A global-stabilizing near-optimal control for real-time trajectory tracking of nonholonomic chained systems

It is well known that any nonlinear optimal control requires a solution to a two-point-boundary-value problem that is solvable only by numerical iteration. In this paper, a new and near-optimal control is proposed for real-time trajectory tracking of any nonholonomic system in the chained form. Design of the proposed control starts with optimal control solutions to two linear subsystems, one time-invariant and the other time varying. The two solutions combined together render a globally stabilizing suboptimal control for the overall system. Then, the optimality condition is invoked to determine the distance between the suboptimal control and the optimal one. Consequently, an improved control can be obtained by modifying the suboptimal control in such a way that the distance aforementioned is minimized as much as possible in closed form. The new control is real-time implementable, globally and exponentially stabilizing, and it is near optimal since its closeness to the optimal control (attainable only off-line) can be measured, monitored on line, and has been minimized. Simulation study of a car-like robot is used to illustrate effectiveness of the proposed design method.