\chapter*[State-of-the-Art]{State-of-the-Art}\stepcounter{chapter}
\label{chpt: state-of-the-art}
\addcontentsline{toc}{chapter}{State-of-the-Art}
\textit{This chapter provides a comprehensive overview of the control strategies developed in literature for wheel slip regulation and the optimization of longitudinal vehicle dynamics. The state-of-the-art review traces the technical evolution from classical linear control, such as industry-standard PID and threshold-based logic, to more sophisticated robust non-linear strategies. Within this context, the investigation focuses on how advanced methods like Sliding Mode Control address the inherent non-linearities and parameter uncertainties of the tire-road interface that simpler linear controllers often fail to manage. Furthermore, the survey extends into modern predictive and intelligent paradigms, evaluating Model Predictive Control and Neural Network-based architectures for their capacity to handle multi-variable optimization and real-time environmental adaptation. By critically analyzing these diverse methodologies, the chapter establishes the theoretical and technical foundation necessary to justify the selection and development of the Adaptive Continuous Integral Sliding Mode Control framework presented in this work.}
\newpage
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\section[Control Objectives and Performance Metrics]{Control Objectives and Performance Metrics}
\label{sec: control objectives and metrics}
The primary objective of a Traction Control System is to regulate the longitudinal slip ratio such that the tire-road friction capability is maintained at its maximum potential value, corresponding to $\mu_{max}$. This task is inherently multi-objective, since it involves a delicate balance between longitudinal performance, directional stability, and drivability.
From a control systems perspective, the goal is to track a time-varying \textit{optimal slip reference}  while rejecting disturbances, such as sudden changes in road surface conditions or vertical load transfers.

The definition of rigorous evaluation criteria is essential to quantify the efficacy of a traction control strategy and to provide a consistent framework for the comparative analysis presented in this chapter. Beyond the primary goal of maintaining the slip ratio at its optimal peak, several key performance indicators act as the analytical lenses through which the subsequent bibliographic review is conducted.

Specifically, the effectiveness of each control paradigm, ranging from traditional to predictive, is qualified by the following performance metrics:
\begin{description}[style=nextline ]
    \item{\textbf{Tracking Accuracy}} Ability of the controller to minimize the steady-state error between the actual and reference slip, particularly under non-linear working conditions.
    \item{\textbf{Transient Response}} Quantifies controller's ability to react to abrupt variations in operating conditions (change in road friction, reference slip, etc...).
    \item{\textbf{Robustness}} System's capability to maintain stability despite parametric uncertainties.
    \item{\textbf{Chattering}} This metric assesses the high-frequency oscillations in the control signal, which can lead to mechanical wear and reduced passenger comfort.
\end{description}

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\section[Traditional Traction Control]{Traditional Traction Control}
\label{sec: traditional tract control}
The most straightforward approach to implementing traction control for anti-slip regulation involves the use of classical feedback loops and heuristic rules, which seek to maintain wheel slip within a stable range through relatively simple control architectures. These methods are often the first choice for industrial applications due to their low computational demand and the ease with which they can be integrated into existing vehicle hardware.

As discussed in reference \cite{ASR_PID}, \textbf{PID controllers} are frequently utilized to regulate the drive torque of in-wheel electric motors by acting directly on the error between the actual and target slip. While this represents a linear and highly accessible approach, it often comes with limitations related to non-linearities of the tire-road interface, as a single set of gains may not remain optimal across diverse road surfaces or during aggressive dynamic transitions. \\
The control law is typically expressed in the time domain as $T_{c}(t) = K_{p} e(t) + K_{i} \int_{0}^{t} e(\tau) d\tau + K_{d} \frac{de(t)}{dt}$, tracking the error $e = \sigma - \sigma_{ref}$. 

To augment these basic structures without significantly increasing mathematical complexity, \textbf{fuzzy logic layers} are often introduced. The work in reference \cite{ANovelFLC} proposes a correctional algorithm that uses FL control to handle the ambiguity of low-friction or irregular road surfaces. By translating human-like expert knowledge into linguistic rules, this method provides an intuitive way to adjust the control response to road roughness without requiring a complex analytical model of the environment.\\
The control action is determined by a set of \textit{if-then rules} that map the slip error and the change in error to a gradient in correctional torque $\Delta T_c$, used to evaluate $T_c = \int \Delta T_c \  dt$.

A primary technical drawback of general linear or threshold-based ASR is high slip ratio overshoot, which can reach 50\% on low-adhesion roads and up to 500\% during abrupt friction transitions \cite{ASR_PID}. This excessive overshoot causes the acceleration curve to jitter, negatively impacting both drivability and the speed of stabilization.

These heuristic methods do not provide a complete resolution for the sudden, transient variation of friction encountered on complex road surfaces. Further focus is required on the relationship between longitudinal force changes and the transient response of the powertrain to fully optimize recovery from slip events.

Moreover, while classical control theory can verify localized stability, these systems struggle to maintain low overshoot during the rapid non-linear transitions typical of high-performance driving.

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\section[Robust Non-Linear Control]{Robust Non-Linear Control}
\label{sec: robust nonlinear control}
The limitations observed in classical PID and fuzzy-based strategies highlight the need for a more rigorous mathematical framework. While heuristic and linear methods attempt to approximate the system's behavior around a nominal operating point, they often fail to account for the global dynamics of the vehicle. In facts, traction control is fundamentally a non-linear tracking problem operating in a highly uncertain environment, therefore a more robust framework is needed to provide guaranteed stability boundaries and consistent performance despite unmodeled dynamics, such as load transfers or varying powertrain response characteristics.

Within the domain of robust non-linear strategies, \textbf{Sliding Mode Control} is distinguished by its ability to maintain invariant performance in the presence of modeling inaccuracies and external disturbances. The design objective involves two primary phases:
\begin{itemize}
    \item Sliding Surface Design: a stable trajectory is defined in the state space such that when the system state is restricted to this surface, the tracking error asymptotically converges to zero.

    \item Control Law Formulation: a discontinuous control law is designed to force the system state onto the sliding surface in finite time and maintain it there.
\end{itemize}

In the reference \cite{TraditionalSMC}, the traction control problem is addressed using a \textbf{Traditional Sliding Mode Control} framework. The primary objective is to drive the system state toward a sliding surface, typically defined as the error between the actual and reference slip ratios, $s = \sigma - \sigma_{ref}$. \\
The core of this strategy is the implementation of a discontinuous control law, which is realized through a switching term, often formulated as $u_{sw} = K \cdot \text{sgn}(s)$, where $K$ is a positive gain and $sgn(s)$ is the sign function. This switching term is technically significant because it characterises the \textit{reaching phase}, forcing the wheel dynamics to move toward the sliding surface in finite time. \\
However, as noted in the reference, this constant switching across the sliding surface is the source of chattering, which results in high-frequency oscillations in the motor torque command. \\
While highly robust against disturbances, this traditional approach can excite unmodeled dynamics in the electric drivetrain.

To address the limitations of the discontinuous law, specifically the steady-state error and the chattering, works in literature, as in reference \cite{TraditionalSMC} and \cite{VTCandApplications}, introduce an augmented version of SMC that incorporates an \textbf{integral term}. In these versions, the sliding surface $s(t)$ is redefined to include the integral of the slip error:\\
 \begin{equation*}
    s(t) = e(t) + \lambda_{i} \int_{0}^{t} e(\tau) d\tau
 \end{equation*}
Unlike the traditional law, which may allow a persistent offset if the switching gain $K$ is too low, the integral term ensures that the tracking error asymptotically converges to zero even in the presence of constant disturbances. It provides global robustness, as the sliding surface can be designed to include the initial conditions of the system, effectively eliminating the reaching phase and its associated vulnerabilities.

Building upon the augmented integral architectures, recent research and the specific methodologies analyzed in the literature review lead to an approach specifically designed to refine the control law by smoothing the response during the most critical phases of traction regulation, named \textbf{Conditional Integral Sliding Mode Control}.\\
In reference \cite{TraditionalSMC}, a solution to the chattering problem related to \acrshort{tsmc} control is proposed, introducing a sophisticated switching logic that manages the integral term, leading to a trade-off solution between tracking precision and transient stability. Unlike a standard integral sliding surface, which is active at all times, the conditional approach operates based on the system's proximity to the sliding surface. In this framework, the integral action is only enabled when the system state enters a predefined boundary layer surrounding the sliding surface.

By deactivating the integrator when the system is far from the sliding surface, such as during a sudden jump in friction or a large initial slip, the controller prevents the \textit{integrator windup} typically responsible for massive overshoots observed in traditional ASR implementations.\\
Within the boundary layer, the integral term takes over the task of rejecting constant disturbances. This allows the discontinuous switching gain to be significantly reduced or replaced with a smoother saturation function, effectively eliminating the high-frequency vibrations in the torque command.

This evolution from the simple switching laws of  control, to the integral-augmented strategies, concludes with the \acrshort{cismc} as the most balanced solution. It provides the capability to handle the non-linearities of the tire-road interface while maintaining the control smoothness required for modern electric drivetrains.


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\section[Predictive Control]{Predictive Control}
\label{sec: predictive control}
Predictive models in the automotive industry have evolved from theoretical frameworks into functional prototypes that significantly outperform traditional control systems. These operate by utilizing a mathematical model of the vehicle's dynamics to predict future behavior and optimize control actions accordingly.

One prominent application is in traction control, where \textbf{Hybrid Model Predictive Control} models manage the nonlinear relationship of tire slip on low-friction surfaces like ice, achieving up to 20\% improvement in slip regulation over standard linear methods. \\
In reference \cite{MPCHybrid}, a \acrshort{mpc} approach specifically designed for traction control is proposed by treating the vehicle's wheel dynamics as a hybrid system. The core of this solution lies in the recognition that tire-road interaction is highly nonlinear, making traditional linear controllers ineffective during sudden changes in surface friction. To manage this, the researchers use a \textbf{Piecewise Affine} (\textbf{PWA}) model, which approximates the complex, nonlinear tire-force curve by dividing it into distinct linear operating regions based on the amount of tire slip. \\
At each sampling interval, the MPC algorithm solves an optimization problem that minimizes a cost function, balancing the deviation from the target slip against the effort required from the actuators, while respecting physical constraints like maximum engine torque and braking limits. Because solving these complex mathematical problems in real time is computationally expensive for standard vehicle hardware, the authors employ \textbf{Explicit MPC}. This technique moves the heavy calculations offline, partitioning the state space into various polyhedral regions where the optimal control law is pre-calculated as an affine function.

Another solution presented in \cite{NMPC} utilizes \textbf{Nonlinear Model Predictive Control} (\textbf{NMPC}) to address the highly complex and nonlinear nature of tire-road interactions. Unlike the hybrid or piecewise linear approaches, \acrshort{nmpc} directly incorporates the nonlinear tire model into its optimization process, which allows it to more accurately represent the physical limits of the vehicle under varying conditions. \\
The controller accomplishes slip regulation by solving a constrained nonlinear optimization problem at every time step, minimizing the error between the actual and the reference slip while also penalizing excessive changes in motor torque, to ensure a smoother control. \\
A critical advantage of this NMPC approach for in-wheel motor configurations is its ability to handle the rapid torque response of electric motors. Simulation results indicated that this solution provides superior tracking performance and robustness compared to standard linear control methods, effectively preventing wheel lock-up and spinning even under extreme driving maneuvers.

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\section[Comparative Analysis]{Comparative Analysis}
\label{sec: comparative analysis}
Evaluating the various control strategies for automotive traction systems reveals a clear progression from simple regulation to complex, predictive optimization. Each method offers a distinct balance of computational demand and performance accuracy.

\acrshort{pid} controllers remain the industry standard due to their simplicity and ease of implementation. However, they struggle with the highly nonlinear and time-varying nature of tire-road friction, often leading to sub-optimal performance in extreme conditions like ice or sudden maneuvers. Linear \acrshort{mpc} improves upon this by using a predictive model and handling system constraints, but its reliance on a single linear model makes it inaccurate when the vehicle operates outside a specific narrow range of slip .

To capture more complex dynamics, \acrshort{mpc} with \acrshort{pwa} models approximate nonlinearities by switching between different linear regimes. This provides a significant boost in accuracy, but also increases the mathematical complexity of the optimization problem. \acrshort{nmpc} sits at the top of the performance hierarchy, directly integrating nonlinear tire models and motor dynamics to provide the most precise tracking and robustness, especially for four-in-wheel motor electric vehicles. \\ The primary drawback of NMPC is its extreme computational cost, which often requires high-end processors or complex simplification techniques to run in real-time.  


Between these extremes lie \acrlong{smc} variants. \acrshort{tsmc} offers fast convergence and finite-time stability but can suffer from chattering. \acrshort{cismc} emerges as the ideal trade-off for modern automotive applications. It combines the robustness of sliding mode control with integral action, effectively rejecting disturbances and eliminating steady-state error without the extreme computational overhead of \acrshort{nmpc}. \acrshort{cismc} provides near-optimal results by maintaining the desired slip ratio even under uncertainty, yet it remains simple enough to be implemented on standard electronic control units without the heavy optimization solvers required by predictive models. This balance of stability and manageable complexity makes \acrshort{cismc} a highly practical choice for mass-market vehicle systems.

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\section[Market Analysis]{Market Analysis}
\label{sec: market analysis}
%\subsection{Mass-Market}
For the vast majority of mass-market vehicles, the primary controller used is the \acrshort{pid} controller. It is the industry workhorse because of its low computational weight and high reliability for standard driving conditions. In these systems, the controller monitors wheel speed sensors and, upon detecting slip, triggers either motor torque reduction or brake intervention.

In the EV segment, which now accounts for a rapidly growing portion of the mass market, the control strategy is shifting. Approximately 60\% of EVs sold in 2024 utilized specialized traction modules designed for the instant torque of electric motors \cite{congruence_traction_2024}.
Because electric motors can adjust torque much faster than a mechanical engine or a hydraulic brake system, suppliers are moving toward more advanced, high-frequency control loops. While many still use sophisticated PID variants, there is a clear trend toward Predictive Traction Systems and AI-enhanced algorithms, which saw a 47\% increase in adoption in 2024.

The current market trend shows a shift from \textit{reactive systems} to \textit{proactive architectures}, led by major OEMs and Tier-1 suppliers:
\begin{itemize}
    \item \textbf{BYD iTAC (Intelligent Torque Adaptation Control)}: BYD has redefined traction logic by moving away from traditional ABS-based wheel sensors. Their iTAC system utilizes the electric motor’s internal speed sensors, which offer significantly higher resolution and faster sampling rates. Unlike conventional PID-based systems that wait for a detected slip threshold to cut power, iTAC "anticipates" slip by monitoring motor speed fluctuations. It stabilizes the wheel in milliseconds by applying "negative torque" or millimetric torque adjustments. In real-world performance, this has demonstrated a 0.7-second improvement in 0–60 km/h acceleration on low-friction surfaces like snow compared to conventional systems.
    \item \textbf{Bosch Integrated Traction Control \& VMM}: As a global leader, Bosch is shifting toward Vehicle Motion Management (VMM), a software suite that orchestrates brakes and motors through a centralized architecture [15]. Their logic increasingly adopts Model Predictive Control (MPC) to calculate required torque based on "road perception"—integrating cloud-based data regarding road friction and slope. This high-level integration is visible in systems like the iBooster, which coordinates slip during regenerative braking. By using predictive algorithms, Bosch avoids unwanted wheel lock-ups during energy recovery without the characteristic vibrations on the brake pedal associated with legacy ABS/TCS interventions.
\end{itemize}




The choice of controller is heavily influenced by cost and regulatory mandates such as Euro NCAP or Bharat NCAP \cite{kings_research_traction_2024}. OEMs prioritize systems that can guarantee safety across a wide range of drivers and road surfaces at the lowest possible cost per unit. This economic reality is why the high-complexity models like NMPC are currently reserved for the premium and motorsport segments, while the reliable, cost-effective PID and basic predictive models continue to rule the mass market.

