*5.2. Analysis and Discussion*

The reference path of the AUV is generated by tracking the sinusoidal shape trajectory, and the initial state of the AUV is set as:*η*<sup>0</sup> = [0; 0; 0; 0; 27 ∗ *π*/180], *ν*<sup>0</sup> = [0.1; 0; 0; 0; 0].

To visually compare the control performance of "MPC", "RTMPC", and "ATMPC", AUV's trajectories during path tracking are shown in Figure 3.

Intuitive path-tracking performance can be visualized in the trajectory of AUV during path tracking, which is the position of the AUV given in Section 3.1. Figure 3 shows a threedimensional view of the AUV's path-tracking control performance of "MPC", "RTMPC", and "ATMPC". The trajectory of "MPC" fails to track the reference trajectory well. Although the path-tracking deviation of "MPC" tends to converge, there are still several obvious position offsets, especially at the beginning of path tracking. Actual trajectories of both "RTMPC" and "ATMPC" can separately track the nominal trajectory. Note that the nominal trajectory of "ATMPC" tracks the reference trajectory better, compared with that of "RTMPC".

**Figure 3.** AUV trajectory during path tracking.

$$\begin{cases} \begin{aligned} \chi\_r &= t \\ y\_r &= 5 \text{sim}(0.1t) \\ z\_r &= 0.01t \\ \theta\_r &= -\text{atan}(\frac{0.01}{\sqrt{1 + 0.5 \cos^2(0.1t)}}) \\ \psi\_r &= \text{atan}(0.5 \cos(0.1t)) \\ u\_{sr} &= 1 \end{aligned} \tag{41}$$

where *xr*, *yr*, and *zr* are the reference positions. *usr* is the reference surge speed.

To compare the path-tracking deviation in detail, the path-tracking deviation and path-tracking integral deviation are respectively shown in Figures 4 and 5. The maximum deviations in position and attitude angles are given in Table 5. Section 3.3 introduced the path-tracking deviation, whose absolute value is used. The definition of path-tracking integral deviation is given in (41). As shown in Figure 4, under sinusoidal external disturbances and parametric uncertainties, position and attitude angle deviations of three methods all have a bounded and convergent tendency over time. Compared with the position and attitude deviations of "MPC", that of "RTMPC" has been all effectively reduced in every moment. As shown in Figure 5, the growth trend of the integral deviation is also much slower. In addition, the maximum position deviation of "RTMPC" can be reduced from 0.38 m to 0.12 m. That is a reduction of about 68%. The maximum pitch angle deviation of "RTMPC" can be reduced from 3.45◦ to 0.58◦. That is a reduction of about 83%. The maximum yaw angle deviation of "RTMPC" can be reduced from 3.45◦ to 1.07◦. That is a reduction of about 69%. It can be seen the "RTMPC" has good robustness against external disturbances and parametric uncertainties.

Compared with "RTMPC", the proposed tube-based event-triggered path-tracking strategy has a smaller position and attitude angle deviations. The maximum position deviation of "ATMPC" can be reduced from 0.12 m to 0.04 m. That is a reduction of about 67%. As shown in Figure 4, compared with the position deviation in the *x* direction and of "RTMPC", that of "ATMPC" is almost the same in every moment. However, after about 20 s, the position deviation of *y* is much smaller in every moment. In addition, the maximum yaw angle deviation can be reduced from 1.07◦ to 0.45◦. That is a reduction of about 58%. As shown in Figure 4, after about 10 s, the yaw angle deviation is almost smaller in every moment. Integral deviations can intuitively show the variation trend of these position and attitude deviations in Figure 5.

**Figure 4.** Deviation of position and attitude angle.

**Figure 5.** Integral deviation of position and attitude angle.

**Table 5.** Maximum deviation of position and attitude angles.


To compare control input smoothness, the range of the AUV's speed and the control input are respectively shown in Figures 6 and 7. AUV's speed and the control input have been given in Section 3.1. As shown in Figure 6, the surge speed of "ATMPC" and "RTMPC" tracks the desired surge speed well. The sway speed, heave speed, pitch speed, and yaw

speed changes in "ATMPC" and "RTMPC" occur more smoothly, compared with those of "MPC".

As shown in Figure 7, compared with the stern thruster force of "MPC", that of "ATMPC" and "RTMPC" have better smoothness, avoiding the high-frequency oscillation phenomenon. As shown in the local zoom-in of Figure 7, the smoothness of the stern thruster force of "ATMPC" is enhanced, compared with that of "RTMPC". It can be seen that the nominal control input of the "ATMPC" is within the upper bound of the tight constraint, and the output value of "ATMPC" is almost the lowest, which may be consistent with the purpose of energy conservation in real-world application.

Like the vertical plane deflection shown in Figure 7, those of "RTMPC" and "ATMPC" can all avoid large periodic changes, compared with that of "MPC". At the beginning of the simulation, the range of the vertical plane deflection of "MPC" has a tendency to be unstable. With the adaptive tight constrain introduced, the vertical plane deflection

of "ATMPC" changes more smoothly. As the simulation time goes on, there is almost no oscillation phenomenon. As shown in Figure 7, the blue line is contained within the gray area, and the trend of the upper and lower limits in the gray area is consistent with the trend of the blue line.

Like the translational plane deflection shown in Figure 7, that of "RTMPC" and "ATMPC" can also avoid large periodic changes, compared with that of "MPC". With the adaptive tight constraint introduced, the translational plane deflection of "ATMPC" changes more smoothly, and tends to stabilize more quickly.

To analyze the real-time performance, the time consumption of different methods is recorded in Table 6. Note that the tightened constraint set of "RTMPC" is calculated offline. It can be explained that the average time consumption and the maximal time consumption of "RTMPC" are almost the same as those of "MPC". The average time consumption and the maximal time consumption of "ATMPC" will not increase by much: the average time consumption increases by 2.91 ms, and the maximal time consumption increases by 3.47 ms.

**Table 6.** Time consumption of different methods.


#### **6. Conclusions**

In this paper, a novel tube-based event-triggered path-tracking strategy against disturbance is proposed, which consists of an LMPC controller and a tube MPC controller. In the LMPC controller, based on the nominal kinematics model of the AUV, a nominal optimal speed control law is obtained to converge the nominal path-tracking deviation. In the tube MPC controller, AUV's available control inputs are separately calculated based on a decoupled model. Considering the nonlinear hydrodynamic characteristics of the AUV, an LMI is formulated to calculate the feedback matrix and tight constraints offline. The terminal region in the tube MPC controller is obtained offline using linear differential inclusion technology. When the surge speed step signal does not exceed the upper bound, the tight constraints become adaptive. Numerical simulation results show that the feedback matrix is successfully used to match the actual trajectory and the nominal trajectory. With the adaptive constraints introduced, the nominal trajectory tracks the reference better. Note that the online computing time of the tube MPC is acceptable, and these corresponding control inputs are also smooth. Therefore, the proposed tube-based event-triggered path-tracking strategy can enhance the path-tracking performance and ensure good real-time performance.

In the MPC controller, the disturbance upper bound needs to be set appropriately. If the bound is too small, the robustness is weak; otherwise, the tube will be too conservative. The RPI set may not be obtained, or the optimal control problem is easy to be infeasible. In numerical simulation, the disturbance upper bound is still easy to set appropriately. In the application of a real-world system, it may be a challenge. The disturbance bound is different for different real-time scenarios, which may be difficult to accurately set. This may lead to degradation of the control performance. In future research, the work will be extended to predict the model mismatches due to parametric uncertainties and external disturbances to improve the accuracy of the nominal model, based on data-driven technology, such as machine learning. The RPI set is used to address the bounded prediction deviation. If the prediction deviation is convergent and bounded, it can effectively solve the problem of setting the disturbance upper bound in real-world applications.

**Author Contributions:** Conceptualization, Y.C. and Y.B.; methodology, Y.C. and Y.B.; software, Y.C.; validation, Y.C. and Y.B.; formal analysis, Y.C. and Y.B.; investigation, Y.C. and Y.B.; resources, Y.C. and Y.B.; data curation, Y.C. and Y.B.; writing—original draft preparation, Y.C.; writing—review and editing, Y.C. and Y.B.; visualization, Y.C. and Y.B.; supervision, Y.B.; project administration, Y.B.; funding acquisition, Y.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Hunan Provincial Natural Science Foundation of China (2021JC0010), and Defense Industrial Technology Development Program (JCKY2021110B024, JCKY2022110C072).

**Data Availability Statement:** Data is unavailable due to privacy or ethical restrictions.

**Acknowledgments:** Haicheng Zhang (Hunan University) and Weisheng Zou (Hunan University) were acknowledged to provide careful guidance.

**Conflicts of Interest:** The authors declare no conflict of interest.
