## 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 \(\mu\)-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  
\[\dot{x}_{p}(t)=A_{p}x(t)+B_{p}\big{[}u(t)+\Delta\big{(}x_{p}(t)\big{)}\big{]}, \tag{35}\]  
In order to prevent control inputs from exceeding the actuator limits, the reference model is modified to include input constraints as follows:  
\[\dot{x}_{m}(t)=A_{m}x_{m}(t)+B_{m}\Big{(}r(t)+\dot{K}_{u}^{\tau}(t)\Delta u_{ ad}(t)\Big{)}, \tag{36}\]  
Figure 4: Schematic illustrating the closed-loop control system architecture. The red arrow indicate the modification of control input and reference model, and the blue arrows represent the flow of adaptive gain updates. Moreover, input constraints should also be incorporated into the predictor dynamics, which can be formulated as follows:  
\[\dot{\hat{x}}(t)=A_{prd}\big{(}\hat{x}(t)-x_{p}(t)\big{)}+B_{m}\Big{(}r(t)+\hat{ K}_{u}^{T}(t)\Delta u_{ad}(t)\Big{)}, \tag{37}\]  
Considering Equations (35)-(37), both the tracking error dynamics and prediction error dynamics are given by  
\[\dot{\hat{e}}(t)=A_{m}\hat{e}(t)+B_{p}\Big{(}\Delta K_{x}^{T}x_{p}(t)+\Delta K _{r}^{T}r(t)-\Delta W^{T}\beta\big{(}x_{p}(t)\big{)}\Big{)}-B_{m}\Delta K_{u}(t )\Delta u_{ad}(t) \tag{38}\]  
\[\dot{\hat{e}}(t)=A_{prd}\hat{e}(t)-B_{p}\Big{(}\Delta K_{x}^{T}x_{p}(t)+\Delta K _{r}^{T}r(t)-\Delta W^{T}\beta\big{(}x_{p}(t)\big{)}\Big{)}+B_{m}\Delta K_{u}(t )\Delta u_{ad}(t) \tag{39}\]  
where \(\Delta K_{u}=\hat{K}_{u}-K_{u}\) 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{\hat{K}}_{x}=-\Gamma_{x}x_{p}(t)\bar{e}B_{p}\\ \dot{\hat{K}}_{r}=-\Gamma_{r}r(t)\bar{e}B_{p}\\ \dot{\hat{W}}=-\Gamma_{d}\beta(x_{p}(t))\bar{e}B_{p}\\ \dot{\hat{K}}_{u}=\Gamma_{u}\Delta u_{ad}(t)\bar{e}B_{m}\end{array}