In this paper, a novel nonsingular finite-time backstepping controller is constructed for trajectory tracking of marine vessel subject to unknown external disturbances. Firstly, in the presence of disturbances, a disturbance observer (DO) is proposed to estimate and compensate the disturbances exactly in finite time. Secondly, a finite-time tracking controller is designed in the classical backstepping procedure, however, the inevitable singularity appears in calculating the derivative of virtual control. Furthermore, for overcoming this singularity, a nonsingular finite-time backstepping controller is designed by adopting a finite-time command filter to estimate the derivative, instead of calculating it directly. Theoretical analysis demonstrates the closed-loop system is finite-time stable. Finally, simulation results and comparisons illustrate the effectiveness of the proposed method. INDEX TERMS Finite-time command filter, disturbance observer (DO), nonsingular finite-time backstepping controller, marine vessel, trajectory tracking.
Considering the reverse logistics system composed of two manufacturers and two recyclers under the cross-competitive take-back mode, which is influenced by multiple factors (industry competition, economies of scale, government subsidies, remanufacturing rate, etc.), a model for remanufacturing reverse logistics system based on Stackelberg game is established. Then, the Nash equilibrium solution of decision variables is solved to obtain the best profit of all participants under the cross-competitive take-back mode. Furthermore, the parameter constraint analysis is carried out, and the monopolistic take-back mode is introduced for comparative analysis. Then, the sensitivity analysis of the model is carried out. At last, a case analysis is carried out based on the current situation of waste electrical and electronic equipment (WEEE) recycling in China. The results show that the cross-competitive take-back mode is more advantageous to all participants in the reverse logistics system than the monopolistic take-back mode. Recyclers should actively sign contracts with multiple manufacturers to recycle waste products, making full use of the advantages of cross-competitive take-back mode to maximize the profits of all participants, so as to encourage them to recycle waste products and achieve sustainable development.
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