# Optimal Multi-Objective Adaptive Coordinated Control via Hierarchical MPC Framework for Autonomous Vehicles

Fadel Tarhini, Ghewa Masry, Reine Talj, Moustapha Doumiati

► To cite this version:

Fadel Tarhini, Ghewa Masry, Reine Talj, Moustapha Doumiati. Optimal Multi-Objective Adaptive Coordinated Control via Hierarchical MPC Framework for Autonomous Vehicles. European Control Conference (ECC 2025), Jun 2025, Thessaloniki, Greece. pp.2229-2236, 10.23919/ECC65951.2025.11187014. hal-04988190

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Submitted on 8 Dec 2025

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# Optimal Multi-Objective Adaptive Coordinated Control via Hierarchical MPC Framework for Autonomous Vehicles

Fadel Tarhini, Ghewa Masry, Reine Talj and Moustapha Doumiati

***Abstract***— This paper introduces a novel minimum-order, multi-objective centralized MPC control architecture for autonomous vehicles. The contributions are fourfold. First, we propose a hierarchical architecture that integrates path-tracking, speed control, and stability control based on a minimum-order predictive model. Second, stability control is dynamically activated and relaxed using an adaptive weight, based on a stability index. Third, speed control employs an adaptive weight to improve robustness, reduce overshoot and oscillations, and enhance energy efficiency. Fourth, path-tracking control is enhanced with an adaptive weighting scheme that considers lateral error, road adherence, and curvature to ensure smooth convergence and prevent oscillations, while also improving accuracy under uncertain and low-adherence conditions. The architecture is validated in a joint simulation between Simulink/Matlab and SCANeR Studio vehicle dynamics simulator. Our findings demonstrate the effectiveness of the architecture in enhancing stability and comfort at low runtime, while maintaining path-tracking precision and speed control robustness, at high speeds, high curvature, and low adhesion.

## I. INTRODUCTION

The rapid advancement of autonomous vehicle (AV) technology is transforming the landscape of modern mobility, necessitating innovative control strategies to ensure seamless navigation, safety, and energy efficiency in increasingly complex driving environments. The architecture of AVs typically comprises four core components: perception, decision-making, trajectory planning, and control [1]. Among these, Model Predictive Control (MPC) has proven to be a powerful optimal control framework, driving advancements in decision-making [2], collision avoidance [3], trajectory planning [4], connected vehicles [5], control [6], and actuator coordination [7].

Lateral control necessitates sustained regulation over an extended time horizon, rendering precise path tracking a complex yet critical objective for AVs [8]. Ensuring accurate path tracking while maintaining vehicle stability is particularly challenging due to the inherent complexities of nonlinear vehicle dynamics, including tire slip, varying vehicle speeds, and parameter uncertainties. Consequently, control strategies capable of addressing complex multi-variable control problems and managing system constraints are essential to achieving effective path tracking. An MPC-based approach for path-tracking control is proposed in [9], utilizing adaptive dynamic programming to solve the problem. The model employed is a fourth-order system designed solely for the objective of path tracking. Similarly, [10] introduces a Flatness-MPC approach to achieve exact system linearization while preserving nonlinear characteristics and mitigating modeling errors. In the same context, [11] proposed an MPC-based

path-tracking control for four-wheel drive autonomous vehicles. However, these studies considered only path-tracking control. Other studies have explored the integration of speed control with path-tracking within a unified MPC framework. For instance, [12] introduced a linear parameter-varying (LPV) path-following model, where speed variation is treated as a parameter perturbation. Similarly, [13] proposed a coupled longitudinal and lateral control strategy using an offset-free MPC based on a fourth-order model. While these methods demonstrate effectiveness, their predictive models are relatively high in order. Increasing the order of the predictive model introduces additional complexity and significantly increases the computational demands placed on the MPC controller.

Other works employed MPC to target chassis control objectives which includes stability, maneuverability, and suspension control. For example, [14] proposed a real-time nonlinear MPC for stability control in distributed drive vehicles. A unified MPC controller is proposed in [15] for lateral stability and rollover prevention. Similarly, [16] presented an MPC-based coordinated control strategy for yaw and roll stability that considers the dynamic safety requirements. However, these methods do not incorporate path-tracking control. The path-tracking control considering vehicle stability based on MPC has been studied by recent works [17]. The authors in [18] presented a fourth-order MPC controller for path-tracking and yaw stabilization control. Similarly, [19] proposed a robust MPC for coordinated path tracking and direct yaw moment control in in-wheel autonomous vehicles. [20] introduced a nonlinear MPC for path-tracking, incorporating multiple dynamic constraints to address various safety and stability requirements. However, despite utilizing a fourth-order MPC model, these studies do not integrate speed control.

The literature reveals a gap in the development of a unified MPC controller with minimal order that combines path tracking, speed control, and stability control. When independent controllers are designed for these objectives, the stability of the vehicle may be compromised under extreme conditions. In this paper, we present a novel centralized MPC architecture that integrates path tracking, speed control, and stability control using the minimum-order predictive model necessary to address all three objectives. A novel adaptive weighting mechanism is developed for the three control objectives, ensuring: 1) stability control is promoted in critical driving situations; 2) speed control remains robust to initial conditions, despite assumptions in the speed control model, while achieving smooth convergence, mitigating oscillations and overshoots, and incorporating an indirect energy-saving criterion; and 3) path-tracking control maintains high accuracy with minimal steady-state error at high velocities, preventing oscillations and overshoots, and ensuring robustness against uncertainties, low adhesion, and high curvature roads. The proposed architecture is benchmarked against existing

Fadel Tarhini, Ghewa Masry, and Reine Talj are with Université de Technologie de Compiègne, CNRS, Heudiasyc UMR 7253, CS 60 319, 60 203 Compiègne, France. email: name.surname@hds.utc.fr.

Moustapha Doumiati is with ESEO-IREENA lab UR 4642, Nantes Université, 10 Bd Jeanneteau, 49100 Angers, France. email: name.surname@eseo.fr.
MPC approaches in the literature, as well as against robust nonlinear Super-Twisting Sliding Mode Control (STSMC) and robust linear LPV/H∞ control, as detailed in [21]. While STSMC and LPV/H∞ approaches offer robustness and effective performance, they yield unconstrained and non-optimal control inputs, potentially limiting energy efficiency and actuator longevity compared to the optimal control capabilities of the proposed architecture.

The paper contributions can be outlined as:

* Development of a novel minimum-order, unified centralized MPC controller that integrates multiple objectives, including path-tracking, speed control, and stability control.

* Introduction of a novel multi-criteria adaptive weighting mechanism for the three control objectives. The stability control weights are defined based on a stability index, the speed control weights are derived from longitudinal error, and the path-tracking weights are a function of lateral error, road adherence, and road curvature.

* Validation of the proposed architecture on the SCANeR Studio vehicle dynamics simulator with a fully dynamic nonlinear vehicle model. The findings demonstrate the effectiveness of the controller in enhancing stability, comfort, and energy economy, while ensuring accurate and robust path-tracking and speed control, with a runtime less than 3 ms.

The rest of the paper is structured as follows: Section II introduces the hierarchical centralized MPC architecture. Section III presents the validation tests and discussion, followed by a conclusion in Section IV.

## II. HIERARCHICAL CENTRALIZED MPC ARCHITECTURE

### A. Global Control Architecture

Fig. 1 provides an overview of the global control architecture, which is structured hierarchically into three layers: the decision layer, control layer, and actuator layer. The decision layer anticipates vehicle driving conditions by monitoring the lateral Stability Index (SI), lateral error $e_y$, and longitudinal velocity error $e_x$, to generate time-varying adaptive weights for the MPC controller. The parameters $Q_β$, $Q_{e_x}$, and $Q_{\dot{\psi}}$ represent the respective weights on the states: side-slip angle $β$, longitudinal error $e_x$, and yaw rate $\dot{\psi}$, while $R_{M_z}$, $R_{T_m}$, and $R_δ$ denote the control input weights for yaw moment $M_z$, total torque $T_m$, and steering angle $δ_c$. Additionally, road adherence and road curvature are fed to the decision layer, where the road adherence coefficient $μ$ is considered accessible. Road curvature $ρ$ is calculated at the trajectory planning level, presented in our previous work [1].

At the control layer, the centralized Multi-Input-Multi-Output (MIMO) MPC controller is developed to generate three control inputs corresponding to the three objectives. Path-tracking is achieved by controlling $\dot{\psi}$ to track a reference generated yaw rate $\dot{\psi}_{ref}$ (1) by generating a steering angle $δ_c$.

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

where $v_x$ is the longitudinal velocity, $Ls$ is the look-ahead distance [8], and $e_{y,Ls}$ is the lateral error calculated at the look-ahead distance. Speed control is implemented by regulating the longitudinal velocity error $e_x$ to converge to zero, achieved through the generation of a total torque $T_m$. In this manner, the actual velocity $v_x$ tracks the desired velocity profile $v_{x,des}$, which is constructed at the trajectory planning

<figure><img src="image_2.png" /><figcaption>Fig. 1: Hierarchical adaptive centralized MIMO MPC architecture.</figcaption></figure>

level [1]. Lastly, stability control is ensured by controlling $β$ to converge to its reference value $β_{ref}$ when the vehicle encounters critical driving conditions.

At the actuator layer, the generated control inputs are realized via physical actuators. This work considers the vehicle equipped with four independent in-wheel motors. Hence, $δ_c$ is realized using Active Front Steering (AFS), while $T_m$ and $M_z$ are achieved using a torque allocation method considering dynamic load distribution, presented in our work [21]. The steering angle and the four wheel torques are ultimately fed to the full nonlinear vehicle model in the SCANeR Studio professional simulator.

### B. Extended Lateral and Longitudinal Model

The single-track linear bicycle model is employed [21], with extensions to include longitudinal dynamics terms. Modeling errors in vehicle dynamics are inevitable due to tire nonlinearities, parametric uncertainties, and time-varying parameters. However, the adaptive nature of the developed weighting functions ensures that the controller remains robust in the presence of such errors. The roll, pitch, and suspension dynamics are neglected, and the steering angles of the left and right wheels are assumed to be identical. The 2-DoF bicycle model is expressed as (2).

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

where $M$ is the vehicle mass and $I_z$ is the vehicle yaw inertia. $F_{yf}$, $F_{yr}$ are the front and rear lateral tire forces respectively, and $l_r$, $l_f$ represent the distance between the vehicle centroid and the rear and front axles respectively.

The lateral tire forces are defined in (3) in terms of the front and rear cornering stiffness $C_f$, $C_r$ respectively, and the slip angles.

$$ \begin{cases} F_{yf} = \mu C_{yf} (\delta - \beta - \frac{l_f \dot{\psi}}{v_x}) \\ F_{yr} = \mu C_{yr} (-\beta + \frac{l_r \dot{\psi}}{v_x}) \end{cases} \qquad (3) $$

In order to prevent complex coupling effects that could introduce nonlinearity and substantially increase the controller computational load, a simplified model for longitudinal ve-
locity tracking is defined in terms of the longitudinal velocity error $e_x = v_x - v_{x,des}$.

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

where $r$ is the tire effective radius and $a_{x,des}$ is the desired longitudinal acceleration. The value of $a_{x,des}$ is derived from the required acceleration to transition from the current velocity $v_x$ to the target velocity $v_{x,des}$ with a sampling time $T_s$, and is constrained within the range $[a_{x,min} a_{x,max}] m/s^2$. Assuming small $\delta$, and combining (2), (3), and (4), the extended lateral and longitudinal model can be formulated as (5).

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

As three distinct objectives are considered in the controller, each with a corresponding control input, the extended predictive model (5) constitutes the minimum-order system necessary to achieve these objectives. Each objective requires at least one dedicated state to be controlled, necessitating a minimum of three state variables to adequately capture these interactions. Any reduction in model order below three would omit critical dynamic information, rendering 3 as the lowest-order system capable of fulfilling all control objectives effectively.

### C. MPC Problem Formulation

The state-space representation for the extended lateral and longitudinal model can be represented as in (6).

$$ \begin{aligned} \dot{x} &= Ax + Bu + w \\ y &= Cx \end{aligned} \quad (6) $$

where the state vector $x = [\dot{\psi} \ \beta \ e_x]^T$, the control inputs $u = [\delta_c \ M_z \ T_m]^T$, and $w = [0 \ 0 \ -a_{x,des}]^T$ modeled as an exogenous input. The system matrix, control matrix, and the output matrix are defined as (7).

$$ A = \begin{bmatrix} -\mu \frac{l_r^2 C_r + l_f^2 C_f}{I_z v_x} & \mu \frac{l_r C_r - l_f C_f}{I_z} & 0 \\ -1 + \mu \frac{l_r C_r - l_f C_f}{M v_x^2} & -\mu \frac{C_r + C_f}{M v_x} & 0 \\ 0 & 0 & 0 \end{bmatrix}; \quad (7) $$

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

The continuous model (6) is discretized for the MPC design as follows:

$$ \begin{aligned} x(k+1) &= A_d x(k) + B_d u(k) + T_s w(k) \\ y(k) &= C_d x(k) \end{aligned} \quad (8) $$

where $A_d$, $B_d$ represent the discrete state equations obtained from A, B using the Zero-Order Hold method, and $C_d = C$.

$$ \begin{aligned} A_d &= e^{A T_s} \\ B_d &= \int_0^{T_s} e^{A \tau} d\tau \ B \end{aligned} \quad (9) $$

where $T_s$ is the sampling time of the MPC controller and set to 0.02 s. Consequently, the MPC optimization function

is defined as (10).

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

where $N_p$ is the prediction horizon. $Q, R$, and $S$ represent the diagonal weight matrices of the system states, control inputs and control input increments respectively, and $y_{ref} = [\dot{\psi}_{ref} \ \beta_{ref} \ 0]^T$ is the desired system output. $\beta_{ref}$ in this work is set to 0. The weight matrices are defined as:

$$ \begin{aligned} Q &= \begin{bmatrix} Q_{\dot{\psi}} & 0 & 0 \\ 0 & Q_{\beta} & 0 \\ 0 & 0 & Q_{e_x} \end{bmatrix}; \quad R = \begin{bmatrix} R_{\delta} & 0 & 0 \\ 0 & R_{M_z} & 0 \\ 0 & 0 & R_{T_m} \end{bmatrix} \\ S &= \begin{bmatrix} S_{\delta} & 0 & 0 \\ 0 & S_{M_z} & 0 \\ 0 & 0 & S_{T_m} \end{bmatrix} \end{aligned} \quad (11) $$

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 designed in the next section.

The MPC optimization problem is defined as follows:

$$ \min_u J \quad (12a) $$

subject to

$$ x(k+1) = A_d x(k) + B_d u(k) + T_s w(k) \quad (12b) $$

$$ y(k) = C_d x(k) \quad (12c) $$

$$ \delta_{\min} \le \delta \le \delta_{\max} \quad (12d) $$

$$ M_{z,min} \le M_z \le M_{z,max} \quad (12e) $$

$$ T_{m,min} \le T_m \le T_{m,max} \quad (12f) $$

$$ \Delta \delta_{\min} \le \delta_k - \delta_{k-1} \le \Delta \delta_{\max} \quad (12g) $$

$$ \Delta M_{z,min} \le M_{z_k} - M_{z_{k-1}} \le M_{z,max} \quad (12h) $$

$$ \Delta T_{m,min} \le T_{m_k} - T_{m_{k-1}} \le \Delta T_{m,max} \quad (12i) $$

where $[\delta_{\min} \ \delta_{\max}]$, $[M_{z,min} \ M_{z,max}]$, $[T_{m,min} \ T_{m,max}]$, are the ranges of the control inputs $\delta$, $M_z$, and $T_m$ respectively. Similarly, the ranges of the control input increments are respectively, $[\Delta \delta_{\min} \ \Delta \delta_{\max}]$, $[\Delta M_{z,min} \ \Delta M_{z,max}]$, and $[\Delta T_{m,min} \ \Delta T_{m,max}]$.

The optimization problem is transformed into a QP problem and solved using the OSQP solver. The OSQP solver uses the Alternating Direction Method of Multipliers (ADMM) algorithm to solve the problem in the form [22]:

$$ \min \frac{1}{2} z^T \mathcal{P} z + \mathcal{Q}^T z \quad (13) $$

subject to $\mathcal{L} \le A_c z \le \mathcal{U}$

where $z \in \mathbb{R}^{n_z}$ represents the decision variables, which include control inputs and predicted states over the horizon. The cost function is quadratic, defined by the Hessian matrix $\mathcal{P} \in \mathbb{R}^{n_z \times n_z}$, and the gradient vector $\mathcal{Q} \in \mathbb{R}^{n_z}$. $A_c$ is the constraint matrix, and $\mathcal{L}$ and $\mathcal{U}$ are the lower and upper bounds, respectively.

$$ \begin{aligned} \mathcal{P} &= \text{blkdiag}(\text{kron}(I_{N_p}, Q), \text{kron}(I_{N_p}, R), \text{kron}(I_{N_p-1}, S)) \\ \mathcal{Q} &= [-(Q y_{ref})^T, \dots, -(Q y_{ref})^T, 0, \dots, 0]^T \end{aligned} \quad (14) $$

The constraint matrix $A_c$ is constructed as $A_c = [A_{eq} \ A_{ineq}]^T$ where $A_{eq}$ represents the equality constraints arising from the state-space dynamics, formulated as $A_{eq}z = b_{eq}$, while $A_{ineq}$ represents the inequality constraints imposed on the control inputs and their increments,
<figure><img src="image_3.png" /><figcaption>Fig. 2: Weight adaptive mechanism for control objectives coordination: (a) stability control, (b) speed control, (c) path-tracking, and (d) intermediate function of road adherence (μ) and road curvature (ρ) for path-tracking.</figcaption></figure>

given by $A_{ineq}z \le b_{ineq}$. The lower and upper bounds on the states and control inputs $(L, \mathcal{U})$ are expressed in terms of $b_{eq}$ and $b_{ineq}$. The term $b_{eq}$ consists of an array of the exogenous input $w$, while $b_{ineq}$ encapsulates the upper and lower bounds on the states $(x_{max}, x_{min})$, as well as the corresponding bounds on the control inputs $(u_{max}, u_{min})$ and their increments $(\Delta u_{max}, \Delta u_{min})$. The control input bounds are explicitly given by $u_{min} = [\delta_{min} M_{z,min} T_{m,min}]^T$ and $u_{max} = [\delta_{max} M_{z,max} T_{m,max}]^T$. More information about the OSQP solver can be seen in [13].

### D. Adaptive Weighting Mechanism

The weights on the control objectives are adapted to several parameters based on multiple criteria.

#### 1) Weights on Stability Control

The stability control objective is promoted/attenuated according to the driving conditions. A critical driving situation is identified when the Stability Index ($SI$) (15) surpasses the threshold value of 1 [21].

$$SI = |2.49\dot{\beta} + 9.55\beta| \quad (15)$$

Hence, to activate the stability objective, the weight on the state $\beta$ is anticipated prioritized with an increasing transient state in the range $[SI, \overline{SI}] \le 1$. To this end, define the weight on the side-slip angle $Q_{\beta}$ as follows:

$$Q_{\beta} = q_{\beta,1} + \frac{q_{\beta,2}}{1 + e^{-\frac{2\gamma_1}{SI - SI}(SI - \frac{\overline{SI} + SI}{2})}} \quad (16)$$

where $q_{\beta,1}, q_{\beta,2}$ represent the $Q_{\beta}$ values at the extremities of its transient state, $SI$ is the lower threshold where the stability objective starts activating, and $\overline{SI}$ is the upper threshold where the function saturates. $\gamma_1$ signifies the intensity of the transient-state centroid of $Q_{\beta}$.

The weight on the control input $M_z$ is decreased to attenuate the penalization of the DYC actuator when $SI \ge \overline{SI}$. Hence, $R_{M_z}$ is defined as follows:

$$R_{M_z} = \left[ r_{M,1} - \frac{(r_{M,1} - r_{M,2})}{1 + e^{-\frac{2\gamma_1}{SI - SI}(SI - \frac{\overline{SI} + SI}{2})}} \right] + \varepsilon \quad (17)$$

where $r_{M,1}, r_{M,2}$ are the values of $R_{M_z}$ at the extremities of its transient state, and $\varepsilon > 0$. Therefore, for $SI \ge \overline{SI}$, the stability objective is completely activated, where $Q_{\beta} = q_{\beta,2}$ is maximum and $R_{M_z} = \varepsilon$ is minimum since $r_{M,2}$ is set to 0. The weights on stability control are shown in Fig. 2-(a).

#### 2) Weights on Speed Control

The output value of the control input $T_m$ is an image of the desired longitudinal acceleration $a_{x,des}$. If the vehicle's initial speed is assumed to match the reference speed, as in [6], adaptive speed control weights may not be required. However, if this condition is not met, adaptive weighting

becomes essential to improve the controller's robustness by assigning a mapping between $a_{x,des}$ and the desired output for $T_m$. The weight $Q_{e_x}$ on the state $e_x$ is adapted in accordance with the value of $|e_x|$. $Q_{e_x}$ is increased when the value of $|e_x|$ increases, to reach its maximum $q_{T,2}$ when $a_{x,des} = a_{x,max}$. Hence, $Q_{e_x}$ is prioritized when the longitudinal velocity error is significant and gradually reduced as $e_x$ approaches zero. This adaptive behavior enables the MPC controller to efficiently handle disturbances and uncertainties, akin to the response of STSMC [21] to dynamic system variations.

For this purpose, define $Q_{e_x}$ as follows:

$$Q_{e_x} = q_{e,1} + \frac{(q_{e,2} - q_{e,1})}{1 + e^{-\frac{2\gamma_2}{e_x - e_x}(|e_x| - \frac{e_x + e_x}{2})}} \quad (18)$$

where $q_{e,1}, q_{e,2}$ denote the values of $Q_{e_x}$ at the extremities of the transient-state, $e_x$, $\overline{e_x}$ are the lower and upper threshold of $e_x$, representing the boundaries of the transient-state, and $\gamma_2$ as $\gamma_1$. As $e_x$ increases, the weight on the control input $T_m$ must be reduced to attenuate penalizing the actuator. Hence, define $R_{T_m}$ as follows:

$$R_{T_m} = r_{T,1} - \frac{(r_{T,1} - r_{T,2})}{1 + e^{-\frac{2\gamma_2}{e_x - e_x}(|e_x| - \frac{e_x + e_x}{2})}} \quad (19)$$

where $r_{T,1}, r_{T,2}$ signify the $R_{T_m}$ values at the extremities of its transient state. The weights on speed control are depicted in Fig. 2-(b).

In addition to enhancing the controller robustness, the designed weights ensure smooth convergence of $v_x$ to $v_{x,des}$, preventing overshoots and reducing oscillations. For large values of $|e_x|$, the controller responds quickly and efficiently to minimize the speed error with fast actuator response. Furthermore, by increasing $R_{T_m}$ and decreasing $Q_{e_x}$ for small $|e_x|$, the controller improves energy efficiency. Energy consumption is further reduced by the smooth profile of the control input $T_m$, achieved by incorporating $S_{T_m}$ and imposing constraints on the control input variation. The effect of the adaptive weights are depicted in Fig. 3-(a). It can be observed that the integration of the adaptive weights effectively mitigates oscillations and averts overshoots.

#### 3) Weights on Path-Tracking Control

The path-tracking control objective holds the highest priority, as it directly influences the safety of the autonomous vehicle. Consequently, it is dynamically adjusted in response to the lateral error $|e_y|$, road curvature $\rho$, road adherence $\mu$, and the interaction with other control objectives. To swiftly and effectively minimize the lateral error $e_y$ while enhancing the robustness of the controller, the weight on the yaw rate $Q_{\psi}$ is increased for large $|e_y|$. This weight is gradually attenuated as $|e_y|$ decreases, ensuring smooth
<figure><img src="image_4.png" /><figcaption>Fig. 3: Effect of adaptive weights on path-tracking and speed control: (a) large initial speed error and (b) significant initial lateral displacement from the reference lane. Fixed weights correspond to the extrema of the adaptive weight range.</figcaption></figure>

convergence without inducing oscillations or overshoots. Hence, the weight on the yaw rate state is defined as follows:

$$Q_{\psi} = q_{\psi,1} + \frac{(q_{\psi,2} - q_{\psi,1})}{1 + e^{-\frac{2\gamma_3}{e_y - e_{y_0}}} (|e_y| - \frac{e_{y_0} + e_y}{2})} \quad (20)$$

where $q_{\psi,1}$, $q_{\psi,2}$ represent the values of $Q_{\psi}$ at the extremities of the transient-state, $e_y$, $e_{y_0}$ are the lower and upper boundaries of the transient-state, and $\gamma_3$ as $\gamma_2$. When a risk of instability is detected and the stability objective is prioritized, the generation of the yaw moment can adversely affect path-tracking performance. Hence, $Q_{\psi}$ is increased for $SI \ge SI$ by defining $q_{\psi,2}$ in terms of $SI$ as follows:

$$q_{\psi,2} - q_{\psi,1} = \frac{\xi_1}{1 + e^{-\frac{2\gamma_1}{SI - \frac{SI+SI}{2}}} (SI - \frac{SI+SI}{2})} \quad (21)$$

where $\xi_1 \in \mathbb{R}^+$. Path tracking becomes increasingly challenging as the road adhesion coefficient $\mu$ decreases, particularly on high-curvature roads with low adhesion. Hence, $Q_{\psi}$ is further increased in accordance with decreasing $\mu$ and increasing $\rho$, by defining $q_{\psi,1}$ as follows:

$$q_{\psi,1} = \xi_2 + q_{\mu,\rho} \mathcal{F}(\mu) \quad (22a)$$

$$\mathcal{F}(\mu) = \begin{cases} (1 - \mu)\lambda & \text{if } \mu \le 1 \\ 0 & \text{if } \mu > 1 \end{cases} \quad (22b)$$

where $q_{\mu,\rho}$ is defined in terms of $\mu$ and $\rho$ as follows:

$$q_{\mu,\rho} = \frac{\zeta_1}{1 + e^{-\frac{2\gamma_3}{\bar{\rho} - \rho}(|\rho| - \frac{\bar{\rho} + \rho}{2})}} - \frac{\zeta_2}{1 + e^{-\frac{2\gamma_3}{\bar{\mu} - \mu}(|\mu| - \frac{\bar{\mu} + \mu}{2})}} + \zeta_3 \quad (23a)$$

$$\zeta_2 = (q_{\mu,3} - q_{\mu,2}) \frac{(1 + e^{-\gamma_3})(1 + e^{\gamma_3})}{e^{-\gamma_3} - e^{\gamma_3}} \quad (23b)$$

$$\zeta_3 = \frac{1}{e^{\gamma_3} - e^{-\gamma_3}} \left[ -(1 + e^{-\gamma_3})q_{\mu,1} + (1 + e^{\gamma_3})q_{\mu,2} - \frac{e^{-\gamma_3} - e^{\gamma_3}}{1 + e^{\gamma_3}}\zeta_2 \right] \quad (23c)$$

$$\zeta_1 = (1 + e^{-\gamma_3})(q_{\mu,1} + \frac{1}{1 + e^{\gamma_3}}\zeta_2 - \zeta_3) \quad (23d)$$

where $\xi_2$ is the minimum weight for path-tracking in normal adhesion, and $\lambda \ge 1$. The parameters $\rho$, $\mu$, $\bar{\rho}$, $\bar{\mu}$ are respectively the lower and upper boundaries of the transient-state of the subfunctions of $q_{\mu,\rho}$ in terms of $\rho$ and $\mu$. $q_{\mu,1}$, $q_{\mu,2}$, and $q_{\mu,3}$ are the desired values of the additional state weight $q_{\mu,\rho}$ at its extremities. The form of $q_{\mu,\rho}$ and its parameters are depicted in Fig. 2-(d).

The weight on the control input $R_\delta$ is increased with decreasing $|e_y|$ to enable smooth convergence and mitigate

<table><caption>TABLE I: Nonlinear vehicle model parameters</caption><thead><tr><th>Symbol</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td><i>M</i></td><td>Vehicle's total 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>Vehicle front surface</td><td>2.31 [m<sup>2</sup>]</td></tr><tr><td><i>C</i><sub>d</sub></td><td>Aerodynamic constant</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</td><td>0.7615 [m]</td></tr><tr><td><i>l</i><sub>f</sub></td><td>Distance between CG and the front axle</td><td>1.195 [m]</td></tr><tr><td><i>l</i><sub>r</sub></td><td>Distance between CG and the 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>Wheel effective 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 tires cornering stiffness</td><td>85275 [N/rad]</td></tr></tbody></table>

oscillations. Conversely, $R_\delta$ is decreased as $|e_y|$ increases to reduce penalizing the AFS actuator. Hence, $R_\delta$ is defined as follows:

$$R_{\delta} = r_{\delta,1} - \frac{(r_{\delta,1} - r_{\delta,2})}{1 + e^{-\frac{2\gamma_3}{e_y - e_{y_0}}} (|e_y| - \frac{e_{y_0} + e_y}{2})} \quad (24)$$

where $r_{\delta,1}$, $r_{\delta,2}$ denote the values of $R_\delta$ at the extremities of its transient state. The adaptive weights on path-tracking are presented in Fig. 2-(c). Fig. 3-(b) depicts the effect of the designed weights on path-tracking when the vehicle initial position is far from the reference lane.

### III. SIMULATION TESTS

The control architecture is implemented in Simulink/ MatLab, tested on a full nonlinear vehicle model [23], and validated on the professional SCANeR Studio vehicle dynamics simulator, on a high-fidelity vehicle model. The MPC controller is running at a frequency of 50 Hz. The look-ahead distance $Ls$ (1) is fixed at 5 m. The parameters of the vehicle model are given in Table I.

The effectiveness of the control architecture is supported by the following performance variables:

* Stability Index ($SI$) (15)

* Maximum lateral error ($max(|e_y|)$)

* Root mean square of the lateral error ($rms(e_y)$)

* Energy consumption $E_m = \int_0^t \sum_{i=1}^4 \frac{T_i \omega_i}{\eta_i^{sign}(T_i)}$

where $T_i$, $\omega_i$, $\eta_i$ are respectively the output torque, rotational velocity, and efficiency of the in-wheel motor $i$. The torques $T_i$ are distributed according to a dynamic load distribution-based torque allocation strategy considering $T_m$ and $M_z$ [21]. The motor efficiency is estimated using real-world efficiency data for both motoring and regenerative braking modes. For further details, refer to the authors' previous work [21].

To provide deeper insights into the contributions, a case study is conducted encompassing three distinct scenarios. The first scenario simulates an urban environment with medium-to-high velocities. The second scenario evaluates the controller's steady-state performance under challenging road geometries. The third scenario involves high-velocity driving on an urban map to assess vehicle stability. Additionally, a series of robustness tests are performed to evaluate the controller's resilience to model parameter uncertainties and low-adhesion conditions.

The proposed centralized MPC architecture using the OSQP formulation (denoted by mpc - osqp) is compared against the control strategies: 1) an MPC controller with the same formulation solved using the SQP algorithm (labeled as
<figure><img src="image_5.png" /><figcaption>Fig. 4: Scenario 1 results: (a) vehicle trajectory, (b) lateral errors, (c) longitudinal velocities, and (d) energy consumption comparison.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5: Scenario 2 results: (a) vehicle trajectory, (b) lateral errors, (c) longitudinal velocities, and (d) energy consumption comparison.</figcaption></figure>

$mpc - sqp$); 2) the decentralized robust nonlinear STSMC architecture presented in [21] (denoted by SM); and 3) the semi-centralized robust linear $LPV/H_\infty$ architecture (labeled as $LPV/H_\infty$), also presented in [21]. Note that all the control architectures share the same control objectives. The proposed architecture is further compared with the coupled longitudinal/lateral MPC controller in [13].

### A. Scenario 1: Urban map with medium-to-high speed

The first scenario is executed on the track shown in Fig. 4-(a). The lateral errors of the strategies are presented in Fig. 4-(b), along with velocity profiles in Fig. 4-(c), and energy consumption in Fig. 4-(d). It can be observed that the lateral error, $e_y$, is closest to zero for the $mpc - osqp$ strategy, followed by the $mpc - sqp$ strategy, the SM, and finally the $LPV/H_\infty$ strategy. Moreover, unlike the other strategies, the steady-state error of the $mpc - osqp$ converges to 0, as observed between $t = 40s$ and $t = 50s$. Referring to the velocity profiles, the designed weights enable the MPC controller to track the desired profile with a quick response, followed by smooth convergence as it approaches the target. The small deviations observed at certain instances are attributed to the generation of $M_z$. Furthermore, since the speed controller in the $LPV/H_\infty$ architecture operates independently of the path-tracking and stability controllers, its velocity is observed to occasionally exceed the reference value. The design of the weights for the MPC controller, coupled with the incorporation of the matrix S, constraints on control input increments, and the optimization of control inputs within a centralized MIMO MPC framework, has shown a major reduction in energy consumption compared to the alternative strategies.

Table II presents the performance metrics of the control architectures. The proposed $mpc - osqp$ architecture demonstrates the lowest values for both $max(|e_y|)$ and $rms(e_y)$, showing significant improvement over the alternative architectures.

<table><caption>TABLE II: Performance metrics of the control architectures</caption><thead><tr><th rowspan="2">Scenario cases</th><th rowspan="2">Performance variables</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>mpc - sqp</th><th>mpc - osqp</th></tr></thead><tbody><tr><td rowspan="3">Scenario 1</td><td>max(|e<sub>y</sub>|) [m]</td><td>0.162</td><td>0.245</td><td>0.094</td><td><b>0.073</b></td></tr><tr><td>rms(e<sub>y</sub>) [m]</td><td>0.061</td><td>0.089</td><td>0.0189</td><td><b>0.0118</b></td></tr><tr><td>max(E<sub>m</sub>) [kJ]</td><td>1315.1</td><td>1364.57</td><td>1213.14</td><td><b>1183.65</b></td></tr><tr><td rowspan="3">Scenario 2</td><td>max(|e<sub>y</sub>|) [m]</td><td>0.115</td><td>0.14</td><td>0.079</td><td><b>0.02</b></td></tr><tr><td>rms(e<sub>y</sub>) [m]</td><td>0.086</td><td>0.073</td><td>0.021</td><td><b>0.004</b></td></tr><tr><td>max(E<sub>m</sub>) [kJ]</td><td>557.04</td><td>469.7</td><td>410.76</td><td><b>404.36</b></td></tr><tr><td rowspan="3">Scenario 3</td><td>max(|e<sub>y</sub>|) [m]</td><td><b>0.226</b></td><td>0.568</td><td>0.513</td><td><b>0.243</b></td></tr><tr><td>rms(e<sub>y</sub>) [m]</td><td>0.075</td><td>0.145</td><td>0.116</td><td><b>0.061</b></td></tr><tr><td>max(E<sub>m</sub>) [kJ]</td><td>2013.6</td><td>2410.97</td><td>1972.48</td><td><b>1717.28</b></td></tr></tbody></table>

<table><caption>TABLE III: Enhancements of $mpc - osqp$ compared to the other strategies</caption><thead><tr><th rowspan="2">Scenario cases</th><th rowspan="2">Performance variables</th><th colspan="3">Enhancements of mpc - osqp compared to:</th></tr><tr><th>SM</th><th>LPV/H<sub>∞</sub></th><th>mpc - sqp</th></tr></thead><tbody><tr><td rowspan="2">Scenario 1</td><td>Relative ratio of rms(e<sub>y</sub>) (%)</td><td>80.4</td><td>86.8</td><td>37.3</td></tr><tr><td>Relative ratio of max(E<sub>m</sub>) (%)</td><td>9.9</td><td>13.2</td><td>2.4</td></tr><tr><td rowspan="2">Scenario 2</td><td>Relative ratio of rms(e<sub>y</sub>) (%)</td><td>94.4</td><td>93.3</td><td>76.2</td></tr><tr><td>Relative ratio of max(E<sub>m</sub>) (%)</td><td>27.4</td><td>13.9</td><td>1.5</td></tr><tr><td rowspan="2">Scenario 3</td><td>Relative ratio of rms(e<sub>y</sub>) (%)</td><td>18.1</td><td>57.5</td><td>46.9</td></tr><tr><td>Relative ratio of max(E<sub>m</sub>) (%)</td><td>14.7</td><td>28.7</td><td>12.9</td></tr></tbody></table>

These enhancements are further highlighted through relative ratios between the proposed $mpc-osqp$ approach and the comparison strategies in Table III. The relative ratio of $max(E_m)$ quantifies the energy savings percentage achieved by the $mpc-osqp$ method over the other strategies.

### B. Scenario 2: Infinity loop with medium velocity

The second scenario is conducted on the track illustrated in Fig. 5-(a), with the lateral error for each architecture shown in Fig. 5-(b). The difference in steady-state convergence between the $mpc - osqp$ and other strategies is notable: while the SM and $LPV/H_\infty$ architectures exhibit a steady-state lateral error exceeding 0.1, the $mpc-osqp$ architecture achieves an error below 0.01, demonstrating a tenfold improvement. As shown in Tables II and III, the $mpc-osqp$ maintains a near-zero $rms(e_y)$, underscoring its significant performance advantage in relative ratio over alternative strategies.

The longitudinal velocity profiles are depicted, along the reference speed profile, in Fig. 5-(c). The speed profiles
<figure><img src="image_7.png" /><figcaption>Fig. 6: Scenario 3 results: (a) lateral errors, (b) longitudinal velocities, (c) stability index, and (d) energy consumption comparison.</figcaption></figure>

exhibited similar behaviors for all the strategies, with minor oscillations observed in the *SM* architecture and slight overshoots in the *LPV/H*<sub>∞</sub> strategy. As shown in Fig. 5-(d), the *mpc-osqp* architecture achieves the lowest energy consumption among the compared strategies. The *SM* approach, however, occasionally encounters singularities in control solutions, often generating inputs that exceed actuator limits and may lead to saturation. This, coupled with the independent controller design, leads to its higher energy demands. In contrast, the centralized MIMO formulation of the *mpc-osqp* optimally generates control inputs, resulting in a notable 27.4% reduction in energy consumption compared to *SM*, while revealing better performance.

### C. Scenario 3: Stability testing on a high speed profile

The third test unfolds on the same track as in Fig. 4-(a), with a higher desired speed profile in this case. The high velocity of the vehicle, especially during cornering maneuvers, coupled with the nonlinearities of the tire model, highly solicit the chassis stability. The critical stability regions further impact path-tracking and speed control, placing additional load on the motors and resulting in higher energy consumption. The lateral errors, speed profiles, Stability Index, and energy consumption are respectively depicted in Fig. 6-(a), (b), (c), and (d). A fifth strategy is examined here, corresponding to path-tracking and speed control with the stability objective permanently relaxed (denoted by “stability off”). It is evident that the architecture employing the “stability off” strategy has completely lost stability ($SI > 6$), and eventually diverged from the road.

The high-speed profile has resulted in an increase in $max(|e_y|)$ across all strategies (see Table II). The maximum error is observed at $t \approx 90$ s, coinciding with the highest Stability Index $SI$. To restore stability and reduce $SI$, the stability objective is promoted, which inversely impacts path-tracking performance for all strategies. The weight designed for path-tracking in the *mpc-osqp* architecture, influenced by $SI$, effectively attenuates $max(|e_y|)$ while mitigating $SI$ compared to the other approaches. In the case of the *SM* approach, the $SI$ is notably the lowest at this instant, primarily due to the high magnitude generation of $M_z$, which leads to actuator saturation and the potential inability to achieve this in a physical actuator. This factor contributes to higher energy consumption in the *SM* strategy relative to the *mpc-osqp* approach. The *LPV/H*<sub>∞</sub> architecture demonstrates the highest energy consumption, as it frequently exceeds the speed profile due to its independent speed controller, which operates separately from the path-tracking and stability controller. Notably, the *mpc-osqp*

<figure><img src="image_8.png" /><figcaption>Fig. 7: Robustness of the architecture to 20 % and 50 % uncertainties in cornering stiffness and mass.</figcaption></figure>

method achieves lower energy consumption compared to the *mpc-osqp* approach while maintaining similar speed profiles, attributable to its capacity for more globally optimal solutions and improved path-tracking performance.

### D. Recap and further tests

The performed scenarios have demonstrated the exceptional effectiveness of the designed centralized *mpc-osqp* architecture in path-tracking performance, speed control, and energy efficiency over the decentralized nonlinear STSMC and the semi-centralized linear *LPV/H*<sub>∞</sub> architectures. However, while STSMC and *LPV/H*<sub>∞</sub> methods are highly robust to parameter uncertainties, with STSMC being practically and highly independent of model accuracy, MPC controllers generally rely heavily on precise model parameters. However, the adaptive weights are designed to enhance performance while ensuring robustness. To assess the robustness of the proposed architecture, uncertainties of 20% and 50% are introduced in key parameters, including vehicle mass and front/rear cornering stiffness. A total of 100 Monte Carlo simulations are performed, incorporating random uncertainty sampling on these parameters, as illustrated in Fig. 7-(a,b). These tests follow the configuration of Scenario 2, with the corresponding performance metrics depicted in Fig. 7-(c). It is evident that even under 50% uncertainty, the path-tracking accuracy remains high, with *rms(e_y)* around 0.02 *m* and $max(e_y)$ below 0.035 *m*. This robustness is primarily attributed to the proposed adaptive weights, which effectively enhance stability in both path-tracking and speed control despite model uncertainties.

The computational efficiency of the *mpc-osqp* architecture is examined, where the simulations are conducted under an Intel Core i9-12950HX CPU 2.3-GHz laptop. The computational time (in *ms*) of the *mpc-osqp* and *mpc-sqp* strategies is given in Fig. 8. For the *mpc-sqp* strategy, the maximum iterations of the optimization problem have been restricted to ensure control input generation within $\le 20$ *ms*. The root mean square of computational time
<figure><img src="image_9.png" /><figcaption>Fig. 8: Computational time required for the control inputs generation (ms).</figcaption></figure>

is 12.3 ms for mpc-sqp and 2.8 ms for the proposed mpc-osqp approach, highlighting a significant improvement in computational efficiency.

The performance of the proposed mpc-osqp architecture is further contrasted to the coupled longitudinal/lateral offset-free MPC approach developed in [13]. The test conducted corresponds to a Double Lane Change (DLC) maneuver with a reference velocity of 80 km/h and an adherence coefficient of $\mu = 0.85$. Performance evaluation is based on the cost metrics $\max(|e_y|)$ and $rms(e_y)$, where the results are summarized in Table IV. It can be observed that the

<table><caption>TABLE IV: Performance under DLC test: $v_{x,des} = 22.2$ m/s, $\mu = 0.85$</caption><thead><tr><th rowspan="2">Metric</th><th colspan="2">Controller</th></tr><tr><th>mpc-osqp (ours)</th><th>MPC in [13]</th></tr></thead><tbody><tr><td>max(|e<sub>y</sub>|)</td><td><strong>0.019</strong></td><td>0.1</td></tr><tr><td>rms(e<sub>y</sub>)</td><td><strong>0.007</strong></td><td>0.04</td></tr></tbody></table>

proposed mpc-osqp architecture demonstrated superior performance, attributed to the stability objective inclusion, novel designed weights, and integrating $\mu$ and $\rho$ in the path-tracking weights. Compared to [13], the proposed mpc-osqp is a lower-order system (with 3 states instead of 5) while incorporating an additional stability objective, achieving enhanced computational efficiency (2.8 ms vs 5 ms).

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/N3OKyjUV2t8"

## IV. CONCLUSION

This work presents a novel minimum-order centralized MPC control architecture for autonomous vehicles, integrating coordinated path-tracking, speed control, and stability control. A novel adaptive weighting mechanism has been formulated to dynamically balance these control objectives based on multiple criteria. The proposed architecture demonstrates substantial improvements in path-tracking accuracy, stability, and speed control, while maintaining computational efficiency and robustness under high-velocity, low-adherence conditions, and model uncertainties. Future research endeavors aim to extend the method to an LPV framework for enhanced adaptability to speed variations and to validate the control architecture on an experimental vehicle.

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