A robust optimal control approach in the weighted Sobolev space for underactuated mechanical systems

This paper proposes new robust nonlinear optimal control formulations in the weighted Sobolev space, here called nonlinear W-2 and W(infinity )controllers, to handle with two classes of underactuated mechanical systems: the reduced ones with a reduced number of DOF; and the entire underactuated mechanical systems with input coupling. The optimal control problems are formulated via dynamic programming and particular solutions are presented for the resulting Hamilton-Jacobi equations with the corresponding stability analysis. Also, the concepts of W-m,W-p,W-sigma-stability and W-m,W-p,W-sigma-gain for a general class of systems are established, with the demonstration for the particular case of study. The novel W-2 and W-infinity controllers are synthesized for an unmanned aerial vehicle benchmark. The results demonstrate that these controllers provide better transient performance with faster response against external disturbances in comparison with a classic nonlinear H-infinity controller, in addition to have a simple design. (C) 2021 Elsevier Ltd. All rights reserved.