*5.1. Parameters Set*

Note that the influence of parameters on control performance in the optimal control problem is significant. To focus on evaluating the control performance of the proposed path-tracking strategy, in the contrasting simulations, these same parameters in different methods are all set to the same value. Then, the parameters in the numerical simulation are given as follows. Following [10], the external sinusoidal disturbance term is set as *τ<sup>d</sup>* = [1.25sin(*t*); 0.785sin(*t*); 0.485sin(*t*); 0.0325sin(*t*); 0.325sin(*t*)]. The upper bound of the surge speed step signal Δ*us* is set as 0.05. Following [10], the parametric uncertainties are reflected by the percentage of the hydrodynamic term. Then, Δ*FCD* is set as Δ*FCD* = 0.2(*C*(*ν*) + *D*(*ν*)).

Note that the proposed path-tracking strategy consists of a LMPC controller and a tube MPC controller. The LMPC controller is used to calculate the speed control law to converge the path-tracking deviation, and the tube MPC controller is used to track the speed control law.

For the LMPC controller, these parameters in (24–29) are listed in Table 1. In the LMPC controller, weighting matrix *Q<sup>η</sup>* and *Q<sup>ν</sup>* are for minimizing the path-tracking deviation *eη*. The weight matrix *R<sup>ν</sup>* is for the smooth change in AUV's speed. With these weight matrices set appropriately, the speed control law can efficiently converge the path-tracking deviation, avoiding abrupt changes in AUV's speed.


**Table 1.** Parameter value in the LMPC controller.

Note that the tube MPC controller is used for surge speed control, heading control, and depth control, respectively, based on these decoupled models (19), (20), and (22). These corresponding parameters of each controller in (24)–(29) and (38) are listed in Tables 2–4. Δ*Fx* is the increment of the stern thruster force. Δ*δ<sup>r</sup>* is the increment of the vertical plane deflection. Δ*δ<sup>s</sup>* is increment of the translation plane deflection. In the tube MPC controller, the weighting matrices play a similar role. With the appropriate *QT* and *RT* set, the control input of the AUV can change smoothly to track the nonmail speed control law. The RPI set in Definition 1 is used to obtain the tight constraint in nominal system dynamics to ensure that the deviation *z* (9) also contained in the RPI set. The feedback matrix is used to converge the deviation. As mentioned in Section 4.2, with appropriate parameters *λ*, *μ* and *PR* obtained, the tube MPC controller can efficiently track the speed control law.


**Table 2.** Parameter value in the tube MPC controller for surge speed control.

**Table 3.** Parameter value in the tube MPC controller for heading control.


**Table 4.** Parameter value in the tube MPC controller for depth control.


When the adaptive flexible tube is used, two decision variables are used to dynamically adjust these tight constraints. Parameter *s* represents the upper bound of the tight constraints. Parameters *w*, *ρ*, and nonlinear function <sup>∼</sup> *<sup>w</sup>δ*(s*k*|*t*) are used to represent the variation in the tight constraint.
