Abstract
In this paper, the time-optimal control of the wave equation is derived in closed form. A frequency domain approach is used to obtain the time-optimal solution which is bang-off-bang. The system studied in this paper is a uniform flexible rod with a control input at each end, whose dynamics in axial vibration is represented by the wave equation. In order to verify the optimality of the control profile derived for the distributed parameter system, the system is discretized in space and a series of time-optimal control problems are solved for the finite dimensional model, with increasing number of flexible modes. In the limit, the controllers show the convergence of the first and final switch of the bang-bang controller of the finite dimensional system to the first and final switch of the bang-off-bang controller of the distributed parameter system, in addition to the convergence of the maneuver time. The number of switches in between the first and final switch is a function of the order of the finite dimensional system. The maneuver time of the distributed parameter system is compared to that of an equivalent rigid system and the coincidence of the time-optimal controller for the flexible and rigidized systems is illustrated for certain maneuvers.
Method to Determine the Time-Optimal Control of a Linear System with State Inequality Constraint
