Adaptive Longitudinal Speed Control for Heavy-Duty Vehicles Considering Actuator Constraints and Disturbances Using Simulation Validation

Abstract

Heavy-duty vehicles (HDVs), such as buses and commercial trucks, display unique dynamic characteristics due to their high mass and specific actuator properties. These factors make HDVs particularly sensitive to changes in vehicle load and road gradient, which significantly affect their longitudinal control performance. In other words, such variations present considerable challenges in maintaining stable and efficient longitudinal control of HDVs. To address these challenges, this study proposes a model reference adaptive control (MRAC) framework explicitly designed for HDVs. The control system utilizes a state predictor to mitigate actuator load problems caused by high-frequency components in the adaptive control input. In addition, when input constraints are present, the reference model is modified using the μ-modification technique. The system satisfies Lyapunov stability conditions and ensures stable longitudinal control performance across a range of driving conditions. The proposed closed-loop longitudinal control system was evaluated by implementing the controller using the vehicle dynamics simulation software IPG TruckMaker 12.0.1 and integrated with MATLAB/Simulink R2022b. The test scenarios included repetitive speed change maneuvers, which accounted for uncertainties such as road gradients, headwinds, and vehicle load conditions. The simulation results show that the control system not only effectively suppresses disturbances but also enables stable longitudinal speed tracking by considering actuator load and constraints, outperforming conventional MRAC. These results suggest that the proposed closed-loop longitudinal control system can be effectively applied to HDVs. The findings suggest that the proposed closed-loop longitudinal control system can be effectively applied to HDVs, ensuring improved stability and performance under real-world driving conditions.

1. Introduction

Recently, there has been increasing interest in applying and enhancing autonomous driving technologies for commercial vehicles, particularly in the field of logistics and freight transportation [1,2,3]. Commercial vehicles often operate over long distances and are exposed to many different dynamic driving environments, which makes the development of reliable and robust autonomous driving systems essential. In particular, long-haul operations can lead to profound driver fatigue, which is one of the major causes of serious traffic accidents involving large commercial vehicles [4,5,6]. To mitigate these risks, the development of Level 4 autonomous driving technology for commercial vehicles is receiving significant attention, as it can effectively reduce the need for continuous driver intervention and improve overall driving safety [7,8].

Autonomous driving for commercial vehicles relies heavily on the fusion of multiple sensor inputs to accurately perceive the surrounding environment, identify drivable space, and generate and execute safe and efficient driving plans [9,10]. One of the key requirements for the successful deployment of autonomous driving systems in commercial vehicles is precise longitudinal speed tracking, especially under varying road and traffic conditions. Accurate speed control is key not only to maintaining safe distances from surrounding vehicles but also to ensuring stable vehicle behavior during various driving scenarios. Examples include highway cruising, stop-and-go traffic, and uphill or downhill driving.

Due to the inherent characteristics of commercial vehicles, including high mass, time-varying dynamic properties, and non-linear relationships between control inputs and vehicle responses, conventional fixed-parameter controllers often struggle to maintain consistent speed-tracking performance [11,12,13]. In order to address these challenges, this study proposes an adaptive control framework that can adapt in real time to external disturbances and variations in vehicle dynamics. This approach improves the robustness and tracking accuracy of longitudinal speed control for commercial vehicles.

Cruise control systems demand high-performance speed-tracking capabilities to follow target speeds accurately, even when external disturbances are present. However, the longitudinal dynamics of commercial vehicles exhibit strong nonlinear characteristics due to factors such as variations in vehicle mass, road gradients, and disturbances like headwinds [14,15,16,17]. In addition, the physical limitations imposed by actuator dynamics further complicate the design of precise speed control systems [18,19,20]. According to [21], extensive research has been conducted to overcome these challenges by exploring robust control techniques suitable for commercial vehicle applications.

Several different control methodologies have been proposed to address these limitations, with model reference adaptive control MRAC [22,23,24,25,26,27,28] considered a promising approach for enhancing the longitudinal control performance of HDVs. This is due to its ability to handle external disturbances and actuator limitations.

For example, the author of [29] introduced a model reference adaptive velocity-control system for autonomous vehicles. This approach combines PID control with MRAC and enables robust vehicle control across various road conditions and environments. In [30], a conventional MRAC framework for an adaptive cruise control (ACC) system was proposed. This framework is based on a longitudinal dynamic model that considers uncertainties in the state and input matrices. The study demonstrated through simulations that MRAC outperforms linear state feedback controllers, even in uncertain environments.

While these studies highlight the effectiveness of MRAC in longitudinal velocity control for autonomous vehicles, they do not explicitly address problems related to the high-frequency noise and input constraints inherent in MRAC. In dynamic environments such as those encountered by HDVs, limitations in control input can lead to degraded tracking performance.

The author of [31] introduced a state predictor to address the trade-off between fast adaptation and smooth control signals in conventional MRAC systems. The proposed approach incorporates a predictor-error term into the adaptation law, which makes it possible to effectively suppress high-frequency noise in the control signal caused by the high learning rate of MRAC. As a result, smoother and more stable control signals can be achieved. However, because this approach does not consider the actuator characteristics of the vehicle, tracking performance can significantly deteriorate in the presence of input constraints.

In addition, the author of [32] reported an MRAC control framework designed to handle scenarios where input constraints are present in the control system. This approach adaptively modifies the reference model to suppress actuator saturation caused by the control law, thereby enhancing the practicality and robustness of MRAC, even under input-constrained conditions. However, while this approach addresses problems when input constraints exist, it does not account for instabilities such as high-frequency noise in the control signals, which arise due to the high learning rate of the adaptive law.

These limitations reveal that existing MRAC-based approaches do not simultaneously address the challenges of high-frequency oscillations and actuator saturation—both critical factors for achieving safe and practical speed control in HDVs.

To address these issues, this study proposes three key contributions. First, an MRAC strategy is introduced to suppress external disturbances and enable robust real-time adaptation under dynamic conditions. Second, input constraint-aware model reference design is employed to reduce actuator saturation while maintaining desired performance. Third, a low-pass filter is incorporated into the adaptation law to minimize high-frequency control noise, thereby ensuring smoother control input and mitigating actuator load. These contributions are tailored specifically for HDVs, which are highly sensitive to both control effort and external disturbance due to their large mass and nonlinear dynamics.

In summary, this study presents a novel adaptive control method that improves the speed-tracking performance of HDVs during long-haul operations and in dynamic environments by improving the existing MRAC. Specifically, a state predictor is introduced to address the trade-off between fast learning and smooth control signals. In addition, a control framework that considers input constraints is proposed to suppress actuator saturation. This research leverages the benefits of MRAC, such as disturbance compensation and adaptability to system changes, to enable more precise and stable speed control for autonomous driving systems in HDVs. The proposed closed-loop control system was developed in MATLAB/Simulink R2022b and the vehicle dynamics simulation software IPG TruckMaker 12.0.1. The validation scenario involved repeatedly changing the driving speed on a straight road with varying slopes and headwinds. Four different scenarios were considered based on the vehicle’s mass. The efficiency of the proposed model is explained by comparing the tracking performance of the target speed and the stability of the control inputs with those of two control systems, LQR and conventional MRAC.

The paper is structured as follows: Section 2 provides a detailed description of the vehicle’s longitudinal dynamics, which forms the foundation for the proposed control system. Section 3 introduces the conventional MRAC framework and presents the enhanced design. Section 4 describes the closed-loop control system, incorporating the proposed longitudinal adaptive controller. Section 5 discusses the simulation environment, validation scenarios, and analysis of the simulation results. Finally, Section 6 concludes the paper by summarizing the key contributions and outlining directions for future research.

2. Longitudinal Dynamics of Vehicles

In this study, a longitudinal dynamics model was developed to describe the longitudinal behavior of an HDV that takes into account the vehicle’s forward motion and the external forces acting on it. According to [33], the model was formulated based on simplified equations of motion that capture the relationship between vehicle mass and external forces, allowing for effective use in controller design and performance analysis.

The longitudinal dynamics of a vehicle is shown in Figure 1. This longitudinal model has been widely used as a fundamental model to analyze the longitudinal behavior of HDVs, and many research groups have applied it to the design and validation of different controllers [34,35].

The equation of motion of a vehicle moving on an inclined load along the vehicle’s longitudinal axis is represented based on Newton’s second law:

$$m{a}_x=F_T-F_R,$$

(1)

where the parameter m denotes the vehicle mass, $a_x$ represents the vehicle’s longitudinal acceleration, $F_T$ is the total longitudinal traction force, and $F_R$ signifies the resistance force, which can be further expressed as

$$F_R=F_g+F_r+F_a,$$

(2)

$$where \left\{\begin{array}{c}{F}_g=mg\mathit{sin}\theta \\ F_r={\displaystyle \frac{C_{r}mg\mathit{cos}\theta }{r_{eff}}}\\ F_a=0.5\rho C_d{A}_f{(V_x-V_{air})}^2\end{array}\right. ,$$

(3)

Here, $F_g$ comprises the gravitational force, $F_r$ is the rolling resistance, and $F_a$ denotes the aerodynamic resistance. The parameter $g$ represents the gravitational acceleration, $\theta$ is the road slope, $C_r$ indicates the rolling resistance coefficient, $r_{eff}$ represents the effective tire radius, $\rho$ denotes the air density, $C_d$ indicates the aerodynamic drag coefficient, $A_f$ denotes the frontal area of the vehicle, $V_x$ is the absolute vehicle speed, and $V_{air}$ is the absolute speed of the wind.

According to [36], the acceleration performance of a fully loaded HDV has a maximum acceleration of 0.05 $\mathrm{m}/{\mathrm{s}}^2$ on a flat road when the vehicle speed exceeds 25 mph. Because HDVs have relatively low driving forces, performance can be compromised even under loading conditions that are not a problem for passenger cars. In addition, variations in road gradient and payload may exceed the vehicle’s overall operational limits. Control commands that do not consider these physical limitations are likely to result in engine power saturation, which not only reduces the acceleration performance of the vehicle but also decreases the accuracy of speed tracking. Furthermore, actuator saturation can negatively impact vehicle stability, particularly in platooning scenarios where multiple vehicles follow each other in a coordinated formation, often using automated systems to maintain safe distances, where precise control is essential.

To address these challenges, the authors of [11] created an artificial saturation point to impose an acceleration limit that prevents the vehicle’s actuator from reaching saturation:

$$a_{x, act}\left(t\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left[a_x\left(t\right),a_{x, Lim}\left(m\left(t\right),F_g\left(t\right)\right)\right],$$

(4)

Here, $a_{x, act}$ denotes the actual acceleration and $a_{x, Lim}$ denotes the limitation of acceleration. From Equation (4), it can be seen that the vehicle mass and the road grade both significantly influence the longitudinal dynamics of the vehicle. These factors are time-varying and must be considered in the control design to ensure stable and efficient operation.

Furthermore, the actuation pathway from the control command to the actual vehicle acceleration introduces system lag. This delay arises from the throttle response, transmission dynamics, and engine inertia. According to [37], the actual acceleration of the vehicle can be considered to follow the desired acceleration with a delay, represented as a time delay $\tau$:

$$a_x={\displaystyle \frac{1}{\tau s+1}}{a}_{x,des},$$

(5)

Here, $a_{x,des}$ denotes desired longitudinal acceleration, and $\tau$ represent the time constant, which depends on the HDV’s control interface, actuator response, and dynamics of the system.

Consequently, based on Equations (1)–(5), the state-space model of the longitudinal motion of the HDV can be represented as

$${\dot{x}}_p\left(t\right)=A_p{x}_p\left(t\right)+B_{p}u\left(t\right)+d\left(t\right),$$

(6)

$$where \left\{\begin{array}{l}{x}_p\left(t\right)=\left[v_x\left(t\right)\hspace{1em}{a}_x\left(t\right)\right]\\ u\left(t\right)=a_{des}\left(t\right)\\ d\left(t\right)=F_R\left(t\right)+\varphi \left(t\right)\end{array}\right. ,$$

(7)

$$A_p=\left[\begin{array}{cc}0& 1\\ 0& -{\displaystyle \frac{1}{\tau }}\end{array}\right],$$

(8)

$$B_p=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{\tau }}\end{array}\right],$$

(9)

where $x_p$ is the state vector, which consists of the HDV’s longitudinal velocity, denoted as $v_x$, and the HDV’s longitudinal acceleration, denoted as $a_x$; $u$ is the control input, which represents the desired longitudinal acceleration; and $d$ is the known and unknown model uncertainty of the dynamical system, which consists of the HDV’s longitudinal resistance force and the unmatched nonlinear system model uncertainty, denoted as $\varphi$.

Equation (4) defines the limits of the acceleration input of the HDV due to the vehicle’s mass and road slope. The uncertainties in the model, external disturbances, and input limits can degrade the longitudinal control performance of the HDV, which highlights the need for adaptive control techniques.

3. Overview of the Improved Model Reference Adaptive Control Framework

Ensuring consistent speed-tracking performance in the longitudinal controller of an HDV is crucial for the vehicle’s fuel efficiency and safety. This requires addressing model uncertainties, real-time performance, disturbances, and input regulation. This section introduces the overall concept of MRAC and two extended models: state predictor-based adaptive control and positive $\mu$-modification adaptive control. MRAC enhances longitudinal tracking performance by suppressing model uncertainties and disturbances. This state predictor mitigates high-frequency noise in the control inputs, while the $\mu$-modification addresses actuator saturation caused by adaptive control inputs.

Let the nonlinear uncertain dynamical system be given by

$$\dot{x}\left(t\right)=A_{p}x\left(t\right)+B_p\left[u\left(t\right)+\Delta \left(x\left(t\right)\right)\right],\hspace{1em}x\left(0\right)=x_0,$$

(10)

where $x\left(t\right)\in {\mathbb{R}}^n$ denotes the system state vector available for feedback, $u\left(t\right)\in {\mathbb{R}}^m$ represents the control input, $A_p\in {\mathbb{R}}^{n\times n}$ and $B_p\in {\mathbb{R}}^{n\times m}$ are the known matrices of the actual system, and $\Delta :{\mathbb{R}}^n\to {\mathbb{R}}^m$ is the matched system uncertainty. Consider the ideal reference system that specifies a desired closed-loop dynamics system given by

$${\dot{x}}_m\left(t\right)=A_m{x}_m\left(t\right)+B_{m}r\left(t\right),\hspace{1em}{x}_m\left(0\right)=x_{m0},$$

(11)

where $x_m\left(t\right)\in {\mathbb{R}}^n$ is the reference state vector, $r\left(t\right)\in {\mathbb{R}}^r$ denotes the given uniformly continuous bounded command, $A_m\in {\mathbb{R}}^{n\times n}$ is a Hurwitz matrix, and $B_m\in {\mathbb{R}}^{n\times r}$ is the command input matrix.

The matched uncertainty in Equation (10) can be linearly parameterized as

$$\Delta \left(x\right)=W^T\beta \left(x\left(t\right)\right),$$

(12)

where $W\in {\mathbb{R}}^{s\times m}$ is an unknown constant weighting matrix, and $\beta :{\mathbb{R}}^n\to {\mathbb{R}}^m$ denotes the basis function driven by the state vector $x\left(t\right)$.

3.1. Standard Model Reference Adaptive Control (MRAC)

The control objective of MRAC is to design a feedback controller that ensures that the state variables $x$ of the system asymptotically follow the state variables $x_m$ of the reference model.

If the unknown parameters $A_p$ and $B_p$ are known, then $u\left(t\right)$ can be an ideal fixed gain control law expressed as

$$u\left(t\right)=K_x^{T}x\left(t\right)+K_r^{T}r\left(t\right)-W^T\beta \left(x\left(t\right)\right),$$

(13)

where $K_x\in {\mathbb{R}}^{n\times m}$ is the ideal value of the feedback gain, $K_r\in {\mathbb{R}}^{r\times m}$ denotes the ideal value of the feedforward gain, and $W\in {\mathbb{R}}^{s\times m}$ is the ideal value of the matched uncertainty.

Therefore, using Equations (10), (11) and (13), the following matching conditions are given by

$$A_m=A_p+B_p{K}_x^T, B_m=B_p{K}_r^T,$$

(14)

Applying the control law defined in Equation (13) to Equation (10), it can be seen that the closed-loop system is exactly the same as the reference model defined in Equation (11). Therefore, any bounded command input in Equation (13) provides an asymptotic tracking performance. However, $A_p$ is unknown, and the previously ideal gains $K_x,K_r$, and $W$ cannot be selected. Nevertheless, assuming that such ideal gains exist, as indicated by the Lyapunov analysis, the adaptive feedback control law is given by

$$u_{ad}\left(t\right)={\widehat{K}}_x^T\left(t\right)x\left(t\right)+{\widehat{K}}_r^T\left(t\right)r\left(t\right)-{\widehat{W}}^T(t)\beta \left(x\left(t\right)\right),$$

(15)

Here, $u_{ad}\left(t\right)\in {\mathbb{R}}^m$ denotes adaptive control signal, and ${\widehat{K}}_x\left(t\right), {\widehat{K}}_r\left(t\right),$and ${\widehat{W}}^T(t)$ are estimates of $K_x,K_r$, and $W$. To ensure that the control input $u_{ad}\left(t\right)$ adapts to system uncertainties and maintains stability, let $\mathrm{e}\left(t\right)=x_p\left(t\right)-x_m\left(t\right)$ be the tracking error. Then, the tracking error dynamics can be expressed as follows:

$$\dot{e}\left(t\right)=A_m{e}_m\left(t\right)+B_p\left(\Delta K_x^T{x}_p\left(t\right)+\Delta K_r^{T}r\left(t\right)-\Delta W^T\beta \left(x\left(t\right)\right)\right),$$

(16)

Here, the parameter estimation errors are denoted as $\Delta K_x={\widehat{K}}_x\left(t\right)-K_x$, $\Delta K_r={\widehat{K}}_r\left(t\right)-K_r$, and $\Delta W=\widehat{W}\left(t\right)-W$. To ensure that $\mathrm{e}\left(t\right)$ asymptotically converges to zero as $\mathrm{t}\to \mathrm{\infty }$ and to guarantee the boundedness of all closed-loop signals [38,39,40], the following adaptive laws are proposed for parameter estimates:

$$\left\{\begin{array}{l}{\dot{\widehat{K}}}_x=-{\Gamma }_{x}x\left(t\right)e^T\left(t\right)P{B}_p \\ {\dot{\widehat{K}}}_r={-\Gamma }_{r}r\left(t\right)e^T\left(t\right)P{B}_p \\ \dot{\widehat{W}}={\Gamma }_{\beta }\beta \left(x\left(t\right)\right)e^T\left(t\right)P{B}_p\end{array}\right.,$$

(17)

Here, ${\Gamma }_x\in {\mathbb{R}}^{n\times n}$, ${\Gamma }_r\in {\mathbb{R}}^{r\times r}$, and ${\Gamma }_{\beta }\in {\mathbb{R}}^{s\times s}$ denote positive definite learning rates, and $P\in {\mathbb{R}}^{n\times n}$ is a positive definite solution of the algebraic Lyapunov equation:

$$0=A_m^{T}P+P{A}_m+Q,$$

(18)

where $Q\in {\mathbb{R}}^{n\times n}$ is a given positive definite matrix. Because $A_m$ is Hurwitz, the converse Lyapunov theorem guarantees the existence of a unique positive definite matrix $P$ that satisfies the condition in Equation (18) for a given $Q$. By applying Barbalat’s lemma [41] to the Lyapunov candidate function and using the error dynamics in (16), it can be shown that the tracking error $\mathrm{e}\left(t\right)$ asymptotically converges to zero, which ensures the stability of the overall closed-loop system.

MRAC enables precise tracking performance by adaptively tuning the control parameters so that the system follows the reference model despite system uncertainties or external disturbances. This method makes it particularly attractive for nonlinear or uncertain environments, such as those encountered in HDV applications.

However, in the practical implementation of HDVs, several critical challenges arise. First, actuator saturation becomes a significant problem. The actuators in HDVs—namely, the powertrain and braking system—have strict physical limits in terms of maximum driving torque and braking force. When the control inputs calculated using MRAC exceed physical constraints, the vehicle cannot perform the commanded action. This results in degraded tracking accuracy, increased steady-state errors, and potential instability. Furthermore, excessive demands on actuators may compromise hardware safety and durability. Second, while MRAC’s high adaptation gain is beneficial for rapid response, it can introduce high-frequency oscillations in the control input. These oscillations may propagate through the vehicle system, leading to undesirable effects, such as reduced ride comfort, increased powertrain vibrations, and premature wear of mechanical components, including the brakes and suspension system. In heavy vehicles, where ride stability, component longevity, and energy efficiency are crucial, such high-frequency behaviors are particularly problematic.

In other words, while MRAC provides a strong theoretical foundation for adaptive control, its direct application to HDVs requires additional considerations. Practical constraints, such as actuator limits and system robustness against high-frequency adaptation, must be explicitly addressed through modifications to the standard MRAC structure. This need for adaptation highlights the necessity of HDV-specific changes to MRAC, such as the incorporation of input saturation handling mechanisms, which will be discussed in the following section.

3.2. μ-Modification in Adaptive Control

In order to address the actuator saturation in HDVs, μ-modification is introduced as an adaptive control law. This modification ensures that the actuator can effectively track external commands while respecting input constraints. The architecture of the positive $\mu$-modification control system is shown in Figure 2. In a system following the dynamical model in Equation (10), consider a static actuator model with the following input constraints [42].

$$u\left(t\right)=u_{max}\mathrm{s}\mathrm{a}\mathrm{t}\left({\displaystyle \frac{u_c\left(t\right)}{u_{max}}}\right),$$

(19)

where $u_c\left(t\right)\in {\mathbb{R}}^m$ is the commanded control input, and $u_{max}$ denotes the maximum allowable input limit. In other words,

$$u\left(t\right)=\left\{\begin{array}{l}{u}_c\left(t\right),\left|u_c\left(t\right)\right|\le u_{max},\\ u_{max}sgn\left(u_c\left(t\right)\right), \left|u_c\left(t\right)\right|>u_{max},\end{array}\right.$$

(20)

Therefore, it is essential to find an adaptive control law that considers the input constraints of an actuator with input limitations, such as HDVs. In addition, this control system can protect the adaptive input signal from position saturation. The control architecture with μ-modification is defined as

$$u_c\left(t\right)=u_{ad}\left(t\right)+\mu \Delta u_c\left(t\right),$$

(21)

$$u_{ad}\left(t\right)={\widehat{K}}_x^T(t)x\left(t\right)+{\widehat{K}}_r^T(t)r\left(t\right),$$

(22)

where $\mu$ is the constant, and $\Delta u_c\left(t\right)$ is represented as

$$\Delta u_c\left(t\right)=u_{max}^{\delta }\mathrm{s}\mathrm{a}\mathrm{t}\left({\displaystyle \frac{u_c\left(t\right)}{u_{max}^{\delta }}}\right)-u_c\left(t\right),$$

(23)

Here, $u_{max}^{\delta }=u_{max}-\delta$ when a constant value $0<\delta <u_{max}$ is chosen. In addition, $\Delta u_c\left(t\right)$ represents the control deficiency due to the virtual bound $u_{max}^{\delta }$. By assuming the existence of ideal gains as justified by the Lyapunov analysis, we define the following modified reference model:

$${\dot{x}}_m\left(t\right)=A_m{x}_m\left(t\right)+B_m\left(r\left(t\right)+{\widehat{K}}_u^T\left(t\right)\Delta u_{ad}\left(t\right)\right),$$

(24)

where ${\widehat{K}}_u\left(t\right)$ is the adaptive gain and $\Delta u_{ad}\left(t\right)$ denotes the deficiency of the adaptive control signal. This deficiency is defined by

$$\Delta u_{ad}=u\left(t\right)-u_{ad}\left(t\right),$$

(25)

The respective adaptive laws are as follows:

$$\left\{\begin{array}{l}{\dot{\widehat{K}}}_x=-{\Gamma }_{x}x\left(t\right)e^T\left(t\right)P{B}_p \\ {\dot{\widehat{K}}}_r=-{\Gamma }_{r}r\left(t\right)e^T\left(t\right)P{B}_p \\ {\dot{\widehat{K}}}_u=-{\Gamma }_u\Delta u_{ad}\left(t\right)e^T\left(t\right)P{B}_m\end{array}\right.$$

(26)

But Barbalat’s lemma does not apply in this context. This problem was also addressed by Lavretsky and Hovakimyan [32], and the same limitation applies in this study due to the adaptive nature of the reference model. Therefore, to establish the asymptotic convergence of the tracking error, it is also necessary to prove that at least one of the system states, $x\left(t\right)$ or $x_m\left(t\right)$, remains bounded under the control input.

To establish such boundedness, we introduce an auxiliary Lyapunov function defined as

$$W\left(x\right)=x^T{P}_{A}x,$$

(27)

$$0=x^T{P}_A+P_{A}x+Q_A,$$

(28)

where $P_A=P_A^T>0$ satisfies the Lyapunov equation for a positive definite matrix $Q_A$. By differentiating Equation (27), we derive the following Lyapunov derivative function:

$$\dot{W}\left(x\right)={\dot{x}}^T{P}_{A}x+x^T{P}_A\dot{x}=-x^T(A_P^T{P}_A+P_A{A}_P)+2x^T{P}_A{B}_{P}u(t),$$

(29)

Since the input vector $u_c\left(t\right) \in [-u_{max}, u_{max}]$ satisfies the bounded condition, Equation (28) can be transformed into the following upper bound condition:

$$\dot{W}\left(x\right)\le -{\lambda }_{max}\left(Q_A\right){‖x‖}^2‖P_A{B}_P‖u_{max},$$

(30)

As a result, due to this upper bound condition, when the absolute value of the system state vector becomes larger than a certain threshold, $\dot{W}(x)<0$, ensuring that the system state does not diverge beyond a certain bound and that the tracking error asymptotically converges to zero.

By applying μ-modification to MRAC, the system can maintain stability while effectively handling input constraints caused by actuator saturation. In the case of HDVs, acceleration constraints may arise due to variations in total mass or road gradients, which increases the likelihood of control signals reaching actuator saturation. This study proposes a method to address these problems with the help of μ-modification.

3.3. State Predictor in Adaptive Control

In the MRAC framework described above, high adaptation gains can lead to excessive high-frequency oscillations. The state predictor mitigates this problem by incorporating the prediction error in conjunction with the tracking error—effectively reducing noise in the control input. The architecture of the state predictor in adaptive control is shown in Figure 3. The following state predictor dynamics are presented to compute the prediction error [40]:

$$\dot{\widehat{x}}(t)=A_{prd}\left(\widehat{x}\left(t\right)-x\left(t\right)\right)+A_{m}x(t)+B_{m}u(t),$$

(31)

Here, $A_{prd}\in {\mathbb{R}}^{n\times n}$ is a Hurwitz matrix, and $\widehat{x}\in {\mathbb{R}}^n$ is the predicted state. The prediction error is defined as $\widehat{e}\left(t\right)=\widehat{x}\left(t\right)-x\left(t\right)$. Consequently, the tracking-error and prediction-error dynamics can be represented as follows:

$$\left\{\begin{array}{c}\dot{e}\left(t\right)=A_m{e}_m\left(t\right)+B_p\left(\Delta K_x^T{x}_p\left(t\right)+\Delta K_r^{T}r\left(t\right)-\Delta W^T\beta \left(x\left(t\right)\right)\right)\\ \dot{\widehat{e}}\left(t\right)=A_{prd}\widehat{e}\left(t\right)-B_p\left(\Delta K_x^T{x}_p\left(t\right)+\Delta K_r^{T}r\left(t\right)-\Delta W^T\beta \left(x\left(t\right)\right)\right)\end{array}\right.,$$

(32)

In order to regulate the parameter updates using both error signals, the training-error signal is defined as

$$\overline{e}=\left(e^T\left(t\right)P-{\widehat{e}}^T\left(t\right)P_{prd}\right),$$

(33)

where $P_{prd}>0$ is the is a positive definite solution of the algebraic Lyapunov equation. This signal can be interpreted as a filtered version of the tracking error. While the conventional MRAC update law relies solely on $e^{T}P{B}_P$, the state predictor in the adaptive control structure augments this by subtracting the contribution of the prediction error. Consequently, the adaptive controller is less sensitive to high-frequency components in the tracking error and effectively serves as a low-pass filter. This adjustment makes the control input smoother and improves robustness against measurement noise and fast transients. The state predictor in adaptive control framework uses this prediction-error signal to define the following adaptive laws:

$$\left\{\begin{array}{l}{\dot{\widehat{K}}}_x=-{\Gamma }_{x}x\left(t\right)\overline{e}{B}_p \\ {\dot{\widehat{K}}}_r=-{\Gamma }_{r}r\left(t\right)\overline{e}{B}_p \\ \dot{\widehat{W}}={\Gamma }_{\beta }\beta \left(x\left(t\right)\right)\overline{e}{B}_p\end{array}\right.,$$

(34)

This structure enables the parameter update mechanism to account for both tracking and prediction performance, which ensures sufficient stability even under fast adaptation. The predictor, typically designed to be faster than the reference model, helps mitigate undesirable rapid parameter fluctuations and prevents control chattering.

Lavretsky et al. [31] showed (through Lyapunov analysis) that all error signals in this framework are bounded, and both tracking and prediction errors asymptotically converge to zero. This shows that the predictor-based adaptive control system remains stable despite model uncertainties and parameter variations.

In the context of HDVs, where the vehicle may experience large mass variations, steep road gradients, and actuator saturation, maintaining smooth and robust control is crucial. State predictors are particularly well-suited for such environments, as they maintain high-performance tracking while suppressing high-frequency oscillations. This ultimately improves ride comfort, reduces actuator wear, and enhances robustness in real-world driving conditions.

4. Proposed MRAC for a Longitudinal Control System

In this section, a model that integrates the two approaches discussed in the previous section, $\mu$-modification and state predictor, is presented. In order to effectively apply adaptive control in an environment with unknown disturbance conditions affecting the HDV, this study proposes an architecture that combines these two models. This integrated architecture suppresses input constraints and high-frequency control inputs, as illustrated in Figure 4.

This integrated approach is designed to enhance the robustness of the adaptive control system while ensuring smooth longitudinal control for HDVs under varying load conditions and external disturbances. By incorporating input constraints into the reference model, the system prevents excessive control inputs that can lead to actuator saturation, while the adaptation law mitigates high-frequency oscillations to ensure stable performance in real-world scenarios. In particular, artificial acceleration limits—formulated based on the physical constraints of HDVs—are embedded within the reference model via μ-modification. This structure dynamically adjusts the reference trajectory, preventing infeasible commands under varying payloads and road gradients.

The following sections present the formulation of the reference model considering HDV-specific constraints, the adaptation law integrating $\mu$-modification with the state predictor, and the stability analysis validating the proposed approach. The longitudinal dynamics of an HDV can be expressed as

$${\dot{x}}_p\left(t\right)=A_{p}x\left(t\right)+B_p\left[u\left(t\right)+\Delta \left(x_p\left(t\right)\right)\right],$$

(35)

In order to prevent control inputs from exceeding the actuator limits, the reference model is modified to include input constraints as follows:

$${\dot{x}}_m(t)=A_m{x}_m\left(t\right)+B_m\left(r(t)+{\widehat{K}}_u^T\left(t\right)\Delta u_{ad}(t)\right),$$

(36)

Moreover, input constraints should also be incorporated into the predictor dynamics, which can be formulated as follows:

$$\dot{\widehat{x}}\left(t\right)=A_{prd}\left(\widehat{x}\left(t\right)-x_p\left(t\right)\right)+B_m\left(r\left(t\right)+{\widehat{K}}_u^T\left(t\right)\Delta u_{ad}\left(t\right)\right),$$

(37)

Considering Equations (35)–(37), both the tracking error dynamics and prediction error dynamics are given by

$$\dot{\mathrm{e}}\left(t\right)=A_m\mathrm{e}\left(t\right)+B_p\left(∆K_x^T{x}_p\left(t\right)+\Delta K_r^{T}r\left(t\right)-\Delta W^T\beta \left(x_p\left(t\right)\right)\right)-B_m∆K_u(t)∆u_{ad}(t)$$

(38)

$$\dot{\widehat{e}}\left(t\right)=A_{prd}\widehat{e}\left(t\right)-B_p\left(\Delta K_x^T{x}_p\left(t\right)+\Delta K_r^{T}r\left(t\right)-\Delta W^T\beta \left(x_p\left(t\right)\right)\right)+B_m∆K_u(t)∆u_{ad}(t)$$

(39)

where $∆K_u={\widehat{K}}_u-K_u$ denotes the parameter estimation error. Together with Equation (33), the adaptation law is designed to ensure robust performance by suppressing high-frequency oscillations and maintaining stability despite disturbances. Consequently, the modified adaptation law is as follows:

$$\left\{\begin{array}{l}{\dot{\widehat{K}}}_x=-{\Gamma }_x{x}_p\left(t\right)\overline{e}{B}_p \\ {\dot{\widehat{K}}}_r=-{\Gamma }_{r}r\left(t\right)\overline{e}{B}_p \\ \begin{array}{l}\dot{\widehat{W}}=-{\Gamma }_d\beta \left(x_p\left(t\right)\right)\overline{e}{B}_p\\ {\dot{\widehat{K}}}_u={\Gamma }_u\Delta u_{ad}\left(t\right)\overline{e}{B}_m\end{array}\end{array}\right.,$$

(40)

To verify the stability of the proposed control system, we define the Lyapunov function candidate as

$$V=e^{T}P\mathrm{e}+{\widehat{e}}^T{P}_{prd}\widehat{e}+tr\left(\Delta K_x^T{\Gamma }_x^{-1}\Delta K_x\right)+tr(\Delta K_r^T{\Gamma }_r^{-1}\Delta K_r)+tr(\Delta K_u^T{\Gamma }_r^{-1}\Delta K_u)+tr(\Delta W^T{\Gamma }_d^{-1}\Delta W)>0$$

(41)

The Lyapunov derivative function is

$$\begin{gathered} \dot{V}=e^T\left(A_m^{T}P+P{A}_m\right)\mathrm{e}+{\widehat{e}}^T\left(A_{prd}^T{P}_{prd}+P_{prd}{A}_{prd}\right)\widehat{e}+2tr\left(\Delta K_x^{T}x{\overline{e}}^T\right)-2tr\left(\Delta K_x^T{\Gamma }_x^{-1}{\dot{\widehat{K}}}_x\right)+2tr\left(\Delta K_r^{T}r{\overline{e}}^T\right) \\ -2tr(\Delta K_r^T{\Gamma }_r^{-1}{\dot{\widehat{K}}}_r)+2tr(\Delta K_u^T\Delta u_{ad}{\overline{e}}^T)-2tr(\Delta K_u^T{\Gamma }_u^{-1}{\dot{\widehat{K}}}_u)+2tr(\Delta W^T\beta \left(x_p\left(t\right)\right){\overline{e}}^T)-2tr(\Delta K_d^T{\Gamma }_d^{-1}{\dot{\widehat{K}}}_d) \end{gathered}$$

(42)

By substituting the adaptive law in Equation (40), the Lyapunov derivative function can be simplified as follows:

$$\dot{V}=-e^T\left(t\right)Qe\left(t\right)-{\widehat{e}}^T\left(t\right)Q_{prd}\widehat{e}\left(t\right)\le 0,$$

(43)

This indicates that $\mathrm{e},\widehat{\mathrm{e}}, K_x, K_r, K_u, W$ remain bounded. However, since tracking and prediction errors exhibit nonlinearity due to input saturation, it cannot be guaranteed that the tracking and prediction errors converge asymptotically to zero. Therefore, to complete the proof, it is necessary to additionally show that the system state vector remains bounded.

To prove the boundedness of the system state vector, the following Lyapunov candidate function is defined:

$$L\left(x_p\right)={x_p}^{T}Q{x}_p,$$

(44)

$$0={x_p}^{T}P+P{x}_p+Q,$$

(45)

where $P=P^T>0$ satisfies the Lyapunov equation for a positive definite matrix $Q$. By differentiating Equation (44), we derive the following Lyapunov derivative function:

$$\dot{L}\left(x_p\right)={{\dot{x}}_p}^{T}P{x}_p+{x_p}^{T}P{\dot{x}}_p=-{x_p}^T(A_P^{T}P+P{A}_P)+2{x_p}^{T}P{B}_{P}u\left(t\right),$$

(46)

Likewise, due to the input constraint $u\left(t\right)=[-u_{max}, u_{max}]$, the following upper bound condition can be derived:

$$\dot{L}\left(x_p\right)\le -{\lambda }_{max}\left(Q\right){‖x_p‖}^2‖P{B}_P‖u_{max},$$

(47)

As a result, the state vector $x_p$ is proven to be bounded according to the Lyapunov stability condition, and both the tracking error $e$ and the prediction error $\widehat{e}$ are shown to asymptotically converge to zero.

To effectively apply the proposed control method to commercial vehicles, the state-space equations are formulated based on Equation (6). Specifically, to maintain baseline longitudinal control performance while effectively handling disturbances and acceleration limits, the following two design conditions are introduced:

• Baseline-informed MRAC Initialization
• Limitation of artificial acceleration

First, the MRAC system is initialized using a linear quadratic regulator (LQR) as the baseline controller [43]. The feedback gain from the LQR is designed to preserve nominal longitudinal control performance and to provide a stable starting point for the MRAC adaptation process. This initialization mitigates excessive control effort during the early adaptation phase and enables the adaptive law to respond effectively to disturbances by referencing the LQR structure. Such an approach ensures that the adaptive control system starts from a reliable baseline, which reduces transient instability during initialization.

Second, artificial input constraints are defined, based on Equation (4), in accordance with the physical characteristics of commercial vehicles, such as vehicle mass variations and road slope. These constraints are designed to reflect realistic acceleration limits without requiring explicit parameter estimation and contribute to the overall stability of the system.

By incorporating these two conditions, the proposed closed-loop control system achieves enhanced robustness through the integration of baseline and adaptive control strategies. Furthermore, the explicit definition of meaningful acceleration limits enables the μ-modification technique to be more effectively tailored to real-world heavy-duty vehicle applications. This approach prevents overly aggressive control inputs and enables the system to reject disturbances under practical constraints.

5. Results of Simulation

In order to validate the proposed closed-loop longitudinal control system, simulations were conducted using the software IPG TruckMaker 12.0.1—a high-fidelity simulation environment widely used to assess vehicle dynamics and driver assistance systems—and MATLAB and Simulink R2022b. The vehicle dynamics were modeled using Mercedes-Benz Antos, as provided by TruckMaker12.

Table 1. System parameters used in the vehicle model.

Symbol	Value	Units
m	18,200	kg
$A_f$	12.48	m2
$C_r$	0.4	-
$C_d$	0.7	-
$\tau$	0.5	s

lists the parameters of the longitudinal vehicle model. All parameters listed in Table 1 are based on the default specification of this TruckMaker model, which reflects the typical characteristics of a real-world heavy-duty vehicle. Table 1 lists the parameters of the longitudinal vehicle model.

Furthermore, the simulation environment (IPG TruckMaker12) incorporates detailed physical modeling of key powertrain components, such as engine torque response, gear shifting logic, braking dynamics, and driveline efficiency. These high-fidelity models ensure that actuator behaviors—including limitations and delays—are realistically reproduced, enabling reliable validation of the proposed control strategy under real-world conditions.

For the validation of the control system, four test scenarios were defined based on vehicle mass—the baseline mass (18,200 kg), a 9100 kg load, and an 18,200 kg load—and headwinds. In each scenario, the vehicle experienced speed variations during operation.

To evaluate the longitudinal speed-tracking performance of the HDV, a time-varying speed profile, ranging from 35 to 65 km/h, was defined, as shown in Figure 5. This speed profile was designed to evaluate the controller’s responsiveness under dynamically changing commands and external conditions.

The simulations were conducted on a straight road with inclines to assess the robustness of the control system. The road friction coefficient was set to 0.7, and the sampling time was 10 ms to ensure real-time performance. For the longitudinal control of the vehicle, model uncertainties were introduced through external disturbances—specifically, road slope and headwind. The road slope conditions are shown in Figure 6 and Figure 7.

This setup allows for the verification of the controller’s effectiveness in responding to continuous speed variations in a dynamic environment with common disturbances such as road gradients, headwinds, and changes in vehicle mass.

Moreover, each scenario exhibits clearly defined acceleration limits. Consequently, the proposed control system was used to evaluate the vehicle’s stability and tracking performance across different scenarios.

As noted in [31], the predictor dynamics should be faster than the reference dynamics. We consider the following parameters of predictor dynamics:

$$A_{prd}=\alpha A_p; P_{prd}=\alpha P,$$

(48)

The control parameters for the longitudinal control system are outlined in

Table 2. Parameters used in the control system.

Symbol	Value	Units
$k_{x, 0}$	${\left[1.8257, 1.4817\right]}^T$	-
$k_{r,0}$	${\left[1.8257\right]}^T$	-
$W_0$	${\left[0, 0\right]}^T$	-
$k_{u,0}$	0.3	-
${\Gamma }_x$	$0.1\times I_{2\times 2}$	-
${\Gamma }_r$	$0.1$	-
${\Gamma }_{\beta }$	$0.01\times I_{2\times 2}$	-
${\Gamma }_u$	0.1	-
$\mathrm{\delta }$	0.3$u_{max}$	-
$\mathrm{\mu }$	10	-
$\alpha$	10	-

. The initial feedback gain parameters $k_{x,0}$, $k_{r,0}$ were selected by referring to the optimal gain values obtained from a baseline LQR controller. This ensures that the MRAC framework starts from a reasonable baseline with desirable closed-loop dynamics. The learning rates, represented by the adaptation gains ${\Gamma }_x$, ${\Gamma }_r$, and ${\Gamma }_d$, were chosen to ensure the boundedness of all closed-loop signals and to guarantee stable adaptation behavior under varying driving conditions and external disturbances.

To ensure fair comparison between the proposed method and the conventional MRAC, the same values for the initial gains $k_{x,0}$ and $k_{r,0}$ and learning rates ${\Gamma }_x$, ${\Gamma }_r$, and ${\Gamma }_d$ were applied in both control systems. This allows the performance difference to be attributed solely to structural improvements, such as the use of the state predictor and the input constraint handling mechanism, rather than to parameter tuning.

In order to evaluate the longitudinal control performance of the HDV, the ability to reliably and accurately track the target speed is assessed under conditions that involve model uncertainties, disturbances, and actuator limitations. This section analyzes the tracking performance of four different control systems using previously defined scenarios.

The first system is the LQR controller, the second is the standard MRAC, and the third is the proposed control system. Furthermore, a comparison between the reference model of the standard MRAC and the modified reference model of the proposed control system demonstrates the adjustments made to account for actuator limitations. This analysis is facilitated by the simulation results from the four different scenarios presented in

Table 3. Scenario conditions.

No.	Road Grade [+0.67°]	Payload Condition	Sudden Headwinds [−20, −30 m/s]
Scen 1	O	X	X
Scen 2	O	X	O
Scen 3	O	+9100 kg	O
Scen 4	O	+18,200 kg	O

.

Each simulation scenario incorporates physically meaningful external disturbances—road gradient, headwind, and payload variation—quantitatively modeled based on the force components introduced in Section 2. The road slope was defined with reference to the Korean Road Traffic Act to ensure that the simulated inclines reflected realistic road design standards. The payload variations (up to +100%) correspond to the typical maximum load conditions observed in HDV operations. In addition, headwind disturbances ranging from −20 to −30 m/s were included to simulate severe but physically plausible environmental conditions, such as those encountered during hurricane-level weather. These conditions are directly translated into gravitational, aerodynamic, and rolling resistance forces that influence the vehicle’s longitudinal acceleration. This setup enables a comprehensive and realistic evaluation of how effectively the proposed closed-loop control system maintains robust speed tracking under complex and adverse operating conditions.

Figure 8 and Figure 9 show the results of the longitudinal speed-tracking performance for three different control strategies. The top graph in each figure displays the vehicle velocity. The second graph in each figure displays the vehicle acceleration, and the third graph in each figure displays the vehicle jerk. Figure 8a shows the first scenario, an ideal condition where no actuator limitations are observed. Figure 8b presents the second scenario, where sudden headwinds are applied, which cause slight actuator responses. Figure 9a shows the third scenario, where a payload of +9100 kg was added together with road grade and headwind disturbances, which led to clear signs of actuator saturation. Figure 9b depicts the most challenging scenario, involving a payload of +18,200 kg, together with road grade and headwind disturbances, where actuator limitations become more pronounced.

In the first scenario, the top graph in Figure 8a shows that the LQR, MRAC, and proposed controller successfully track the reference input. Since no disturbances occur in this case, all controllers exhibit ideal speed-tracking performance. The middle graph confirms the absence of an acceleration limit, as actuator saturation does not appear, which results in smooth acceleration responses. The bottom graph shows that the proposed controller effectively suppresses high-frequency noise compared to MRAC, particularly between 40 s and 60 s. This indicates smoother and more stable control inputs. Moreover, MRAC exhibits a peak jerk value of approximately 1.0 $\mathrm{m}/{\mathrm{s}}^3$ during the acceleration phase, while the proposed controller maintains a lower peak jerk of about 0.8 $\mathrm{m}/{\mathrm{s}}^3$. This clearly demonstrates improved smoothness in control action.

In the second scenario, where sudden headwinds occur, the top graph in Figure 8b shows that the LQR, MRAC, and proposed controllers successfully reach the desired speed. However, as seen in the middle graph in Figure 8b, a slight acceleration limit occurs during the control process. This finding indicates that actuator saturation begins to appear in this scenario; however, it does not significantly hinder overall speed tracking. In contrast to the ideal condition in Scenario 1, the reference and modified reference models diverge slightly due to the imposed acceleration limit. As shown in the bottom graph in Figure 8b, the proposed controller demonstrates reduced jerk responses compared to the MRAC. Specifically, the peak jerk value of the standard MRAC reached approximately 0.9 $\mathrm{m}/{\mathrm{s}}^3$, while the proposed controller achieved a lower maximum jerk of around 0.7 $\mathrm{m}/{\mathrm{s}}^3$. Moreover, during the 40–60 s period, the standard MRAC exhibited noticeable high-frequency oscillations, whereas the proposed controller effectively suppressed them, indicating enhanced control smoothness and stability.

In the third scenario, where the actuator limit occurs, the top graph in Figure 9a shows that both the LQR controller and the standard MRAC cannot reach the desired speed between 70 s and 110 s. This error is likely due to a lack of control input caused by the acceleration limit. As seen in the middle graph of Figure 9a, the reference model reduces the acceleration-control input before the vehicle reaches the target speed. In contrast, the proposed control system successfully reached the desired speed. This is because the modified reference model (which accounts for the vehicle’s acceleration limit) continues to apply acceleration input even when the vehicle has not yet reached the desired speed. As shown in the bottom graph of Figure 9a, the standard MRAC shows consistently higher peak jerks compared to the proposed control system throughout the acceleration phase—with a maximum jerk of approximately 0.7 $\mathrm{m}/{\mathrm{s}}^3$. In contrast, the proposed control system reaches a significantly lower maximum jerk (0.6 $\mathrm{m}/{\mathrm{s}}^3$). Noticeable oscillations are observed in the standard MRAC during the 50 to 60 s period, while the proposed control system effectively suppressed these oscillations, which indicates improved stability.

In the fourth scenario, where a more severe actuator limit occurs, the top graph in Figure 9b shows that both the LQR controller and standard MRAC struggle to reach the desired speed over time. This error is also likely due to a lack of control input because of the acceleration limit. In this case, the proposed control system also successfully reached the desired speed. As seen in the middle graph of Figure 9b, the vehicle’s acceleration does not exceed 0.5 m/s², and in the case of standard MRAC, the lack of acceleration prevents the vehicle from reaching the target speed. In contrast, the proposed control system adjusts the acceleration input to account for the acceleration limit, which allows the vehicle to reach its target speed. Therefore, unlike the standard MRAC, the proposed control system made it possible to reach the target speed. As shown in the bottom graph of Figure 9b, the standard MRAC exhibits consistently higher peak jerk values compared to the proposed control system throughout the acceleration phase—with a maximum jerk of approximately 0.8 $\mathrm{m}/{\mathrm{s}}^3$. In contrast, the proposed control system achieved a significantly lower maximum jerk (approximately 0.5 $\mathrm{m}/{\mathrm{s}}^3$). Furthermore, noticeable oscillations are observed in the standard MRAC during the 50 to 60 s period, while the proposed control system effectively suppressed these oscillations, which confirms improved stability.

Figure 8 and Figure 9 clearly demonstrate that as the operating environment becomes increasingly severe, the tracking performance of conventional control methods gradually deteriorates. In particular, the standard MRAC struggles to handle increased vehicle mass, which results in hitting acceleration limits. Consequently, the actual plant fails to follow the reference model, which increases the tracking error. This, in turn, affects the adaptive gain and leads to instability in the control input. As a result, not only does the plant fail to track the desired speed, but the control input also suffers from high-frequency noise, which further degrades control performance. In contrast, the proposed control system actively modified the reference model when acceleration limits were detected, which ensured that the plant could feasibly follow the reference model under the given physical constraints. This modified reference model not only helped the plant track the desired speed more effectively but also mitigated instability in the adaptive gain, ultimately suppressing high-frequency noise in the control input. As shown in Figure 9, these benefits become even more pronounced under severe operating conditions, which highlights the superior robustness and adaptability of the proposed method.

To quantitatively evaluate the performance of the proposed adaptive control method, a comparative analysis was conducted using two conventional control strategies: standard MRAC and proposed MRAC. Three key performance indices were selected to assess the responsiveness of the system and the impact of the control inputs on actuator load:

• Raising Time: The time required for the vehicle to reach 90% of the reference speed from 10%, representing the speed of the control response.
• Speed Error Area (Integrated Absolute Error, IAE): The integral of the absolute difference between the actual vehicle speed and the reference speed over time used to evaluate tracking accuracy.
• Maximum Jerk: The peak rate of change in acceleration over time, indirectly reflecting the smoothness of control inputs. Lower jerk values correspond to reduced actuator stress and improved ride comfort.

These indices serve as comprehensive metrics for evaluating speed-tracking ability, disturbance refection, and control input stability—factors closely related to the physical acceleration limitations and high-gain noise sensitivity considered in this study. The results of the evaluation for Scenarios 1 through 4 are summarized in

Table 4. Performance metrics comparison.

No.		Raising Time	Speed Error Area	Max. Jerk
Scen 1	MRAC	5.91 s	0.8569 m	0.9971 $\mathrm{m}/{\mathrm{s}}^3$
	Proposed	5.80 s	0.6465 m	0.8255 $\mathrm{m}/{\mathrm{s}}^3$
Scen 2	MRAC	5.87 s	1.1346 m	0.9527 $\mathrm{m}/{\mathrm{s}}^3$
	Proposed	5.79 s	1.1296 m	0.7979 $\mathrm{m}/{\mathrm{s}}^3$
Scen 3	MRAC	13.49 s	10.4628 m	0.6860 $\mathrm{m}/{\mathrm{s}}^3$
	Proposed	10.29 s	10.1400 m	0.5668 $\mathrm{m}/{\mathrm{s}}^3$
Scen 4	MRAC	15.32 s	14.6642 m	0.8050 $\mathrm{m}/{\mathrm{s}}^3$
	Proposed	12.17 s	14.1275 m	0.5008 $\mathrm{m}/{\mathrm{s}}^3$

.

Based on overall comparison results, the proposed longitudinal adaptive controller demonstrates faster raising time even under physical input constraints, thanks to its ability to reshape the reference model to accommodate acceleration limits. It also achieves the lowest cumulative speed error, indicating improved disturbance rejection and more precise tracking of the reference speed.

Moreover, the proposed controller consistently produces lower maximum jerk values compared to standard MRAC, confirming its ability to generate smoother control inputs and reduce mechanical stress on actuators. This outcome is primarily attributed to the incorporated predictor-based filtering mechanism, which effectively suppresses high-frequency components in the control signal.

In conclusion, the proposed control framework overcomes the key limitations of standard MRAC by combining reference model modification for input-constrained environments with filtering techniques for noise suppression. As a result, it achieves both fast convergence and stable control inputs, thereby offering superior longitudinal control performance for HDV applications.

6. Conclusions

This study proposed a design methodology for the model reference longitudinal speed control of HDVs, in which a predefined reference model guides the vehicle’s behavior to achieve the desired dynamic performance. The proposed method explicitly considers uncertainties and dynamic characteristics encountered in complex driving environments. By integrating μ-modification adaptive control, the developed closed-loop control system effectively addresses the input limitations that conventional MRAC systems fail to handle. In addition, the system incorporates a state predictor to suppress high-frequency noise. This feature reduces the actuator load, which increases its suitability for HDVs. The stability of the proposed closed-loop control system was analytically verified using Lyapunov stability conditions. Robustness and performance were further validated through various experimental scenarios that simulated realistic, complex environmental and dynamic conditions.

The primary contribution of this study was the identification and mitigation of the limitations associated with conventional control approaches, such as LQR and standard MRAC, especially in environments where HDV acceleration inputs are highly sensitive to external disturbances and vehicle conditions.

Unlike fixed-gain control methods such as PID and LQR, which struggle to maintain performance under rapidly changing conditions, the proposed adaptive approach dynamically adjusts control parameters in real time. This adaptability is critical in HDV applications where system dynamics are heavily influenced by time-varying payloads and environmental disturbances.

The proposed controller modifies the control input in real time, explicitly considering acceleration input limits arising from variations in vehicle mass and environmental conditions. This capability ensures accurate tracking of target speeds. Furthermore, the controller effectively suppresses sudden changes in control inputs and high-frequency noise. Consequently, the actuation commands remain smooth, which reduces actuator overload and wear.

The effectiveness of the proposed control system was demonstrated in three practical scenarios: hill climbing, headwind conditions, and variations in vehicle mass. Across all scenarios, the proposed system consistently showed superior performance in terms of target speed tracking and control input stability.

Most existing studies have primarily focused on improving tracking performance without explicitly addressing actuator limitations. However, actuator constraints significantly influence the stability and durability of HDV control systems in practical applications. Therefore, the development of adaptive control strategies that explicitly account for actuator dynamics and constraints is essential for ensuring both reliable speed-tracking and long-term system robustness of HDV control systems.

Despite the demonstrated effectiveness of the proposed approach, several limitations remain. First, the current control framework focuses solely on longitudinal dynamics and does not consider integration with lateral control strategies, which would be essential for handling complex maneuvers, such as curved road tracking or lane changes. Second, although the simulation scenarios include realistic environmental disturbances—such as road slopes, variable payloads, and headwinds—they do not account for sensor-level uncertainties, such as measurement noise, parameter estimation errors, or data delays, which can significantly impact real-world implementations.

Future research will address these aspects to further enhance the robustness and practical applicability of the proposed control system. In particular, we plan to implement and test the proposed control system on a real HDV to validate its performance under real-world conditions, including varying road gradients, payload levels, and wind disturbances. In addition, the proposed control framework will be extended to vehicle-platooning systems, which require simultaneous speed tracking and inter-vehicle spacing control under severe environmental conditions. Furthermore, variations in vehicle mass affect not only longitudinal but also lateral dynamics. Hence, future studies will explore adaptive control strategies that incorporate input constraints and uncertainties into lateral control systems as well.