Time optimal control of triple integrator with input saturation and full state constraints

Multiple integrator systems with input saturation and state constraints ubiquitously exist in practical problems such as trajectory planning in CNC (computer numerical control) systems, robots, automotive systems and industrial processes. Without state constraints, the optimal control of the multiple integrator with only input saturation is well solved in classic optimal control theory. However, if state constraints are considered, it is still challenging to obtain a global time optimal analytical solution for multiple integrator systems with even higher than second order, e.g., a double integrator plant. In this paper, the global time optimal control law for triple integrator with input saturation and full state constraints is considered. The system has a serial structure of three integral elements, while the control input and system states are within box constraints. A bang-singular-bang time optimal control law is synthesized according to Pontryagin minimum principle. Then, costates and jump conditions are analyzed in detail. Based on Bellman's principle of optimality, switching surfaces and curves in phase space are obtained through dynamic programming method. The time optimal control law is then obtained in analytical state feedback form, and its global convergence property is proved. Through the comparison with numerical solutions, the effectiveness of the proposed control strategy is verified. The proposed scheme will be useful for trajectory planning under input saturation and full state constraints in practical applications.