## 3 Overview of the Improved Model Reference Adaptive Control Framework
### 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 \(u(t)\) can be an ideal fixed gain control law expressed as  
\[u(t)=K_{x}^{T}x(t)+K_{r}^{T}r(t)-W^{T}\beta(x(t)), \tag{13}\]  
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  
\[A_{m}=A_{p}+B_{p}K_{x}^{T},\ B_{m}=B_{p}K_{r}^{T}, \tag{14}\]  
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  
\[u_{ad}(t)=\hat{K}_{x}^{T}(t)x(t)+\hat{K}_{r}^{T}(t)r(t)-\hat{W}^{T}(t)\beta(x (t)), \tag{15}\]  
Here, \(u_{ad}(t)\in\mathbb{R}^{m}\) denotes adaptive control signal, and \(\hat{K}_{x}(t),\ \hat{K}_{r}(t)\), and \(\hat{W}^{T}(t)\) are estimates of \(K_{x},K_{r}\), and \(W\). To ensure that the control input \(u_{ad}(t)\) adapts to system uncertainties and maintains stability, let \(\mathrm{e}(t)=x_{p}(t)-x_{m}(t)\) be the tracking error. Then, the tracking error dynamics can be expressed as follows:  
\[\dot{e}(t)=A_{m}e_{m}(t)+B_{p}\Big{(}\Delta K_{x}^{T}x_{p}(t)+\Delta K_{r}^{T} r(t)-\Delta W^{T}\beta(x(t))\Big{)}, \tag{16}\]  
Here, the parameter estimation errors are denoted as \(\Delta K_{x}=\hat{K}_{x}(t)-K_{x}\), \(\Delta K_{r}=\hat{K}_{r}(t)-K_{r}\), and \(\Delta W=\hat{W}(t)-W\). To ensure that \(\mathrm{e}(t)\) asymptotically converges to zero as \(\mathrm{t}\rightarrow\infty\) and to guarantee the boundedness of all closed-loop signals [38; 39; 40], the following adaptive laws are proposed for parameter estimates:  
\[\left\{ \begin{array}{l}\dot{\hat{K}}_{x}=-\Gamma_{x}x(t)\varepsilon^{T}(t)PB_{p }\\ \dot{\hat{K}}_{r}=-\Gamma_{r}r(t)e^{T}(t)PB_{p}\\ \dot{\hat{W}}=\Gamma_{\beta}\beta(x(t))e^{T}(t)PB_{p}\end{array} \right., \tag{17}\]  
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:  
\[0=A_{m}^{T}P+PA_{m}+Q, \tag{18}\]  
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 [PERSON]'s lemma [41] to the Lyapunov candidate function and using the error dynamics in (16), it can be shown that the tracking error \(\mathrm{e}(t)\) 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.