# Holistic adaptive energy-efficient MPC architecture for multi-objective control in over-actuated autonomous vehicles

Fadel Tarhini, Reine Talj, Moustapha Doumiati

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Fadel Tarhini, Reine Talj, Moustapha Doumiati. Holistic adaptive energy-efficient MPC architecture for multi-objective control in over-actuated autonomous vehicles. Control Engineering Practice, 2025, 164, pp.106464. 10.1016/j.conengprac.2025.106464. hal-05139487

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# Holistic Adaptive Energy-Efficient MPC Architecture for Multi-Objective Control in Over-Actuated Autonomous Vehicles

Fadel Tarhini, Reine Talj, and Moustapha Doumiati

**Abstract**—This paper presents two novel MPC architectures for autonomous over-actuated in-wheel vehicles, targeting enhanced energy efficiency, stability, and control performance. The first architecture introduces a hierarchical centralized MPC framework that utilizes a minimal-order model to integrate path-tracking, speed control, and stability control objectives. The second architecture extends the hierarchical framework into a holistic MPC design, incorporating direct energy-efficient torque allocation and a tire stability criterion. Energy efficiency is significantly improved by minimizing total power consumption and enforcing operational constraints to maximize motor efficiency. Central to both architectures is a novel multi-criteria adaptive weighting mechanism that dynamically reconciles conflicting objectives by adjusting control priorities based on real-time error magnitudes and driving conditions. This mechanism not only resolves potential conflicts between objectives but also enhances robustness to modeling inaccuracies, uncertainties, disturbances, and variations in road adhesion, while significantly improving control performance. Validation is conducted through joint simulations in Simulink/Matlab and the SCANeR Studio vehicle dynamics simulator. The findings demonstrate that both architectures achieve substantial energy savings while maintaining computational efficiency, with improved stability, comfort, and precision in path-tracking and speed control under challenging conditions, including high-speed, high-curvature, and low-adhesion scenarios.

**Index Terms**—Model Predictive Control (MPC), Holistic Architecture, Path-Tracking, Speed Control, Stability Control, Energy-Economy.

## I. INTRODUCTION

Autonomous Vehicles (AVs) are redefining the paradigm of modern mobility by enabling precise, efficient, and adaptive navigation in complex and dynamic driving environments. The architecture of these systems is built upon four fundamental pillars: perception, decision-making [1], trajectory planning, and control [2]. Within this framework, Model Predictive Control (MPC) has emerged as a pivotal methodology, driving progress in decision-making [3], collision avoidance [4], trajectory planning [5], connected vehicles [6], control implementation [7], and actuators coordination [8].

Following the generation of a desired path and speed profile, advanced control systems are employed to guide the AV along the trajectory while maintaining the specified speed.

These control objectives, including path-tracking and speed regulation, are integrated and coordinated within a unified control architecture, potentially incorporating additional performance criteria such as stability and handling [9]. The control architectures are commonly organized in a cascaded two-level structure, with the upper level tasked with generating control inputs, while the lower level ensures their physical realization through the actuators [10], [11]. Transforming the two-level structure into a single-level framework that directly links the controlled states with the actuator commands results in a holistic architecture [12].

Among the key control objectives for AVs, path-tracking has been extensively studied and continues to present significant challenges. Lateral control, for path-tracking, demands continuous regulation over an extended time horizon and necessitates controlling the future state of the vehicle [13]. Achieving accurate and robust path-following while ensuring vehicle stability is particularly difficult due to the inherent complexities of non-linear vehicle dynamics, including tire slip, variable vehicle speed, and uncertainties in system parameters. To enhance robustness, path-tracking accuracy, and energy efficiency, [13] proposed a super-twisting sliding mode controller (STSMC) for lateral control, explicitly formulating the look-ahead distance. Other works have addressed robustness against uncertainties through various strategies. For example, an adaptive inverse controller was developed in [14] to mitigate steering system backlash, while [15] introduced a gain-scheduling lateral control approach based on Linear Parameter Varying (LPV)/$H_\infty$. In [16], an LPV/$H_\infty$ linear robust controller was compared with a nonlinear STSMC for lateral motion and stability control.

While these approaches offer robustness and effective performance, they often result in unconstrained and non-optimal control inputs, which can compromise energy efficiency and reduce actuator durability compared to optimal lateral controllers. As a result, optimal control strategies that address complex multi-variable problems and incorporate system constraints have been developed to improve path-tracking performance. In the context of MPC-based path-tracking, [17] introduced a tire parameter online adaptation module to enhance robustness, while [18] solved the problem using adaptive dynamic programming. Both controllers employed a fourth-order model focused solely on path tracking. Similarly, [19] proposed a flatness-MPC approach via exact system linearization to mitigate modeling errors while preserving nonlinear dynamics. A modified tube-based MPC was presented in [20] to improve disturbance rejection and robustness against uncertainties.

These studies focused exclusively on the path-tracking

This work is carried out within the framework of the V3EA project "Electric, Energy Efficient, and Autonomous Vehicle" (2021-2025), funded by the Research National Agency (ANR) of the French government.
Fadel Tarhini and Reine Talj are with Université de Technologie de Compiègne, CNRS, Heudiasyc UMR 7253, Compiègne, France (email: fadel.tarhini@hds.utc.fr, reine.talj@hds.utc.fr).

Moustapha Doumiati is with École Supérieure d'Électronique de l'Ouest (ESEO)-IREENA UR 4642, Angers, France (email: moustapha.doumiati@eseo.fr).
objective. In general, each control objective in an MPC framework requires at least one regulated state. Although regulating multiple states for a single objective is sometimes unavoidable, it often leads to redundancy and increased design complexity. Hence, reducing the model order, when feasible, improves efficiency while maintaining performance. Incorporating parameter coupling and nonlinearities can enhance model accuracy but significantly increases complexity and computational burden, particularly as the prediction horizon expands. Many studies therefore increase the model order and introduce coupling and nonlinearities to address single or dual objectives [21]. In contrast, this work develops a minimal-order MPC controller that addresses multiple objectives while preserving robustness against inevitable modeling errors.

In addition to path-tracking, several studies have addressed the integration of speed control within a unified MPC framework. For example, [22] proposed a LPV path-following model, treating speed variation as a parameter perturbation. Similarly, [23] developed a coupled longitudinal and lateral MPC strategy by introducing an offset-free criterion to minimize the steady-state lateral error. However, these approaches did not incorporate stability control. Although effective, they rely on predictive models with relatively high orders, which increase complexity and substantially amplify the computational burden. Further control objectives have been addressed using MPC, focusing on chassis-related aspects such as stability, handling, and suspension control [24], [25]. Stability enhancement has been achieved through nonlinear and centralized MPC approaches targeting yaw and roll dynamics. Although effective, these methods typically do not integrate path-tracking alongside stability, and often concentrate on specific dynamic safety requirements without a comprehensive control framework.

To alleviate the computational burden of MPC controllers, many studies adopt Linear Time-Varying (LTV) or LPV models, avoiding the inclusion of nonlinear terms. However, linear models eventually suffer from high modeling errors, particularly at high speeds or during cornering. As a result, integrating stability control objectives alongside path-tracking has become a common practice. This integration has been achieved through decentralized controllers based on STSMC [9], centralized LPV/H∞ controllers [16], and MPC-based strategies. For instance, [26] presented a coordinated path-tracking and yaw stabilization MPC using a fourth-order model. In [27], a robust dual-objective MPC combining path-tracking and Direct Yaw moment Control (DYC) was implemented on over-actuated in-wheel vehicles. Similarly, [28] proposed an MPC-based framework addressing path-tracking and yaw control while considering multiple dynamic constraints. Despite employing fourth-order models and addressing path-tracking and stability, these studies do not incorporate speed control within a unified control framework.

The separation of states-to-control inputs and control inputs-to-actuator commands in a dual-level architecture provides a pragmatic and modular framework, allowing each layer to be optimized independently. This approach is particularly advantageous for over-actuated vehicles, where the lower-level controller often focuses on optimally distributing wheel torques [29]. The in-wheel motor system further enables precise torque control, improved efficiency, streamlined design, and optimized regenerative braking, enhancing dynamic

<figure><img src="image_2.png" /><figcaption>Fig. 1: Schematic diagram of the hierarchical and the holistic architectures.</figcaption></figure>

performance and adaptability [30]. At the upper level, [31] proposed a supervisory control strategy coordinating vehicle handling and lateral stability, combined with lower-level energy efficiency optimization. Similarly, [32] introduced a 7-degree-of-freedom MPC controller paired with a quadratic optimization for combined safety and energy saving. In [33], a two-level distribution strategy was developed to reduce energy consumption while maintaining handling stability. To further improve torque allocation in over-actuated systems, learning-based methods have been explored for energy-efficient distribution [34], alongside MPC-based approaches specifically targeting energy savings at the lower level [35], [36].

While hierarchical architectures provide modularity and focus, their layered structure often introduces latency and coordination challenges, which can hinder performance in highly dynamic scenarios. To address these limitations, a holistic control architecture offers a unified alternative by directly linking controlled states to actuator commands (see Fig. 1). This approach eliminates intermediate layers, enabling the integration of multiple control objectives, including path-tracking, stability, and energy efficiency, within a single optimization framework. Moreover, holistic architectures are inherently globally optimal, as they directly couple energy-efficient torque allocation with state feedback. This seamless integration allows the control system to achieve superior energy optimization by continuously aligning actuator outputs with the dynamic requirements of the vehicle states. A holistic architecture was proposed in [12] to coordinate multiple control objectives. However, the design is not minimal in order and energy efficiency is not considered.

This paper addresses several key gaps in the existing literature: 1) in the development of a unified MPC controller with minimal order that integrates path-tracking, speed control, and stability control; 2) in the enhancement of robustness against disturbances, uncertainties, and dynamic conditions through the introduction of novel adaptive weighting mechanism; and 3) in the optimal torque allocation to improve energy efficiency and stability within a holistic framework. The proposed architecture consolidates these control objectives into a single controller, enabling globally optimal torque distribution that enhances energy economy, maximizes tire stability, and ensures motor operation within high-efficiency regions. Consequently, the holistic approach improves the
overall performance and adaptability of AVs, providing a pathway to meet the stringent demands of advanced vehicle control. Two novel control architectures are developed in this paper for autonomous over-actuated in-wheel vehicles, based on hierarchical and holistic frameworks. First, a hierarchical centralized MPC architecture is proposed, integrating path-tracking, speed control, and stability control using the minimal system order required. The hierarchical approach demonstrates superior performance compared to existing strategies. This architecture is then extended into a holistic MPC framework, preserving the same control objectives while introducing an energy-saving criterion through motor power minimization and high-efficiency operating constraints. A tire stability criterion is also incorporated to optimize load usage. The holistic architecture achieves superior control performance and significant energy savings compared to the hierarchical structure and existing methods. A schematic comparison of the two architectures is provided in Fig. 1.

At the core of the proposed architectures, a novel adaptive weighting mechanism is introduced to enhance coordination and robustness. Specifically, the mechanism: 1) enables dynamic coordination among multiple control objectives to ensure balanced performance under diverse driving conditions; 2) continuously adjusts the weights of state and control input variables based on real-time error magnitudes and driving context, improving robustness and control performance under uncertainty and critical situations; and 3) prioritizes control objectives in response to situational demands to suppress overshoot, reduce oscillations, and enhance disturbance rejection and compensation for model inaccuracies.

The paper contributions can be outlined as:

* Development of a novel hierarchical centralized MPC architecture based on a minimal-order model, integrating path-tracking, speed control, and stability control objectives.

* Design of a novel holistic MPC architecture incorporating energy-efficiency and tire-stability allocation criteria. Energy efficiency is achieved by minimizing total power consumption through weighted torque control inputs and enforcing high-efficiency operating constraints.

* Introduction of a multi-criteria adaptive weighting mechanism for dynamic coordination of the control objectives. Stability control weights are formulated using a stability index, speed control weights are derived from longitudinal speed and its error, and path-tracking weights are adapted based on lateral error, road adherence, and curvature.

* Comprehensive validation of the proposed architectures using the SCANeR Studio vehicle dynamics simulator with a fully dynamic nonlinear vehicle model. The results demonstrate that the holistic architecture significantly enhances vehicle stability, comfort, and energy efficiency, while maintaining precise and robust path-tracking and speed control with low computational runtime.

The rest of the paper is structured as follows: Section II introduces the hierarchical centralized MPC architecture. The architecture is transformed into a holistic one and thoroughly explained in Section III. The system validation is presented in Section IV through simulation results and discussion, followed by a conclusion in Section V.

<figure><img src="image_3.png" /><figcaption>Fig. 2: An overview on the global hierarchical architecture.</figcaption></figure>

## II. HIERARCHICAL MPC ARCHITECTURE

The hierarchical architecture is introduced first, followed by its extension to the holistic design. This structure ensures clarity and facilitates understanding of the progression to the more advanced holistic framework, both of which share similar concepts including the control objectives and the adaptive state weights.

### A. Global Hierarchical Architecture

Figure 2 illustrates a schematic diagram of the proposed hierarchical MPC framework. At the forefront, a state-error-aware module predicts vehicle driving conditions by analyzing the lateral Stability Index (SI), lateral error $e_y$, and longitudinal velocity error $e_x$. This module adjusts time-varying adaptive weights for the MPC controller, categorized into two groups: (1) state weights, $Q_β$, $Q_{e_x}$, and $Q_{\dot{\psi}}$, which correspond to the side-slip angle $β$, longitudinal speed error $e_x$, and yaw rate $\dot{\psi}$, respectively; and (2) control input weights, $R_{M_z}$, $R_{T_m}$, and $R_δ$, which represent the weights for yaw moment $M_z$, total torque $T_m$, and steering angle $δ_c$. Additionally, road adherence and curvature information are supplied to this module, with the adherence coefficient $μ$ assumed to be accessible, and the road curvature $ρ$ calculated at the trajectory planning stage as detailed in [2].

The adaptive weights are received by the centralized Multi-Input-Multi-Output (MIMO) MPC controller to generate three control inputs corresponding to the three objectives. Path-tracking is accomplished by controlling the yaw rate $\dot{\psi}$ to follow its reference yaw rate $\dot{\psi}_{ref}$, through the generation of a steering angle $δ_c$. $\dot{\psi}_{ref}$ is defined as

$$ \dot{\psi}_{ref} = -\frac{2v_x e_{y,Ls}}{Ls^2}, \qquad (1) $$

where $v_x$ represents the longitudinal velocity, $Ls$ denotes the look-ahead distance [13], and $e_{y,Ls}$ corresponds to the lateral error evaluated at the look-ahead distance. Speed control is achieved by minimizing the longitudinal velocity error $e_x$, ensuring its convergence to zero through the generation of
a total torque $T_m$. This allows the actual velocity $v_x$ to effectively track the desired velocity profile $v_{x,des}$, which is predefined at the trajectory planning stage [2]. Finally, stability control is maintained by regulating the side-slip angle $\beta$, ensuring it converges to its reference value $\beta_{ref}$ under critical driving conditions. This is achieved through the generation of the yaw moment $M_z$. While $T_m$ and $M_z$ may sometimes conflict, the proposed adaptive weights dynamically reconcile them by adjusting their relative influence on the MPC cost function based on driving condition.

The generated control inputs are realized by a coordination of designated actuators, including four independent in-wheel motors and small electric motor for the Active Front Steering (AFS). The actuators are modeled using first-order transfer functions [16]. Given the negligible energy consumption of the AFS actuator compared to the in-wheel motors, its energy use is excluded from consideration in this study. The steering command $\delta_c$ is executed via the AFS system, while the total torque $T_m$ and yaw moment $M_z$ are distributed using a torque allocation strategy. Ultimately, the steering angle and individual wheel torques are applied to the full nonlinear vehicle model within the SCANeR Studio professional simulation environment.

## B. Extended Lateral and Longitudinal Model

The single-track linear bicycle model [16], is utilized with additional terms to account for longitudinal dynamics. While modeling errors are inevitable due to factors such as tire nonlinearities, parametric uncertainties, and time-varying parameters, the proposed adaptive weighting functions incorporated into the controller enhance its robustness against such discrepancies. Simplifications are made by neglecting roll, pitch, and suspension dynamics, and it is assumed that the steering angles of the left and right wheels are identical. The resulting model is described as

$$ egin{cases} M v_x (\dot{\beta} + \dot{\psi}) = F_{yf} \cos(\delta) + F_{yr} \\ I_z \ddot{\psi} = F_{yf} l_f \cos(\delta) - F_{yr} l_r + M_z \end{cases} \quad (2) $$

where $M$ represents the vehicle mass, and $I_z$ denotes the yaw moment of inertia. The lateral tire forces at the front and rear wheels are $F_{yf}$ and $F_{yr}$, respectively. $l_f$ and $l_r$ indicate the distances from the vehicle's center of gravity to the front and rear axles, respectively.

The lateral tire forces, $F_{yf}$ and $F_{yr}$, are defined by

$$ egin{aligned} F_{yf} &= \mu C_{yf} (\delta - \beta - \frac{l_f \dot{\psi}}{v_x}) \\ F_{yr} &= \mu C_{yr} (-\beta + \frac{l_r \dot{\psi}}{v_x}) \end{aligned} \quad (3) $$

where $C_f$ and $C_r$ are the front and rear cornering stiffness coefficients. While literature methods tend to introduce coupling and tire nonlinearities to enhance the prediction model and performance of the MPC controller, this work adopts a minimal-order linear time-varying model. Modeling errors, uncertainties, and disturbances are compensated through the proposed adaptive weighting strategy. To avoid the complexities and nonlinearities associated with coupling dynamics, which can significantly increase computational load, the longitudinal velocity tracking model is simplified and expressed in terms

of the longitudinal velocity error, $e_x = v_x - v_{x,des}$. The corresponding rate of change is

$$ \dot{e}_x = a_x - a_{x,des} = \frac{T_m}{Mr} - a_{x,des} \quad (4) $$

where $r$ represents the effective tire radius, and $a_{x,des}$ denotes the desired longitudinal acceleration. The value of $a_{x,des}$ is calculated based on the acceleration required to transition from the current velocity $v_x$ to the target velocity $v_{x,des}$ within the control sampling time $T_s$. Assuming small $\delta$, and combining (2), (3), and (4), the extended lateral and longitudinal model can be formulated as:

$$ egin{cases} \ddot{\psi} = -\mu \frac{l_r^2 C_r + l_f^2 C_f}{I_z v_x} \dot{\psi} + \mu \frac{l_r C_r - l_f C_f}{I_z} \beta + \mu \frac{l_f C_f}{I_z} \delta + \frac{1}{I_z} M_z \\ \dot{\beta} = (-1 + \mu \frac{l_r C_r - l_f C_f}{M v_x^2}) \dot{\psi} - \mu \frac{C_r + C_f}{M v_x} \beta + \mu \frac{C_f}{M v_x} \delta \\ \dot{e}_x = \frac{T_m}{Mr} - a_{x,des} \end{cases} \quad (5) $$

Given the three distinct control objectives, each associated with a dedicated control input, the extended predictive model (5) constitutes the minimum-order system necessary to achieve these objectives. Each objective requires at least one state variable, resulting in a three-state model that captures the essential system dynamics and interactions. Any further reduction would omit critical information, compromising the ability to satisfy all control objectives.

**Remark 1.** The desired longitudinal acceleration $a_{x,des}$ serves as a proxy parameter for torque mapping, directly influencing the total torque $T_m$ rather than representing the actual vehicle acceleration. Its bounds define the permissible range within which $T_m$ can adapt, rather than corresponding to the physical acceleration limits of the vehicle. The control input $T_m$ is mapped proportionally to $a_{x,des}$, with its variation governed by the predefined speed control weights. Consequently, $a_{x,des}$ is designed to ensure that $v_x$ tracks $v_{x,des}$ while adhering to the rate constraints defined by its bounds. It can be defined according to (6a) or (6b), where a smooth profile is employed to ensure a continuous control input. To prevent excessive acceleration when the initial velocity deviates significantly from the desired velocity, $a_{x,des}$ is constrained within a predefined range [$a_{x,min}, a_{x,max}$].

$$ a_{x,des} = \frac{v_{x,des} - v_x}{T_s}; \quad \text{subj. to, } a_{x,des} \in [a_{x,min}, a_{x,max}] \quad (6a) $$

$$ a_{x,des} = \begin{cases} \frac{a_{x,max}}{1+e^{-\frac{s}{a_d-a_d}(e_x-\frac{a_d+a_d}{2})}} & \text{if } e_x \ge 0 \\ \frac{a_{x,min}}{1+e^{-\frac{s}{a_d+a_d}(e_x-\frac{a_d-a_d}{2})}} & \text{if } e_x < 0 \end{cases} \quad (6b) $$

where $T_s$ is the control sampling time step. $\overline{a_d}$ dictates how aggressively the vehicle responds for large $e_x$, while $\overline{a_d}$ influences the lower acceleration threshold for low $e_x$. Consequently, $\overline{a_d}$, $\overline{a_d}$, $a_{min}$, $a_{max}$ are selected to balance the desired responsiveness, aggressiveness, and smoothness of the actual velocity response.

## C. Hierarchical MPC Problem Formulation

The extended lateral and longitudinal model can be transformed into a state-space representation as

$$ egin{aligned} \dot{x} &= Ax + Bu + w \\ y &= Cx \end{aligned} \quad (7) $$
The system, control, and output matrices are defined as (8), where $x = [\dot{\psi} \ \beta \ e_x]^T$ is the state vector, $u = [\delta_c \ M_z \ T_m]^T$ is the control inputs vector, and $w = [0 \ 0 \ -a_{x,des}]^T$ modeled as an exogenous input.

$$A = \begin{bmatrix} -\mu \frac{l_r^2 C_r + l_f^2 C_f}{l_r C_r - l_f C_f} & \mu \frac{l_r C_r - l_f C_f}{C_r + C_f} & 0 \\ -1 + \mu \frac{l_r C_r - l_f C_f}{M v_x^2} & -\mu \frac{I_z}{M v_x} & 0 \\ 0 & 0 & 0 \end{bmatrix}; \quad (8)$$

$$B = \begin{bmatrix} \mu \frac{l_f C_f}{I_z} & \frac{1}{I_z} & 0 \\ \mu \frac{C_f}{M v_x} & 0 & 0 \\ 0 & 0 & \frac{1}{M r} \end{bmatrix}; \quad C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$

The continuous-time state-space model (7) is discretized for the MPC design as

$$\begin{aligned} x(k+1) &= A_d x(k) + B_d u(k) + T_s w(k) \\ y(k) &= C_d x(k) \end{aligned} \quad (9)$$

where $A_d$ and $B_d$ are the discrete-time state-space matrices obtained from A and B using the Zero-Order Hold (ZOH) method, and $C_d = C$. Consequently, the cost function of the MPC is defined as

$$\begin{aligned} J = & \sum_{k=1}^{N_p} \left( (y_k - y_{ref})^T Q (y_k - y_{ref}) + u_k^T R u_k \right) \\ & + \sum_{k=2}^{N_p} (u_k - u_{k-1})^T S (u_k - u_{k-1}) \end{aligned} \quad (10)$$

where $N_p$ is the prediction horizon, $Q, R$, and $S$ represent the diagonal weight matrices associated with the system states, control inputs, and control input increments, respectively. The desired system output is denoted as $y_{ref} = [\dot{\psi}_{ref} \ \beta_{ref} \ 0]$, where $\beta_{ref}$ is set to 0 in this work. The weight matrices are defined as $Q = \text{diag}(Q_{\dot{\psi}}, Q_{\beta}, Q_{e_x})$, $R = \text{diag}(R_{\delta}, R_{M_z}, R_{T_m})$, and $S = \text{diag}(S_{\delta}, S_{M_z}, S_{T_m})$, where $Q_{\dot{\psi}}, Q_{\beta}, Q_{e_x}, R_{\delta}, R_{M_z}, R_{T_m}, S_{\delta}, S_{M_z},$ and $S_{T_m}$ are scalar weighting coefficients. The optimization problem is expressed as

$$\min_{u} J \quad (11a)$$

subject to

$$x(k+1) = A_d x(k) + B_d u(k) + T_s w(k) \quad (11b)$$

$$y(k) = C_d x(k) \quad (11c)$$

$$\delta_{min} \le \delta_k \le \delta_{max} \quad (11d)$$

$$M_{z,min} \le M_{z_k} \le M_{z,max} \quad (11e)$$

$$T_{m,min} \le T_{m_k} \le T_{m,max} \quad (11f)$$

$$\Delta \delta_{min} \le \delta_k - \delta_{k-1} \le \Delta \delta_{max} \quad (11g)$$

$$\Delta M_{z,min} \le M_{z_k} - M_{z_{k-1}} \le \Delta M_{z,max} \quad (11h)$$

$$\Delta T_{m,min} \le T_{m_k} - T_{m_{k-1}} \le \Delta T_{m,max} \quad (11i)$$

where $[\delta_{min} \ \delta_{max}]$, $[M_{z,min} \ M_{z,max}]$, $[T_{m,min} \ T_{m,max}]$, respectively represent the ranges of the control inputs $\delta, M_z$, and $T_m$. Similarly, $[\Delta \delta_{min} \ \Delta \delta_{max}]$, $[\Delta M_{z,min} \ \Delta M_{z,max}]$, and $[\Delta T_{m,min} \ \Delta T_{m,max}]$ denote the ranges of the control input increments, in the same order. Note that the control input limits are dictated by the physical actuator constraints, while the control input increment limits are designed to prevent abrupt changes, ensuring smooth control action and enhancing actuator longevity.

The MPC optimization problem is formulated as a Quadratic Programming (QP) problem and solved using the OSQP solver [37]. More details on the problem formulation and solver

implementation can be found in [23]. The current problem transformation to meet the OSQP demands significantly enhances computational efficiency by ensuring fast convergence, as will be demonstrated later.

## D. Adaptive Weighting Mechanism

The weights assigned to the control objectives are adapted based on multiple criteria linked to several parameters. In addition to improving the performance of the control objectives, the adaptive weighting mechanism enhances the robustness of the MPC controller. The functional forms of the weights are carefully designed to satisfy the following conditions: 1) boundedness, with minimum values corresponding to low errors and maximum values to high errors; 2) smooth transitions with adjustable transient behavior; and 3) continuity and differentiability to avoid discontinuities in the generated control inputs. For the hierarchical case, the weights on states and control inputs are jointly discussed from the perspective of achieving the control objectives.

### 1) Weights on Stability Control

The weights associated with stability control, i.e. weight on the state $\beta$ and on the control input $M_z$, are dynamically adjusted based on the stability assessment. The emphasis on the stability control objective is modulated in response to varying driving conditions. A critical driving scenario is recognized when the Stability Index (SI) (12) exceeds the threshold value of 1, where the constants in (12) correspond to normal adherence ($\mu = 1$) [16].

$$SI = |2.49\dot{\beta} + 9.55\beta|. \quad (12)$$

To ensure an anticipatory emphasis on the stability objective under critical conditions, the stability objective is activated by prioritizing the weight on $\beta$, with a smoothly increasing transition within the range $[SI \ \overline{SI}] \le 1$. Accordingly, the weight on the side-slip angle is defined as

$$Q_{\beta} = q_{\beta,1} + \frac{q_{\beta,2}}{1 + e^{-\frac{2\gamma_1}{\overline{SI} - SI}(SI - \frac{\overline{SI} + SI}{2})}}, \quad (13)$$

where $q_{\beta,1}$ and $q_{\beta,2}$ represent the extrema values of $Q_{\beta}$ within its transient state, corresponding to the lower and upper bounds, respectively. $SI$ marks the lower threshold where the stability objective starts activating, while $\overline{SI}$ denotes the upper threshold where $Q_{\beta}$ saturates. $\gamma_1$ controls the intensity of the transient-state centroid of $Q_{\beta}$. The weight $Q_{\beta}$ is shown in Fig. 3-(i)-(a). Within the same range, the weight applied to the control input $M_z$ is reduced to attenuate the penalization of the DYC actuator when $SI \ge \overline{SI}$. Consequently, $R_{M_z}$ is expressed as follows:

$$R_{M_z} = r_{M_{z,2}} - \frac{(r_{M_{z,2}} - r_{M_{z,1}})}{1 + e^{-\frac{2\gamma_1}{\overline{SI} - SI}(SI - \frac{\overline{SI} + SI}{2})}}, \quad (14)$$

where $r_{M_{z,2},r_{M_{z,1}}}$ are respectively the upper and lower extremities of the transient state of $R_{M_z}$. Hence, the stability objective is entirely activated for $SI \ge \overline{SI}$, where $Q_{\beta} = q_{\beta,2}$ is maximum and $R_{M_z} = r_{M_{z,1}} = \varepsilon > 0$ is minimum, and vice versa. The shape of $R_{M_z}$ is depicted in Fig. 3-(ii).

### 2) Weights on Speed Control

Adaptive weighting in speed control is crucial for mitigating disturbances, compensating for modeling errors, and enhancing controller robustness. These factors are amplified
<figure><img src="image_4.png" /><figcaption>Fig. 3: Weight adaptive mechanism for control objectives coordination: (i) weights on states $Q$, (ii) weights on control inputs $R$.</figcaption></figure>

if the initial error on speed is large, contrary to the case of matching the initial speed with the desired one, as in [12]. The output value of the control input $T_m$ is an image of the desired longitudinal acceleration $a_{x,des}$. Hence, the adaptive weights aid in assigning an effective mapping between $a_{x,des}$ and the corresponding desired output for $T_m$. The weight $Q_{e_x}$ on the state $e_x$ is adapted primarily according to $|e_x|$ and as a function of $v_x$. $Q_{e_x}$ is augmented as the value of $|e_x|$ increases, to reach its maximum ($Q_{e_x}^{max}$) when the desired acceleration is extremum ($a_{max}$ or $a_{min}$). Hence, $T_m$ is saturated for extremum value of $a_{x,des}$ and varies according to the designed mapping in the ranges $[0 \ a_{max}]$ and $[a_{min} \ 0]$.

Therefore, $Q_{e_x}$ is prioritized when $|e_x|$ is significant and gradually attenuated as $|e_x|$ approaches zero. The maximum value, $Q_{e_x}^{max}$, is also adjusted according to $v_x$, such that $Q_{e_x}$ is increased with a transient state over the range $[\underline{v_x} \ \overline{v_x}]$ with steepness $\chi$. The adaptation based on $v_x$ ensures that the weighting of the speed error state increases with $v_x$, reflecting its growing prioritization at higher velocities. For this purpose, define $Q_{e_x}$ as follows:

$$Q_{e_x} = \sigma_1 + \frac{(\sigma_2 - \sigma_1)}{1 + e^{-\frac{2\gamma_2}{\overline{e_x} - \underline{e_x}}(|e_x| - \frac{\overline{e_x} + \underline{e_x}}{2})}}, \quad (15a)$$

$$\sigma_1 = q_{e_x,1} + \frac{(q_{e_x,2} - q_{e_x,1})}{1 + e^{-\frac{2\chi}{\overline{v} - \underline{v}}(v_x - \frac{\overline{v} + \underline{v}}{2})}}, \quad (15b)$$

$$\sigma_2 = q_{e_x,3} + \frac{(q_{e_x,4} - q_{e_x,1})}{1 + e^{-\frac{2\chi}{\overline{v} - \underline{v}}(v_x - \frac{\overline{v} + \underline{v}}{2})}}, \quad (15c)$$

where $\{q_{e_x,1}, q_{e_x,3}, q_{e_x,2}, q_{e_x,4}\}$ represent the desired values at the extremities of $Q_{e_x}$ at the points $A_1(\underline{v}, e_x)$, $B_1(\overline{v}, \overline{e_x})$, $C_1(\overline{v}, e_x)$, $D_1(\underline{v}, \underline{e_x})$ respectively. $[\underline{e_x} \ \overline{e_x}]$ and $[\underline{v} \ \overline{v}]$ are the ranges of the transient state in one-dimensional fixed values for $v_x$ and $e_x$ respectively. $\gamma_2$ and $\chi$ are steepness parameters as $\gamma_1$. The form of $Q_{e_x}$, along with its parameters, are depicted in Fig. 3-(i)-(b). Over the same defined ranges, the weight on the control input $T_m$ is reduced as $e_x$ increases, to attenuate penalizing the actuator. Hence, define $R_{T_m}$ as follows:

$$R_{T_m} = r_{T_m,2} - \frac{(r_{T_m,2} - r_{T_m,1})}{1 + e^{-\frac{2\gamma_2}{\overline{e_x} - \underline{e_x}}(|e_x| - \frac{\overline{e_x} + \underline{e_x}}{2})}}, \quad (16)$$

where $r_{T_m,1}, r_{T_m,2}$ signify the values of $R_{T_m}$ at the extremities of its transient state (see Fig. 3-(ii)).

In addition to reinforcing the controller robustness, the designed weight functions facilitate the smooth convergence of $v_x$ to $v_{x,des}$, effectively minimizing overshoots and mitigating oscillatory behavior. For large $|e_x|$, the controller responds

swiftly and efficiently, reducing the speed error with rapid actuator adjustments. Furthermore, by increasing $R_{T_m}$ and decreasing $Q_{e_x}$ for small $|e_x|$, the controller enhances energy efficiency. The energy consumption is further optimized through the smooth profile of the control input $T_m$, achieved by the inclusion of $S_{T_m}$ and the imposition of constraints on the rate of change of the control input.

Validation of the proposed adaptive weighting strategy demonstrates its effectiveness in attenuating oscillations, preventing overshooting, and improving tracking accuracy.

**Remark 2.** The proposed adaptive behavior enhances the controller robustness by dynamically adjusting the control priorities based on the magnitude of the error. As the error increases, the weight on the relevant states and control inputs is prioritized/relaxed to improve performance and stability. This adaptive behavior allows the MPC controller to more effectively and smoothly handle varying disturbances and uncertainties, similar to how sliding mode control adapts to changes in system dynamics. Consequently, the proposed adaptive weighting methodology can be generalized to other systems and models, offering enhanced robustness and performance across diverse applications.

### 3) Weights on Path-Tracking Control

The path-tracking control objective is essential for ensuring the safety of the autonomous vehicle. Accordingly, its weighting is dynamically adapted based on the lateral error $|e_y|$, road curvature $\rho$, road adhesion $\mu$, and its interconnection with other control objectives. Following a similar approach to that employed in speed control, the weight on the yaw rate ($Q_{\dot{\psi}}$) is increased for large $|e_y|$ to swiftly and effectively minimize the lateral error $e_y$ while enhancing the robustness of the controller. As $|e_y|$ decreases, $Q_{\dot{\psi}}$ is gradually attenuated, ensuring smooth convergence while avoiding oscillations and overshoots.

Hence, $Q_{\dot{\psi}}$ is expressed as follows:

$$Q_{\dot{\psi}} = q_{\dot{\psi},1} + \frac{(q_{\dot{\psi},2} - q_{\dot{\psi},1})}{1 + e^{-\frac{2\gamma_3}{\overline{e_y} - \underline{e_y}}(|e_y| - \frac{\overline{e_y} + \underline{e_y}}{2})}}, \quad (17)$$

where $q_{\dot{\psi},1}, q_{\dot{\psi},2}$ denote the lower and upper extremity values of the transient-state of $Q_{\dot{\psi}}$, respectively. $[\overline{e_y} \ \underline{e_y}]$ is the range of lower and upper thresholds of $e_y$ during the transient state, and $\gamma_3$ as $\gamma_2$.

In the case where a risk of instability is detected and
the stability objective is prioritized, the generation of the yaw moment may compromise path-tracking performance. To maintain precise path-tracking while simultaneously addressing stability, the weight $Q_{\dot{\psi}}$ is increased for $SI \geq SI$, with $q_{\dot{\psi},2}$ defined as a function of SI as follows:

$$q_{\dot{\psi},2} - q_{\dot{\psi},1} = \frac{\xi_1}{1 + e^{-\frac{2\gamma_1}{SI-SI}(SI-\frac{SI+SI}{2})}}, \quad (18)$$

where $\xi_1 \in \mathbb{R}^+$. The complexity of path-tracking control significantly increases as the road adhesion coefficient $\mu$ decreases, particularly on high-curvature roads with reduced traction. Hence, $Q_{\dot{\psi}}$ is further augmented in accordance with decreasing $\mu$ and increasing $\rho$. Conversely, abrupt variations in the desired speed profile result in a sharp increase in $Q_{e_x}$. To address this demand and maintain accurate path-tracking performance, $Q_{\dot{\psi}}$ is dynamically adapted in response to changes in $Q_{e_x}$. Hence, $q_{\dot{\psi},1}$ is defined as follows:

$$q_{\dot{\psi},1} = \xi_2 + \frac{Q_{e_x}}{\xi_3} + q_{\mu,\rho} \mathcal{F}(\mu), \quad (19)$$

where $\xi_2$ is the minimum path-tracking weight under normal adhesion, and $\xi_3 > 0$. $\mathcal{F}(\mu) = 1_{\{\mu \le 1\}}(\mu)$ is an indicator function. The term related to $Q_{e_x}$ can be relaxed by assigning $\xi_3$ by a sufficiently large value. The intermediate function for path-tracking, $q_{\mu,\rho}$, is defined in terms of $\mu$ and $\rho$ as follows:

$$q_{\mu,\rho} = \frac{\zeta_1}{1 + e^{-\frac{2\gamma_3}{\bar{\rho}}(|\rho| - \frac{\bar{\rho}+\rho}{2})}} - \frac{\zeta_2}{1 + e^{-\frac{2\gamma_3}{\bar{\mu}}(|\mu| - \frac{\bar{\mu}+\mu}{2})}} + \zeta_3, \quad (20a)$$

$$\zeta_2 = (q_{\mu,3} - q_{\mu,2}) \frac{(1 + e^{-\gamma_3})(1 + e^{\gamma_3})}{e^{-\gamma_3} - e^{\gamma_3}}, \quad (20b)$$

$$\zeta_3 = \frac{1}{e^{\gamma_3} - e^{-\gamma_3}} \left[ -(1 + e^{-\gamma_3})q_{\mu,1} + (1 + e^{\gamma_3})q_{\mu,2} - \frac{e^{-\gamma_3} - e^{\gamma_3}}{1 + e^{\gamma_3}} \zeta_2 \right], \quad (20c)$$

$$\zeta_1 = (1 + e^{-\gamma_3})(q_{\mu,1} + \frac{1}{1 + e^{\gamma_3}}\zeta_2 - \zeta_3), \quad (20d)$$

where $\rho, \mu, \bar{\rho}, \bar{\mu}$ denote respectively the lower and upper boundaries of $q_{\mu,\rho}$ in two dimensions, with respect to $\rho$ and $\mu$. Referring to Fig. 3-(i)-(d), the desired values of the reshaped function at $A_2(\mu, \rho)$, $B_2(\mu, \bar{\rho})$, and $C_2(\bar{\mu}, \rho)$ are respectively $q_{\mu,3}$, $q_{\mu,2}$, and $q_{\mu,1}$. The shapes of $Q_{e_x}$, $q_{\mu,\rho}$, and their parameters are shown in Figures 3-(i)-(c) and 3-(i)-(d).

The weight on the steering angle is increased as $|e_y|$ decreases to facilitate smooth convergence and suppress oscillations. Conversely, as $|e_y|$ increases, $R_\delta$ is decreased to reduce the penalization on the AFS actuator. Consequently, $R_\delta$ is defined as follows:

$$R_{\delta} = r_{\delta,2} - \frac{(r_{\delta,2} - r_{\delta,1})}{1 + e^{-\frac{2\gamma_3}{e_y}(|e_y| - \frac{e_y+e_x}{2})}}, \quad (21)$$

where $r_{\delta,1}$, $r_{\delta,2}$ signify the values at the extremities of the transient state of $R_\delta$. The shaping of the weight $R_\delta$ is presented in Fig. 3-(ii). A test where the vehicle starts with an initial lateral deviation shows that the adaptive weighting scheme enables faster, smoother, and more stable convergence to the reference lane, while the fixed-weight baseline exhibits overshooting, border collisions, and slow oscillatory convergence.

**Remark 3.** The weight functions are general, scalable, and

straightforward to implement, with fully tunable parameters. 1) The boundary values (e.g., $q_{\dot{\psi},1}$, $q_{\dot{\psi},2}$) determine the control action at low and high error levels for performance variables such as SI and $e_y$. 2) The lower and upper thresholds (e.g., $e_y$, $\bar{e}_y$) define the range over which the control action activates and fully engages, influencing responsiveness (rate of error correction), aggressiveness (control intensity), and smoothness. 3) The steepness parameters (e.g., $\gamma_1$) govern the sharpness of the transition, with higher values resulting in more abrupt weight adjustments.

### III. HOLISTIC MPC ARCHITECTURE

#### A. Global Holistic Architecture

The global holistic control architecture comprises a single MIMO MPC controller that processes state errors as inputs, incorporating time-varying weights and constraints to preserve tire stability and promote high-efficiency operation. The state weights in the holistic architecture are consistent with those used in the hierarchical case. The control input weights, however, are designed to 1) achieve the control objectives and 2) minimize the total power consumption of the four in-wheel motors. Constraints on the torque control inputs are dynamically updated based on a joint criterion that balances tire stability with the requirement to operate the motors within high-efficiency zones.

The fulfillment of control objectives follows the same approach as in the hierarchical case: path-tracking is achieved by regulating the yaw rate $\dot{\psi}$ to follow the reference $\dot{\psi}_{ref}(1)$; speed control is accomplished by minimizing the longitudinal speed error $e_x = v_x - v_{x,des}$; and stability control is ensured by regulating the side-slip angle $\beta$ to track the reference $\beta_{ref}$. The controller outputs, comprising the steering angle and four individual wheel torques, are directly transmitted to the physical actuators. The full nonlinear vehicle model is implemented in the SCANeR Studio professional vehicle dynamics simulator.

#### B. Extended Holistic Model

The holistic design is formulated by integrating the four torques, along with the steering angle, through direct linkage to the states. The total torque $T_m$ is then expressed in terms of the four individual torques as follows:

$$T_m = T_1 + T_2 + T_3 \cos(\delta) + T_4 \cos(\delta), \quad (22)$$

where the sub-indices {1, 2, 3, 4} respectively correspond to the rear-left (rl), rear-right (rr), front-right (fr), and front-left (fl) wheels. The corrective yaw moment $M_z$ is expressed in terms of $T_i$ as follows:

$$M_z = \frac{-T_2}{r} t_r + \frac{-T_3}{r} t_r \cos(\delta) + \frac{T_1}{r} t_r + \frac{T_4}{r} t_r \cos(\delta), \quad (23)$$

where $t_r$ is the half rear track. Assuming a small steering angle $\delta$, combining (22), (23), with (5), and manipulating the combined model into a state-space form, the extended holistic model can be expressed as

$$\frac{\partial}{\partial t} \begin{bmatrix} \dot{\psi} \\ \beta \\ e_x \end{bmatrix} = A_{[3*3]} \begin{bmatrix} \dot{\psi} \\ \beta \\ e_x \end{bmatrix} + B_{h[3*5]} \begin{bmatrix} \delta \\ T_1 \\ T_2 \\ T_3 \\ T_4 \end{bmatrix} + w \quad (24)$$
Therefore, the discretized state-space model for the holistic
design is expressed as:

$$
\begin{align}
x(k+1) &= A_d x(k) + B_{d,h} u_h(k) + T_s w(k) \tag{25} \\
y(k) &= C_d x(k) \nonumber
\end{align}
$$

where $u_h = [\delta T_{rl} \ T_{rr} \ T_{fr} \ T_{fl}]^T$, and the discretized control matrix is given as

$$
\begin{align}
B_{d,h} &= \int_0^{T_s} e^{A\tau} d\tau \, B_h \nonumber \\
B_h &= \begin{bmatrix}
\mu l_f \frac{C_f}{\frac{1}{I_z r}} & \frac{t_r}{I_z r} & -\frac{t_r}{I_z r} & -\frac{t_r}{I_z r} & \frac{t_r}{I_z r} \\
\mu C_f \frac{M v_x}{M v_x} & 0 & 0 & 0 & 0 \\
0 & \frac{1}{M r} & \frac{1}{M r} & \frac{1}{M r} & \frac{1}{M r}
\end{bmatrix} \tag{26}
\end{align}
$$

As in the hierarchical framework, $x = [\dot{\psi} \ \beta \ e_x]^T$ is the state vector, and $w = [0 \ 0 \ -a_{x,des}]^T$ represents the exogenous input. The matrices $A$ and $C$ are similar to (8).

C. Individual Torque Constraints

In addition to achieving the control objectives, the torque control inputs are regulated to ensure motor operation within high-efficiency zones while preserving tire stability. This is accomplished by enforcing appropriate constraints on the control inputs. The efficiency of the in-wheel motor $\eta$ is a function of the generated torque $T$ and angular velocity $\omega$, expressed as $\eta = f(T, \omega)$. This relationship is represented by the motor efficiency map. Fig. 4 illustrates the efficiency maps for the PD18 DC electric in-wheel motor [16], covering both traction and regenerative braking operations.

For each motor *i*, and for a given rotational velocity $\omega_i$, there exists a specific torque range within which the motor operates at maximum efficiency. To eliminate the computational overhead of online searching, an offline approach is proposed wherein an explicit function is derived using a regression technique. For each $\omega$ within the motor's operational range [0, 1500] rpm, two torque values are identified that correspond to the maximum achievable efficiency. These values define the lower and upper torque constraints for traction operation, as well as an additional pair of constraints for regenerative braking (see Fig. 4). Consequently, four explicit polynomial functions are derived corresponding to: lower traction/driving constraint $T_{cons}^{d,l}$, upper traction constraint $T_{cons}^{d,u}$, lower braking constraint $T_{cons}^{b,l}$, and upper braking constraint $T_{cons}^{b,u}$.

$$
T_{\text{cons}}^{p,q} = \sum_{k=1}^{7} \vartheta_{p,q,k} \omega^{7-k}, \quad p \in \{d, b\}, \quad q \in \{l, u\} \quad (27)
$$

Each function in (27) is represented as a sixth-order polyno-
mial, with coefficients $\vartheta_{p,q,k}$ identified via regression. Con-
sequently, the motor operates within its maximum efficiency
zone, either in traction or braking, provided that the combined
grade $(T, \omega)$ remains within the upper and lower polynomial
constraint boundaries (or in their vicinity).

The stability control objective, addressed through $\beta$, ensures
the recovery of vehicle stability in scenarios of stability loss.
However, in the presence of tire slip, achieving overall chassis
stability may be hindered if the tire-road interaction lacks
sufficient adhesion. Hence, constraints are imposed on the
torque at each tire *ij*, where *i* ∈ {r, f} and *j* ∈ {l, r}, to

<figure><img src="image_5.png" /><figcaption>Fig. 4: Upper and lower torque constraints for high efficiency zones.</figcaption></figure>

ensure that the Tire Working-load Usage (TWU) (28) remains
below 1.

$$
TWU_{ij} = \frac{F_{x,ij}^2 + F_{y,ij}^2}{\mu^2 F_{z,ij}^2} \quad (28)
$$

where $F_{z,ij}$ denotes the vertical load on tire *ij*. By manipulating $F_{x,ij}$ in (28) into torques, the stability-based torque constraints for each motor are given by:

$$
T_{\text{cons},ij}^{\text{st}} = r \sqrt{\mu^2 F_{z,ij}^2 - F_{y,ij}^2}, \quad i \in \{r, f\}, \ j \in \{l, r\} \quad (29)
$$

Therefore, for each motor *ij*, two potential zones for torque
constraints are enforced through a joint criteria of high-
efficiency and tire stability as follows:

$$
[T_{c,ij}^{d,l} \quad T_{c,ij}^{d,u}] = \begin{cases} [T_{cons,ij}^{d,l} \quad T_{cons,ij}^{d,u}], & T_{cons,ij}^{st} \ge T_{cons,ij}^{d,u} \\ [T_{cons,ij}^{d,l} \quad T_{cons,ij}^{st}], & T_{cons,ij}^{st} \le T_{cons,ij}^{d,u} \\ [0 \quad T_{cons,ij}^{st}], & T_{cons,ij}^{st} < T_{cons,ij}^{d,l} \end{cases} \quad (30a)
$$

$$
[T_{c,ij}^{b,u} \quad T_{c,ij}^{b,l}] = \begin{cases} [T_{cons,ij}^{b,u} \quad T_{cons,ij}^{b,l}], & -T_{cons,ij}^{st} \le T_{cons,ij}^{b,u} \\ [T_{cons,ij}^{b,u} \quad -T_{cons,ij}^{st}], & T_{cons,ij}^{b,u} \le -T_{cons,ij}^{st} \le T_{cons,ij}^{b,l} \\ [-T_{cons,ij}^{st} \quad 0], & -T_{cons,ij}^{st} > T_{cons,ij}^{b,l} \end{cases} \quad (30b)
$$

where the efficiency constraints $T_{cons,ij}^{d,l}$, $T_{cons,ij}^{d,u}$, $T_{cons,ij}^{b,l}$, and $T_{cons,ij}^{b,u}$ are interpolated online using (27) for each motor *ij*.

D. Holistic MPC Problem Formulation

Denoting $Q_h$, $R_h$, and $S_h$ respectively as the state, control input, and control input increment cost matrices for the holistic framework, the cost function is defined as (31).

$$
\begin{equation}
\begin{split}
J_h = \sum_{k=1}^{N_p} & \left( (y_k - y_{ref})^T Q_h (y_k - y_{ref}) + u_k^T R_h u_k + s_k^T P_h s_k \right) \\
& + \sum_{k=2}^{N_p} (u_k - u_{k-1})^T S_h (u_k - u_{k-1})
\end{split}
\tag{31}
\end{equation}
$$

where $P_h = \text{diag}(P_\delta, P_{T_1}, P_{T_2}, P_{T_3}, P_{T_4})$ is the cost matrix on the slack variable $s$, and $y_{ref}$ is identical to the hierarchical case. The optimization problem in the holistic MPC is defined as follows:

$$
\min_{u} J_{h} \tag{32a}
$$
<figure><img src="image_6.png" /><figcaption>Fig. 5: State weights on longitudinal speed error $e_x$: two possible approaches.</figcaption></figure>

subject to

$$x_{(k+1)} = A_d x_{(k)} + B_{d,h} u_{h_{(k)}} + T_s w_{(k)} \quad (32b)$$

$$y_{(k)} = C_d x_{(k)} \quad (32c)$$

$$u_{min} - s_{(k)} \le u_{h_{(k)}} \le u_{max} + s_{(k)} \quad (32d)$$

$$\Delta u_{min} < u_{h_{(k)}} - u_{h_{(k-1)}} < \Delta u_{max} \quad (32e)$$

$$s_{(k)} \ge 0 \quad (32f)$$

To ensure feasibility under the imposed constraints, a slack variable is introduced. The cost matrix $P_h$ penalizes the slack variable, activating it only when necessary to maintain a feasible solution. The constraints on the control input are defined as $u_{min} = [\delta_{min} T_{rl,min} T_{rr,min} T_{fr,min} T_{fl,min}]^T$ and $u_{max} = [\delta_{max} T_{rl,max} T_{rr,max} T_{fr,max} T_{fl,max}]^T$. Similarly, the constraints on the control input increments are defined as $\Delta u_{min} = [\Delta \delta_{min} \Delta T_{rl,min} \Delta T_{rr,min} \Delta T_{fr,min} \Delta T_{fl,min}]^T$ and $\Delta u_{max} = [\Delta \delta_{max} \Delta T_{rl,max} \Delta T_{rr,max} \Delta T_{fr,max} \Delta T_{fl,max}]^T$. The constraints on each motor torque are divided into two distinct ranges. Accordingly, the minimum and maximum torque constraints are defined as

$$T_{ij,min} = \begin{cases} T_{c,ij}^{d,l}, & \text{if } z_{ij} = 1 \\ T_{c,ij}^{b,u}, & \text{if } z_{ij} = 0 \end{cases} \quad (33a)$$

$$T_{ij,max} = \begin{cases} T_{c,ij}^{d,u}, & \text{if } z_{ij} = 1 \\ T_{c,ij}^{b,l}, & \text{if } z_{ij} = 0 \end{cases} \quad (33b)$$

where $z_{ij}$, with $ij \in \{rl, rr, fr, fl\}$, is a Boolean variable indicating the motor operating mode: $z_{ij} = 1$ for traction and $z_{ij} = 0$ for braking.

The cost matrices are defined as follows: $Q_h = Q$ as in the hierarchical case, $R_h = \text{diag}(R_\delta, R_{T_1}, R_{T_2}, R_{T_3}, R_{T_4})$, and $S_h = \text{diag}(S_\delta, S_{T_1}, S_{T_2}, S_{T_3}, S_{T_4})$. Finally, the optimization problem is formulated as a quadratic program and solved using the OSQP solver.

### E. Adaptive Weighting Mechanism

The state weights are consistent with the hierarchical design, comprising $Q_\beta$ (13) for the side-slip angle, $Q_{e_x}$ (15) for the longitudinal speed error, and $Q_{\dot{e}_x}$ (17) for the yaw rate. Conversely, the control input weight matrix $R_h$ is adaptively tuned to various parameters, ensuring the coordination of the in-wheel motors for achieving the control objectives while optimizing energy efficiency.

**Remark 4.** Speed control can be achieved by adapting the weight on the speed error $Q_{e_x}$ (15), using the exogenous input $w$, and mirroring the weighting function for $e_x \ge 0$ and $e_x < 0$. This approach is illustrated in Fig. 5-(a), where $q_{e_x,1}$, $q_{e_x,2}$, $q_{e_x,3}$, $q_{e_x,4} \ge 0$, and $[\underline{e_x}, \overline{e_x}] \ge 0$. An alternative approach defines $Q_{e_x}$ over the range $[\underline{e_x}, \overline{e_x}]$ with $\underline{e_x} \times \overline{e_x} < 0$, ensuring $q_{e_x,2} = -q_{e_x,1}$, as shown in

Fig. 5-(b). In this case, $\sigma_1 = q_{e_x,1}$ and $\sigma_2 = q_{e_x,2}$ are substituted into (15), and the exogenous input is modified to $w_2 = [0, 0, -|a_{x,des}|]^T$. Both approaches have been evaluated and yield comparable performance. However, in the second case, when the state error approaches zero, oscillations may arise in $Q_{e_x}$, potentially inducing oscillations in the speed profile. To mitigate this, a filter is applied to smooth $Q_{e_x}$ around zero.

The control objectives are effectively and robustly achieved through the joint adaptation of state and control input weights. For the control input weights $R_h$, assigning the same adaptive function to $R_{T_1}$, $R_{T_2}$, $R_{T_3}$, and $R_{T_4}$ ensures the fulfillment of speed and stability objectives with uniform torque distribution among the four motors. Adjusting the value of $R_{T_i}$ allows for targeted torque allocation, with a reduced $R_{T_i}$ corresponding to an increased torque output for motor $i$, and vice versa.

Neglecting battery power losses, the output power consumption is expressed as (34) [16], where $\eta_i$ denotes the efficiency of motor $i$, estimated using the maps in Fig. 4. To allocate the four torques while minimizing energy consumption, a quadratic form of (34) is minimized.

$$P_o = \sum_{i=1}^{4} \frac{\omega_i}{\eta_i^{sgn(T_i)}} T_i \quad (34)$$

For $T_i \in [-T_{max}, T_{max}]$, direct minimization of $P_o$ tends to favor higher negative torques, potentially causing excessive braking during speed control. This requires large traction torques for compensation, leading to increased overall energy consumption. To address this, a quadratic form is employed to: 1) minimize the magnitude of $T_i$, thereby preventing excessive torque demand in both traction and braking, and 2) maintain the quadratic structure of the cost function, enabling its direct incorporation into the control input weights. Hence, the energy term quadratic form is expressed as (35), which can be observed as $T^T R T$, with $T = [T_1 \ T_2 \ T_3 \ T_4]^T$ represents the four torque control inputs, and $R$ is the diagonal scaling matrix.

$$\begin{aligned} \min_{T_i} J_e &= \frac{\omega_1}{\eta_1^{sgn(T_1)}} T_1^2 + \frac{\omega_2}{\eta_2^{sgn(T_2)}} T_2^2 + \frac{\omega_3}{\eta_3^{sgn(T_3)}} T_3^2 + \frac{\omega_4}{\eta_4^{sgn(T_4)}} T_4^2 \\ &= [T_1 \ T_2 \ T_3 \ T_4] \begin{bmatrix} \frac{\omega_1}{\eta_1^{sgn(T_1)}} & 0 & 0 & 0 \\ 0 & \frac{\omega_2}{\eta_2^{sgn(T_2)}} & 0 & 0 \\ 0 & 0 & \frac{\omega_3}{\eta_3^{sgn(T_3)}} & 0 \\ 0 & 0 & 0 & \frac{\omega_4}{\eta_4^{sgn(T_4)}} \end{bmatrix} [T_1 \ T_2 \ T_3 \ T_4] \end{aligned} \quad (35)$$

From (35), it is evident that $T$ is a subset of $u_h$, while $R$ can be directly incorporated into the control weight matrix $R_h$. Integrating $R$ alone into $R_h$ ensures minimization of power consumption but neglects the weighting requirements for the control objectives. To address this, a combined criterion is proposed to adapt the weights in $R_h$, ensuring effective speed and stability control while simultaneously optimizing energy consumption. Consequently, a reference weight for the control input torques is derived as a function of $e_x$:

$$R_{\tau} = r_{\tau,2} - \frac{(r_{\tau,2} - r_{\tau,1})}{1 + e^{-\frac{2\gamma_4}{\overline{e_x} - \underline{e_x}}(|e_x| - \frac{\overline{e_x} + e_x}{2})}} \quad (36)$$

where $r_{\tau,2}$ and $r_{\tau,1}$ are respectively the upper and lower saturated boundaries of $R_\tau$, and $\gamma_4$ as $\gamma_3$. This reference weight ensures the effective achievement of control objectives
<figure><img src="image_7.png" /><figcaption>Fig. 6: Adaptive weights on control inputs in the holistic architecture.</figcaption></figure>

while maintaining robustness and preventing oscillations and overshoots. To preserve the order (ascending or descending) of the scheduled weights $\omega_i/\eta_i^{sgn(T_i)}$ while operating within the scale of $R_\tau$ (which is much smaller than the range of $\omega/\eta^{sgn(T_i)}$), the weights on the four torques are defined as in (37), where $\kappa_1 \in (0, 1]$ and $\kappa_2 > 1$ are scaling factors.

$$ \Xi = \left\{ \frac{\omega_1}{\eta_1^{sgn(T_1)}}, \frac{\omega_2}{\eta_2^{sgn(T_2)}}, \frac{\omega_3}{\eta_3^{sgn(T_3)}}, \frac{\omega_4}{\eta_4^{sgn(T_4)}} \right\} \quad (37a) $$

$$ \lambda_i^{sc} = \left[ \kappa_1 + (\kappa_2 - \kappa_1) \frac{\frac{\omega_i}{\eta_i^{sgn(T_i)}} - \min(\Xi)}{\max(\Xi) - \min(\Xi)} \right] \quad (37b) $$

$$ R_{T_i} = R_{\tau} \times \lambda_i^{sc}, \quad i \in \{1, 2, 3, 4\} \quad (37c) $$

The form of the torque weights and their corresponding parameters are illustrated in Fig. 6. The weight on the steering angle $R_\delta$ follows a similar structure to the hierarchical case as in (21). From (36) and (37), the structure of $R_\tau$, integrated into $R_{T_i}$, ensures effective control. The scaling parameters $\lambda_i^{sc}$ minimize energy consumption by preserving the order of $\omega_i/\eta_i^{sgn(T_i)}$, assigning higher torques to motors with lower ratios.

## IV. VALIDATION AND DISCUSSION

The proposed architectures are implemented in Simulink/Matlab, tested on a full nonlinear vehicle model [38], and validated using the SCANeR Studio professional vehicle dynamics simulator. SCANeR Studio is widely adopted in the automotive industry, providing high-fidelity and reliable validation. The MPC controller in both architectures operates at a frequency of 50 Hz, with a prediction horizon of $N_p = 8$ s. The parameters of the full nonlinear vehicle model are listed in Table I.

The performance effectiveness of the architectures is evaluated using the following metrics: Stability Index ($SI$) (12); maximum lateral error ($max(|e_y|)$); root mean square of the lateral error ($rms(e_y)$); and energy consumption $E_m = \int_0^t \sum_{i=1}^4 T_i \omega_i / \eta_i^{sgn(T_i)} dt$ [16]. Additional energy consumption metrics are considered, including equivalent mass reduction and battery State of Charge (SoC) [31], computed using Coulomb counting. The equivalent mass reduction represents the estimated decrease in vehicle mass required for the reference strategy to match the energy efficiency of the optimal strategy (see [31]).

To evaluate the effectiveness of the proposed architectures, a case study is conducted initially under three representative driving scenarios: steady-state operation on complex road

<table><caption>TABLE I: Parameters of the nonlinear vehicle dynamics model</caption><thead><tr><th>Symbol</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td><i>M</i></td><td>Total vehicle mass</td><td>1719 [kg]</td></tr><tr><td><i>m</i><sub>ij</sub></td><td>Wheel mass</td><td>12.25 [kg]</td></tr><tr><td><i>ρ</i><sub>air</sub></td><td>Air density</td><td>1.3 [kg/m<sup>3</sup>]</td></tr><tr><td><i>s</i><sub>surf</sub></td><td>Frontal area</td><td>2.31 [m<sup>2</sup>]</td></tr><tr><td><i>C</i><sub>d</sub></td><td>Aerodynamic drag coefficient</td><td>0.314</td></tr><tr><td><i>t</i><sub>f</sub>, <i>t</i><sub>r</sub></td><td>Front and rear half-track width</td><td>0.7615 [m]</td></tr><tr><td><i>l</i><sub>f</sub></td><td>Distance from CG to front axle</td><td>1.195 [m]</td></tr><tr><td><i>l</i><sub>r</sub></td><td>Distance from CG to rear axle</td><td>1.513 [m]</td></tr><tr><td><i>I</i><sub>z</sub></td><td>Yaw moment of inertia</td><td>3300 [kg·m<sup>2</sup>]</td></tr><tr><td><i>I</i><sub>w</sub></td><td>Wheel rotational inertia</td><td>1.83 [kg·m<sup>2</sup>]</td></tr><tr><td><i>r</i></td><td>Effective wheel radius</td><td>0.316 [m]</td></tr><tr><td><i>C</i><sub>f</sub>, <i>C</i><sub>r</sub></td><td>Front and rear tire cornering stiffness</td><td>85275 [N/rad]</td></tr></tbody></table>

geometries, urban driving at medium-to-high speeds, and peri-urban environments at very high speeds. The hierarchical and holistic MPC architectures are compared against decentralized Super-Twisting Sliding Mode (STSM) and semi-centralized LPV/$H_\infty$ controllers [16]. The latter employs an STSM controller for longitudinal speed control and LPV/$H_\infty$ for path-tracking and lateral stability. The look-ahead distance $Ls$ (1) is fixed at $5$ m in these scenarios to isolate the impact of the control strategies. Subsequent tests then employ the adaptive $Ls$ formulation developed in [13].

Second, the low-level performance is evaluated, focusing on motor energy consumption and tire stability. For this analysis, the hierarchical MPC is denoted as “mpc - hi”, the holistic MPC without incorporating the high-efficiency constraints (30) as “mpc - ho”, and the holistic MPC with the joint constraints (30) as “mpc - ho - high η”. To illustrate the effectiveness of the holistic MPC in minimizing energy consumption, its performance is compared with architectures employing hierarchical MPC at the upper level with torque allocation methods from the literature targeting energy efficiency. Specifically, the comparison includes the online and offline optimization allocation strategies presented in [16] and the analytical modeling strategy, referred to as “AM”, in [30].

Third, the performance of the hierarchical and holistic MPC architectures is benchmarked against state-of-the-art MPC strategies. The evaluation replicates the scenarios outlined in previous studies, including Double Lane Change (DLC) tests and an obstacle avoidance maneuver at very high speeds under low adhesion conditions. The comparative analysis draws on methods presented in [12], [20], [23], and [26]. Other scenarios include a sudden intrusion under mixed-traction conditions and city driving at intersections. To further validate the robustness of the proposed adaptive weighting mechanism, tests are conducted under parameter uncertainties and supported by 100 Monte Carlo simulations for statistical reliability.

### A. High-Level Performance: Accuracy and Effectiveness

The first scenario unfolds on the track given in Fig. 7-(a). The road curvature and the corresponding reference lateral acceleration, $a_{y,ref} = \rho v_x^2$, are illustrated in Fig.7-(b). The prescribed speed profile on this track poses a significant challenge to vehicle stability, as $a_{y,ref}$ exceeds $6$ m/s$^2$,
<figure><img src="image_8.png" /><figcaption>Fig. 7: Scenario 1: Steady-state performance evaluation on complex road geometries.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8: Scenario 2: Urban driving performance evaluation at medium-to-high speeds.</figcaption></figure>

providing a demanding test for evaluating the accuracy of path-tracking and speed control. The lateral errors and speed profiles of the four controllers are presented in Figs. 7-(c) and 7-(d), respectively. While the *SM* and *LPV/H*<sub>∞</sub> architectures exhibited steady-state lateral errors exceeding 0.1 *m*, the hierarchical MPC achieved approximately 0.02 *m*, and the holistic MPC further reduced this to below 0.01 *m*, reflecting a tenfold improvement. This enhancement is further supported by the *rms(e)*<sub>y</sub> and *max(|e*<sub>y</sub>|) metrics detailed in Table II. The hierarchical MPC demonstrated an accuracy of 9 *mm*, while the holistic MPC achieved a higher precision with a 4 *mm* accuracy. For speed control, both the hierarchical and holistic MPC controllers demonstrated comparable performance, achieving superior steady-state error compared to the *SM* controller. In contrast, the *LPV/H*<sub>∞</sub> controller exhibited a slight tendency to overshoot, due to the decoupling of the speed controller from the centralized path-tracking and stability controller.

The second scenario takes place on the track shown in Fig. 8-(a). The desired speed profile is deliberately shifted upwards to assess the accuracy over comfort in scenarios with varying speed and curvature. As illustrated in Fig. 8-(b), the road curvature and reference lateral acceleration frequently reach 6 *m/s*<sup>2</sup>, imposing slight demands on vehicle stability. The lateral errors are depicted in Fig. 8-(c), where the *SM* and *LPV/H*<sub>∞</sub> control strategies exhibit noticeable steady-state errors. In contrast, the proposed hierarchical and holistic MPC architectures maintain near-zero lateral errors, demonstrating their superior path-tracking accuracy. The longitudinal speed profiles in Fig. 8-(d) indicate comparable performance across all strategies, except for the *LPV/H*<sub>∞</sub> controller, which exhibits notable overshoot.

The third scenario is conducted on the track shown in Fig.

8-(a), but with significantly higher velocities to solicit chassis and tire stability under extreme conditions. The Stability Index for all strategies, including a baseline without the stability objective activated, is presented in Fig. 9-(a). The road curvature and reference lateral acceleration are shown in Fig. 9-(b), lateral errors in Fig. 9-(c), and longitudinal velocities in Fig. 9-(d). Without the stability objective activated, the vehicle loses stability and veers off the road, highlighting its critical importance. The high velocity profile pushes *a*<sub>y,ref</sub> to approximately 9 *m/s*<sup>2</sup>, placing the vehicle on the verge of instability. The hierarchical and holistic MPC architectures outperform the *SM* and *LPV/H*<sub>∞</sub> strategies in speed tracking, as reflected in their reduced velocity deviations. Abrupt velocity drops, particularly at *t* = 90 *s*, have a pronounced impact on lateral errors. Therefore and according to Table II, although the *max(|e*<sub>y</sub>|) is slightly higher in the MPC architectures than *SM*, both of them exhibited lower *rms(e)*<sub>y</sub> than the other strategies, with a better accuracy for the holistic one. Hence, in a highly complex and critical situation, and with abrupt velocity drops, the proposed MPC architectures demonstrated superior accuracy in path-tracking and speed control.

The performed scenarios examine the accuracy and effec-
tiveness of the four control architectures for a constant look-
ahead distance *Ls*. However, the variation of *Ls* significantly
affects path-tracking accuracy and vehicle stability [13]. To
address this, the explicit formulation of *Ls* developed in [13]
is employed to examine the lateral errors under adaptive *Ls*.
The second scenario is re-executed under identical conditions
with adaptive *Ls*, and the results are summarized in Table III.
The findings demonstrate a notable improvement across all
control strategies. Specifically, the *SM* controller achieved a
reduction exceeding 50 % in both *max(|e*<sub>y</sub>|) and *rms(e*<sub>y</sub>),
<figure><img src="image_10.png" /><figcaption>Fig. 9: Scenario 3: Peri-urban driving performance evaluation at high speeds and aggressive lateral dynamics.</figcaption></figure>

<table><caption>TABLE II: Performance metrics of the control architectures</caption><thead><tr><th rowspan="2">Scenario cases</th><th rowspan="2">Performance metrics</th><th colspan="4">Performance variables of the control architectures in different scenarios</th></tr><tr><th>SM</th><th>LPV/H<sub>∞</sub></th><th>Hierarchical mpc</th><th>Holistic mpc</th></tr></thead><tbody><tr><td rowspan="2">Scenario 1</td><td>max(|e<sub>y</sub>|) (m)</td><td>0.115</td><td>0.14</td><td>0.025</td><td><b>0.018</b></td></tr><tr><td>rms(e<sub>y</sub>) (m)</td><td>0.086</td><td>0.073</td><td>0.009</td><td><b>0.004</b></td></tr><tr><td rowspan="2">Scenario 2</td><td>max(|e<sub>y</sub>|) (m)</td><td>0.162</td><td>0.245</td><td><b>0.097</b></td><td>0.107</td></tr><tr><td>rms(e<sub>y</sub>) (m)</td><td>0.061</td><td>0.089</td><td><b>0.012</b></td><td><b>0.012</b></td></tr><tr><td rowspan="2">Scenario 3</td><td>max(|e<sub>y</sub>|) (m)</td><td><b>0.226</b></td><td>0.568</td><td>0.269</td><td>0.266</td></tr><tr><td>rms(e<sub>y</sub>) (m)</td><td>0.075</td><td>0.145</td><td>0.048</td><td><b>0.044</b></td></tr></tbody></table>

while the $LPV/H_{\infty}$ approach showed an enhancement of approximately 66 %. The MPC architectures outperformed both, achieving over 30 % reduction in $rms(e_y)$, with the holistic MPC delivering superior overall accuracy.

## B. Low-Level Performance: Energy and Efficiency

In order to examine the strategy presented for energy economy in the holistic architecture, the widely known NEDC and WLTP cycles are employed [33]. In this test scenario, the vehicle operates on a highway, following a straight path. At the upper-level control, the hierarchical MPC is tested against $SM$ and $LPV/H_{\infty}$ in the viewpoint of energy consumption. Due to the inherent optimality of MPC, adaptive weight adjustments, and constraints on control input increments, the hierarchical MPC achieved the lowest energy consumption. Hence, to compare energy consumption in a concise manner, the mentioned literature torque allocation strategies [16], [30] are integrated with the hierarchical MPC. The $SM$ controller in the upper-level, combined with the dynamic load distribution method presented in [16], serves as the baseline for comparison with the remnant strategies. The dynamic load distribution strategy determines torque allocation for the four wheels based on tire load variations, which are influenced by acceleration, mass, and vehicle parameters, without relying on optimization.

The NEDC velocity cycle, illustrated in Fig. 10-(a), is employed to evaluate the speed profiles of the $SM$, hierarchical, and holistic MPC architectures. The corresponding energy consumption results for the baseline strategy, the hierarchical MPC combined with various torque allocation strategies, and the holistic MPC with and without energy-efficiency constraints (27) are presented in Fig. 10-(b). While the baseline strategy marked the highest energy consumption, the hierarchical MPC with the torque allocation strategies demonstrated substantial energy savings. Referring to Table IV, while highly efficiency strategies like "$mpc-hi$ + online" and "$mpc-hi + AM$" exhibited up to 40 % and 45 % energy savings respectively, the holistic $mpc-ho$ approach achieved up to 58 %. Energy savings increased to 70 % with the

<table><caption>TABLE III: Performance under an adaptive look-ahead distance</caption><thead><tr><th rowspan="2">Controller</th><th colspan="2">SM</th><th colspan="2">LPV/H<sub>∞</sub></th><th colspan="2">Hierarchical mpc</th><th colspan="2">Holistic mpc</th></tr><tr><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th></tr></thead><tbody><tr><td>Fixed Ls</td><td>0.162</td><td>0.061</td><td>0.245</td><td>0.089</td><td><b>0.097</b></td><td><b>0.012</b></td><td>0.107</td><td><b>0.012</b></td></tr><tr><td>Adaptive Ls [13]</td><td>0.071</td><td>0.026</td><td>0.187</td><td>0.029</td><td><b>0.081</b></td><td>0.0084</td><td>0.088</td><td><b>0.0082</b></td></tr><tr><td>Relative ratio</td><td>55.9%</td><td>55.6%</td><td>23.3%</td><td>66.8%</td><td>17.3%</td><td>31.4%</td><td>18.1%</td><td>34.5%</td></tr></tbody></table>

"mpc-ho" - high $\eta$" strategy, which enforces motor operation within high-efficiency zones while also achieving the highest equivalent mass reduction and SoC values. The combined operating points ($T, \omega$) of the four in-wheel motors are mapped onto their efficiency maps in Fig. 11. The output torques are analyzed for the holistic MPC architecture with and without the energy-efficiency constraints defined in (27). The analysis confirms that the imposed constraints effectively guide the motors to operate within the predefined polynomial boundaries during both traction and braking phases, wherever feasible. In situations where high-efficiency operation conflicts with the control objectives, the slack variables in (32) allow controlled constraint relaxation, maintaining system performance while accommodating operational limitations.

The WLTP cycle, illustrated in Fig. 12-(a), is employed to further evaluate the performance of the proposed control architectures. The energy consumption for each strategy is shown in Fig. 12-(b), while the corresponding energy savings are summarized in Table IV. The efficient hierarchical architectures demonstrated notable energy savings, achieving reductions of 20 %, 22 %, and 27 % for the offline, online, and AM torque allocation strategies combined with the hierarchical MPC, respectively. In comparison, the holistic $mpc-ho$ delivered approximately 45 % energy savings. This value was further enhanced to 49 % by enforcing the energy-efficiency constraints, demonstrating the superior potential of the holistic approach in reducing energy consumption. Additionally, this approach effectively preserves battery SoC while achieving the highest equivalent mass reduction compared to the reference. The motor efficiencies corresponding to the in-wheel torque outputs are shown in Fig. 13. The proposed method for incorporating high-efficiency polynomial constraints demonstrates its effectiveness in guiding motor operation within the targeted high-efficiency regions.

The holistic architecture achieves significant energy savings compared to existing approaches, though this optimality comes at the cost of increased computational complexity. Table V summarizes the performance, energy consumption under the NEDC cycle, energy savings, and computational time, offering a direct comparison with various literature strategies. While
<table><caption>TABLE IV: Comparison via direct energy consumption metrics for each strategy under the NEDC and WLTP cycles.</caption><thead><tr><th rowspan="2">Cycles</th><th rowspan="2">Metrics</th><th colspan="8">Control Strategy</th></tr><tr><th>Reference SM</th><th>mpc - hi</th><th>mpc - hi + offline</th><th>mpc - hi + online</th><th>mpc - hi + AM</th><th>mpc - ho</th><th>mpc - ho - high η</th></tr></thead><tbody><tr><td rowspan="4">NEDC cycle</td><td>Total (E<sub>m</sub>) (kJ)</td><td>5806.8</td><td>3723.6</td><td>3506.6</td><td>3468.7</td><td>3173.7</td><td>2397.2</td><td><strong>1731.5</strong></td></tr><tr><td>Relative ratio w.r.t reference SM</td><td>reference</td><td>35.8 %</td><td>39.6 %</td><td>40.2 %</td><td>45.3 %</td><td>58.7 %</td><td><strong>70.1 %</strong></td></tr><tr><td>Energy consumption (kWh)</td><td>1.613</td><td>1.034</td><td>0.974</td><td>0.963</td><td>0.882</td><td>0.666</td><td><strong>0.481</strong></td></tr><tr><td>Equivalent weight reduction (kg)</td><td>reference</td><td>616.7</td><td>680.9</td><td>692.2</td><td>779.5</td><td>1009.4</td><td><strong>1206.4</strong></td></tr><tr><td rowspan="4">WLTP cycle</td><td>SoC (starting from 0.8)</td><td>0.7266</td><td>0.7529</td><td>0.7557</td><td>0.7562</td><td>0.7599</td><td>0.7697</td><td><strong>0.7781</strong></td></tr><tr><td>Total (E<sub>m</sub>) (kJ)</td><td>12361.8</td><td>10488</td><td>9837.1</td><td>9537.9</td><td>9013.1</td><td>6683.4</td><td><strong>6257.3</strong></td></tr><tr><td>Relative ratio w.r.t reference SM</td><td>reference</td><td>15.15 %</td><td>20.4 %</td><td>22.8 %</td><td>27.1 %</td><td>45.9 %</td><td><strong>49.3 %</strong></td></tr><tr><td>Energy consumption (kWh)</td><td>3.434</td><td>2.913</td><td>2.733</td><td>2.645</td><td>2.504</td><td>1.856</td><td><strong>1.738</strong></td></tr><tr><td rowspan="4">WLTP cycle</td><td>Equivalent weight reduction (kg)</td><td>reference</td><td>260.5</td><td>351.1</td><td>392.6</td><td>465.6</td><td>789.6</td><td><strong>848.8</strong></td></tr><tr><td>SoC (starting from 0.8)</td><td>0.6439</td><td>0.6675</td><td>0.6757</td><td>0.6795</td><td>0.6862</td><td>0.7156</td><td><strong>0.7209</strong></td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>(a) Longitudinal velocities</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Energy consumption</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(a) Longitudinal velocities</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b) Energy consumption</figcaption></figure>

<caption>Fig. 10: Energy evaluation under the NEDC cycle.</caption>

<figure><img src="image_15.png" /><figcaption>Fig. 11: Operating points and efficiencies of in-wheel motors under the NEDC cycle.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13: Operating points and efficiencies of in-wheel motors under the WLTP cycle.</figcaption></figure>

the holistic controller delivers superior tracking accuracy and substantial energy reduction, it incurs a higher computational cost, highlighting the trade-off between optimality and complexity. In addition, a comparison with the recent energy-oriented MPC framework in [39] under the WLTP cycle shows that the proposed holistic MPC achieves 3.64 kJ/kg, compared to 9.88 kJ/kg reported in [39], corresponding to a 63 % improvement in energy efficiency.

**Remark 5.** While the holistic MPC controller achieves substantial energy savings of 70%, this comparison is made against the SM controller with a classical torque allocation strategy based on dynamic load distribution, both of which are not optimal control approaches. However, Table IV shows that, compared to the hierarchical MPC with advanced energy-saving allocation strategies, the holistic controller achieves a 50 % reduction relative to [16] and 45 % relative to [30] under the NEDC cycle, and 34 % and 30 % reductions under the WLTP cycle. Hence, although the upper-level and lower-

<table><caption>TABLE V: Performance, energy, and computational trade-offs.</caption><thead><tr><th rowspan="2">Strategy</th><th colspan="4">Metrics</th></tr><tr><th>rms(e<sub>g</sub>) [m]</th><th>Energy [kWh]</th><th>Energy Saving [%]</th><th>Comp. Time [ms]</th></tr></thead><tbody><tr><td>Strategy in [30]</td><td>0.05</td><td>0.88</td><td>-</td><td><strong>0.00038</strong></td></tr><tr><td>Strategy in [16]</td><td>0.08</td><td>0.96</td><td>-</td><td>2.8</td></tr><tr><td>Strategy in [31]</td><td>0.1</td><td>1.44</td><td>-</td><td>0.06</td></tr><tr><td>Holistic (ours)</td><td><strong>0.01</strong></td><td><strong>0.48</strong></td><td>45 vs. [30]<br>50 vs. [16]<br>66 vs. [31]</td><td>6.8</td></tr></tbody></table>

Note: Energy consumption is evaluated under the NEDC cycle.

level controllers are individually optimized, global optimality is not achieved without a unified design. This highlights the importance of holistic optimality and its significant impact on overall energy efficiency.

### C. Performance Comparison with State-of-Art MPC Methods

The proposed MPC architectures are benchmarked against state-of-the-art MPC methods under identical test conditions.
<figure><img src="image_17.png" /><figcaption>Fig. 14: DLC test at $v_x = 22.2 \text{ m/s}$ with $\mu = 0.85$.</figcaption></figure>

<table><caption>TABLE VI: Performance under DLC test: $v_x = 22.2 \text{ m/s}$, $\mu = 0.85$</caption><thead><tr><th rowspan="2">Metric</th><th colspan="4">Controller</th></tr><tr><th>Hierarchical MPC</th><th>Holistic MPC</th><th>MPC in [23]</th><th>MPC in [26]</th></tr></thead><tbody><tr><td>max(|e<sub>y</sub>|)</td><td>0.019</td><td><b>0.014</b></td><td>0.1</td><td>0.587</td></tr><tr><td>rms(e<sub>y</sub>)</td><td>0.007</td><td><b>0.006</b></td><td>0.04</td><td>0.188</td></tr></tbody></table>

<table><caption>TABLE VII: Performance for obstacle avoidance: $v_x = 33.3 \text{ m/s}$, $\mu = 0.3$</caption><thead><tr><th rowspan="2">Metric</th><th colspan="3">Controller</th></tr><tr><th>Hierarchical MPC</th><th>Holistic MPC</th><th>MPC in [12]</th></tr></thead><tbody><tr><td>max(|e<sub>y</sub>|)</td><td><b>0.41</b></td><td>0.42</td><td>0.8</td></tr><tr><td>rms(e<sub>y</sub>)</td><td>0.18</td><td><b>0.15</b></td><td>0.28</td></tr><tr><td>Relative ratio w.r.t [12]: max(|e<sub>y</sub>|)</td><td>48.8 %</td><td>46.5 %</td><td>-</td></tr><tr><td>Relative ratio w.r.t [12]: rms(e<sub>y</sub>)</td><td>35.7 %</td><td>45.5 %</td><td>-</td></tr></tbody></table>

The comparison focuses on path-tracking accuracy, replicating the scenarios from the respective literature to ensure consistency. The first evaluation considers a Double Lane Change (DLC) maneuver at a reference speed of 80 km/h and an adherence coefficient of $\mu = 0.85$. The resulting vehicle trajectory and lateral errors are illustrated in Fig. 14. The coupled longitudinal-lateral MPC controller from [23] targets offset-free path-tracking, while [26] introduces a path-tracking MPC with lateral stability control. Table VI reports the lateral error metrics for the proposed hierarchical and holistic MPC architectures alongside these existing approaches. The offset-free controller in [23] surpasses [26] in path-tracking accuracy. However, the proposed MPC architectures outperform both, achieving an accuracy of 7 mm for the hierarchical approach and 6 mm for the holistic architecture. The holistic architecture yields the highest path-tracking accuracy overall.

The second test involves a sudden obstacle avoidance scenario under extreme conditions, with a snowy road surface ($\mu = 0.3$) and a reference speed of 120 km/h. The reference lateral acceleration exceeds 6.5 $m/s^2$, driven by the combined effects of road curvature and desired speed. The low adhesion further increases the complexity of the scenario. The vehicle trajectory and lateral errors are shown in Fig. 15. In this test, the proposed hierarchical and holistic MPC architectures are compared with the holistic MPC from [12], with the corresponding metrics summarized in Table VII. Both proposed architectures achieve comparable $max(|e_y|)$ values, with the proposed holistic architecture demonstrating improved rms(e_y) accuracy and outperforming [12]. The inclusion of the stability control objective with the adaptive Ls, and the adaptive weighting mechanism, particularly under varying $\mu$,

<figure><img src="image_18.png" /><figcaption>Fig. 15: Obstacle avoidance test at $v_x = 33.3 \text{ m/s}$ with $\mu = 0.3$.</figcaption></figure>

<table><caption>TABLE VIII: Performance of the controllers under different road adhesion</caption><thead><tr><th rowspan="2">Controller</th><th colspan="2">μ = 0.7</th><th colspan="2">μ = 0.6</th><th colspan="2">μ = 0.5</th></tr><tr><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th><th>max(|e<sub>y</sub>|)</th><th>rms(e<sub>y</sub>)</th></tr></thead><tbody><tr><td>MPC in [20]</td><td>0.274</td><td>0.105</td><td>0.569</td><td>0.187</td><td>0.641</td><td>0.22</td></tr><tr><td>Hierarchical MPC</td><td>0.012</td><td>0.005</td><td>0.018</td><td>0.009</td><td>0.031</td><td>0.014</td></tr><tr><td>Relative ratio w.r.t [20]</td><td>95.3 %</td><td>94.5 %</td><td>96.7 %</td><td>95.1 %</td><td>95.2 %</td><td>93.5 %</td></tr><tr><td>Holistic MPC</td><td><b>0.009</b></td><td><b>0.004</b></td><td><b>0.017</b></td><td><b>0.008</b></td><td><b>0.029</b></td><td><b>0.014</b></td></tr><tr><td>Relative ratio w.r.t [20]</td><td>96.4 %</td><td>95.7 %</td><td>96.9 %</td><td>95.2 %</td><td>95.4 %</td><td>93.6 %</td></tr></tbody></table>

contributes to the enhanced path-tracking performance under these extreme conditions. Compared to [12], the proposed holistic architecture shows over a 45 % improvement in path-tracking metrics.

The third evaluation considers a series of DLC maneuvers at a reference speed of 54 km/h, during which the friction coefficient gradually decreases from 0.7 to 0.5. These tests replicate the conditions presented in [20]. A summary of the lateral error metrics is provided in Table VIII. Significant improvements are observed with both proposed architectures, with relative reductions exceeding 93 % in rms(e_y) and max(|e_y|) across all cases. The holistic architecture achieves the highest accuracy overall.

### D. Sudden Intrusion under Mixed Traction and City Driving

To rigorously evaluate the performance and robustness of the proposed adaptive MPC architectures, a sudden intrusion scenario is introduced, involving a high-speed emergency lane change under mixed-traction conditions (Fig. 16 and Fig. 17). In this scenario, the road surface transitions abruptly from dry asphalt ($\mu = 1$) to icy ($\mu = 0.2$), and then to a semi-wet surface ($\mu = 0.8$). As depicted in Fig. 17, a surrounding vehicle suddenly cuts into the ego vehicle's lane, prompting an evasive maneuver. The initial part of the lane change is executed under normal traction. However, mid-maneuver, the vehicle encounters a patch of ice before regaining partial grip on a semi-wet surface. The maneuver occurs at a high speed of 130 km/h, forcing an extreme situation that challenges the ability of the controller to maintain stability, path-tracking accuracy, and robust performance under abrupt changes in road friction.

Fig. 16 illustrates the real-time evolution of the adaptive weights applied to states and control inputs in the hierarchical and holistic architectures. The lateral tracking errors shown in Fig. 17 highlight the performance of the holistic MPC, the hierarchical MPC, and two baseline variants: a non-adaptive hierarchical controller where the path-tracking weights do not vary with adhesion coefficient $\mu$ ("non-adaptive w.r.t. $\mu$"), and
<figure><img src="image_19.png" /><figcaption>Fig. 16: Real-time profiling of the weights on states and control inputs in the sudden intrusion scenario with mixed traction.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 17: Sudden intrusion scenario under mixed traction.</figcaption></figure>

a hierarchical controller with the stability objective deactivated (“stability off”).

Throughout the maneuver, the adaptive weights evolve smoothly and continuously in real time, preventing abrupt variations that could destabilize the controller. The weight on the speed error state ($Q_{e_x}$) is initially high to promote rapid convergence to the desired velocity and gradually decreases as the target speed is approached. Simultaneously, the weight on the torque input ($R_{T_m}$) starts low to enable prompt actuation and progressively increases as the demand for control effort declines. In the “stability off” case, initial tracking accuracy is high due to exclusive prioritization of path-following. When the friction coefficient drops, the absence of stability regulation results in significant deviation, approaching the lateral safety margins. In the “non-adaptive w.r.t. $\mu$” case, the controller performs adequately under nominal conditions but fails to respond to the sudden loss of adhesion, leading to loss of control and lane departure. Meanwhile, the adaptive MPCs effectively modulate the weight on yaw rate ($Q_{\psi}$), increasing it during lateral motion and peaking during the low-traction segment, before tapering off as grip recovers. The corresponding input weight on steering angle ($R_{\delta}$) is simultaneously reduced to maintain responsiveness. The adaptive strategy relaxes path-tracking objectives ($Q_{\psi} \perp R_{\delta}$) when stability must take precedence, allowing the vehicle to navigate adverse conditions with controlled tracking errors. This dynamic adjustment enables the controller to: 1) activate stability control precisely when necessary, preventing excessive deviation caused by skidding at the cost of a modest, anticipated increase in lateral error, and 2) respond to changes in adhesion by balancing path-tracking, stability, and actuation effort, maintaining robust performance and preserving traction under extreme and conflicting conditions.

<figure><img src="image_21.png" /><figcaption>Fig. 18: City intersection driving and corresponding box plot of lateral errors. A total of 12 tests are conducted, with three maneuvers (straight, left turn, and right turn) performed from each of the four approach directions: east, north, west, and south.</figcaption></figure>

The holistic architecture further improves adaptability by modulating torque input weights based on motor torque demand, rotational speed, and efficiency (see Fig. 16-(c)). A lower $R_{T_m}$ reduces the penalization of the associated motor torque, allowing increased output when dynamically advantageous. This adaptive weighting improves energy economy and preserves tire stability without compromising control authority.

To evaluate the controller and the adaptability of the adap-
<figure><img src="image_22.png" /><figcaption>Fig. 19: Percentage of 20% and 50% random uncertainty on cornering stiffness and mass. 100 Monte Carlo simulations are performed in each case.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 20: Box plot showing the effect of uncertainty in key vehicle parameters on the performance metrics. The effect of the adaptive weights is shown in (a), (b), and (c), compared to fixed weights in (d).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 21: Computational time comparison (in ms).</figcaption></figure>

tive weights in city driving, several tests are conducted on an intersection map, as shown in Fig. 18. From each direction (west in panel (a), north in panel (b), east in panel (c), and south in panel (d)), three maneuvers are performed: straight, right turn, and left turn. The scenarios reflect typical daily driving patterns, with speed profiles ranging from stops at intersections to slow turning and gradual acceleration. The lateral tracking errors are summarized as a box plot in panel (e). Both architectures achieve comparable performance, with an average lateral tracking error of approximately 2 cm, confirming the effectiveness, adaptability, flexibility, and robustness of the adaptive weighting across a wide range of operating conditions.

## E. Robustness and Computational Analysis with a Recap

The predictive model is formulated with minimum order, excluding lateral-longitudinal coupling and nonlinearities, while adaptive weights are designed to enhance performance and improve robustness. To assess robustness, uncertainties of 20% and 50% are introduced into key parameters, including vehicle mass and front and rear cornering stiffness. A total of 100 Monte Carlo simulations are conducted with random uncertainty sampling applied to these parameters, as shown in Fig. 19.

The performance objective errors are depicted using box plots in Fig. 20. Panels (a), (b), and (c) showcase results with adaptive weights, while panel (d) illustrates the outcomes with fixed weights, highlighting the impact of adaptive weighting on robustness. The adaptive weights significantly enhance robustness across all metrics. Under a 20% uncertainty, rms($e_y$) and $\max(|e_y|)$ improve from approximately 0.6 m and 2 m (Fig. 20-(d)) to below 0.013 m and 0.025 m (Fig. 20-(a)),

respectively. Similarly, for speed control, rms($e_x$) decreases from over 3 m/s to below 0.43 m/s in both architectures. Even under a 50% uncertainty, the robustness of the proposed controllers remains evident, with rms($e_y$) around 0.02 m for the hierarchical architecture and approximately 0.012 m for the holistic one (Fig. 20-(b)). In contrast, fixed weights result in significant control deterioration, leading to road divergence under a 50% uncertainty level. This confirms the effectiveness of adaptive weighting in preserving control robustness under varying levels of parameter uncertainty.

The computational efficiency of each architecture is evaluated by comparing the hierarchical MPC with a lower-level torque allocation strategy against the holistic architecture. Simulations are conducted using Simulink Real-Time on a laptop equipped with an Intel Core i9-12950HX CPU at 2.3 GHz. Computational times are presented in Fig. 21, demonstrating that all approaches achieve real-time execution at 50 Hz. The hierarchical MPC combined with the analytical modeling-based torque allocation strategy [30] stands out for its computational efficiency, ensuring tire stability and achieving energy savings. Although the holistic architecture requires higher computation time, it outperforms the hierarchical approach in overall performance and energy efficiency.

Overall, the holistic architecture integrates a single controller based on a minimum-order predictive model, capable of improving path-tracking and speed control accuracy, enhancing robustness, maintaining vehicle and tire stability, and ensuring high-efficiency motor operation. It yields significant energy savings while performing effectively under challenging conditions, including high speeds, sharp curvatures, and low-adherence roads. Additional tests conducted on the SCANeR Studio simulator, showcasing variations in road adherence and performance on snowy roads, are available in a video at: "https://youtu.be/4nF5zanIO4Ahttps://youtu.be/4nF5zanIO4A".

**Remark 6.** The employed high-fidelity vehicle model accurately captures real-world dynamics, including all relevant nonlinearities, and all control methods are evaluated under identical conditions. Consequently, the relative improvement of the proposed holistic controller over existing robust controllers is more meaningful than the absolute rms($e_y$) values. Compared to other MPC-based approaches, the proposed holistic controller achieves significantly higher accuracy. Several factors contribute to this enhancement. The holistic MPC architecture integrates direct energy-efficient torque allocation, minimizing abrupt control actions that degrade path-tracking. Its computational efficiency enables real-time operation at 50 Hz, allowing finer trajectory corrections and reduced tracking errors. The most significant factor is the proposed
adaptive weight mechanism, which improves path-tracking accuracy and robustness, particularly when stability objectives are included. Additionally, the look-ahead distance contributes to improved tracking precision. In real-world conditions, autonomous vehicles are subject to sensor noise, road irregularities, GPS errors, sensor drift, and latency. Consequently, actual implementations may exhibit slightly higher errors due to external disturbances. Nevertheless, the adaptive weighting ensures robustness, and the proposed strategy remains fundamentally superior to existing methods.

## V. CONCLUSION

This study presents two novel MPC architectures for autonomous over-actuated in-wheel vehicles, addressing energy efficiency, stability, and control performance under demanding situations. The hierarchical centralized MPC integrates path-tracking, speed control, and stability objectives using a minimal-order predictive model, complemented by an adaptive weighting mechanism that dynamically prioritizes stability in critical conditions. The holistic MPC extends the hierarchical design by embedding energy-efficient torque allocation and tire stability constraints, further improving energy savings and operational efficiency. Comprehensive validation demonstrates that the holistic architecture achieves substantial energy savings while maintaining high accuracy and robustness in path-tracking and speed control, even under high-speed, high-curvature, and low-adhesion conditions. Future work will integrate fault-tolerant strategies to ensure consistent performance in the presence of actuator faults and extend validation to hardware-in-the-loop testing and real-world experiments.

## REFERENCES

[1] R. Quirynen, S. Safaoui, and S. Di Cairano, "Real-time mixed-integer quadratic programming for vehicle decision-making and motion planning," *IEEE Transactions on Control Systems Technology*, vol. 33, no. 1, pp. 77–91, 2025.

[2] F. Tarhini, R. Talj, and M. Doumiati, "Safe and energy-efficient jerk-controlled speed profiling for on-road autonomous vehicles," *IEEE Transactions on Intelligent Vehicles*, pp. 1–16, 2024.

[3] M. Ammour, R. Orjuela, and M. Basset, "A mpc combined decision making and trajectory planning for autonomous vehicle collision avoidance," *IEEE Transactions on Intelligent Transportation Systems*, vol. 23, no. 12, pp. 24 805–24 817, 2022.

[4] H. D. Nguyen, D. Kim, Y. S. Son, and K. Han, "Linear time-varying mpc-based autonomous emergency steering control for collision avoidance," *IEEE Transactions on Vehicular Technology*, vol. 72, no. 10, pp. 12 713–12 727, 2023.

[5] T. Qie, W. Wang, C. Yang, Y. Li, Y. Zhang, W. Liu, and C. Xiang, "An improved model predictive control-based trajectory planning method for automated driving vehicles under uncertainty environments," *IEEE Transactions on Intelligent Transportation Systems*, vol. 24, no. 4, pp. 3999–4015, 2023.

[6] E. Landolfi, A. Salvi, A. Troiano, and C. Natale, "Robustness analysis of a distributed adaptive model predictive control for connected and automated vehicles against delays," *IEEE Transactions on Control Systems Technology*, vol. 32, no. 5, pp. 1540–1554, 2024.

[7] Z. Zheng, X. Zhao, S. Wang, Q. Yu, H. Zhang, Z. Li, H. Chai, and Q. Han, "Extension coordinated control of distributed-driven electric vehicles based on evolutionary game theory," *Control Engineering Practice*, vol. 137, p. 105583, 2023.

[8] N. Guo, X. Zhang, Y. Zou, B. Lenzo, T. Zhang, and D. Göhlich, "A fast model predictive control allocation of distributed drive electric vehicles for tire slip energy saving with stability constraints," *Control Engineering Practice*, vol. 102, p. 104554, 2020.

[9] F. Tarhini, R. Talj, and M. Doumiati, "Multi-objective control architecture for an autonomous in-wheel driven electric vehicle," *IFAC-PapersOnLine*, vol. 56, no. 2, pp. 11 470–11 476, 2023.

[10] Y. Tian, X. Cao, X. Wang, and Y. Zhao, "Four wheel independent drive electric vehicle lateral stability control strategy," *IEEE/CAA Journal of Automatica Sinica*, vol. 7, no. 6, pp. 1542–1554, 2020.

[11] Y. Ma, J. Chen, J. Wang, Y. Xu, and Y. Wang, "Path-tracking considering yaw stability with passivity-based control for autonomous vehicles," *IEEE Transactions on Intelligent Transportation Systems*, vol. 23, no. 7, pp. 8736–8746, 2022.

[12] Y. Liang, Y. Li, A. Khajepour, and L. Zheng, "Holistic adaptive multi-model predictive control for the path following of 4wid autonomous vehicles," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 1, pp. 69–81, 2021.

[13] F. Tarhini, R. Talj, and M. Doumiati, "Dynamic and real-time continuous look-ahead distance for autonomous vehicles: an explicit formulation," *Vehicle System Dynamics*, vol. 62, no. 9, pp. 2210–2236, 2024.

[14] X. Zhou, Z. Wang, H. Shen, and J. Wang, "Robust adaptive path-tracking control of autonomous ground vehicles with considerations of steering system backlash," *IEEE Transactions on Intelligent Vehicles*, vol. 7, no. 2, pp. 315–325, 2022.

[15] Y. T. Xuanhao Cao, Tao Xu and X. Ji, "Gain-scheduling lpv synthesis $h_\infty$ robust lateral motion control for path following of autonomous vehicle via coordination of steering and braking," *Vehicle System Dynamics*, vol. 61, no. 4, pp. 968–991, 2023.

[16] F. Tarhini, R. Talj, and M. Doumiati, "Dual-level control architectures for over-actuated autonomous vehicle's stability, path-tracking, and energy economy," *IEEE Transactions on Intelligent Vehicles*, vol. 9, no. 1, pp. 287–303, 2024.

[17] G. Chen, J. Yao, H. Hu, Z. Gao, L. He, and X. Zheng, "Design and experimental evaluation of an efficient mpc-based lateral motion controller considering path preview for autonomous vehicles," *Control Engineering Practice*, vol. 123, p. 105164, 2022.

[18] C. Hu, Z. Wang, X. Bu, J. Zhao, J. Na, and H. Gao, "Optimal tracking control for autonomous vehicle with prescribed performance via adaptive dynamic programming," *IEEE Transactions on Intelligent Transportation Systems*, vol. 25, no. 9, pp. 12 437–12 449, 2024.

[19] Z. Wang, J. Zha, and J. Wang, "Autonomous vehicle trajectory following: A flatness model predictive control approach with hardware-in-the-loop verification," *IEEE Transactions on Intelligent Transportation Systems*, vol. 22, no. 9, pp. 5613–5623, 2021.

[20] W. Wang, Y. Zhang, C. Yang, T. Qie, and M. Ma, "Adaptive model predictive control-based path following control for four-wheel independent drive automated vehicles," *IEEE Transactions on Intelligent Transportation Systems*, vol. 23, no. 9, pp. 14 399–14 412, 2022.

[21] M. Dalboni, D. Tavernini, U. Montanaro, A. Soldati, C. Concari, M. Dhaens, and A. Sorniotti, "Nonlinear model predictive control for integrated energy-efficient torque-vectoring and anti-roll moment distribution," *IEEE/ASME Transactions on Mechatronics*, vol. 26, no. 3, pp. 1212–1224, 2021.

[22] Y. Deng, Y. Wang, H. Wang, and Z. Zuo, "Model predictive lateral control for unmanned ground vehicles with speed coupling: A terminal constraint-free approach," *IEEE Transactions on Systems, Man, and Cybernetics: Systems*, vol. 54, no. 2, pp. 925–935, 2024.

[23] L. Ge, Y. Zhao, F. Ma, and K. Guo, "Towards longitudinal and lateral coupling control of autonomous vehicles using offset free mpc," *Control Engineering Practice*, vol. 121, p. 105074, 2022.

[24] P. Wang, H. Liu, L. Guo, L. Zhang, H. Ding, and H. Chen, "Design and experimental verification of real-time nonlinear predictive controller for improving the stability of production vehicles," *IEEE Transactions on Control Systems Technology*, vol. 29, no. 5, pp. 2206–2213, 2021.

[25] Y. Liang, S. Zhang, W. Zhao, C. Wang, K. Xu, and W. Liang, "Coordinated control of yaw and roll stability in heavy vehicles considering dynamic safety requirements," *Control Engineering Practice*, vol. 148, p. 105963, 2024.

[26] G. Park and S. B. Choi, "A model predictive control for path tracking of electronic-four-wheel drive vehicles," *IEEE Transactions on Vehicular Technology*, vol. 70, no. 11, pp. 11 352–11 364, 2021.

[27] H. Peng, W. Wang, Q. An, C. Xiang, and L. Li, "Path tracking and direct yaw moment coordinated control based on robust mpc with the finite time horizon for autonomous independent-drive vehicles," *IEEE Transactions on Vehicular Technology*, vol. 69, no. 6, pp. 6053–6066, 2020.

[28] L. Zhai, C. Wang, Y. Hou, and C. Liu, "Mpc-based integrated control of trajectory tracking and handling stability for intelligent driving vehicle driven by four hub motor," *IEEE Transactions on Vehicular Technology*, vol. 71, no. 3, pp. 2668–2680, 2022.

[29] X. Cao, Y. Tian, X. Ji, and B. Qiu, "Fault-tolerant controller design for path following of the autonomous vehicle under the faults in braking actuators," *IEEE Transactions on Transportation Electrification*, vol. 7, no. 4, pp. 2530–2540, 2021.

[30] F. Tarhini, R. Talj, and M. Doumiati, "On analytical modeling for fast multi-objective torque allocation in over-actuated iwms vehicles," *IEEE/CAA Journal of Automatica Sinica*, pp. 1–19, 2025.

[31] N. Guo, X. Zhang, Y. Zou, B. Lenzo, G. Du, and T. Zhang, "A supervisory control strategy of distributed drive electric vehicles for coordinating handling, lateral stability, and energy efficiency," *IEEE*
*Transactions on Transportation Electrification*, vol. 7, no. 4, pp. 2488–2504, 2021.

[32] H. Peng, W. Wang, C. Xiang, L. Li, and X. Wang, "Torque coordinated control of four in-wheel motor independent-drive vehicles with consideration of the safety and economy," *IEEE Transactions on Vehicular Technology*, vol. 68, no. 10, pp. 9604–9618, 2019.

[33] X. Hu, H. Chen, Z. Li, and P. Wang, "An energy-saving torque vectoring control strategy for electric vehicles considering handling stability under extreme conditions," *IEEE Transactions on Vehicular Technology*, vol. 69, no. 10, pp. 10787–10796, 2020.

[34] H. Wei, N. Zhang, J. Liang, Q. Ai, W. Zhao, T. Huang, and Y. Zhang, "Deep reinforcement learning based direct torque control strategy for distributed drive electric vehicles considering active safety and energy saving performance," *Energy*, vol. 238, p. 121725, 2022.

[35] J. Liang, F. Wang, J. Feng, M. Zhao, R. Fang, D. Pi, and G. Yin, "A hierarchical control of independently driven electric vehicles considering handling stability and energy conservation," *IEEE Transactions on Intelligent Vehicles*, vol. 9, no. 1, pp. 738–751, 2024.

[36] X. Hu, P. Wang, Y. Hu, and H. Chen, "A stability-guaranteed and energy-conserving torque distribution strategy for electric vehicles under extreme conditions," *Applied Energy*, vol. 259, p. 114162, 2020.

[37] B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, "Osqp: an operator splitting solver for quadratic programs," *Mathematical Programming Computation*, vol. 12, no. 4, p. 637–672, 2020.

[38] A. Chebly, R. Talj, and A. Charara, "Coupled longitudinal/lateral controllers for autonomous vehicles navigation, with experimental validation," *Control Engineering Practice*, vol. 88, pp. 79–96, 2019.

[39] H. Wei, Q. Ai, W. Zhao, and Y. Zhang, "Modelling and experimental validation of an ev torque distribution strategy towards active safety and energy efficiency," *Energy*, vol. 239, p. 121953, 2022.