**5. Concluding remarks**

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**Figure 3.** The simulation results of the proposed controller when *dθ*(*t*) = 0.5 cos(0.1*t*) (solid line – simulation results, dash

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308 Adaptive Robust Control Systems

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line – desired trajectory).

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desired trajectory).

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**Figure 4.** The simulation results of the proposed controller when *dθ*(*t*) = 1.5*e*<sup>−</sup>*<sup>t</sup>*

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0 200 Considering unknown persistent perturbations in unactuated dynamics, this chapter designs an observer-based robust control method for underactuated crane systems. Specifically, a reduced-order augmented-state observer is designed to recover the lumped perturbation terms in unactuated dynamics. Further, based on the observer, a new sliding manifold is constructed to improve the robust performance of the control system. Then, the state variables are made to stay on the manifold by applying a designed robust control law in the presence of non-vanishing perturbations in unactuated dynamics. Finally, the convergence is proved in this chapter theoretically by using Lyapunov control theories. Moreover, the proposed observer-based robust controller is verified to be effective and robust by numerical simulation results.
