Exact time-optimal control of the wave equation

Abstract

The time-optimal control of a distributed parameter system is derived in closed form. The class of systems studied in this work is distributed parameter systems whose dynamics are governed by the wave equation. A frequency domain approach is utilized to arrive at the time-optimal solution that is bang-off-bang. To corroborate the optimally of the control profile derived for the distributed parameter system, the system is discretized in space and a series of time-optimal control problems is solved for the finite dimensional model, with an increasing number of flexible modes. The limiting controller shows the convergence of the first and last switch of the bang-bang controller of the finite dimensional system to the first and last 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 last switch is a function of the order of the finite dimensional system. The maneuver time of the distributed parameter system is compared with that of an equivalent rigid system, and it is shown for certain maneuvers that the bang-bang control profile of the rigid system is also the time-optimal control of the distributed system. N recent years, interest in the study of large space structures has grown and is reflected by the publication of numerous papers dealing with the modeling and control aspects of these structures. The effects of flexibility of these structures have to be taken into account in designing controllers. Most of the papers arrive at tractable models by spatial discretization of the structure using finite element or the assumed mode method. Designs of time-optimal controllers for these reduced-order models have been shown to be bang-bang. Singh et al.1 solve for the time-optimal control profile of a flexible slewing beam by deriving a set of algebraic equations that are satisfied by the switch times and the final time. A homotopy approach is used to solve for the switch times and the maneuver time. BenAsher et al. 2 arrive at the time-optimal control of a slewing beam using a parameter optimization technique. The model they use to represent the dynamics of the system is arrived at by discretizing the system by the assumed mode method. They also show that the control profile is antisymmetric about the midmaneuver time for a system that is undamped. Singh and Vadali3 present a frequency domain approach to arrive at the time-optimal control profile for a system represented by a system of ordinary differential equations. A parameter optimization problem is formulated to minimize the maneuver time subject to the constraint that the time-delay filter used to generate the bang-bang control profile cancels all of the poles of the system. Bennighof and Boucher4 solve the minimum-effort problem for a system with a finite number of modes with the objective of minimizing the excitation of the uncontrolled higher modes. They show that the spillover energy cannot be decreased for time less than a constant that corresponds to the time required for waves to travel through the structure. They infer from this result that the minimum time control of a structure is a function of the time required for the wave to travel between actuators. Bennighof and Boucher5 confirm their conjecture that the wave speed defines the minimum time required for a maneuver to be completed. They solve the minimum time control problem for a rest-to-rest maneuver of a one-dimensional second-order distributed parameter system that is driven by two control inputs. They use the traveling wave formulation to solve the time-optimal controller exactly and show that it is bang-off-bang and not bang-bang,