Optimal Guaranteed Cost Hierarchical Sliding Mode Control for Under-Actuated Nonlinear Systems

This paper devotes to solving the adaptive optimal guaranteed cost control problem for switched under-actuated nonlinear systems (SUANS) in nominal form. An actor-critic (AC) network architecture with hierarchical sliding-mode surface (HSMS) is first established. As the system moves on the HSMS, the original system stability issue turns into the issue of designing approximate optimal control policies. Subsequently, a HSMS-based Hamilton-Jacobi-Bellman (HJB) equation is constructed by utilizing Bellman's principle of optimality, and the approximate solution of the HSMS-based HJB equation is gained via employing AC neural networks (NNs). It is noted that the AC updating laws of this paper are synchronously tuned, which achieves the simplification of the ADP process. Afterwards, a HSMS-based approximate optimal guaranteed cost control scheme is developed under AC dual NNs architecture. Finally, the practicality of the developed adaptive optimal guaranteed cost control scheme is verified by using a practical ball-arc system.