## 6 Numerical Results  
In this section, we present a set of two simulation results. The simulations were carried out by using the Matlab\({}^{\text{\tiny{b}}}\) software environment. The first simulation consists of tracking the reference of an angular position for the yaw's dynamic and a depth reference in the Z axis. In the second one, the underwater vehicle has the task of tracking a 3D helical trajectory with radius of 5 meters. In this simulation analysis, we present a comparison between the DNCS, a standard feedback controller designed by using Linear Quadratic Regulator (LQR) [55], and a Proportional Derivative (PD) controller implementing an online auto tuning method to set its parameters by using the successive approximation method (SAM) [56]. Hence, the self-tuned PD controller applies an adaptive approach to tackle the AUV control problem in order to perform a comparative performance analysis against the DNCS controller under fair conditions. Certainly, there exist other nonparametric control schemes or sliding mode techniques, which incorporates adaptive properties, that could be suitable alternatives for the system class subject of our analysis. However, a number of them need some gain adjustment, which is no longer required for the proposed DNCS controller. To conduct a more realistic simulation, in both scenarios, external disturbances were added to the dynamic model. Table 1 presents the parameters used for the dynamic model given in Equation (8). These parameters are taken from [57], derived by several experimental tests on the BlueROV2 vehicle [41]. These parameter values are also used for derivation of the referred LQR feedback controller. For the control strategy presented in Equation (42) and the adaptive laws given by Equations (46) and (47), the parameters used in the simulation are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(X_{u}\) & 3.03 kg/s & \(X_{u}\) & 5.5 kg \\ \(Y_{\text{p}}\) & 3.21 kg/s & \(Y_{\text{p}}\) & 12.7 kg \\ \(Z_{w}\) & 8.18 kg/s & \(Z_{w}\) & 14.57 kg \\ \(N_{r}\) & 0.07 kgm/s & \(N_{r}\) & 0.12 kgm\({}^{2}\) \\ \(X_{u|u|}\) & 4.03 kg/m & \(m\) & 11.5 kg \\ \(Y_{\text{p}|x}\) & 6.21 kg/m & \(W\) & 112.8 N \\ \(Z_{w|w|}\) & 5.18 kg/m & \(B\) & 114.8 N \\ \(N_{r|r}\) & 0.07 kgm/s & \(l_{zz}\) & 0.16 kgm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Underwater vehicle parameters used in simulation. It is worth mentioning that, for control purposes, the DNCS was developed based on the simplified dynamics given by Equations (21)-(23). However, it is important to stress that, in simulation experiments, we introduce the complete AUV 4 DOF dynamical Equations (17)-(20) that considers a completely coupled AUV system (without any simplification). The simulation framework used in the experiments of the second scenario considers the four controllers (\(x,y,z\), and yaw) acting at the same time. Implementation of the DNCS algorithm stabilizes to the yaw dynamic Equation (20), ensuring a fast controller response providing a rapid convergence to the origin. The previous action guarantees that the translational (\(x\),\(y\)) dynamics in Equations (17) and (18) can be rapidly reduced to the ones described in Equations (21) and (22), simplifying the trigonometric coupling functions involving the yaw angle. Therefore, the DNCS algorithm is applied to the horizontal translational dynamic Equations (21) and (22) and for the z-axis dynamic Equation (23) separately to track desired 3D trajectories in the x-y-z space. However, the simulation were conducted by using the nominal model given by Equations (17) and (20). In order to preserve the controllability condition for the estimated AUV system, it is necessary to guarantee the boundedness of \(u(\xi)\) in Equation (42) by modifying the adaptive law Equation (47), so the estimates \(\hat{b}\) cannot reach values arbitrarily close or equal zero.