## 3 Overview of the Improved Model Reference Adaptive Control Framework
### \(\mu\)-Modification in Adaptive Control  
In order to address the actuator saturation in HDVs, \(\mu\)-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]. \[u(t)=u_{max}\text{sat}\bigg{(}\frac{u_{c}(t)}{u_{max}}\bigg{)}, \tag{19}\]  
where \(u_{c}(t)\in\ \mathbb{R}^{m}\) is the commanded control input, and \(u_{max}\) denotes the maximum allowable input limit. In other words,  
\[u(t)=\left\{ \begin{array}{ll}u_{c}(t),&|u_{c}(t)|\leq u_{max},\\ u_{max}sgn(u_{c}(t)),&|u_{c}(t)|>u_{max},\end{array} \right. \tag{20}\]  
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 \(\mu\)-modification is defined as  
\[u_{c}(t)=u_{ad}(t)+\mu\Delta u_{c}(t), \tag{21}\]  
\[u_{ad}(t)=\hat{K}_{x}^{T}(t)x(t)+\hat{K}_{r}^{T}(t)r(t), \tag{22}\]  
where \(\mu\) is the constant, and \(\Delta u_{c}(t)\) is represented as  
\[\Delta u_{c}(t)=u_{max}^{\delta}\text{sat}\bigg{(}\frac{u_{c}(t)}{u_{max}^{ \delta}}\bigg{)}-u_{c}(t), \tag{23}\]  
Here, \(u_{max}^{\delta}=u_{max}-\delta\) when a constant value \(0<\delta<u_{max}\) is chosen. In addition, \(\Delta u_{c}(t)\) 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:  
\[\dot{x}_{m}(t)=A_{m}x_{m}(t)+B_{m}\Big{(}r(t)+\hat{K}_{u}^{T}(t)\Delta u_{ad} (t)\Big{)}, \tag{24}\]  
Figure 2: \(\mu\)-modification in adaptive control. The red arrows represent the flow of updated gains based on the adaptation laws. where \(\hat{K}_{u}(t)\) is the adaptive gain and \(\Delta u_{ad}(t)\) denotes the deficiency of the adaptive control signal. This deficiency is defined by  
\[\Delta u_{ad}=u(t)-u_{ad}(t), \tag{25}\]  
The respective adaptive laws are as follows:  
\[\left\{ \begin{array}{l}\hat{\hat{K}}_{x}=-\Gamma_{x}x(t)e^{T}(t)PB_{p}\\ \hat{\hat{K}}_{r}=-\Gamma_{r}r(t)e^{T}(t)PB_{p}\\ \hat{\hat{K}}_{u}=-\Gamma_{u}\Delta u_{ad}(t)e^{T}(t)PB_{m}\end{array} \right. \tag{26}\]  
But [PERSON]'s lemma does not apply in this context. This problem was also addressed by [PERSON] and [PERSON] [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, \(x(t)\) or \(x_{m}(t)\), remains bounded under the control input. To establish such boundedness, we introduce an auxiliary Lyapunov function defined as  
\[W(x)=x^{T}P_{A}x, \tag{27}\]  
\[0=x^{T}P_{A}+P_{A}x+Q_{A}, \tag{28}\]  
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:  
\[\dot{W}(x)=\dot{x}^{T}P_{A}x+x^{T}P_{A}\dot{x}=-x^{T}\Big{(}A_{p}^{T}P_{A}+P_ {A}A_{p}\Big{)}+2x^{T}P_{A}B_{p}u(t), \tag{29}\]  
Since the input vector \(u_{c}(t)\ \in[-u_{max},\,u_{max}]\) satisfies the bounded condition, Equation (28) can be transformed into the following upper bound condition:  
\[\dot{W}(x)\leq-\lambda_{max}(Q_{A})\|x\|^{2}\|P_{A}B_{p}\|u_{max}, \tag{30}\]  
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.