\chapter*{Simulations and Results}\label{chpt: simulations&results}
\stepcounter{chapter}
\addcontentsline{toc}{chapter}{Simulations and Results}
\textit{This chapter presents the virtual validation framework and the comparative performance analysis of the proposed wheel slip controllers. \\ To achieve seamless integration with the IPG Automotive CarMaker simulation environment, specific software adaptation blocks were developed to handle motor torque commands and state variable routing.} 

\textit{Subsequently, the controllers' performance is evaluated and compared through a series of critical maneuvers under varying friction conditions, including abrupt road surface transitions ($\mu$-jumps) and asymmetrical surfaces ($\mu$- split). Finally, the simulation results are presented and analyzed with a specific focus on slip tracking accuracy, motor torque smoothness, and the suppression of powertrain torsional oscillations, highlighting the distinct performance advantages offered by the CISMC strategy.}

\newpage

\section{Simulation Environment}
To evaluate the proposed control algorithms under realistic conditions, a high-fidelity simulation environment is established by interfacing MATLAB/Simulink with IPG Automotive CarMaker. Within this framework, CarMaker handles the complex, non-linear physical plant dynamics, including the multi-body chassis, a detailed powertrain configuration, aerodynamic forces, and dynamic tire models, while the complete control logic runs entirely within Simulink.

\subsection{CarMaker for Simulink}
The validation environment is built as a co-simulation architecture using CarMaker for Simulink (CM4SL).\\
CM4SL establishes a real-time co-simulation loop where the high-fidelity multi-body chassis and tire dynamics run natively within CarMaker, while the customized control algorithms execute entirely inside MATLAB/Simulink. Communication is synchronized at each simulation time step using dedicated S-Functions, allowing the control loop to read real-time kinematic states directly from the CarMaker Data Dictionary (via \texttt{Read CM Dict} blocks). Concurrently, the calculated controller torque commands are directly injected back into the virtual powertrain assembly using a Model Plug-in layer (via \texttt{Write CM Dict} blocks), overriding the nominal vehicle actuators to close the loop under realistic unmodeled physical constraints.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{images/CM4SL_picture.png}
    \caption{CM4SL environment.}
    \label{fig: CM4SL_pic}
    
\end{figure}
Within this co-simulation architecture, the developed slip control logic is embedded directly inside the \texttt{IPGVehicle > Powertrain} subsystem. Rather than delegating torque generation to CarMaker’s native internal powertrain or driver models, the customized Simulink plug-in bypasses the default actuator commands, overwriting target torque values for the electric motor, whenever the controller state is active.
\newpage
Simulations within CarMaker4Simulink environment have been carried out with a timestep of 0.001 s.
\subsection{Actuator Target Assignment and Modified Powertrain Subsystem}\label{subsec: EMTargetAssignment_SimandRes}
Once the core tracking loop within the Traction Control System computes the raw corrective action, the output is routed to the EM Target Assignment layer ($T_{EM}^{req} = \tfrac{T_{drive}^{req}}{\eta_{EM} \cdot i_g}$). The theoretical controller output is scaled and saturated according to the physical operational boundaries of the electric motors. The validated electromagnetic request is then explicitly assigned to the dictionary variable \texttt{PT.Control.Motor.Trq\_trg}.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.8\textwidth]{images/CM4SL_scheme.png}
    \caption{Actuator target assignment and Modified Powertrain subsystem}
    \label{fig: torque_trg_mod_pt_scheme}   
\end{figure}

However, to ensure that the overall vehicle plant model seamlessly executes the controller's request, the default torque transmission paths inside CarMaker must be bypassed. This is achieved by routing the torque target through a customized \textbf{Modified Powertrain} block configuration located between the native \textbf{PowerTrain} and \textbf{PowerTrain Upd} interfaces.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.85\textwidth]{images/ModPTSimulink.png}
    \caption{Modified Powertrain block within Simulink.}
    \label{fig: ModPT_outscheme}
\end{figure}

Within this modified powertrain stage, the calculated motor torque ($T_{EM}^{trg}$) is re-elaborated: the block automatically models the driveline transformations, such as the reduction gear efficiency and transmission ratios to derive the net driving torque applied directly at the wheel axle ($T_{driving}$). 

\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/ModPTSimulink_detail.png}
    \caption{Detail on Modified Powertrain block.}
    \label{fig: ModPT_inscheme}
\end{figure}

Controller's activity state is defined by reading a flag which constantly observes the distance between the actual wheels' slipping state and the optimal slip value: control-loop is enabled whenever $\sigma_j > \sigma_{th,j} = 0.9\cdot \sigma_{opt,j}$, in order to initiate the control action just before entering an unstable condition. \\
A dedicated decision block continuously reads the functional state of the controller (ON or OFF):
\begin{itemize}
    \item Controller OFF: The decision block passes the nominal, unmodified variables directly from the default CarMaker Powertrain Interface.
    \item Controller ON: The loop instantly switches to inject the modified, controller-derived kinetic and kinematic quantities directly into the PowertrainUpd Interface.
\end{itemize}

By establishing this overriding sequence, the simulation ensures that the high-fidelity chassis model updates its longitudinal tire forces based strictly on the optimized slip control requests, preventing any internal algebraic loops or parameter conflicts from disrupting the co-simulation layer.
\section{Selected Test Maneuvers and Simulation Scenarios}
The scenarios tested to evaluate and compare the performance of the proposed controllers have been selected to replicate operational conditions that typically challenge vehicle stability and traction control systems.

Three distinct test benchmarks have been designed:
\begin{description}[style=nextline]
    \item[Low Friction] Full gas pedal maneuver starting from approximately 10 km/h, followed by a braking maneuver. Road friction coefficient is $\mu_{road} = 0.3$.
    \item[Mu-jump] Constant gas pedal ($\approx 70\%$), starting from approximately 10 km/h. Road friction coefficient varies from $\mu_{road,high} = 1$ to $\mu_{road,low} = 0.3$. Speed of the vehicle is not over 20 Km/h when the change in road friction is encountered.
    \item[Mu-split] Constant gas pedal ($\approx 70\%$), starting from approximately 10 km/h. Road friction coefficient varies from $\mu_{road,high} = 1$ to $\mu_{R_{road,low}} = 0.3$ for the right wheels and $\mu_{L_{road,low}} = 0.1$ for the left wheels. Speed of the vehicle is not over 20 Km/h when the change in road friction is encountered.
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{Low Friction Maneuver}\label{sec:low_mu}
The first test profile begins with an initial constant-speed cruising phase at $V_{vhcl}\approx10$ km/h ($\approx 2.8$ m/s), followed by a 100\% full gas pedal step input initiated at approximately 4.5 seconds after simulation starts, demanding maximum available acceleration torque. Finally at $t\approx14.5$ s a braking demand is commanded. \\
The road is characterized by uniform low friction coefficient $\mu_{road} = 0.3$, representative of packed snow or extremely wet cobblestones, and the Reference Slip Estimator identifies a constant optimal longitudinal slip peak of $\sigma_{opt} = 0.1038$ for both rear wheels, during the traction phase, switching to a negative sign target $\sigma_{opt} = -0.0976$ during the braking phase.\\
Results shown in this section refer to the simulation being stopped as soon as $V_{vhcl}=0.8$ m/s after the braking phase, to avoid loss of information given by poor mathematical definitions fidelity at such low speed.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Traction Control System OFF}
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/LowMu/TCSOFF/TCSOFF - Speeds.png}
    \caption{Vehicle and driving wheels' speed with traction control OFF.}
    \label{fig: LowMu/TCSOFF/TCSOFF - Speeds}
\end{figure}
Without an active traction control system, the full-throttle tip-in causes immediate, severe spin on both wheels.

The wheels accelerate exponentially, peaking at a peripheral speed of over 62 m/s, while the actual vehicle velocity increases at a slower rate due to the loss of effective force transmission.

Braking phase is not initiated due to complete loss of adherence at the tire-ground contact.

Comparing actual longitudinal slip against reference optimal value for both wheels (Fig. \ref{fig: LowMu/TCSOFF/TCSOFF - Longitudinal Slip}), slip increase is visible as soon as pedal input is applied ($t\approx4.5$ s): it bypasses the stable linear region and jumps straight to an initial spike reaching full spin ($\sigma = 1$) and maintaining a value above 0.8 afterwards.  

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{images/LowMu/TCSOFF/TCSOFF - Longitudinal Slip.png}
    \caption{Actual vs. reference longitudinal slip with traction control OFF.}
    \label{fig: LowMu/TCSOFF/TCSOFF - Longitudinal Slip}
\end{figure}

The consequences of uncontrolled slip are confirmed by the utilized friction coefficient plot in Figure \ref{fig: LowMu/TCSOFF/TCSOFF - Friction}, showing  how the adherence capabilities of the tire-road interface are never efficiently exploited ($|\mu_{ul}|<|\mu_{road}|$) as the tire is operating deep within the unstable sliding region.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{images/LowMu/TCSOFF/TCSOFF - Friction.png}
    \caption{Utilized vs. road friction coefficient with traction control OFF.}
    \label{fig: LowMu/TCSOFF/TCSOFF - Friction}
\end{figure}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{PI Controller}
Figure \ref{fig: LowMu/PI/PI Controller - Activity} illustrates the binary operational state of the PI activation block. As expected, the controller switches from an idle state (0) to an active state (1) immediately at $t\approx4.5$ s as the wheel slip crosses the activation threshold ($\sigma_j>\sigma_{th,j}$). The PI controller remains fully engaged throughout both the traction and braking phases.
\begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/PI/PI Controller - Activity.png}
        \caption{PI controller activity.}
        \label{fig: LowMu/PI/PI Controller - Activity}
\end{figure}

Under active PI control, the vehicle’s longitudinal behavior is significantly improved. As plotted in Figure \ref{fig: LowMu/PI/PI Controller - Speeds}, the driving wheel speeds no longer diverge into uncontrolled spin; instead, they are held at a regulated distance above the vehicle speed curve in the V-t plane. This tight velocity coupling matches the target optimal slip ($\sigma_{opt}$) velocity profile. \\
The system remains stable during steady-state acceleration, and the control loop successfully prevents wheel lock-up during the sudden traction-to-braking transition at $t \approx 14.5$ s.

    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/PI/PI Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with PI.}
        \label{fig: LowMu/PI/PI Controller - Speeds}
    \end{figure}

The slip tracking performance is verified in Figure \ref{fig: LowMu/PI/PI Controller - Longitudinal Slip}. The PI controller successfully contains the initial slip spike, bringing the longitudinal slip magnitude to match the optimal value of 0.1038 during the traction phase, and cleanly transitions to -0.0976 during the deceleration phase.

Looking closely at the controller’s tracking performance (Fig \ref{fig: LowMu/PI/PI Controller - Tracking Error}), longitudinal slip tracking error is kept to zero for the whole simulation, despite a negligible overshoot to $e_{max} < 0.02$ in correspondence of the transition phase ($t\approx 14.5$). Wheel’s rotational speed tracking error ($\omega_j-\omega_{ref,j}$) follows the same trend, with magnitude of overshoot ranging from 0.3 to $-0.9$ rad/s approximately. This behavior is caused by the physical wind-up of the integral channel before the controller can adapt to the rapid change in torque direction.
        
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/LowMu/PI/PI Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with PI.}
        \label{fig: LowMu/PI/PI Controller - Longitudinal Slip}
        \vspace{0.5em}

        \centering
        \includegraphics[width=0.9\textwidth]{images/LowMu/PI/PI Controller - Tracking Error.png}
        \caption{PI controller tracking error.}
        \label{fig: LowMu/PI/PI Controller - Tracking Error}   
    \end{figure}

Controller’s capability to enhance friction utilization at the tire-ground contact is reported in the Figure \ref{fig: LowMu/PI/PI Controller - Friction}, where $|\mu_{ul}|\ge|\mu_{road}|$ for both wheels throughout the whole controller’s intervention. Unlike the TCS OFF configuration, the system fully exploits the available adherence of the tire-road interface during both acceleration and deceleration, drastically improving overall safety and acceleration metrics.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.85\textwidth]{images/LowMu/PI/PI Controller - Friction.png}
        \caption{Utilized vs. road friction coefficient with PI.}
        \label{fig: LowMu/PI/PI Controller - Friction}
    \end{figure}

\subsubsection{Controller's Output}
As previously discussed in Sec \ref{sec: PID_design}, the control logic outputs a driving torque request at the wheel, which is further manipulated for electric motors target setting. \\
The calculated torque request generated by the PI control logic is reported in Figure \ref{fig: LowMu/PI/PI Controller - Controller Output Torques} and Figure \ref{fig: LowMu/PI/PI Controller - Controller Output Torques - ZOOM}. The total command torque is mathematically decomposed into the primary Control Torque and an auxiliary Compensation Torque serving as an anti-windup layer. \\
As highlighted in the zoomed-in version of controller’s output plot in Figure \ref{fig: LowMu/PI/PI Controller - Controller Output Torques - ZOOM}, the compensation torque remains strictly zero during regular operations. Even, during the extreme reference sign inversion at $t\approx 14.5$ s, the control torque never hits its saturation bounds, hence the anti-windup compensation loop doesn't activate.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.95\textwidth]{images/LowMu/PI/PI Controller - Controller Output Torques.png}
        \caption{PI controller output torques.}
        \label{fig: LowMu/PI/PI Controller - Controller Output Torques}

        \vspace{0.5em}

        \centering
        \includegraphics[width=0.95\textwidth]{images/LowMu/PI/PI Controller - Controller Output Torques - ZOOM.png}
        \caption{PI controller output torques, detail.}
        \label{fig: LowMu/PI/PI Controller - Controller Output Torques - ZOOM}
    \end{figure}

Finally, target electric motor torque commands and actual physical motor torque outputs are detailed in Figure \ref{fig: LowMu/PI/PI Controller - EM Torques}.

The resulting target torque for the electric motor is a smooth steady profile, and is easily tracked by the EM torque.
    \begin{figure}[H]
            \centering
            \includegraphics[width=1\textwidth]{images/LowMu/PI/PI Controller - EM Torques.png}
            \caption{Electric motors torque with PI controller.}
            \label{fig: LowMu/PI/PI Controller - EM Torques}
    \end{figure}
However, during the torque reversal phase at $t \approx 14.5$ s, the reference signal demands an instantaneous change in sign: this step change forces the target torque to change at a frequency out of the maximum physical motor envelope, hence the actual motor torque tries to track this extreme demand with an instantaneous step response, according the hardware capability limits, inevitably resulting in a short-time oscillation, before stabilizing into a steady braking profile.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/LowMu/PI/PI Controller - EM Torques - ZOOM.png}
        \caption{Electric motors torque with PI controller, detail.}
        \label{fig: LowMu/PI/PI Controller - EM Torques - ZOOM}
    \end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage    
\subsection{TSM Controller}
The primary objective of this architecture is to exploit the robust, non-linear discontinuous reaching laws of sliding mode theory to ensure faster transient response and absolute invariance to model uncertainties compared to the baseline PI controller.

The controller utilizes an identical switching threshold mechanism to the PI layout, mapping the idle state to 0 and full engagement to 1. The loop triggers instantaneously at $t\approx1$ s, remaining active throughout the entire high-slip regime.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Activity.png}
    \caption{TSM controller activity.}
    \label{fig: LowMu/TSMC/TSM Controller - Activity.png}
\end{figure}

Figure \ref{fig: LowMu/TSMC/TSM Controller - Speeds} illustrates the vehicle longitudinal speed and the tangential wheel velocities.
\newpage
The application of the non-linear switching manifold completely alters the transient response compared to the PI baseline.

The TSMC limits the initial spin surge instantaneously, locking the wheel velocity tightly onto the tracking line prescribed by the optimal reference slip.

However, a close inspection of the wheel speed curves during steady-state traction reveals high-frequency, low-amplitude oscillations directly superimposed onto the velocity trajectories.\\
This kinematic ripple is the physical manifestation of high-frequency actuator switching.

\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Speeds.png}
    \caption{Vehicle and driving wheels' speed with TSMC.}
    \label{fig: LowMu/TSMC/TSM Controller - Speeds}
\end{figure}

\newpage
This tracking behavior is further confirmed by the longitudinal slip development plotted in Figure \ref{fig: LowMu/TSMC/TSM Controller - Longitudinal Slip}.

TSMC demonstrates a remarkably steep reaching phase, forcing the longitudinal slip to hit the optimal reference boundary ($\sigma_{opt} = 0.1038$) within milliseconds, even during the aggressive torque reversal at $t \approx14.5$ s, where the slip profile switches instantly to match the negative reference.

Unlike the PI controller, which suffered from integral wind-up lag, the TSMC crosses the zero-slip line with practically zero transient delay. Despite this exceptional speed, the slip never fully stabilizes at a smooth constant value; instead, it exhibits continuous high-frequency chatter directly around the reference optimal slip curve.

\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Longitudinal Slip.png}
    \caption{Actual vs. reference longitudinal slip with TSMC.}
    \label{fig: LowMu/TSMC/TSM Controller - Longitudinal Slip}
\end{figure}

To quantify this tracking capability, Figure \ref{fig: LowMu/TSMC/TSM Controller - Tracking Error.png} isolates the raw longitudinal slip tracking error.
\newpage

The tracking error metrics highlight the trade-offs of the traditional sliding mode approach. \\
During the severe traction-to-braking step transition at $t\approx14.5$ s, the peak slip tracking error magnitude is again approximately $0.2$.

Because the traditional sliding control law employs a discontinuous sign function ($sign(s)$), the tracking error does not asymptote to a clean zero. Instead, it enters a continuous, high-frequency chattering around the zero-error axis, with a persistent average residual oscillation amplitude of approximately ±0.05.

Moreover, it's visible how the intensity of chattering is inversely proportional to vehicle speed. While at low speeds ($V_{vhcl}<5$ m/s) the maximum error amplitude is in the order of 0.14, at high speeds ($V_{vhcl}>12$ m/s) the system is more stable, guaranteeing a lower amplitude of oscillations ($e_{max} \approx 0.02$).
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Tracking Error.png}
        \caption{TSM controller tracking error.}
        \label{fig: LowMu/TSMC/TSM Controller - Tracking Error.png}   
    \end{figure}
\newpage
As reported in Figure \ref{fig: LowMu/TSMC/TSM Controller - Friction}, ensuring that the longitudinal slip is restricted to the peak zone ($\sigma = 0.1038$), the TSMC maximizes force transmission, maintaining the utilized friction coefficient close or higher than the road nominal one. \\ Utilized friction coefficient magnitude is always higher than road theoretical friction's one, and also higher than in the PI case, confirming a higher capability of the TSM controller of maximizing adherence at tire-ground contact.

The rapid tracking response during the torque reversal guarantees that the tire contact patch switches from maximum acceleration force to maximum braking force almost instantaneously. However, matching the slip fluctuations, the calculated friction forces exhibit high-frequency noise, which can introduce mechanical vibrations.

\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Friction.png}
    \caption{Utilized vs. road friction coefficient with TSMC.}
    \label{fig: LowMu/TSMC/TSM Controller - Friction}
\end{figure}
\newpage
\subsubsection{Controller's Output}
The underlying cause of the high-frequency kinematic fluctuations is identified by examining the controller output variables. Figure \ref{fig: LowMu/TSMC/TSM Controller - Controller Output Torques} illustrates the computed command torque request.

To counteract unmodeled drivetrain dynamics and external disturbances, the switching gain must be set to a high value. Consequently, the controller command signal switches back and forth between its extreme operational limits.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Controller Output Torques.png}
    \caption{TSM controller output torques.}
    \label{fig: LowMu/TSMC/TSM Controller - Controller Output Torques} 
\end{figure}
This severe chattering behavior is continuously sustained during both the steady-state traction phase ($t=5 s \div 14$ s) and the braking phase ($t=14 s \div 15$ s).
\begin{figure}[H]
    \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSMC Controller - EM Torques.png}
    \caption{Electric motors torque with TSM controller.}
    \label{fig: LowMu/TSMC/TSMC Controller - EM Torques} 
\end{figure}
Figure \ref{fig: LowMu/TSMC/TSMC Controller - EM Torques} displays the target electromagnetic motor torque command set for the electric motors.

Because modern permanent magnet synchronous motors possess high-bandwidth power electronics, the physical actuator actively attempts to execute this high-frequency chattering profile.\\
As a result, the actual motor torque output continuously oscillates over a massive envelope. While this switching enforces absolute mathematical robustness on the slip tracking layer, it is physically unacceptable in a production vehicle.

In a real powertrain layout, high-frequency torque chattering of this magnitude would excite the torsional resonant modes of the drive shafts, generate severe acoustic noise, accelerate mechanical fatigue of the reduction gears, and cause high thermal losses in the power inverter stages.

\subsubsection{Controller's Stability}
To confirm the theoretical stability proofs discussed in Section \ref{subsec: lyapunov_TSMC} with direct numerical simulation, Figure \ref{fig: LowMu/TSMC/TSM Controller - Controller Stability.png} illustrates the real-time tracking of the sliding manifold surface $s$ alongside its corresponding computed time derivative $\dot V (s) = s \ \dot s$ for both rear wheels.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/TSMC/TSM Controller - Controller Stability.png}
        \caption{TSM controller's stability analysis.}
        \label{fig: LowMu/TSMC/TSM Controller - Controller Stability.png}
    \end{figure}
Before controller's intervention, both variables obviously rest at zero. The severe torque demand caused by gas pedal input induces an instantaneous perturbation in wheel kinematics, forcing the sliding surface variable to spike away from the ideal manifold ($s=0$). Concurrently, the Lyapunov derivative exhibits an immediate, sharp negative impulse, plunging deeply into the negative-definite region.\\ 
This behavior strictly validates the reaching condition: it proves that the non-linear control law reacts to the tracking error by generating an immediate corrective action that forces the system trajectory back toward the sliding surface.

After this reaching phase, rather than settling smoothly at absolute zero, both $s$ and $\dot V$ exhibit continuous, high-frequency chattering around the zero axis. Because the controller relies on a discontinuous $sign(s)$ term, it overcorrects at every sampling interval. Consequently, the sliding surface oscillates rapidly across zero with a bounded, residual amplitude. \\
Matching this behavior, the Lyapunov derivative oscillates rapidly in phase with the switching control law. Because $\dot s$ switches signs every time $s$ crosses zero, their product chatters symmetrically around the zero line.

While the system remains macroscopically stable and bounded within this tight limit cycle, this continuous chattering represents high-frequency energy that directly causes the torque oscillations seen in the actuator outputs.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\subsection{CISM Controller}
The core objective of this advanced architecture is to overcome the physical limitations of the previous two setups, aiming to match the instantaneous, robust tracking response of the TSMC while completely eliminating the high-frequency actuator chattering that compromises powertrain integrity.

The operational profile of the CISMC loop is shown in Figure \ref{fig: LowMu/CISMC/CISM Controller - Activity}
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/CISMC/CISM Controller - Activity.png}
        \caption{CISM controller activity.}
        \label{fig: LowMu/CISMC/CISM Controller - Activity}   
    \end{figure}
Mirroring the previous control configurations, the threshold switch registers an immediate transition from an idle state (0) to full enforcement (1) at $t\approx4.5$ s when the driver's full-throttle step input occurs.\\
The controller remains continuously active, handling the localized slip dynamics through both traction and braking regimes.
\newpage
Figure \ref{fig: LowMu/CISMC/CISM Controller - Speeds} shows vehicle longitudinal speed alongside individual rear wheels tangential velocities.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/CISMC/CISM Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with CISMC.}
        \label{fig: LowMu/CISMC/CISM Controller - Speeds}
    \end{figure}
Speed profiles are altered by the introduction of the conditional integral sliding surface.

At $t\approx4.5$ s, the rear wheels velocities rise smoothly above the vehicle velocity baseline with almost zero overshoot or transient spin spikes.

Crucially, unlike the TSMC speed profiles, the velocity lines under CISMC are perfectly smooth and continuous during steady-state acceleration ($t=5 s \div 14$ s). The high-frequency ripple is entirely suppressed, demonstrating that the boundary layer smoothing successfully translates to clean kinematic wheel behavior.
\newpage
This tracking capability is confirmed by the longitudinal slip development reported in Figure \ref{fig: LowMu/CISMC/CISM Controller - Longitudinal Slip}.

The CISMC achieves excellent tracking performance. When the full throttle step is applied, the actual slip converges exponentially onto the optimal target ($\sigma_{opt} = 0.1038$), despite an istantaneous overshoot to 0.17. Once the sliding manifold is reached, the slip remains tightly locked on the target reference without any of the high-frequency oscillatory chatter seen in the TSMC.

During the sharp torque inversion at $t\approx14.5$ s, the tracking remains stable. Thanks to the conditional integration rule, the integral term is frozen the moment the actuator saturates during the rapid torque reversal, preventing integration wind-up. When the system exits saturation to track the braking target ($\sigma_{opt} = -0.0976$), the slip converges cleanly to the new value with negligible lag and zero undershoot.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/CISMC/CISM Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with CISMC.}
        \label{fig: LowMu/CISMC/CISM Controller - Longitudinal Slip}
    \end{figure}
\newpage
Because the controller stabilizes the tire slip ratio precisely at the absolute peak of the road profile ($\sigma = 0.1038$), as shown in Figure \ref{fig: LowMu/CISMC/CISM Controller - Friction}, utilized friction coefficient is held continuously close to and above its theoretical physical maximum.

By avoiding the tracking fluctuations of the PI and the high-frequency chattering forces of the TSMC, the CISMC maintains a steady force application at the contact patch, ensuring predictable, maximized acceleration performance.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/CISMC/CISM Controller - Friction.png}
        \caption{Utilized vs. road friction coefficient with CISMC.}
        \label{fig: LowMu/CISMC/CISM Controller - Friction}
    \end{figure}

\newpage
\subsubsection{Controller's Output}
Chattering elimination can be visualized by analyzing the controller's internal outputs of Figure \ref{fig: LowMu/CISMC/CISM Controller - Controller Outputs}.

As discussed in Section \ref{sec: CISMC_ControlLaw}, the control input, $u_{ref}$, corresponds to a driving torque request at the wheel axle. This signal is saturated accordingly to the physical limit for maximum driving torque (see Eq. \eqref{eq: TrqDriveMAX_CISMC}) and translated into a target torque command for the electric motor, as seen in detail in Section \ref{subsec: EMTargetAssignment_SimandRes}.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/CISMC/CISM Controller - Controller Outputs.png}
        \caption{CISM controller's output torques.}
        \label{fig: LowMu/CISMC/CISM Controller - Controller Outputs}
    \end{figure}
As explained in Section \ref{chpt: Final Controller}, the CISMC logic avoids chattering by replacing the discontinuous sign function with a continuous saturation function inside a predefined boundary layer ($\varphi$). The output torque profile displayed in detail in Figure \ref{fig: LowMu/CISMC/CISM Controller - EM Torques} reflects this design, providing clean, smooth changes in torque demand during both the steady acceleration and steady braking regimes.

    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/LowMu/CISMC/CISM Controller - EM Torques.png}
        \caption{Electric motors torque with CISM controller.}
        \label{fig: LowMu/CISMC/CISM Controller - EM Torques}
    \end{figure}

The target torque command sent via the \texttt{PT.Control.Motor.Trq\_trg} is entirely free of high-frequency switching noise. \\
During  traction ($t=5 s\div 14$ s), the torque command remains a continuous signal below 100 Nm per wheel.\\
During the  traction-to-braking transition at $t\approx14.5$ s, the controller demands a smooth, rapid drop to full negative braking saturation before settling into a stable deceleration torque profile. Since the command signal is smooth, the actual physical torque delivered by the in-wheel motors matches the target exactly.
\newpage
\subsubsection{Controller's Stability}
To evaluate how the CISMC resolves the physical limitations of the traditional formulation, Figure \ref{fig: LowMu/CISMC/CISM Controller - System Inverse Dynamics} and \ref{fig: LowMu/CISMC/CISM Controller - Controller Stability} isolates the internal components of the system inverse dynamics in the control law (functions $f$ and $g$), and the time-domain evolution of the sliding surface $s$.

Considering the operational time window within which the controller's state is ON ($t\gtrapprox 4.5$), despite the initial sudden torque surge perturbing the wheel plant kinematics, causing a brief, localized deviation, the sliding surface is held at zero. \\
Because the CISMC replaces the discontinuous sign function with a continuous saturation interpolation function inside a predefined boundary layer, the control law smoothly handles the perturbation, forcing the tracking error to converge exponentially back toward the ideal trajectory ($s=0$) within milliseconds of the initial tip-in.

By allowing the control law to interpolate continuously within the boundary layer, the high-frequency switching phenomenon is completely eliminated. Within this layer, the system behaves similarly to a high-gain linear controller where the conditional integral term continuously estimates and compensates for the unmodeled dynamics and external disturbances. This structural change ensures that the sliding surface tracking remains smooth and precise, preventing the high-frequency ripple from reaching the wheel speed trajectories.

During the rapid transition from full acceleration to an aggressive braking demand at $t \approx 4.5$ s, the reference slip target shifts instantly from a positive to a negative value. This step change creates a large tracking error, causing $s$ to experience a transient spike. \\
The sliding variable briefly exits the boundary layer ($|s|>\varphi$), which automatically switches the controller back to its robust sliding mode reaching law ($sign(\tfrac{s}{\varepsilon})$) to drive the error down rapidly. \\
Simultaneously, the conditional logic detects that the motor actuators have hit their physical saturation limits and temporarily freezes the integral state. This prevents integration wind-up and ensures that as soon as $s$ is pulled back inside the boundary layer, the tracking variable hooks cleanly back onto the zero axis without any undershoot or secondary oscillations.

While function $f$ represents the autonomous, unforced dynamics of the wheel slip state, which scales based on the current vehicle velocity and estimated wheel deceleration trends, function $g$ maps how effectively the electromagnetic motor torque translates into slip acceleration. Both functions match the macroscopic physical phases of the vehicle maneuver. They change smoothly during steady acceleration and steady braking, remaining entirely free of high-frequency noise.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/LowMu/CISMC/CISM Controller - System Inverse Dynamics.png}
        \caption{CISMC system inverse dynamics functions.}
        \label{fig: LowMu/CISMC/CISM Controller - System Inverse Dynamics}
    \vspace{0.5em}
    \centering
        \includegraphics[width=0.9\textwidth]{images/LowMu/CISMC/CISM Controller - Controller Stability.png}
        \caption{CISM controller sliding surface.}
        \label{fig: LowMu/CISMC/CISM Controller - Controller Stability}
    \end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{Mu-jump Maneuver}\label{sec: mu_jump}
The second testing scenario evaluates the transient response, adaptability, and stability of the vehicle when encountering a sudden decrease in road surface adherence. The maneuver is simulated under a constant 70\% gas pedal input, initiated from an initial cruising velocity of $V_{vhcl}\approx10 km/h$.
\\
To evaluate the worst-case transient response, the surface step transition occurs precisely while the vehicle velocity is low ($V_{vhcl}\approx20 km/h$), maximizing the potential for severe wheel spin if the control loop fails to instantly re-stabilize the plant.
\\
The road is characherized by an initial dry asphalt section followed by snowy/wet cobbleston, and again wet asphalt. Inherently, friction coefficient varies in the range $\mu_{road} \in [1, 0.3, 0.5]$, and the Reference Slip Estimator identifies optimal longitudinal slip peaks in the range $\sigma_{opt} \approx 0.10 \div 0.14$

\subsection{Traction Control System OFF}
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/MuJump/TCSOFF/TCSOFF - Speeds.png}
    \caption{Vehicle and driving wheels' speed with traction control OFF.}
    \label{fig: MuJump/TCSOFF/TCSOFF - Speeds}
\end{figure}
When the maneuver is tested with the Traction Control System completely deactivated, on the initial high-adherence surface, the constant 70\% throttle command produces stable vehicle acceleration, with the wheel speeds tracking the vehicle velocity curve.

However, the instant the rear wheels cross the friction bump at $t\approx 1.5$ s, the abrupt reduction in available tire-road friction lowers the admissible longitudinal force, causing the rear tires to exceed the adhesion limit.

Since the electric motor continues to output an unconstrained, high driving torque, the excess energy goes entirely into accelerating the wheel, hence the tangential wheels' speeds spike exponentially, shooting from under 24 km/h (6.6 m/s) to a severe spinning state over 100 km/h (28 m/s) within a few seconds, up to almost 160 km/h (44 m/s).

This divergence can be visualized analysing longitudinal slip behavior in Figure \ref{fig: MuJump/TCSOFF/TCSOFF - Longitudinal Slip}.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/MuJump/TCSOFF/TCSOFF - Longitudinal Slip.png}
    \caption{Actual vs. reference longitudinal slip with traction control OFF.}
    \label{fig: MuJump/TCSOFF/TCSOFF - Longitudinal Slip}
\end{figure}
\newpage
Ignoring initial spikes in slip given by maneuver start, prior to the transition, the longitudinal slip is bounded within the stable region ($\sigma<0.01$). At the friction jump ($t\approx2.6$ s), the slip breaks through the stable limit and rises monotonically, saturating near a full spinning state of $\sigma \approx 0.76$. At this operating point, the tire is completely decoupled from efficient force generation.

The severe consequences of this transition on force transmission is highlighted in Figure \ref{fig: MuJump/TCSOFF/TCSOFF - Friction}, which tracks the  utilized friction coefficient.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/MuJump/TCSOFF/TCSOFF - Friction.png}
    \caption{Utilized vs. road friction coefficient with traction controll OFF.}
    \label{fig: MuJump/TCSOFF/TCSOFF - Friction}
\end{figure}
Upon crossing the $\mu$-jump, the utilized friction drops abruptly: since the slip simultaneously shoots up to the unstable region, the tire operates on the downward tail of the low-friction curve, utilizing only a fraction ($\mu_{ul} \approx 0.23$) of the road physical limit.
\newpage
\subsection{PI Controller}
PI activation history is illustrated in Figure \ref{fig: MuJump/PI/PI Controller - Speeds}.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/PI/PI Controller - Activity.png}
        \caption{PI controller activity.}
        \label{fig: MuJump/PI/PI Controller - Activity} 
    \end{figure}
On the initial dry asphalt section ($\mu=1.0$), the high adherence allows the tire contact patches to transfer the demanded powertrain torque within the stable linear elastic micro-slip zone. Consequently, the longitudinal slip remains below the activation threshold, and the PI status is OFF. \\
The sudden reduction in road adherence causes an immediate slip departure, consequently the control loop registers this threshold breach instantly, switching to full engagement (1).

Figures \ref{fig: MuJump/PI/PI Controller - Speeds} and \ref{fig: MuJump/PI/PI Controller - Longitudinal Slip} display the resulting wheel peripheral speeds, vehicle longitudinal velocity, and longitudinal slip comparison to optimal target. \\
Upon entering the low-friction zone, the Reference Slip Estimator updates the optimal slip reference from $\sigma_{opt}^{\mu = 1} = 0.1410$ to $\sigma_{opt}^{\mu = 0.3} = 0.1038$. \\
Because the linear PI lacks of predictive action, it relies entirely on the feedback error accumulation. As a result, the driving wheel speeds experience a transient surge: the longitudinal slip exhibits a visible overshoot, peaking at $\sigma = 0.12$. \\
The controller takes approximately 26 ms to stabilize the slip onto the target.
\newpage
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/PI/PI Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with PI.}
        \label{fig: MuJump/PI/PI Controller - Speeds}
    \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/PI/PI Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with PI.}
        \label{fig: MuJump/PI/PI Controller - Longitudinal Slip}
    \end{figure}
    \newpage
When entering the intermediate road friction segment ($\mu = 0.5$), the supervisor updates the target slip to $\sigma_{opt}^{\mu = 0.5} = 0.1370$ . Due to the sudden increase in road capacity, an instantaneous drop in wheel speed and a corresponding undershoot in longitudinal slip down to $\sigma=0.1342$ is visible. This happens because PI control loop briefly ramps up the torque to exploit the higher friction limit, achieving then steady-state convergence onto the new reference line.

Figure \ref{fig: MuJump/PI/PI Controller - Tracking Error} details the isolated wheel speed error.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuJump/PI/PI Controller - Tracking Error.png}
        \caption{PI controller tracking error}
        \label{fig: MuJump/PI/PI Controller - Tracking Error}
    \end{figure}
Considering the operating window of the controller ($t \approx 1.5 \div 15$ s), since the first severe $\mu$-jump, the longitudinal slip tracking error is held constant at zero for the whole maneuver. \\ The only exception is made for a slight rise in longitudinal slip error of $e_{\sigma} \approx 0.002$, corresponding to an overshoot in rotational wheel speed error of $e_{\omega} \approx 0.2$ m/s, in coincidence of the rise in road friction coefficient from $\mu = 0.3$ to $\mu = 0.5$.

\newpage
Figure \ref{fig: MuJump/PI/PI Controller - Friction} displays the utilized friction coefficient generated at the driving wheels.

The PI controller provides a massive improvement compared to the unstable, open-loop configuration. \\
During the steady-state windows of both the low-friction zone ($t=1.5 \div 8.5$ s) and intermediate-friction zone ($t = 8.5 \div 15$ s), the utilized friction coefficient overcomes the road nominal friction coefficient, guaranteeing a maximization of tire traction capability.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/MuJump/PI/PI Controller - Friction.png}
    \caption{Utilized vs. road friction coefficient with PI.}
    \label{fig: MuJump/PI/PI Controller - Friction}
\end{figure}
\newpage
\subsubsection{Controller's Output}
Figure \ref{fig: MuJump/PI/PI Controller - Controller Output Torques} reports the internal parts of the calculated torque command for both rear wheels. \\
While accelerating on dry asphalt ($\mu = 1$), for the majority of the time the wheel slip stays below the activation threshold, hence the control loop is inactive, so the command torque is zero.

Once a threshold breach is detected, while approaching the end of the dry asphalt segment, the controller activates, injecting a torque command ($\approx 500$ Nm) to bring the wheel to the desired slipping condition ($\sigma_{opt} = 0.1410$). After few milliseconds there is the jump in surface friction down to $\mu = 0.3$, which lowers the target slip down to $\sigma_{opt} = 0.1038$.  This makes the torque command decrease to finally stabilize to $T_{cmnd}\approx 450$ Nm for the whole low-friction segment.

When the driving wheels cross the boundary from the low-friction surface to the higher adherence intermediate section ($\mu=0.5$) at $t = 8.5$ s, the road can instantly support a higher longitudinal tractive force. The sudden increase in road grip acts as an external mechanical load, quickly pulling the wheel speeds down. This causes the actual longitudinal slip to instantly drop below the reference target (see Fig. \ref{fig: MuJump/PI/PI Controller - Longitudinal Slip}). \\
This kinematic drop creates a spike in tracking error (see Fig. \ref{fig: MuJump/PI/PI Controller - Tracking Error}). Because a standard PI controller is purely reactive, it responds directly to this localized error signal, hence as a result, the target torque jumps from its previous low-friction holding value of approximately 400 Nm and climbs toward a new steady-state value of roughly 750 Nm. Once the wheel speeds accelerate enough to match the new optimal slip target ($\sigma_{opt} = 0.1370$), the tracking error returns to zero, and the torque command stabilizes cleanly.

As the $T_{ctrl}$ coming from the PI never overcomes saturation limits, $T_{cmps} = 0$ for the whole simulation.

The behavior described is obviously matched by the EM torque signals, as displayed in Figure \ref{fig: MuJump/PI/PI Controller - EM Torques}.

As the torque profile requested by the controller is smooth and within actuator's feasibility limits, the electric motors easily manage to match the target imposed.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/PI/PI Controller - Controller Output Torques.png}
        \caption{PI controller's output torques.}
        \label{fig: MuJump/PI/PI Controller - Controller Output Torques}
        \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/PI/PI Controller - EM Torques.png}
        \caption{Electric motors torque with PI controller.}
        \label{fig: MuJump/PI/PI Controller - EM Torques} 
    \end{figure}
    \newpage
\subsection{TSM Controller}
The operational status of TSM control loop is tracked in Figure \ref{fig: MuJump/TSMC/TSM Controller - Activity}.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuJump/TSMC/TSM Controller - Activity.png}
        \caption{TSM controller activity.}
        \label{fig: MuJump/TSMC/TSM Controller - Activity} 
    \end{figure}
Again, for the very first portion of the dry asphalt section, the controller's status sits flat at 0. As soon as the slip ratio  crosses the engagement threshold, the status flag steps cleanly to 1, and remains locked at full engagement for the remainder of the simulation window.

Figure \ref{fig: MuJump/TSMC/TSM Controller - Speeds} and \ref{fig: MuJump/TSMC/TSM Controller - Longitudinal Slip} illustrate the vehicle and wheel velocities alongside the longitudinal slip tracking trajectories.

TSM controller reacts more aggressively than the PI. \\
The wheel velocities exhibit negligible macroscopic overshoot, completely suppressing the severe wheel spinout seen in the open-loop configuration, and matching the vehicle speed profile with a fixed distance between curves, corresponding to the target slip.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/TSMC/TSM Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with TSMC.}
        \label{fig: MuJump/TSMC/TSM Controller - Speeds}
    \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/TSMC/TSM Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with TSMC.}
        \label{fig: MuJump/TSMC/TSM Controller - Longitudinal Slip}
    \end{figure}
During controller's intervention window, the actual slip curves never form a perfectly flat line. Instead, they exhibit a continuous, dense, high-frequency oscillation around the reference targets. This is a direct visual indicator of sliding mode chattering, confirming that the state trajectories are continuously crossing back and forth over the sliding surface as the control law switches between its upper and lower bounds.

The actual longitudinal slip curves for the rear wheels converge onto the updated target within milliseconds, completely suppressing the severe wheel spinout seen in the open-loop configuration. \\
Since the controller is activated fews milliseconds after the start of the simulation, for $t<1$ s the longitudinal slip profile displays severe chattering with high amplitude of oscillations. This is mainly caused by the fact that the vehicle is travelling at low speed, influencing stability of mathematical definitions in the model, and that the controller is trying to bring the wheels at a condition of higher slip to match the reference $\sigma_{opt} = 0.1410$.

As the road surface switches from $\mu = 1$ to $\mu = 0.3$, the reference slip target steps down to 0.1038, and the actual longitudinal slip curves for both the rear-left and rear-right wheels track this change rapidly, with only a small tracking undershoot down to approximately 0.05.

When transitioning to the intermediate-friction section ($\mu = 0.5$), the reference slip target steps up to 0.1370. Unlike a linear controller which would suffer from tracking delays, the non-linear switching law forces the actual slip to jump upward instantly.

The tracking error plot directly reflects the underlying longitudinal slip dynamics, revealing a reduced tracking capability at lower vehicle velocities. Within this initial low-speed regime, the system exhibits a maximum tracking overshoot of 0.20 and undershoot of $-0.05$. As the vehicle velocity increases, the amplitude of these slip oscillations progressively decays, indicating enhanced tracking stability.

By ensuring instant convergence to the optimal slip zones, the TSMC maximizes longitudinal force transmission. The utilized friction coefficient drops directly from its asphalt value to stabilize on a flat plateau above 0.3 on the low-grip surface. The moment the vehicle crosses the boundary at $t = 8.5$ s, steps upward instantly to hit a flat plateau above 0.5. Because the slip tracking is tight, the friction allocation is remarkably improved, though it is superimposed with a low-amplitude, high-frequency noise ripple matching the slip chattering.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/TSMC/TSM Controller - Tracking Error.png}
        \caption{TSM controller's tracking error.}
        \label{fig: MuJump/TSMC/TSM Controller - Tracking Error} 
    \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/TSMC/TSM Controller - Friction.png}
        \caption{Utilized vs. road friction coefficient with TSMC.}
        \label{fig: MuJump/TSMC/TSM Controller - Friction}   
    \end{figure}
\subsubsection{Controller's Output}
The internal torque calculation plot in Figure \ref{fig: MuJump/TSMC/TSM Controller - Controller Output Torques} that the primary Control Torque command switches violently between its upper and lower bounds to maintain the sliding state.

When the vehicle hits the intermediate-friction surface ($\mu=0.5$), the controller shifts its macroscopic operating point to match the higher tractive capability. However, the high-frequency switching remains active.

As illustrated in Figure \ref{fig: MuJump/TSMC/TSMC Controller - EM Torques}, the Target Torque command and the Actual Torque output of the electric motors oscillate rapidly across a huge peak-to-peak switching band between tires and motors feasibility limits.

While this aggressive switching behavior enforces mathematical tracking invariance against road variations, it represents an unacceptable physical penalty in a real vehicle application,  directly justifying the need for a smoothed, chattering-free sliding mode variant.

    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuJump/TSMC/TSM Controller - Controller Output Torques.png}
        \caption{TSM controller output torques.}
        \label{fig: MuJump/TSMC/TSM Controller - Controller Output Torques} 
    \end{figure}
    \begin{figure}[H]
        \includegraphics[width=1\textwidth]{images/MuJump/TSMC/TSMC Controller - EM Torques.png}
        \caption{Electric motors torque with TSM controller.}
        \label{fig: MuJump/TSMC/TSMC Controller - EM Torques} 
    \end{figure}


\subsubsection{Controller's Stability}
To evaluate the stability and switching behavior of the TSMC, Figure \ref{fig: MuJump/TSMC/TSM Controller - Controller Stability} tracks the sliding surface $s$ and the calculated Lyapunov function derivative ($\dot V (s) = \dot s s$) for both rear wheels.

For the portion of controller's activity along the high-friction surface, sliding surface reaches its maximum overshoot (0.05) and undershoot (-0.2) values, and chatters around an equilibrium value of approximately 0.03, an offset from the ideal condition $s=0$ caused by system instability at low speeds. \\
The instant the vehicle hits the low-friction boundary, the driving wheels accelerate rapidly, forcing the sliding surface to spike violently up to a positive peak of approximately 0.02, from this point on chattering around the equilibrium $s = 0$. \\
As travelling speed increases, the amplitude of chattering oscillations decreases, as the system and control loop stabilize.

    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuJump/TSMC/TSM Controller - Controller Stability.png}
        \caption{TSM controller's stability analysis.}
        \label{fig: MuJump/TSMC/TSM Controller - Controller Stability}    
    \end{figure}

Lyapunov function derivative plot confirms that the non-linear control law generates an aggressive corrective energy response that forces the perturbed state trajectories back toward the sliding surface. Because the discontinuous control law (sign(s)) switches signs at every solver step, the derivative $\dot s$ alters its direction continuously. As a result, $\dot V$ exhibits rapid, symmetric high-frequency chattering directly across the zero trajectory, between 0.15 and $-0.15$.

\newpage
\subsection{CISM Controller}
The dynamic activation state history of the CISMC loop is recorded in Figure \ref{fig: MuJump/CISMC/CISM Controller - Activity}.
Identical to the baseline PI configuration, At $t \approx 1.5$ s the sudden loss of road friction forces the slip ratio to cross the activation threshold, shifting the controller's state instantly to 1 (active). The CISMC remains fully engaged for the rest of the simulation window to manage the changing road friction.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - Activity.png}
        \caption{CISM controller activity.}
        \label{fig: MuJump/CISMC/CISM Controller - Activity} 
    \end{figure}

Figure \ref{fig: MuJump/CISMC/CISM Controller - Speeds} and \ref{fig: MuJump/CISMC/CISM Controller - Longitudinal Slip} illustrate the vehicle and wheel velocities alongside the resulting longitudinal slip tracking trajectories.

When entering the low-friction zone ($\mu=0.3$), the reference slip target steps down to 0.1038. The CISMC layout handles this severe transition with good damping, despite an instantaneous overshoot up to 0.18. The driven wheel velocities experience negligible macroscopic overshoot or uncontrolled spinout. The actual longitudinal slip curves for both rear wheels converge exponentially onto the reference line within approximately 0.5 seconds.

When transitioning to the intermediate-friction surface, the controller manages to track the higher slip reference (0.1370) with zero overshoot and null delay, locking cleanly onto the new target condition.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with CISMC.}
        \label{fig: MuJump/CISMC/CISM Controller - Speeds}
    \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with CISMC.}
        \label{fig: MuJump/CISMC/CISM Controller - Longitudinal Slip}
    \end{figure}
Crucially, unlike the fuzzy and wide oscillation bands seen under traditional SMC, both the wheel velocities and longitudinal slip lines remain perfectly thin, smooth, and ripple-free during controller's active phase ($t = 1.5 \div 15$). This confirms the succesful chattering suppression performance of the CISMC.

The tracking performance of the controller is plotted in Figure \ref{fig: MuJump/CISMC/CISM Controller - Tracking Error}. \\
At the initial friction drop ($t \approx 1.5$ s), the longitudinal slip tracking error experiences an isolated, brief positive peak limited to a maximum of 0.06, matching the robust transient performance of the TSMC but without its steady-state noise.\\
Once the transient perturbations are rejected, the slip tracking error converges directly to a flat, noise-free absolute zero line. By allowing the integral term to operate within the boundary layer, the controller completely eliminates the persistent chattering error observed in the traditional SMC layout.

Figure \ref{fig: MuJump/CISMC/CISM Controller - Friction} illustrates the evolution of the utilized friction coefficient at the rear-left and rear-right wheels. \\
Within controller's intervention time window ($t= 1.5 \div 15$ s) the utilized friction coefficient is always higher than road nominal one.

The moment the vehicle crosses onto the $\mu = 0.3$ section, the available road friction drops instantly. Under CISMC control, the utilized friction coefficient drops directly to a transient low of 0.29 for a fraction of a second before recovering sharply, stabilizing onto a perfectly flat, noise-free line at 0.32.

When transitioning to the section at $\mu=0.5$, the road capacity steps upward and $\mu_{ul}$ displays a fast and sharp transient response, shooting up to 0.54, where it stablizes.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - Tracking Error.png}
        \caption{CISM controller's tracking error.}
        \label{fig: MuJump/CISMC/CISM Controller - Tracking Error} 
        \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - Friction.png}
        \caption{Utilized vs. road friction coefficient with CISMC.}
        \label{fig: MuJump/CISMC/CISM Controller - Friction}  
    \end{figure}


\subsubsection{Controller's Output}
The underlying control inputs driving this performance are reported in Figure \ref{fig: MuJump/CISMC/CISM Controller - Controller Outputs}.

The control input ($u_{ref}$) behaves as a smooth, continuous signal, as the conditional integration block manages the actuator bounds safely. \\
At $t\approx8.5$ s, the controller increases its torque command to exploit the intermediate grip surface, stepping up smoothly from a low-friction holding value of approximately 50 Nm to a new steady-state plateau of roughly 100 Nm per wheel.

As illustrated in Figure \ref{fig: MuJump/CISMC/CISM Controller - EM Torques}, the Target Torque command line and the Actual Torque output line overlap perfectly, tracking the demands with no visible lag. Most importantly, the huge high-frequency switching  seen under traditional SMC are completely eliminated. The actual motor torque line remains perfectly smooth and quiet through both the $\mu=0.3$ and $\mu=0.5$ surfaces.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuJump/CISMC/CISM Controller - Controller Outputs.png}
        \caption{CISM controller's output torques.}
        \label{fig: MuJump/CISMC/CISM Controller - Controller Outputs}
    \end{figure}
    \begin{figure}[H]
            \centering
            \includegraphics[width=1\textwidth]{images/MuJump/CISMC/CISM Controller - EM Torques.png}
            \caption{Electric motors torque with CISM controller.}
            \label{fig: MuJump/CISMC/CISM Controller - EM Torques}
    \end{figure}
    \newpage
\subsubsection{Controller's Stability}
As stated previously, the smooth control output achieved by the CISMC relies on the real-time estimation and cancellation of the vehicle’s non-linear dynamics through functions $f$ and $g$.

Function $f$ and $g$ evolves smoothly without high-frequency oscillations, showing step-like transitions exactly synchronized with the surface boundaries.\\
While these model-based terms absorb the deterministic non-linear variations of the plant, the auxiliary switching block is relieved of high-gain duties, ensuring a highly stable control command.

In Figure \ref{fig: MuJump/CISMC/CISM Controller - Controller Stability} tracking metrics for the sliding surface are reported.\\
At the initial high-to-low $\mu$-jump ($t\approx 1.5$s), the sudden acceleration of the driving wheels forces the sliding surface into an istantaneous  positive spike peaking at approximately 0.1.

The most significant improvement over the traditional sliding mode architecture is seen in the fact that, once the transient spike is suppressed, $s$ returns to and stays flat at exactly zero for the whole simulation. \\
The controller completely eliminates the high-frequency chattering seen in traditional SMC, proving that the conditional integral term successfully handles the remaining unmodeled errors asymptotically, guaranteeing a noise-free, highly accurate slip regulation.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - System Inverse Dynamics.png}
        \caption{CISMC system inverse dynamics functions.}
        \label{fig: MuJump/CISMC/CISM Controller - System Inverse Dynamics}
        \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuJump/CISMC/CISM Controller - Controller Stability.png}
        \caption{CISM controller stability.}
        \label{fig: MuJump/CISMC/CISM Controller - Controller Stability}
    \end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mu-split Maneuver}\label{sec: mu_split}
To establish a clear benchmark for evaluating the active control strategies, this section analyzes the vehicle's open-loop response during an asymmetric transition to a $\mu$-split surface.\\
Under a constant 70\% gas pedal demand, the vehicle crosses a road boundary where the left wheels encounter a low-grip profile ($\mu_L = 0.3$) and the right wheels drop onto an ultra-low adherence profile ($\mu_R = 0.1$).
\subsection{Traction Control System OFF}
The vehicle and wheels velocity profiles under open-loop operation are illustrated in Figure \ref{fig: MuSplit/TCSOFF/TCSOFF - Speeds}.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuSplit/TCSOFF/TCSOFF - Speeds.png}
        \caption{Vehicle and driving wheels' speed with traction control OFF.}
        \label{fig: MuSplit/TCSOFF/TCSOFF - Speeds}
    \end{figure}
Prior to the transition, the vehicle accelerates uniformly on dry asphalt. Both the rear-left and rear-right wheel tangential speeds perfectly match the longitudinal velocity of the vehicle body.

The instant the vehicle crosses the friction boundary at low speed ($V_{vhcl}\approx 20$ km/h), the absence of a traction control system causes an immediate, loss of kinematic stability. Because a constant 70\% throttle command continues to inject power into the driving axles, the wheels immediately accelerate beyond the physical limits of the road.

Due to the severe asymmetry of the road, the two driven wheels diverge rapidly. \\
The right wheel, sitting on the most critical $\mu = 0.1$ surface, spins out more violently, peaking at approximately 72 m/s by $t\approx8.5$ s.
The left wheel, experiencing a slightly higher grip level ($\mu=0.3$), also breaks away completely but climbs at a lower rate, reaching a peak near 44 m/s.

Because the wheel speeds diverge so massively from the actual vehicle velocity, the vehicle completely loses stability and simulation is ended prematurely by the software, not managing to complete the maneuver.

Figure \ref{fig: MuSplit/TCSOFF/TCSOFF - Longitudinal Slip} confirms this unstable behavior by mapping the longitudinal slip ratios of both wheels.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuSplit/TCSOFF/TCSOFF - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with traction control OFF.}
        \label{fig: MuSplit/TCSOFF/TCSOFF - Longitudinal Slip}
    \end{figure}
Both slip curves experience an immediate exponential growth at $t = 4.5$ s. \\
Left wheel slip shoots straight to the maximum saturation boundary of 1.0 within less than 4 s from the transition point. Right wheel follows a slightly delayed but equally unstable trajectory, reaching full spinning condition at the same time.

The physical consequences of this uncontrolled spinout on force generation are recorded in Figure \ref{fig: MuSplit/TCSOFF/TCSOFF - Friction}, which tracks the utilized road friction coefficient.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\textwidth]{images/MuSplit/TCSOFF/TCSOFF - Friction.png}
    \caption{Utilized vs. road friction coefficient with traction controll OFF.}
    \label{fig: MuSplit/TCSOFF/TCSOFF - Friction}
\end{figure}
Because the tire operates within the unstable region of the $\mu - \sigma$ curve, the available tractive force continuously degrades. Furthermore, the massive difference in force production between the right and the left track introduces a severe, uncontrolled yaw moment across the chassis. In a physical vehicle, this highly asymmetric force distribution would instantly spin the car out of its lane.
\newpage
\subsection{PI Controller}
This section evaluates the tracking capabilities, transient errors, and structural limitations of the PI architecture during the highly asymmetric $\mu$-split event.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.8\textwidth]{images/MuSplit/PI/PI Controller - Activity.png}
        \caption{PI controller activity.}
        \label{fig: MuSplit/PI/PI Controller - Activity} 
    \end{figure}
On dry asphalt ($\mu$=1.0), both wheels operate below the activation slip threshold, keeping controller's status at 0. The moment the vehicle crosses the friction boundary, the sudden slip departure causes controller's activation at $t\approx 1$.

Figure \ref{fig: MuSplit/PI/PI Controller - Speeds} and \ref{fig: MuSplit/PI/PI Controller - Longitudinal Slip} capture the resulting vehcile and wheels tangential velocities and longitudinal slip profiles. 

For the rear-left wheel ($\mu$ = 0.3), at the friction bump the reference slip target steps down to 0.0935. The actual slip shoots upward dramatically, peaking at an undershoot of 0.07 and overshoot of 0.1290 at $t \approx 1$ s before the integral gain accumulates enough error to decelerate the wheel.

For the rear-right wheel the reference slip target steps down to 0.0568. It undergoes a smaller, yet distinct overshoot up to 0.09. \\


    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/PI/PI Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with PI.}
        \label{fig: MuSplit/PI/PI Controller - Speeds}
    \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/PI/PI Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with PI.}
        \label{fig: MuSplit/PI/PI Controller - Longitudinal Slip}
    \end{figure}

Both slip metrics eventually converge to their reference target by $t \approx 1.4$ s. However, because the slip targets fo the surfaces are unequal, a continuous wheel speed difference develops. By $t = 6$ s, the rear-right wheel travels at roughly 13.7 m/s while the rear-left wheel at 13.2 m/s to maintain that optimal slip ratio on the slicker road patch.

Controller's tracking deviations are reported in Figure \ref{fig: MuSplit/PI/PI Controller - Tracking Error}.\\
Mirroring the slip plots, the longitudinal slip tracking error spikes to +0.017 and $-0.035$ on the left wheel and $-0.033$ on the right wheel. The corresponding physical wheel speed error peaks at 0.375 and $-0.9$ rad/s for rear-left and $-1$ rad/s for rear-right.

Tracking error stabilizes after 0.8 seconds, lying flat for the rest of the simulation.

Figure \ref{fig: MuSplit/PI/PI Controller - Friction} illustrates the resulting utilized friction coefficient generated at each tire contact patch.

Once the PI loop stabilizes the slip onto the optimal target ($t>1.5 $s), the utilized friction curves settle onto two distinct, flat plateaus: rear-left wheel stabilizes perfectly at $\mu=0.30$, fully exploiting the maximum tractive capacity of the snow track, while for the rear-right wheel the controller manages to maximize friction utilization up to $\mu \approx 1$.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/PI/PI Controller - Tracking Error.png}
        \caption{PI controller tracking error.}
        \label{fig: MuSplit/PI/PI Controller - Tracking Error}   
    \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/PI/PI Controller - Friction.png}
        \caption{Utilized vs. road friction coefficient with PI.}
        \label{fig: MuSplit/PI/PI Controller - Friction}
    \end{figure}

\subsubsection{Controller's Output}
The internal torque composition plot in Figure \ref{fig: MuSplit/PI/PI Controller - Controller Output Torques} clearly shows the decoupled operation of the anti-windup loops.

At the instant of activation, the primary Control Torque for both wheels spikes instantly proportionally to the slip error. $T_{cmnd}$ saturates, reaching feasibility limits for both wheels.

Limit overcoming is more aggressive in the case of the left wheel, for which Compensation Torque ($T_{cmps}$) doesn't manage to bring the final Control Torque ($T_{ctrl}$) within the predefined boundaries.\\
The rear-right channel experiences a less severe saturation window, demanding a smaller contribution from its matching anti-windup compensation path.\\
After this brief transition phase, output torque stabilizes to 435 Nm for the left and 143 Nm for the right wheels.

Figure \ref{fig: MuSplit/PI/PI Controller - EM Torques} tracks the resulting target and actual electromagnetic torques.\\
Both electric motors track their respective controller commands with negligible lag.

On the ultra-low-grip right track ($\mu=0.1$), the motor stabilizes to provide a holding torque of approximately only 20 Nm.\\
On the low-grip left track ($\mu=0.3$), the torque must be limited less severely, holding a steady-state value of 50 Nm.

\begin{figure}[H]
    \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/PI/PI Controller - Controller Output Torques.png}
        \caption{PI controller's output torques.}
        \label{fig: MuSplit/PI/PI Controller - Controller Output Torques}
    \vspace{0.5em}
    \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/PI/PI Controller - EM Torques.png}
        \caption{Electric motors torques with PI controller.}
        \label{fig: MuSplit/PI/PI Controller - EM Torques}
\end{figure}
\newpage
\subsection{TSM Controller}
This section investigates the closed-loop performance of the TSMC architecture under the asymmetric $\mu$-split condition.

The activation history of the control loop is shown in Figure \ref{fig: MuSplit/TSMC/TSM Controller - Activity}. \\
On the majority of the dry asphalt section ($t<1$ s), both wheels operate within the stable linear elastic micro-slip regime, holding the activation flags flat at 0 (inactive). At $t \approx 1$ controller switches to full engagement (1) simultaneously on both wheels.
    \begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuSplit/TSMC/TSM Controller - Activity.png}
        \caption{TSM controller activity.}
        \label{fig: MuSplit/TSMC/TSM Controller - Activity} 
    \end{figure}
Figure \ref{fig: MuSplit/TSMC/TSM Controller - Speeds} and \ref{fig: MuSplit/TSMC/TSM Controller - Longitudinal Slip} detail the vehicle speed, wheels tangential speeds and longitudinal slip behaviors.

The moment the vehicle encounters the friction bump ($t\approx1$ s), the TSMC demonstrates its robust sliding invariance. Unlike the PI controller, there is no large macroscopic wheel speed runaway or delayed recovery, and both driving wheels are aggressively constrained by the switching logic.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/TSMC/TSM Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with TSMC.}
        \label{fig: MuSplit/TSMC/TSM Controller - Speeds}
            \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/TSMC/TSM Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with TSMC.}
        \label{fig: MuSplit/TSMC/TSM Controller - Longitudinal Slip}
    \end{figure}

The actual longitudinal slip trajectories for the rear-left and rear-right wheels converge onto their target line within milliseconds, experiencing a very minor, transient overshoot that is quickly contained ($\approx 0.2$).

While macroscopic tracking is tight, both wheel speed lines and slip lines exhibit a distinct high-frequency chattering across the sliding surface $s=0$.

Controller's tracking performance is reported in Figure \ref{fig: MuSplit/TSMC/TSM Controller - Tracking Error}.

At the transition point ($t\approx 1$ s), the slip tracking error begins its reaching phase, exhibiting an isolated positive impulse bounded to a maximum of +0.07 on both tracks, initiating the sliding mode phase after only 0.8 seconds.\\ 
Outside of this initial transition peak, the errors do not settle to a smooth zero line. Instead, the slip tracking error chatters continuously with a persistent average amplitude of roughly ±0.02.

Figure \ref{fig: MuSplit/TSMC/TSM Controller - Friction} reports the resulting utilized friction coefficient on the two asymmetric tracks.

TSMC manages to achieve a maximized road capacity utilization.\\
The rear-left tire drops onto a perfectly flat plateau slightly above 0.30, fully exploiting the maximum tractive capability of the track.\\
The rear-right tire stabilizes onto a higher plateau at approximately 0.8, maximizing adherence in the lowest-friction segment.

Both plateaus are superimposed with a low-amplitude, high-frequency noise ripple matching the underlying slip chattering frequency.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{images/MuSplit/TSMC/TSM Controller - Tracking Error.png}
    \caption{TSM controller tracking error.}
    \label{fig: MuSplit/TSMC/TSM Controller - Tracking Error}   
        \vspace{0.5em}
    \centering
    \includegraphics[width=0.9\textwidth]{images/MuSplit/TSMC/TSM Controller - Friction.png}
    \caption{Utilized vs. road friction coefficient with TSMC.}
    \label{fig: MuSplit/TSMC/TSM Controller - Friction}
\end{figure}
\subsubsection{Controller's Output}
As shown in the torque decomposition plot in Figure \ref{fig: MuSplit/TSMC/TSM Controller - Controller Output Torques}, the primary Control Torque commands switch violently between their upper and lower limits.

When hitting the friction bump, the controller successfully decouples the macroscopic torque operating points to match the unequal road limits.\\
The average torque request is forced into a wide chattering band that oscillates heavily between 6100 and -1900 Nm, with higher frequency of oscillations for the left wheel.

As testified by Figure \ref{fig: MuSplit/TSMC/TSMC Controller - EM Torques}, electric motors are forced to execute these massive, high-frequency oscillations.

While the TSMC effectively manages the asymmetric traction drop and eliminates the severe transient overshoots, the resulting actuator chattering represents an unacceptable physical penalty. \\High-frequency torque switching of this magnitude across both axles would generate heavy drivetrain vibrations and severe mechanical stress on the half-shafts and reduction gears, directly highlighting the need for a boundary-layer smoothing technique.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/TSMC/TSM Controller - Controller Output Torques.png}
        \caption{TSM controller output torques.}
        \label{fig: MuSplit/TSMC/TSM Controller - Controller Output Torques} 
        \vspace{0.5em}
        \includegraphics[width=0.9\textwidth]{images/MuSplit/TSMC/TSMC Controller - EM Torques.png}
        \caption{Electric motors torque with TSM controller.}
        \label{fig: MuSplit/TSMC/TSMC Controller - EM Torques} 
    \end{figure}
    \newpage
\subsubsection{Controller's Stability}
Sliding surface trajectories are collected in Figure \ref{fig: MuSplit/TSMC/TSM Controller - Controller Stability}.

The sliding variable $s$ oscillates symmetrically directly around the $s=0$ axis for the entire duration of the active phase, with amplitude $\pm 0.022$. \\Concurrently, the Lyapunov derivative chatters rapidly across the zero line, alternating between $\pm 0.4$. This confirms that while the system remains globally stable and bounded, the discontinuous $sign(s)$ term prevents asymptotic convergence.
\begin{figure}[H]
        \centering
        \includegraphics[width=1\textwidth]{images/MuSplit/TSMC/TSM Controller - Controller Stability.png}
        \caption{TSM controller stability.}
        \label{fig: MuSplit/TSMC/TSM Controller - Controller Stability}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\subsection{CISM Controller}
This section analyzes the performance of the proposed CISMC strategy under the asymmetric $\mu$-split maneuver.

The activation timing of each wheel controller is tracked in Figure \ref{fig: MuSplit/CISMC/CISM Controller - Activity}.

On the dry asphalt section, both wheels operate within the stable linear  regime, keeping the activation flags flat at 0. At $t\approx1.5$ s, the sudden loss of road friction forces the slip ratio to cross the activation threshold, shifting the flags instantly to 1. Both control channels remain fully engaged for the remainder of the simulation window.
    \begin{figure}[H]
        \centering
        \includegraphics[width=0.8\textwidth]{images/MuSplit/CISMC/CISM Controller - Activity.png}
        \caption{CISM controller activity.}
        \label{fig: MuSplit/CISMC/CISM Controller - Activity} 
    \end{figure}
Figure \ref{fig: MuSplit/CISMC/CISM Controller - Activity} and \ref{fig: MuSplit/CISMC/CISM Controller - Speeds} illustrate the vehicle and wheels tangential velocities and longitudinal slip tracking trajectories.

The CISMC layout handles the split friction with remarkable damping capability. The driven wheel velocities experience almost zero macroscopic overshoot or uncontrolled spinout, safely matching the vehicle velocity trajectory.

As shown in Figure \ref{fig: MuSplit/CISMC/CISM Controller - Longitudinal Slip}, the reference slip targets for both wheels step down to 0.1038 (left) and 0.0604 (right).\\

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - Speeds.png}
        \caption{Vehicle and driving wheels' speed with CISMC.}
        \label{fig: MuSplit/CISMC/CISM Controller - Speeds}
            \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - Longitudinal Slip.png}
        \caption{Actual vs. reference longitudinal slip with CISMC.}
        \label{fig: MuSplit/CISMC/CISM Controller - Longitudinal Slip}
    \end{figure}

Driven by the non-linear sliding law, the actual longitudinal slip trajectories, after initial overshoots up to 0.1958 (for the rear-left) and 0.1878 (for the rear-right), converge exponentially onto the reference lines within a 0.6 seconds.

Crucially, unlike the fuzzy and wide oscillation bands seen under traditional SMC, both the wheel velocities and longitudinal slip lines remain perfectly thin, smooth, and ripple-free during all active phases.

Controller tracking performance is reported in Figure \ref{fig: MuSplit/CISMC/CISM Controller - Tracking Error}.

At the initial friction drop ($t\approx 1.5$ s), the longitudinal slip tracking error experiences an isolated, brief overshoot to 0.0781 on both wheels, matching the robust transient performance of the TSMC but without its steady-state chattering and noise.

The key benefit of the conditional integration block is seen after the initial overshoot. Once the transient perturbations are rejected, the slip tracking error converges directly to a flat, noise-free absolute zero line. \\
By allowing the integral term to operate within the boundary layer, the controller completely eliminates the persistent chattering error observed in the traditional SMC layout.

By keeping the slip tracking tightly locked onto the optimal reference , the CISMC ensures a perfectly smooth friction profile, perfectly matching the road friction coefficients.

    \begin{figure}[H]
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - Tracking Error.png}
        \caption{CISMC controller tracking error.}
        \label{fig: MuSplit/CISMC/CISM Controller - Tracking Error}
            \vspace{0.5em}
        \centering
        \includegraphics[width=0.85\textwidth]{images/MuSplit/CISMC/CISM Controller - Friction.png}
        \caption{Utilized vs. road friction coefficient with CISMC.}
        \label{fig: MuSplit/CISMC/CISM Controller - Friction}
    \end{figure}

\subsubsection{Controller's Output}
The underlying control inputs driving this controller's performance are reported in Figure \ref{fig: MuSplit/CISMC/CISM Controller - Controller Outputs} and \ref{fig: MuSplit/CISMC/CISM Controller - EM Torques}.

The primary Control Torque commands behave as smooth, continuous signals. The conditional integration block manages the actuator bounds safely; during the initial transition window, the torque request spikes to its upper limit, where the anti-windup logic freezes the integrator to prevent numerical saturation growth.\\
Afterwards, torque request stabilizes at 140 Nm for right wheel and 426 Nm for left wheel, coherently to the higher traction capability at $\mu = 0.3$.

The resulting actual electromagnetic torques track the respective controller's target with negligible lag.\\
On the low-grip left track, electric motor torque stabilizes at 50 Nm, while on the ultra-low-grip right track the torque is limited severely, holding a perfectly smooth steady-state value of only 16.5 Nm.

Most importantly, the huge high-frequency switching bands seen under traditional SMC are completely eliminated, as the actual motor torque lines remain perfectly smooth and quiet through the entire maneuver.
    \begin{figure}[H]
    \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - Controller Outputs.png}
        \caption{CISM controller output torques.}
        \label{fig: MuSplit/CISMC/CISM Controller - Controller Outputs}
        \vspace{0.5em}
        \centering
        \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - EM Torques.png}
        \caption{Electric motors torque with CISM controller.}
        \label{fig: MuSplit/CISMC/CISM Controller - EM Torques}
    \end{figure}
\subsubsection{Controller's Stability}
Figure \ref{fig: MuSplit/CISMC/CISM Controller - System Inverse Dynamics} tracks the behavior of the internal system dynamics approximation functions $f$ and $g$.

Both functions mimics the dynamics of the vehicle's maneuver. In correspondence of the friction bump, they adapt smoothly to capture the changing wheel slip acceleration trends on the snow and ice sections. \\ Because these model-based feedforward terms handle the non-linear plant dynamics, the auxiliary sliding law only needs to reject minor residual errors, keeping the total actuator demand stable.

The numerical analysis of controller's stability is visualized in Figure \ref{fig: MuSplit/CISMC/CISM Controller - Controller Stability}, tracking the sliding surface $s$.

Considering the time window during which the controller intervenes, the sudden slip perturbation in correspondence of the friction bump causes an overshoot up to 0.17. This spike is suppressed instantaneously, allowing the tracking states to enter the continuous boundary layer ($|s|\ll \varphi$) so that the discontinuous switching law is smoothly interpolated inside this layer, allowing the controller to completely eliminate the high frequency chattering seen in traditional SMC.
\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - System Inverse Dynamics.png}
    \caption{CISMC system inverse dynamics functions.}
    \label{fig: MuSplit/CISMC/CISM Controller - System Inverse Dynamics}
        \vspace{0.5em}
    \centering
    \includegraphics[width=0.9\textwidth]{images/MuSplit/CISMC/CISM Controller - Controller Stability.png}
    \caption{CISM controller stability.}
    \label{fig: MuSplit/CISMC/CISM Controller - Controller Stability}
\end{figure}
\section{Performance Comparison}
This section provides a comprehensive comparative analysis of the three control architectures evaluated throughout this thesis.\\
While the previous section examined each controller independently under specific transient conditions, this comparative discussion evaluates their global performance across the three selected maneuvers.

The objective is to evaluate the fundamental trade-offs inherent to each control methodology. Specifically, the analysis focuses on the balance between transient robustness under non-linear system variations, steady-state regulation accuracy, and the physical feasibility of the control inputs, together with the overall vehicle's performance improvement. By assessing these characteristics simultaneously, this section demonstrates how the architectural enhancements introduced in the CISMC successfully reconcile the limitations of both classical linear control and conventional non-linear sliding mode techniques.

Table \ref{tab: controllers_parameters} resumes and collects the parameters selected during controller's design, for each architecture implemented.

\begin{table}[htbp]
    \centering
    \caption{Parameters and gains of the implemented controllers (PI, TSMC, and CISMC).}
    \label{tab: controllers_parameters}
    \begin{tabular}{l c c r}
        \toprule
        \textbf{} & \textbf{Symbol} & {\textbf{Value}}\\
        \midrule
        \textbf{PI Controller} & &\\
        Proportional Gain & $k_p$ & 2000\\ % Sostituisci i valori con i tuoi
        Integral Gain     & $k_i$ & 300\\
        \midrule
        \textbf{TSMC Controller} & &\\
        Sliding Gain             & $K$       & 4000\\
        Braking Sliding Gain     & $K_{\text{brake}}$ & 2000\\
        \midrule
        \textbf{CISMC Controller} & & & \\
        Sliding Gain traction           & $K_1$       & 8 \\
        Sliding Gain braking           & $K_{1_{brake}}$     & 350\\
        Reaching Gain traction         & $K_2$    & 15\\
        Reaching Gain braking          & $K_{2_{brake}}$  & 50\\
        Boundary Layer traction         & $\varphi$    & 0.02\\
        Boundary Layer braking         & $\varphi_{brake}$  & 0.06\\
        \bottomrule
    \end{tabular}
\end{table}
\newpage
\subsection{Low Friction Maneuver}
When the full acceleration step is commanded, the open-loop system (TCS OFF) encounters an immediate, catastrophic wheel spinout, with the longitudinal slip rapidly saturating at $\sigma = 1$. This response confirms that the available engine torque vastly exceeds the physical tractive capacity of the road.

When active control is engaged, all three architectures (PI, TSMC, and CISMC) demonstrate high effectiveness in suppressing the slipping runaway state. As evidenced by the longitudinal slip tracking trajectories, each controller successfully intervenes at the exact onset of the acceleration step without any visible operational delay. Both the linear PI loop and the non-linear sliding mode variants force the longitudinal slip curves to intercept and lock onto the optimal target slip ($\sigma_{opt} = 0.1038$).

The traditional sliding mode controller (Fig. \ref{fig: LowMu/TSMC/TSM Controller - Longitudinal Slip}) fails to achieve a smooth, asymptotic convergence. Due to the discontinuous switching nature of the sign function, the actual longitudinal slip profile is trapped in a high-frequency chattering cycle.

\begin{figure}[htbp]
    \centering
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/PI_LowMu_Slip_traction.png} % Sostituisci con la tua immagine
        \caption{PI controller.}
        \label{fig: Comparative/PI_LowMu_Slip_traction}
    \end{subfigure}
    \hfill % Spazio orizzontale elastico per separare le due immagini
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/CISMC_LowMu_Slip_traction.png} % Sostituisci con la tua immagine
        \caption{CISMC controller.}
        \label{fig: Comparative/CISMC_LowMu_Slip_traction}
    \end{subfigure}
    \caption{Zoomed-in longitudinal slip comparison, traction.}
    %\label{fig:confronto_controllori}
\end{figure}

The PI controller (Fig. \ref{fig: Comparative/PI_LowMu_Slip_traction}) provides a cleaner steady-state longitudinal slip signal than the TSMC. Because it is a linear loop relying on continuous error accumulation, it does not induce high-frequency chattering. The slip profile appears as a thin line; however, it exhibits minor low-frequency waves and tracking deviations due to its pure reliance on reactive gains without non-linear plant compensation.

The CISMC (Fig. \ref{fig: Comparative/CISMC_LowMu_Slip_traction}) demonstrates its superiority by combining the advantages of both layouts. The actual longitudinal slip trajectory tracks the reference target as an exceptionally thin, smooth, and ripple-free line. By implementing a  boundary layer dual logic, the high-frequency chattering seen in TSMC is completely suppressed. Simultaneously, the conditional integration block actively suppresses residual unmodeled plant mismatches, forcing the tracking error to a noise-free absolute zero line that outperforms the PI's low-frequency tracking variance.

\begin{figure}[htbp]
    \centering
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/PI_LowMu_Slip_braking.png} % Sostituisci con la tua immagine
        \caption{PI controller.}
        \label{fig: Comparative/PI_LowMu_Slip_braking}
    \end{subfigure}
    \hfill % Spazio orizzontale elastico per separare le due immagini
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/CISMC_LowMu_Slip_braking.png} % Sostituisci con la tua immagine
        \caption{CISMC controller.}
        \label{fig: Comparative/CISMC_LowMu_Slip_braking}
    \end{subfigure}
    \caption{Zoomed-in longitudinal slip comparison, braking.}
    %\label{fig:confronto_controllori}
\end{figure}

In the PI layout (Fig. \ref{fig: Comparative/PI_LowMu_Slip_braking}), the transition down to the braking reference line introduces a visible settling delay. The linear controller requires a finite time to process the sudden error sign inversion and discharge its integrator history, resulting in a slightly wider transient trajectory before locking back onto the negative reference.

In the CISMC layout (Fig. \ref{fig: Comparative/CISMC_LowMu_Slip_braking}), the step-down is executed with extreme precision. The moment the target drops, the sliding surface variable $s$ exits the boundary layer, immediately freezing the integrator and switching the control law back to its high-gain sliding mode reaching state. This non-linear drive pulls the slip down exponentially. Once inside the negative boundary layer, the conditional integrator re-engages smoothly, ensuring that the actual slip locks onto the braking target with zero undershoot or secondary settling waves, maintaining optimal deceleration tracking until the simulation is ended.

The parameters, summarized in Table \ref{tab: LowMu_results}, isolate the specific operational trade-offs across the simulation.

The macroscopic kinematic metrics, specifically \textbf{Top Speed} ($V_{vhcl}^{max}$) and distance required to bring the vehicle speed to 0.8 m/s normalized by the top speed (in the table referenced to as \textbf{Normalized Braking Distance}) highlight how traction management directly translates into vehicle performance.

The impact of integration management is explicitly evident during the deceleration phase initiated at $t\approx14.5$ s. The CISMC minimizes the normalized braking distance yielding a direct safety improvement over the classical PI architecture, with a reduction of 0.26\%.

The quality of the regulation loop is measured via the \textbf{Root Mean Square Error} (RMSE) of the longitudinal slip and the \textbf{Settling Time} evaluated under a strict ±1\% tolerance band.

Even though the CISMC achieves steady-state convergence slower than PI, the linear PI framework requires almost 5s to fully decay its initial setting oscillations and clamp perfectly onto the reference target. \\
TSMC settling time is not specified, as the oscillations never fully lay within the ±1\% tolerance band.

The CISMC minimizes the tracking error, yielding a low RMS value(0.0054) normalized with respect to $\sigma_{peak}$ comparable to the PI (0.0011), and extremely lower than the TSMC (0.0294).

Moreover, \textbf{Torque Standard Deviation} is extremely high for the TSMC, which is a behavior representative of the chattering phenomenon characterizing this control law, while it's kept the lowest for the CISMC.

To qualitatively evaluate how friction generation is managed at the tire-road interface, it's possible to analyze the maximum and average friction deviation indexes (\textbf{Max $\Delta \mu$} and \textbf{Average $\Delta \mu$}). \\
While CISMC manages to achieve results comparable to the PI strategy, the average is strongly influenced by the initial overshoot at the regulation start, which decreases the overall percentage value, bringing the wheel to a higher slip condition, hence loosing adherence at the tire-road interface.


    \begin{table}[H]
        \centering
        {\large\bfseries Low-$\mu$ Maneuver Comparative Results} \\[0.6em]
        \begin{tabular}{l c c c}
            \toprule
            \textbf{Performance Indicator} & \textbf{PI} & \textbf{TSMC} & \textbf{CISMC} \\
            \midrule
            Normalized Braking Distance [m] & 0.7814 & 0.7838 & 0.7794\\
            Top Speed [m/s] & 20.5964 & 21.1458 & 18.5244 \\
            Avg. Longitudinal Acceleration [m/s$^{2}$] & 1.5584 & 1.5720 & 1.5720 \\
            Settling Time [s] & 0.015 & - & 0.2940 \\
            \midrule
            \textit{Wheel Slip \& Friction} & & & \\
            Slip RMS [-] & 0.0011 & 0.0294 & 0.0054 \\
            $\sigma_{peak} [-] $ & 0.1074 & 0.1953 & 0.1707\\
            Max $\Delta\mu$ - Rear Left [\%] & 20.4199 & 17.4330 & 17.5712 \\
            Avg $\Delta\mu$ - Rear Left [\%] & 6.7141 & 8.8810 & 5.7574 \\
            Max $\Delta\mu$ - Rear Right [\%] & 20.4199 & 17.4330 & 17.5712 \\
            Avg $\Delta\mu$ - Rear Right [\%] & 6.7141 & 8.9103 & 5.7574 \\
            \midrule
            \textit{Torque \& Actuation Metrics} & & & \\
            Torque Std. Deviation - Rear Left [Nm] & 4.8792 & 224.2332 & 2.6084 \\
            Torque Std. Deviation - Rear Right [Nm] & 4.8793 & 224.2257 & 2.6084 \\
            Chattering Frequency [Hz] & -- & 409.8814 & -- \\
            Control Effort [$(Nm)^2$] & $3.06\times10^9$ & $1.54\times10^{11}$ & $2.98\times10^9$ \\
            \bottomrule
        \end{tabular}
        \caption{Performance comparison between PI, TSMC, and CISMC controllers during Low-$\mu$ simulation.}
        \label{tab: LowMu_results}
    \end{table}

Finally, with comparable or better results, CISMC guarantees a reduction in \textbf{Control Effort} (energy of the signal computed as $||u||_{L_2}^2$) of $-98\%$ with respect to TSMC and $-2.6\%$ with respect to PI.

\newpage
\subsection{Mu-jump Maneuver}
When the sequential multi-step friction transitions are encountered under a continuous acceleration demand, the open-loop system immediately suffers from severe instability at the first friction dropout boundary. The sudden mismatch between the requested drivetrain torque and the collapsed tractive capability causes the longitudinal slip to uncontrollably diverge to $\sigma = 0.8$.

When active control is engaged, all three architectures (PI, TSMC, and CISMC) demonstrate high adaptability in rejecting these abrupt surface disturbances.

Both the linear PI loop and the CISMC force the longitudinal slip curves to intercept and track the step-changing optimal targets corresponding to each road segment, while the TSMC fails to achieve a smooth convergence.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/PI_MuJump_slip.png} % Sostituisci con la tua immagine
        \caption{PI controller.}
        \label{fig: Comparative/PI_MuJump_slip}
    \end{subfigure}
    \hfill % Spazio orizzontale elastico per separare le due immagini
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/CISMC_MuJump_slip.png} % Sostituisci con la tua immagine
        \caption{CISMC controller.}
        \label{fig: Comparative/CISMC_MuJump_slip}
    \end{subfigure}
    \caption{Zoomed-in longitudinal slip comparison.}
    %\label{fig:confronto_controllori}
\end{figure}

The performance indexes, systematically organized in Table \ref{tab: MuJump_results}, break down the specific control capabilities during non-uniform road travel.

While TSMC strategy allows to reach highest \textbf{Top Speed} and \textbf{Average Acceleration}, PI and CISMC is comparable, with lowest \textbf{Settling Time} for the PI case - difference probably given by the fact that the CISMC is disadvantaged by the presence of a higher overshoot ($\sigma_{peak}$) at the beginning of the regulation.

However, the linear PI regulation relies entirely on continuous error accumulation without non-linear plant compensation, which makes it reactive and prone to prolonged low-frequency setting waves during sharp friction step variations.

Figure \ref{fig: tracking error comparison} highlights the tracking error curve in correspondence of the friction jump, for the PI and CISMC cases. Here it's visible how CISMC manages to keep the tracking error lower under surface adherence variation, manifesting a higher adaptability to system's variability.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.7\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/PI_MuJump_trackerror.png} % Sostituisci con la tua immagine
        \caption{PI controller.}
        \label{fig: Comparative/PI_MuJump_trackerror}
    \end{subfigure}
    \hfill % Spazio orizzontale elastico per separare le due immagini
    \begin{subfigure}[b]{0.48\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Comparative/CISMC_MuJump_trackerror.png} % Sostituisci con la tua immagine
        \caption{CISMC controller.}
        \label{fig: Comparative/CISMC_MuJump_trackerror}
    \end{subfigure}
    \caption{Zoomed-in tracking error comparison.}
    \label{fig: tracking error comparison}
\end{figure}

Since the primary physical limitation of sliding mode strategies is their compatibility with real electromagnetic actuators, this engineering trade-off can be quantified by the \textbf{Torque Standard Deviation}.

As predictable, TSMC displays the highest value among the three controllers, while CISMC manages to keep it the lowest, providing a solution to the structural limitations of the TSMC, and a performance improvement with respect to PI.

To mathematically evaluate how effectively each control loop exploits the available adherence limit across the transient friction steps, the maximum and average friction deviation indexes (\textbf{Max $\Delta \mu$} and \textbf{Average $\Delta \mu$}) are analyzed, certifying how closely the actual tire operational point matches the peak of the local friction curve.

CISMC manages to achieve higher tire-road adherence and friction utilization with respect to the linear PI controller, manifesting more efficient traction exploitation at the road.

\begin{table}[H]
        \centering
        {\large\bfseries $\mu$-Jump Maneuver Comparative Results} \\[0.6em]
        \begin{tabular}{l c c c}
            \toprule
            \textbf{Performance Indicator} & \textbf{PI} & \textbf{TSMC} & \textbf{CISMC} \\
            \midrule
            Top Speed [m/s] & 48.4583 & 60.9005 & 48.0719 \\
            Avg. Longitudinal Acceleration [m/s$^2$] & 2.3289 & 2.9046 & 2.3064 \\
            Settling Time [s] & 0.011 & - & 0.319 \\
            \hline
            \textit{Wheel Slip \& Friction Metrics} & & & \\
            Slip RMS [-] & 1.9161 $\times 10^{-4}$ & 0.0192 & 0.0040 \\
            $\sigma_{peak}$ [-] & 0.1370 & 0.1860 & 0.1796 \\
            Max $\Delta\mu$ - Rear Left [\%] & 13.5266 & 273.5032 & 14.4233 \\
            Avg $\Delta\mu$ - Rear Left [\%] & 7.1305 & 7.8987 & 6.9777 \\
            Max $\Delta\mu$ - Rear Right [\%] & 13.5266 & 273.5033 & 14.4233 \\
            Avg $\Delta\mu$ - Rear Right [\%] & 6.9104 & 7.9701 & 6.7602 \\
            \hline
            \textit{Torque \& Actuation Metrics} & & & \\
            Torque Std. Deviation - Rear Left [Nm] & 24.7233 & 207.0892 & 24.0148 \\
            Torque Std. Deviation - Rear Right [Nm] & 25.8619 & 208.3752 & 25.0774 \\
            Chattering Frequency [Hz] & -- & 409.9878 & -- \\
            Control Effort [$(Nm)^2$] & $1.14\times10^{10}$ & $3.49\times10^{11}$ & $1.12\times10^{10}$ \\
            \bottomrule
            \end{tabular}
            \caption{Performance comparison between PI, TSMC, and CISMC controllers during $\mu$-Jump simulation.}
        \label{tab: MuJump_results}
\end{table}
Finally, with comparable or better results, CISMC guarantees a reduction in \textbf{Control Effort} (energy of the signal computed as $||u||_{L_2}^2$) of $-97\%$ with respect to TSMC and $-1.75\%$ with respect to PI.

These collected quantitative indicators mathematically validate the architectural superiority of the CISMC, which breaks the structural trade-off between non-linear robustness and actuator longevity under highly transient friction conditions.

\newpage
\subsection{Mu-split Maneuver}
In this scenario, the left and right wheels encounter drastically different friction conditions at the road interface, generating a massive destabilizing yaw moment that tests the differential tracking performance of the powertrain.

When the jumps in friction are encountered by the wheels, the open-loop system immediately suffers from severe instability, uncontrollably reaching full spin condition in few seconds.

As evidenced by the longitudinal slip tracking trajectories (Fig. \ref{fig: MuSplit/PI/PI Controller - Longitudinal Slip}, \ref{fig: MuSplit/TSMC/TSM Controller - Longitudinal Slip}, \ref{fig: MuSplit/CISMC/CISM Controller - Longitudinal Slip}), each controller successfully handles the sudden grip variations, preventing runaway wheel spin during drops in friction and avoiding prolonged slip undershoots when the road surface grip partially recovers.

The performance metrics which allow to provide a comparative analysis of the three controllers tested are reported in Table \ref{tab: MuSplit_results}.

As for the simulations discussed previously, the TSMC allows to reach highest top speed, while CISMC vehicle performance is comparable to PI one, also in terms of average acceleration.



\begin{table}[H]
        \centering
        {\large\bfseries $\mu$-Split Maneuver Comparative Results} \\[0.6em]
        \begin{tabular}{l c c c}
            \toprule
            \textbf{Performance Indicator} & \textbf{PI} & \textbf{TSMC} & \textbf{CISMC} \\
            \midrule
            Top Speed [m/s] & 21.7326 & 22.6205 & 19.6712 \\
            Avg. Longitudinal Acceleration [m/s$^2$] & 0.9670 & 0.9561 & 0.9612 \\
            Settling Time [s] & 0.51 & - &  0.29 \\
            \hline
            \textit{Wheel Slip \& Friction Metrics} & & & \\
            Slip RMS [-] & 0.0018$ & 0.0221 & 0.0040 \\
            $\sigma_{peak}$ [-] & 0.1383 & 0.2004 & 0.1958 \\
            Max $\Delta\mu$ - Rear Left [\%] & 19.1808 & 260.1806 & 15.2440 \\
            Avg $\Delta\mu$ - Rear Left [\%] & 8.5150 & 8.3081 & 5.6937 \\
            Max $\Delta\mu$ - Rear Right [\%] & 946.3363 & 701.5474 & 20.7246 \\
            Avg $\Delta\mu$ - Rear Right [\%] & 902.2839 & 642.4977 & 20.7246 \\
            \hline
            \textit{Torque \& Actuation Metrics} & & & \\
            Torque Std. Deviation - Rear Left [Nm] & 29.1013 & 223.5607 & 2.8698 \\
            Torque Std. Deviation - Rear Right [Nm] & 32.4636 & 229.4097 & 0.5686 \\
            Chattering Frequency [Hz] & -- & 411.6896 & -- \\
            Control Effort [$(Nm)^2$] & $6.51\times10^9$ & $1.86\times10^{11}$ & $2.63\times10^9$ \\
            \bottomrule
            \end{tabular}
            \caption{Performance comparison between PI, TSMC, and CISMC controllers during $\mu$-Split simulation.}
        \label{tab: MuSplit_results}
\end{table}

During the $\mu$-Split maneuver the CISMC succeeds in obtaining the lowest settling time amd \textbf{Slip RMS}among the three control strategies.


Best results in terms of friction utilization are achieved by the PI and TSMC, but the CISMC stays comparable, guaranteeing anyway a substantial improvement for traction at the tire-road interface.

Major strength points of the CISMC strategy are reported for what concerns the \textbf{Torque and Activation Metrics}.\\
CISMC allows for the lowest \textbf{Torque Standard Deviation}, which is one order of magnitude lower than the PI and two orders of magnitude lower of the TSMC.\\
This architectural superiority is confirmed also by the \textbf{Control Effort} metrics. CISMC is realized with the lowest control effort between all three control strategies implemented, with a reduction of $-99\%$ with respect to TSMC and $-60\%$ with respect to PI.
\newpage
\section{Robustness to Parameters Variability}
Since in physical automotive applications control robustness against plant parameter variations is a critical factor for vehicle safety and stability, and real-time parameters such as the effective wheel radius, longitudinal tire forces and vertical loads fluctuate continuously due to centrifugal tire growth, tread wear, and load transfer during acceleration or braking, to test the structural resilience and superiority of the proposed CISM control loops, a sensitivity analysis was conducted.

For the three maneuvers exposed in the previous sections, all three controllers have been analyzed under the bounded variability of the following quantities reported in Table \ref{tab: sensitivity_parameters}.

\begin{table}[H]
    \centering
    {\large\bfseries Parametric Variations} \\[0.6em]
    \begin{tabular}{l c c c}
        \toprule
        \textbf{Physical Parameter} & \textbf{Nominal} & \textbf{Minimum} & \textbf{Maximum} \\
        \midrule
        Effective Wheel Radius [m] & $0.343$ & $0.2944$ & $0.3046$ \\
        Longitudinal Tire Force [N] & Dynamic & $-1374$ & $6593$ \\
        Vertical Load [N] & Dynamic & $3252$ & $4487$ \\
        \bottomrule
   \end{tabular}
    \caption{Range of parametric perturbations applied during simulations.}
    \label{tab: sensitivity_parameters}
\end{table}

As demonstrated by the evaluation conducted across Sections \ref{sec:low_mu}, \ref{sec: mu_jump} and \ref{sec: mu_split}, all controllers demonstrated a satisfactory capability to adapt to the abovementioned parameters variability, without any performance degradation. Hence, to further investigate controllers' robustness, the asymmetric $\mu$-split simulation campaign, recognized as the most dynamically critical operational profile due to its severe cross-track disturbance, was replicated for all three controllers, considering an increase of the total vehicle mass from its nominal value of 1600 kg up to an overloaded condition of 2100 kg (a +31.25\% mass increase). This severe unmodeled plant perturbation directly shifts the vehicle's longitudinal inertia, and consequently the system's dynamics and response.

The resulting longitudinal slip profiles are illustrated in Figure \ref{fig: Mass variation/PI Controller - Longitudinal Slip} and \ref{fig: Mass variation/TSM Controller - Longitudinal Slip} for the PI and TSMC, and in Figure \ref{fig: Mass variation/CISM Controller - Longitudinal Slip} for the proposed CISMC solution.

As evidenced by the longitudinal slip curves in the PI simulation plot, the linear architecture undergoes a severe performance degradation under non-nominal mass conditions. Because the PI loop relies solely on fixed feedback gains without inertia compensation, the altered vehicle momentum completely saturates the error-integration capacity.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.9\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Mass variation/PI Controller - Longitudinal Slip.png} 
        \caption{PI controller.}
        \label{fig: Mass variation/PI Controller - Longitudinal Slip}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.9\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Mass variation/TSM Controller - Longitudinal Slip.png}
        \caption{TSM controller.}
        \label{fig: Mass variation/TSM Controller - Longitudinal Slip}
    \end{subfigure}
    \caption{Longitudinal slip profile comparison with $m = 2100$ kg.}
\end{figure}

The actual slip on the low-adherence patch exhibits high-amplitude, low-frequency tracking waves, drifting aggressively away from the optimal reference line and failing to prevent localized wheel spinning. This unstable behavior demonstrates that the linear framework cannot guarantee vehicle drivability when the plant parameters stray significantly from the nominal calibration point.

Conversely, the traditional TSMC proves its inherent non-linear robustness by forcing the longitudinal slip to remain bounded around the optimal target despite the 500 kg mass mismatch.

    \begin{figure}{\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/Mass variation/CISM Controller - Longitudinal Slip.png}
        \caption{Longitudinal slip profile with CISM controller, with $m = 2100$ kg.}
        \label{fig: Mass variation/CISM Controller - Longitudinal Slip}
    \end{figure}

The proposed CISMC achieves the superior control compromise under severe uncertainty. \\
A shown in the CISMC longitudinal slip plot, despite the system operating with an unmodeled mass of 2100 kg, the actual longitudinal slip curves on both the rear-left and rear-right wheels converge seamlessly and rapidly onto the optimal slip targets. The control logic safely absorbs the parameter mismatch without tracking lag or signal distortion. Most importantly, the tracking line remains perfectly clean and noise-free, completely preserving the zero-chattering behavior of the torque output.

\newpage
\section{Robustness to Compliant Driveshafts and Time Delay Costraints}
To thoroughly investigate the industrial feasibility of the proposed CISMC topology, the co-simulation framework was upgraded to include severe hardware-representative non-idealities.

In real-world automotive traction control, assuming an infinitely rigid connection between the electric motor and the wheel hub introduces a significant modeling gap. To prove that the CISMC is suitable for a vehicle-in-the-loop implementation, the control loop was stress-tested by incorporating the following non-idealities:
\begin{itemize}
\item \textbf{Torsional Drivetrain Compliance:} the rear half-shafts were modeled as compliant spring-damper systems with physical torsional stiffness ($k_{DS}$) and torsional damping ($c_{DS}$).
\item \textbf{Actuation and Communication Latency:} a communication delay of 5 ms was introduced, by means of a low-pass filter, to replicate CAN bus propagation bottlenecks and inverter power electronics settling times.
\end{itemize}
The CISMC was subjected to these combined constraints across all three operational scenarios to evaluate its tracking performance and chattering-rejection capabilities.

\begin{table}[H]
    \centering
    {\large\bfseries Real Powertrain Constraints} \\[0.6em]
    \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l c c c@{\extracolsep{\fill}}}
        \toprule
        \textbf{Parameter} &  & \textbf{Value} & \textbf{Unit} \\
        \midrule
        Drive Shaft Torsional Stiffness & $k_{DS}$ & $804583.66$ & $\text{Nm/rad}$ \\
        Drive Shaft Torsional Damping & $c_{DS}$ & $286.48$ & $\text{Nms/rad}$ \\
        Equivalent Filter Time Constant & $\tau_{lpf}$ & $0.005$ & $\text{s}$ \\
        Filter Cut-off Frequency & $f_{{cut-off}_{lpf}}$ & $200$ & $\text{Hz}$ \\
        \bottomrule
    \end{tabular*}
    \caption{Physical constraints and drivetrain non-idealities implemented for the real-world validation framework.}
    \label{tab: drivetrain_real_constraints}
\end{table}
The proposed CISMC demonstrates an exceptional capacity for tracking persistence, establishing a highly robust operational boundary that remains uncompromised across the distinct road-adherence profiles.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.58\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/CISMC with delay/LowMu/CISM Controller - Longitudinal Slip.png} 
        \caption{Low-$\mu$ maneuver.}
        \label{fig: CISMC with delay/LowMu/CISM Controller - Longitudinal Slip}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.58\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/CISMC with delay/MuJump/CISM Controller - Longitudinal Slip.png} 
        \caption{$\mu$-jump maneuver.}
        \label{fig: CISMC with delay/MuJump/CISM Controller - Longitudinal Slip}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.58\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/CISMC with delay/MuSplit/CISM Controller - Longitudinal Slip.png} 
        \caption{$\mu$-split maneuver.}
        \label{fig: CISMC with delay/MuSplit/CISM Controller - Longitudinal Slip}
    \end{subfigure}
    \caption{CISMC longitudinal slip profile comparison with real-world costraints.}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.58\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/CISMC with delay/LowMu/CISM Controller - Controller Outputs.png} 
        \caption{Low-$\mu$ maneuver.}
        \label{fig: CISMC with delay/LowMu/CISM Controller - Controller Outputs}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.58\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/CISMC with delay/MuJump/CISM Controller - Controller Outputs.png} 
        \caption{$\mu$-jump maneuver.}
        \label{fig: CISMC with delay/MuJump/CISM Controller - Controller Outputs}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.58\textwidth}
        \centering
        \includegraphics[width=\textwidth]{images/CISMC with delay/MuSplit/CISM Controller - Controller Outputs.png} 
        \caption{$\mu$-split maneuver.}
        \label{fig: CISMC with delay/MuSplit/CISM Controller - Controller Outputs}
    \end{subfigure}
    \caption{CISM controller outputs comparison with real-world costraints.}
    \label{fig: CISMC_outputs_compratison_driveshafts}
\end{figure}

The controller successfully decouples the wheels' slipping dynamics from the high-frequency parasitic dynamics injected by transmission compliance and actuation lag. \\
The conditional integration mechanism dynamically manages sudden transient errors without entering anti-windup saturation or generating overshoot cascades.

The actuator control effort and corresponding electric motor responses illustrated in Figure \ref{fig: CISMC_outputs_compratison_driveshafts} confirm the chattering-free nature of the CISMC under realistic hardware latencies, with complete absence of high-frequency switching or destructive torque reversals across all configurations, and settling into a smooth, chatter-free steady-state command.

By maintaining highly precise, chatter-free slip regulation across all transient and steady-state events, the CISMC proves that robust non-linear tracking can coexist with actuator longevity.