## 4 Proposed MRAC for a Longitudinal Control System
\right., \tag{40}\]  
To verify the stability of the proposed control system, we define the Lyapunov function candidate as  
\[V=e^{T}P\hat{e}+\bar{\delta}^{T}P_{prd}\hat{e}+tr\Big{(}\Delta K_{x}^{T} \Gamma_{x}^{-1}\Delta K_{x}\Big{)}+tr\Big{(}\Delta K_{r}^{T}\Gamma_{r}^{-1} \Delta K_{r}\Big{)}+tr\Big{(}\Delta K_{u}^{T}\Gamma_{r}^{-1}\Delta K_{u} \Big{)}+tr\Big{(}\Delta W^{T}\Gamma_{d}^{-1}\Delta W\Big{)}>0 \tag{41}\]  
The Lyapunov derivative function is  
\[\dot{V}=e^{T}\big{(}A_{m}^{T}P+PA_{m}\big{)}\hat{e}+\bar{\delta}^{T}\Big{(}A_ {prd}^{T}P_{prd}+P_{prd}A_{prd}\Big{)}\hat{e}+2 tr\Big{(}\Delta K_{x}^{T} \bar{\kappa}\bar{\epsilon}^{T}\Big{)}-2 tr\Big{(}\Delta K_{x}^{T}\Gamma_{x}^{ -1}\hat{K}_{x}\Big{)}+2 tr\Big{(}\Delta K_{r}^{T}\bar{\epsilon}^{T}\Big{)} \tag{42}\]  
By substituting the adaptive law in Equation (40), the Lyapunov derivative function can be simplified as follows:  
\[\dot{V}=-e^{T}(t)Qe(t)-\bar{\epsilon}^{T}(t)Q_{prd}\hat{e}(t)\leq 0, \tag{43}\]  
This indicates that \(\hat{e},\hat{e},\ K_{x},\ K_{r},\ K_{u},\ W\) 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:  
\[L\big{(}x_{p}\big{)}={x_{p}}^{T}Qx_{p}, \tag{44}\]  
\[0={x_{p}}^{T}P+Px_{p}+Q, \tag{45}\]  
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:  
\[\dot{L}\big{(}x_{p}\big{)}={\dot{x}_{p}}^{T}Px_{p}+{x_{p}}^{T}\dot{P}\dot{x}_{ p}=-{x_{p}}^{T}\Big{(}A_{p}^{T}P+PA_{p}\Big{)}+2{x_{p}}^{T}PB_{p}u(t), \tag{46}\]Likewise, due to the input constraint \(u(t)=[-u_{max},\,u_{max}]\), the following upper bound condition can be derived:  
\[\dot{L}(x_{p})\leq-\lambda_{max}(Q)\big{\|}x_{p}\big{\|}^{2}\|PB_{P}\|u_{max}, \tag{47}\]  
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 \(\hat{e}\) 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.