Key Takeaway: The minimum time required for control of flexible structures is related to the time required for waves to travel through the structure, with the minimum time being a fraction of the period of the lowest flexible mode.

Abstract

The minimum time required for accomplishing a rigid-body maneuver of a flexible structure by means of a finite number of unbounded inputs is investigated. The control task is accomplished by driving a finite number of structure modes from an initial state of zero to a desired final state in a given time interval. A minimum-effort control strategy is used so that uncontrolled higher modes will not be excited excessively by the control inputs. It is found that for less than a certain time interval for control, it is not possible to decrease the amount of spillover energy in uncontrolled modes at the end of the control interval by driving more flexible modes to zero. This time interval is identified as the minimum time required for control of flexible structures, and it is related to the time required for waves to travel through the structure. For one-dimensional second-order systems, the minimum time is equal to the time required for waves to travel between adjacent pairs of actuators. A similar result is found for fourth-order systems. Because the period of the lowest flexible mode is equal to the time required for a wave to make a round trip throughout the structure, the minimum time for control is in general some fraction of the period of the lowest flexible mode, which is determined by the number and placement of actuators.