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The dynamics of hydraulic systems are highly nonlinear and the system may be subjected to non-smooth and discontinuous nonlinearities due to directional change of valve opening, friction, etc. Aside from the nonlinear nature of hydraulic dynamics, hydraulic servo systems also have large extent of model uncertainties. To address these challenging issues, a robust state-feedback controller is designed by employing backstepping design technique such that the system output tracks a given signal arbitrarily well, and all signals in the closed-loop system remain bounded. Moreover, a relevant disturbance attenuation inequality is satisfied by the closed-loop signals. Compared with previously proposed robust controllers, this paper's robust controller based on backstepping recursive design method is easier to design, and is more suitable for implementation.

Based on nonlinear trajectory generation (NTG) software package, a general approach (i.e. numerical solution) to trajectory planning for yoyo motion is presented. For the real-time control of such periodical dynamic system, a critical problem is how to implement fast solving the optimal trajectory, so as to meet the real-time demand. However, traditional numerical solution methods are very time-consuming. In this paper, the optimization problem is solved by mapping the problem to a lower-dimension space. And combined with multithread programming technology, the computation time for solving the optimal trajectory is greatly reduced. Simulation results show that the numerical solution is identical to the analytic one, which demonstrates the correctness of the proposed method. The computation time of one cycle of yoyo simulation is about 10 ms, which shows that the proposed numerical method can be applied to the real-time control of yoyo playing.