## 3 Multiple AUVs-UAVs Control Design  
### L1 Adaptive Control with Actuator Time Delay  
In this section, the L1 adaptive control utilized in [26] is used for a single AUV extended with an actuator time delay to stabilize multiple AUVs. The main feature of utilizing the L1 adaptive controller with an actuator time delay is its ability to adapt rapidly and robustly, which augments the performance of the system. This capacity has been achieved by separating adaption from robustness. Utilizing a modified L1 controller, the uncertainties are predestined by using a fast method and the recompensed uncertainties proceed through a filter. Then, the overall output is included within the input signal. There are two key functions to this filter. First, it ensures that the control signal stays within an acceptable frequency range. Second, it separates the robustness from the adaptation. With this filter, we can improve robustness as well as performance. For more details, see [27; 28; 29]. In order to implement the modified L1 controller to the quadrotor/underwater vehicle, a representation of the state space for (5) and (17) is required. \[\dot{x}_{1}(m)=x_{2},\text{with}\ x_{1}(0)=x_{1_{0}}\] \[\dot{x}_{2}(m)=A_{2}x_{2}(m)+f_{2}(m,x(m))+B_{2}wu(m-\tau),\ \text{with}\ x_{2}(0)=x_{2_{0}}\] \[y(m)=Cx(m), \tag{19}\]  
where, the system's states are denoted by \(x(m)=[x_{1}(m),\ x_{2}(m)]^{\top}\in\ \mathbb{R}^{2n}\), the known matrix is represented by \(A_{2}\in\ \mathbb{R}^{n\times n}\), the constant matrix is represented by \(B_{2}\in\ \mathbb{R}^{m\times n}\), \(u(m-\tau)\in\ \mathbb{R}^{m}\) represents the control vector with time delay \(\tau\), \(w\in\ \mathbb{R}\) denote the input's uncertainty, the known constant matrix is represented by \(C\in\ \mathbb{R}^{m\times n}\), the system's output is represented by \(y(m)\in\ \mathbb{R}^{m}\), and an unknown nonlinear function is denoted by \(f_{2}(m)\). We can rewrite (19) as follows [30]:  
\[\dot{x}=Ax(m)+f(m,x)+B_{m}wu(m-\tau), \tag{20}\]with  
\[A=\left[ \begin{array}{cc}0_{n\times n}&I_{n\times n}\\ 0_{n\times n}&A_{2}\end{array} \right],\,f=\left[ \begin{array}{cc}0_{n\times 1 }\\ f_{2}\end{array} \right]\text{and}\ B_{m}=\left[ \begin{array}{cc}0_{n\times m} \\ B_{2}\end{array} \right].\]  
From (17), one can have  
\[\ddot{\eta}=M_{e}(\eta)^{-1}\ [\tau_{e}-C_{e}(\eta,\
u)\dot{\eta}-D_{e}(\
u, \eta)\dot{\eta}-g_{e}(\eta)]. \tag{21}\]  
Let  
\[\eta_{1}=\eta\] \[\eta_{2}=\dot{\eta},\ \ \text{thus}\] \[\dot{\eta_{1}}=\dot{\eta}=\eta_{2}\] \[\dot{\eta_{2}}=\ddot{\eta}. \tag{22}\]  
Therefore, the state space representation can be obtained from (21) and (22) as follows:  
\[\left[ \begin{array}{cc}\dot{\eta}_{1}\\ \dot{\eta}_{2}\end{array} \right]= \left[ \begin{array}{cc}0_{6\times 6}&I_{6\times 6}\\ 0_{6\times 6}&\frac{-D_{e}}{M_{e}}\end{array} \right]\ \left[ \begin{array}{cc}\eta_{1}\\ \eta_{2}\end{array} \right]-\left[ \begin{array}{cc}0_{6\times 1}\\ \frac{g_{e}}{M_{e}}\end{array} \right]+\left[ \begin{array}{cc}0_{6\times 6 }\\ \frac{1}{M_{e}}\end{array} \right]\tau\] \[y(m)= \left[ \begin{array}{cc}I_{6\times 6}&0_{6\times 6}\\ 0_{6\times 6}&I_{6\times 6}\end{array} \right]\ \left[ \begin{array}{cc}\eta_{1}\\ \eta_{2}\end{array} \right]. \tag{23}\]  
Equation (23) can be rewritten as  
\[\dot{x}(m)= A_{m}x(m)+B_{m}(w(m)u_{ad}(m-\tau)+\theta(m)\|x(m)\|_{\infty}\] \[+ \sigma(m)),\text{with}x(0)=x_{0}\] \[y(m)= c^{\top}x(m). \tag{24}\]  
Therefore, the dynamic of the state predictor \(\hat{x}(m)\in\mathbb{R}^{n}\) can be defined as:  
\[\dot{x}(m)= A_{m}\hat{x}(m)+B_{m}(\hat{w}(m)u_{ad}(m-\tau)+\hat{\theta}(m)\|x( m)\|_{\infty}\] \[+ \hat{\sigma}(m))\text{with}\hat{x}(0)=\hat{x}_{0}\] \[\hat{y}(m)= c^{\top}\hat{x}(m), \tag{25}\]  
with the estimated parameters denoted by \(\hat{w}(m)\), \(\hat{\theta}(m)\) and \(\hat{\sigma}(m)\). Hence, the error can be defined as \(\hat{x}(m)=x(m)-\hat{x}(m)\), \(\hat{w}(m)=w(m)-\hat{w}(m)\), \(\hat{\theta}(m)=\theta(m)-\hat{\theta}(m)\), and \(\tilde{\sigma}(m)=\sigma(m)-\hat{\sigma}(m)\).