Higher Order Sliding Mode Controllers With Optimal Reaching

Higher order sliding mode (HOSM) control design is considered for systems with a known permanent relative degree. In this paper, we introduce the Robust Fuller's Problem that is a robust generalization of the Fuller's problem, a standard optimal control problem for a chain of integrators with bounded control. By solving the Robust Fuller's Problem it is possible to obtain feedback laws that are HOSM algorithms of generic order and, in addition, provide optimal finite-time reaching of the sliding manifold. A common difficulty in the use of existing HOSM algorithms is the tuning of design parameters: our methodology proves useful for the tuning of HOSM controller parameters in order to assure desired performances and prevent instabilities. The convergence and stability properties of the proposed family of controllers are theoretically analyzed. Simulation evidence demonstrates their effectiveness.