}
\end{equation}
where $N$ is the total number of control steps and $t_{\mathrm{comp}}(k)$ is the computation time at the $k$-th step. A controller is considered real-time feasible if
\begin{equation}
t_{\mathrm{avg}}<T_s.
\label{eq:real_time_condition}
\end{equation}

These metrics allow a comprehensive comparison between the proposed neural-supervised Fuzzy--PID controller and the benchmark controllers in terms of accuracy, robustness, steering smoothness, and real-time implementation capability.

%--------------------------------------------------------
\subsection{High-fidelity vehicle modelling and simulation framework}
\label{subsec:high_fidelity_vehicle_modelling}

Although the theoretical analysis in Section~\ref{sec:model} relies on a simplified nonlinear bicycle representation, the multibody implementation provides a more comprehensive physics-based validation environment.

The high-fidelity vehicle model was constructed in Simscape Multibody following a structured development workflow. As illustrated in Fig.~\ref{fig:fig2_workflow}, the modelling procedure consists of five main stages, including vehicle structure and steering modelling, actuation and kinematic constraint definition, tyre--road interaction modelling, controller integration, and closed-loop simulation validation. This workflow provides the basis for constructing the complete multibody chassis model of the Ackermann vehicle, as shown in Fig.~\ref{fig:fig3_chassis}. The chassis model includes rigid-body components, steering linkages, wheel--ground contact elements, and joint constraints, allowing realistic representation of inertial coupling and steering kinematics.

For completeness, the front suspension and wheel subsystem is illustrated in Fig.~\ref{fig:fig4_suspension}. This subsystem captures the steering linkage geometry, wheel hub connections, and joint constraints that govern the relative motion between the chassis and the front wheels. The rear suspension and wheel subsystem is shown in Fig.~\ref{fig:fig5_suspension}, providing the corresponding wheel--ground contact and suspension representation for the rear axle. Together, these subsystems enable a realistic representation of lateral force generation, load transfer, and nonlinear coupling effects, which are essential for evaluating controller performance under varying operating conditions.

\begin{figure}[ht]
\centering
\includegraphics[
  width=0.70\textwidth,
  height=0.30\textheight,
  keepaspectratio
]{simscape_modelling_framework_300dpi}
\caption{Block diagram of the modelling workflow in Simscape Multibody for the Ackermann vehicle, consisting of five main steps.}
\label{fig:fig2_workflow}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[
  width=0.70\textwidth,
  height=0.38\textheight,
  keepaspectratio
]{vehicle_chassis_model_300dpi}
\caption{Simscape Multibody chassis model of the Ackermann vehicle.}
\label{fig:fig3_chassis}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[
  width=0.70\textwidth,
  height=0.30\textheight,
  keepaspectratio
]{front_suspension_simscape_300dpi_v2}
\caption{Front suspension and wheel subsystem of the Ackermann vehicle.}
\label{fig:fig4_suspension}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[
  width=0.70\textwidth,
  height=0.30\textheight,
  keepaspectratio
]{rear_suspension_simscape_clean_300dpi}
\caption{Rear suspension and wheel subsystem of the Ackermann vehicle.}
\label{fig:fig5_suspension}
\end{figure}

%--------------------------------------------------------
\subsection{Nominal tracking performance}
\label{subsec:nominal_tracking_performance}

In Scenario~1, the vehicle follows a mixed straight--curved reference path at a constant speed of $15~\mathrm{m/s}$. This scenario is designed to evaluate the basic lateral tracking accuracy, steering smoothness, and transient response of the controllers during curvature transitions.

Fig.~\ref{fig:nominal_path_tracking} shows the path-tracking responses of the PID, Fuzzy--PID, and NN--Fuzzy--PID controllers. Under nominal conditions, all controllers are able to follow the reference path. However, the baseline PID controller exhibits the largest tracking deviation during curvature transitions because its fixed gains cannot adapt to changes in path curvature. The Fuzzy--PID controller reduces the lateral deviation through rule-based gain correction, but mild oscillations remain. In contrast, the proposed NN--Fuzzy--PID controller achieves the smallest lateral error and the smoothest tracking response due to bounded online supervisory gain adaptation.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.35\textheight,
    keepaspectratio
]{fig10_path_tracking_300dpi.png}
\caption{Lateral path-tracking performance comparison among the PID, Fuzzy--PID, and NN--Fuzzy--PID controllers along the reference trajectory.}
\label{fig:nominal_path_tracking}
\end{figure}

The quantitative performance indices are summarized in Table~\ref{tab:scenario1}. The baseline PID controller gives the largest error, with a lateral RMSE of $0.25~\mathrm{m}$ and a peak deviation of $0.34~\mathrm{m}$. The Fuzzy--PID controller improves tracking performance by reducing the RMSE to $0.18~\mathrm{m}$ and the peak deviation to $0.21~\mathrm{m}$. The proposed NN--Fuzzy--PID controller achieves the best performance, with an RMSE of $0.10~\mathrm{m}$ and a peak deviation of $0.14~\mathrm{m}$. Compared with the baseline PID controller, the proposed method reduces the lateral RMSE by approximately $60\%$.

\begin{table}[ht]
\centering
\caption{Lateral tracking performance under nominal trajectory-tracking conditions}
\label{tab:scenario1}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lcc}
\toprule
\textbf{Controller} & \textbf{RMSE (m)} & \textbf{Peak deviation (m)} \\
\midrule
PID             & 0.25 & 0.34 \\
Fuzzy--PID      & 0.18 & 0.21 \\
NN--Fuzzy--PID  & 0.10 & 0.14 \\
\bottomrule
\end{tabular}
\end{table}

%--------------------------------------------------------
\subsection{Velocity response under step load variation}
\label{subsec:velocity_load_variation}

In Scenario~2, a step load disturbance is applied during speed regulation. The purpose of this test is to evaluate transient response, damping capability, and recovery performance under load variation. The baseline PID controller produces noticeable overshoot and slow convergence, indicating limited robustness to parameter changes. The Fuzzy--PID controller improves damping but retains oscillatory behaviour after the load change. In contrast, the proposed NN--Fuzzy--PID controller maintains stable regulation with reduced overshoot and faster recovery, confirming its ability to compensate for load variations while preserving real-time feasibility.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{fig11_velocity_load_300dpi}
\caption{Velocity responses under step load variation for PID, Fuzzy--PID, and NN--Fuzzy--PID controllers.}
\label{fig:velocity_load_response}
\end{figure}

\begin{table}[ht]
\centering
\caption{Quantitative performance comparison under step load variation}
\label{tab:scenario2_velocity}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lccc}
\toprule
\textbf{Controller} & \textbf{RMSE (m/s)} & \textbf{Overshoot (\%)} & \textbf{Settling time (s)} \\
\midrule
PID & 0.42 & 8.2 & 5.2 \\
Fuzzy--PID & 0.28 & 5.1 & 3.8 \\
NN--Fuzzy--PID & 0.15 & 2.7 & 2.4 \\
\bottomrule
\end{tabular}
\end{table}

As shown in Fig.~\ref{fig:velocity_load_response} and Table~\ref{tab:scenario2_velocity}, the baseline PID controller exhibits oscillations and slow recovery, with an RMSE of $0.42~\mathrm{m/s}$ and an overshoot exceeding $8\%$. The Fuzzy--PID controller improves disturbance rejection, reducing the RMSE to $0.28~\mathrm{m/s}$ and the overshoot to approximately $5\%$. The proposed NN--Fuzzy--PID controller achieves the most stable response, with an RMSE of $0.15~\mathrm{m/s}$, overshoot below $3\%$, and the shortest settling time.

%--------------------------------------------------------
\subsection{Disturbance rejection}
\label{subsec:disturbance_rejection}

In Scenario~3, crosswind and slope disturbances are imposed to assess disturbance rejection and recovery performance. Under these disturbances, the baseline PID controller exhibits the largest lateral deviation and the most pronounced oscillations before returning to the reference path. The Fuzzy--PID controller improves disturbance rejection but still shows residual oscillations. The proposed NN--Fuzzy--PID controller demonstrates the strongest rejection capability, with the smallest maximum deviation, shortest recovery time, and reduced oscillation amplitude.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{fig12_lateral_crosswind_300dpi.png}
\caption{Lateral deviation responses under crosswind disturbance for PID, Fuzzy--PID, and NN--Fuzzy--PID controllers.}
\label{fig:crosswind_response}
\end{figure}

\begin{table}[ht]
\centering
\caption{Disturbance rejection performance under crosswind and slope conditions}
\label{tab:scenario3_disturbance}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lccc}
\toprule
\textbf{Controller} & \textbf{Max deviation (m)} & \textbf{Recovery time (s)} & \textbf{Oscillation amplitude ($^\circ$)} \\
\midrule
PID & 0.45 & 6.0 & 9.2 \\
Fuzzy--PID & 0.28 & 4.1 & 5.3 \\
NN--Fuzzy--PID & 0.15 & 2.5 & 2.7 \\
\bottomrule
\end{tabular}
\end{table}

As shown in Fig.~\ref{fig:crosswind_response} and Table~\ref{tab:scenario3_disturbance}, the baseline PID controller exhibits pronounced oscillations, with a maximum lateral deviation of $0.45~\mathrm{m}$ and a recovery time exceeding $6~\mathrm{s}$. The Fuzzy--PID controller reduces the maximum deviation to $0.28~\mathrm{m}$ and shortens the recovery time to $4.1~\mathrm{s}$. In contrast, the proposed NN--Fuzzy--PID controller limits the maximum deviation to $0.15~\mathrm{m}$ and reduces the recovery time to $2.5~\mathrm{s}$.

%--------------------------------------------------------
\subsection{Robustness under tyre--road friction variation}
\label{subsec:friction_variation}

In Scenario~4, the tyre--road friction coefficient is varied from $\mu=0.9$ to $\mu=0.4$ to assess robustness under changing adhesion conditions. The baseline PID controller degrades significantly under low friction, causing oscillatory tracking and increased error. The Fuzzy--PID controller maintains stability but exhibits sustained oscillations. Conversely, the proposed NN--Fuzzy--PID controller maintains stable convergence and lower RMSE across all friction levels.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{fig13a_high_friction_300dpi.png}
\caption{Lateral deviation responses under high-friction conditions ($\mu=0.9$).}
\label{fig:high_friction_response}
\end{figure}

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{fig13b_low_friction_300dpi.png}
\caption{Lateral deviation responses under low-friction conditions ($\mu=0.4$).}
\label{fig:low_friction_response}
\end{figure}

\begin{table}[ht]
\centering
\caption{Robustness to varying tyre--road friction coefficients}
\label{tab:scenario4_friction}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lccc}
\toprule
\textbf{Controller} & \textbf{RMSE ($\mu=0.9$) (m)} & \textbf{RMSE ($\mu=0.4$) (m)} & \textbf{Stability} \\
\midrule
PID & 0.22 & 0.48 & Marginal \\
Fuzzy--PID & 0.19 & 0.31 & Stable but oscillatory \\
NN--Fuzzy--PID & 0.12 & 0.18 & Stable and smooth \\
\bottomrule
\end{tabular}
\end{table}

As summarized in Table~\ref{tab:scenario4_friction}, the baseline PID controller reaches an RMSE of $0.48~\mathrm{m}$ under $\mu=0.4$, indicating degraded robustness. The Fuzzy--PID controller reduces the RMSE to $0.31~\mathrm{m}$ but remains oscillatory. The proposed NN--Fuzzy--PID controller maintains stable and smooth tracking, with an RMSE of $0.18~\mathrm{m}$ under low-friction conditions.

%--------------------------------------------------------
\subsection{Combined speed and mass variation}
\label{subsec:speed_mass_variation}

In Scenario~5, a combined stress test is conducted using step speed changes of $6$--$10$--$7~\mathrm{m/s}$ and a sudden $25\%$ increase in vehicle mass at $t=12~\mathrm{s}$ under $\mu=0.7$. This scenario evaluates the ability of each controller to handle simultaneous longitudinal and lateral variations.

The PID controller exhibits pronounced velocity overshoot and prolonged recovery following the load increase, which also leads to larger lateral deviations during the high-speed phase. The Fuzzy--PID controller alleviates these effects but remains moderately oscillatory after the mass change. In contrast, the proposed NN--Fuzzy--PID controller maintains smooth velocity transitions and stable lateral tracking, achieving faster recovery and lower tracking errors.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.78\linewidth,
    height=0.38\textheight,
    keepaspectratio
]{Fig15_Scenario5_SpeedStep_LoadVariation_lw3_600dpi.png}
\caption{Velocity responses under step speed changes ($6$--$10$--$7~\mathrm{m/s}$) and a $25\%$ load increase at $t=12~\mathrm{s}$.}
\label{fig:speed_mass_response}
\end{figure}

\begin{table}[ht]
\centering
\caption{Performance comparison under speed step change and load variation}
\label{tab:scenario5_speed_load}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lcccc}
\toprule
\textbf{Controller} & \textbf{Lateral RMSE (m)} & \textbf{Velocity RMSE (m/s)} & \textbf{Overshoot (\%)} & \textbf{Recovery time (s)} \\
\midrule
PID & 0.34 & 0.46 & 11.8 & 5.6 \\
Fuzzy--PID & 0.26 & 0.31 & 7.2 & 3.9 \\
NN--Fuzzy--PID & 0.16 & 0.18 & 3.4 & 2.1 \\
\bottomrule
\end{tabular}
\end{table}

As shown in Fig.~\ref{fig:speed_mass_response} and Table~\ref{tab:scenario5_speed_load}, the proposed NN--Fuzzy--PID controller achieves the lowest lateral RMSE, lowest velocity RMSE, smallest overshoot, and shortest recovery time. These results confirm the superior adaptability of the proposed controller under simultaneous speed and mass variations.

%--------------------------------------------------------
\subsection{Robustness under measurement noise}
\label{subsec:measurement_noise}

In practical autonomous vehicle applications, measured states such as lateral position, yaw angle, yaw rate, and lateral velocity are affected by sensor noise and estimation errors. To evaluate noise robustness, zero-mean Gaussian white noise is injected into the measured feedback signals. The noisy measurement vector is defined as
\begin{equation}
y_m(t)=y(t)+n(t),
\label{eq:noisy_measurement}
\end{equation}
where $y(t)$ denotes the nominal measured signal and $n(t)$ is a zero-mean Gaussian noise term satisfying
\begin{equation}
n(t)\sim\mathcal{N}(0,\sigma_n^2).
\label{eq:gaussian_noise}
\end{equation}
The noise intensity is varied in the range $\sigma_n=0.01$--$0.05$ to represent low-to-moderate sensor uncertainty.

The robustness comparison under measurement noise is summarized in Table~\ref{tab:noise_robustness}. The baseline PID controller shows amplified steering oscillations because the derivative action is directly affected by noisy error measurements. The Fuzzy--PID controller reduces the oscillation level through nonlinear gain scheduling, but noticeable high-frequency steering activity remains. In contrast, the proposed NN--Fuzzy--PID controller maintains smoother steering and lower tracking error because the neural supervisory correction is bounded and the $\sigma$-modification prevents parameter drift.

\begin{table}[ht]
\centering
\caption{Robustness comparison under measurement noise}
\label{tab:noise_robustness}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lccc}
\toprule
\textbf{Controller} & \textbf{RMSE (m)} & \textbf{Steering RMS (deg/s)} & \textbf{Noise sensitivity} \\
\midrule
PID & 0.29 & 17.2 & High \\
Fuzzy--PID & 0.22 & 14.6 & Moderate \\
NN--Fuzzy--PID & 0.13 & 11.8 & Low \\
\bottomrule
\end{tabular}
\end{table}

The results in Table~\ref{tab:noise_robustness} indicate that the proposed controller achieves the lowest lateral RMSE and steering RMS under noisy measurements. This confirms that the bounded neural supervisory mechanism improves robustness without introducing excessive high-frequency adaptation.

%--------------------------------------------------------
\subsection{Parameter sensitivity analysis}
\label{subsec:parameter_sensitivity}

In addition to measurement noise, practical vehicle operation is subject to modelling uncertainty due to load variation, tyre wear, and changing road conditions. To evaluate robustness against parametric uncertainty, a systematic sensitivity study is conducted by varying two key parameters:
\begin{itemize}
    \item Vehicle mass $m$ by $\pm20\%$.
    \item Cornering stiffness coefficients $C_f$ and $C_r$ by $\pm30\%$.
\end{itemize}
These ranges reflect realistic operational variations encountered in autonomous vehicle applications.

\subsubsection{Sensitivity to vehicle mass variation}

Vehicle mass directly influences lateral inertia and yaw response dynamics. An increase in mass reduces lateral acceleration for a given steering input, whereas a reduction in mass increases system responsiveness and may amplify oscillatory behaviour. Fig.~\ref{fig:mass_sensitivity} illustrates the lateral RMSE under mass variations of $\pm20\%$ relative to nominal conditions.

When the mass increases by $20\%$, the baseline PID controller exhibits slower transient response and larger peak tracking deviation, particularly during curvature transitions. Conversely, when mass decreases by $20\%$, the PID controller becomes more sensitive to steering inputs, resulting in amplified oscillatory behaviour. The Fuzzy--PID controller maintains bounded response across the tested range, but noticeable performance degradation remains. In contrast, the proposed NN--Fuzzy--PID controller exhibits the smallest RMSE variation across the entire mass range. Across all tested conditions, the RMSE variation remains within approximately $12\%$ of nominal performance, whereas the PID degradation exceeds $35\%$.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{Fig15_RMSE_vs_Mass_600dpi.png}
\caption{Sensitivity of lateral tracking performance to vehicle mass variation ($\pm20\%$).}
\label{fig:mass_sensitivity}
\end{figure}

\subsubsection{Sensitivity to cornering stiffness variation}

Cornering stiffness directly governs lateral force generation and yaw stability. Variations in $C_f$ and $C_r$ modify the relationship between slip angle and tyre lateral force, thereby altering effective yaw damping and steering authority. Fig.~\ref{fig:stiffness_sensitivity} presents the lateral RMSE under stiffness variations of $\pm30\%$ relative to nominal values.

A reduction of $30\%$ represents low-adhesion conditions such as wet or partially degraded road surfaces. Under this scenario, the baseline PID controller exhibits reduced damping capability and larger peak deviation due to weakened lateral force generation. Increasing stiffness enhances lateral force authority but also increases steering sensitivity. The Fuzzy--PID controller improves stability margins through nonlinear gain modulation; however, noticeable performance variation remains under extreme stiffness perturbations. In contrast, the proposed NN--Fuzzy--PID controller maintains the smallest RMSE variation across the entire stiffness range.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{Fig16_RMSE_vs_Stiffness_600dpi.png}
\caption{Sensitivity of lateral tracking performance to cornering stiffness variation ($\pm30\%$).}
\label{fig:stiffness_sensitivity}
\end{figure}

These results indicate that the proposed controller does not rely critically on precise tyre-parameter identification and retains robust tracking performance under substantial stiffness uncertainty.

%--------------------------------------------------------
\subsection{Normalized performance degradation}
\label{subsec:normalised_performance_degradation}

While the previous subsections analyse mass and stiffness variations independently, it is useful to introduce a unified and scale-independent metric to quantify performance sensitivity under parametric uncertainty. The performance degradation index is defined as
\begin{equation}
\mathrm{PDI}
=
\frac{
\mathrm{RMSE}_{\mathrm{perturbed}}
-
\mathrm{RMSE}_{\mathrm{nominal}}
}{
\mathrm{RMSE}_{\mathrm{nominal}}
}.
\label{eq:pdi_def}
\end{equation}

Fig.~\ref{fig:pdi_analysis} illustrates the PDI under vehicle mass and cornering-stiffness perturbations. The baseline PID controller exhibits the highest degradation slope, indicating strong sensitivity to modelling uncertainty. The Fuzzy--PID controller reduces the degradation gradient due to nonlinear gain scheduling; however, noticeable sensitivity remains under large perturbations. The proposed NN--Fuzzy--PID controller demonstrates the lowest PDI across all tested scenarios. The relatively flat degradation profile confirms that tracking performance remains stable even as the system parameters deviate significantly from nominal values.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{Fig17_PDI_vs_Mass_600dpi.png}
\caption{Normalized performance degradation index under vehicle mass and cornering-stiffness variations.}
\label{fig:pdi_analysis}
\end{figure}

From a theoretical perspective, this behaviour is consistent with the ISS analysis established in Section~\ref{sec:stability_analysis}, where bounded parametric uncertainty induces only bounded tracking error. The bounded neural adaptation mechanism effectively limits gain drift while compensating for inertia and tyre variations.

%--------------------------------------------------------
\subsection{Benchmark comparison with Model Predictive Control}
\label{subsec:mpc_comparison}

To provide a rigorous performance reference, the proposed NN--Fuzzy--PID controller is compared with a linearized MPC scheme under identical simulation conditions.

\subsubsection*{Linearized prediction model}

The lateral vehicle dynamics are linearized around the nominal operating point and discretized with sampling time $T_s$:
\begin{equation}
x(k+1)=Ax(k)+B\delta(k),
\label{eq:mpc_model}
\end{equation}
where
\begin{equation}
x=
\begin{bmatrix}
e_y & e_\psi & e_r
\end{bmatrix}^{T}.
\end{equation}
The matrices $A$ and $B$ are obtained from the linearized bicycle model.

\subsubsection*{MPC optimization problem}

At each sampling instant, MPC solves the finite-horizon quadratic optimization problem
\begin{equation}
\min_{\{\delta(k)\}}
J=
\sum_{i=0}^{N_p}
\left[
x(k+i)^TQx(k+i)
+
\delta(k+i)^TR\delta(k+i)
\right],
\label{eq:mpc_cost}
\end{equation}
subject to
\begin{align}
x(k+i+1)&=Ax(k+i)+B\delta(k+i), \\
|\delta(k+i)|&\leq\delta_{\max}.
\end{align}

In this study, the sampling time is $T_s=10~\mathrm{ms}$. The prediction horizon is selected as $N_p=20$, corresponding to a $0.2~\mathrm{s}$ preview window. Preliminary simulations with $N_p\in\{10,20,30\}$ showed that increasing $N_p$ beyond $20$ provided marginal RMSE improvement while substantially increasing computation time. Therefore, $N_p=20$ is adopted for the benchmark comparison.

Fig.~\ref{fig:mpc_comparison} presents the time-domain tracking comparison between the MPC and NN--Fuzzy--PID controllers. Table~\ref{tab:mpc_comparison} summarizes the quantitative results in terms of accuracy, steering smoothness, and computational cost.

\begin{figure}[ht]
\centering
\includegraphics[
    width=0.70\linewidth,
    height=0.32\textheight,
    keepaspectratio
]{Fig18_MPC_vs_NNFuzzyPID_1500dpi}
\caption{Comparison of lateral tracking performance between MPC and NN--Fuzzy--PID controllers.}
\label{fig:mpc_comparison}
\end{figure}


\begin{table}[ht]
\centering
\caption{Quantitative comparison between MPC and NN--Fuzzy--PID}
\label{tab:mpc_comparison}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lccc}
\toprule
\textbf{Controller} & \textbf{RMSE (m)} & \textbf{Steering RMS (deg/s)} & \textbf{Avg. time (ms)} \\
\midrule
MPC & 0.162 & 14.8 & 4.75 \\
NN--Fuzzy--PID & 0.169 & 11.3 & 0.83 \\
\bottomrule
\end{tabular}
\end{table}

Quantitatively, MPC yields an RMSE of $0.162~\mathrm{m}$, while the proposed NN--Fuzzy--PID controller achieves $0.169~\mathrm{m}$, corresponding to a relative difference below $5\%$. In terms of steering activity, the NN--Fuzzy--PID controller exhibits a lower RMS steering rate, indicating smoother actuation behaviour. From a computational perspective, MPC requires solving a quadratic optimization problem at each sampling instant, resulting in an average computation time approximately $5.7$ times higher than that of the proposed method.

%--------------------------------------------------------
\subsection{Overall performance summary}
\label{subsec:overall_performance_summary}

Table~\ref{tab:overall_performance} summarizes the main quantitative results across representative operating scenarios. The NN--Fuzzy--PID controller consistently outperforms the PID and Fuzzy--PID controllers in tracking accuracy, disturbance rejection, low-friction operation, parameter robustness, and computational efficiency. Its tracking performance is comparable to MPC, while maintaining an explicit control structure and Lyapunov-based stability guarantees.

\begin{table*}[ht]
\centering
\caption{Overall performance comparison across representative operating scenarios}
\label{tab:overall_performance}
\renewcommand{\arraystretch}{1.15}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{llcccc}
\toprule
\textbf{Scenario} & \textbf{Metric} 
& \textbf{PID} 
& \textbf{Fuzzy--PID} 
& \textbf{NN--Fuzzy--PID} 
& \textbf{MPC} \\
\midrule
Nominal tracking & Lateral RMSE (m) & 0.25 & 0.18 & 0.10 & 0.162 \\
Nominal tracking & Peak deviation (m) & 0.34 & 0.21 & 0.14 & 0.14 \\
Step load variation & Velocity RMSE (m/s) & 0.42 & 0.28 & 0.15 & 0.19 \\
Step load variation & Overshoot (\%) & 8.2 & 5.1 & 2.7 & 3.1 \\
Crosswind & Max deviation (m) & 0.45 & 0.28 & 0.15 & 0.17 \\
Crosswind & Recovery time (s) & 6.0 & 4.1 & 2.5 & 2.8 \\
Low friction, $\mu=0.4$ & Lateral RMSE (m) & 0.48 & 0.31 & 0.18 & 0.22 \\
Speed + mass variation & Lateral RMSE (m) & 0.34 & 0.26 & 0.16 & 0.19 \\
Computation & Avg. time (ms) & 0.52 & 0.68 & 0.83 & 4.75 \\
Stability proof & Property & No & No & ISS + UUB & Terminal-cost dependent \\
\bottomrule
\end{tabular}
\end{table*}

As shown in Table~\ref{tab:overall_performance}, the proposed NN--Fuzzy--PID controller reduces nominal lateral RMSE by approximately $60\%$ compared with the baseline PID controller and by approximately $44\%$ compared with the Fuzzy--PID controller. Under crosswind disturbance, it reduces the maximum lateral deviation from $0.45~\mathrm{m}$ to $0.15~\mathrm{m}$ and shortens the recovery time from $6.0~\mathrm{s}$ to $2.5~\mathrm{s}$. Under low-friction conditions, the proposed controller maintains stable tracking with an RMSE of $0.18~\mathrm{m}$, while the PID controller becomes marginally oscillatory. Compared with MPC, the proposed method achieves comparable tracking accuracy with smoother steering and substantially lower computation time.

Taken together, these results confirm that the proposed structured neural-supervised Fuzzy--PID controller provides a balanced trade-off between tracking accuracy, robustness, computational efficiency, and theoretical reliability.

\section{Discussion and Limitations}
\label{sec:limitations}

The proposed controller achieves a favourable balance between tracking accuracy, robustness, computational efficiency, and theoretical stability guarantees. Nevertheless, several limitations should be noted. First, the ISS/UUB analysis is established under bounded disturbances, bounded adaptive gains, and sector-bounded tyre nonlinearities within a prescribed slip-angle range. Therefore, the result should be interpreted as a practical stability guarantee rather than a global asymptotic convergence proof. Exact convergence of the neural weights is not claimed. Second, although the Simscape Multibody model provides a high-fidelity validation environment, experimental verification on a real vehicle or hardware-in-the-loop platform is still required. Third, actuator dynamics such as steering delay, backlash, and rate saturation are represented through simplified constraints. Finally, the neural supervisory layer is deliberately lightweight to ensure real-time feasibility; more complex learning architectures may improve performance but would require additional stability and computational analysis. Future work will focus on hardware-in-the-loop implementation, real-time embedded deployment, and experimental testing under combined longitudinal--lateral tyre slip conditions.

\section {Conclusion}

The main contribution of this work is not the independent use of PID, fuzzy logic, or neural networks, but their structured and stability-oriented integration for Ackermann vehicle steering control. The Fuzzy--PID component provides an interpretable nominal control structure, while the neural supervisory layer supplies only bounded residual gain corrections. By combining gain projection, steering saturation, anti-windup implementation, and $\sigma$-modified neural adaptation, the proposed controller avoids uncontrolled neural--fuzzy cross-coupling and remains suitable for Lyapunov-based ISS/UUB analysis. The simulation results confirm that the proposed method improves tracking accuracy, disturbance rejection, parameter robustness, and computational efficiency compared with PID and Fuzzy--PID baselines, while achieving tracking performance comparable to linearized MPC with lower computational demand.

\section*{Data Availability Statement}
The datasets generated and analysed during the current study are available from the corresponding author upon reasonable request.

\section*{Competing Interests}
The authors declare no competing interests.

\section*{Acknowledgements}
The authors gratefully acknowledge the University of Transport and Communications, Vietnam, for supporting this work.

\section*{Author Contributions Statement}

Vo Thanh Ha: Conceptualization, Methodology, Formal analysis, Supervision, Project administration, and Writing--original draft.

Nguyen Minh Huy: Software, Simulation implementation, Data curation, Validation, and Visualization.

Nguyen Van Hai: Vehicle modelling, Simscape Multibody implementation, Investigation, Validation, and Writing--review and editing.

Nguyen Tam Thanh: Literature review, Data organization, Comparative analysis, Visualization, and Writing--review and editing.

Vo Quang Vinh: Methodological consultation, Stability-analysis review, Technical validation, Formal discussion, and Writing--review and editing.

All authors have read and approved the final manuscript.


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