Near-minimum time open-loop slewing of flexible vehicles

Abstract

Minimum time, open-loop, optimal controls are calculated for single-axis maneuvers of a flexible structure. By shaping the control profiles with two independent parameters, a wide variety of control histories can be pro- duced. Based on the dynamics of the model, with a normalized time scale, the resulting Pontryagin's necessary conditions yield a nonlinear fixed final time, fixed final state, two-point boundary value problem with the maneuver time as a control parameter. Upon generating numerical solutions to the problem, the final maneuver time and residual flexural energy are compared to the bang-bang solution as a measure of the success of a given maneuver. Examples presented illustrate near-minimum time maneuvers with control of flexible modes in addi- tion to the rigid body modes, as well as the qualitative and quantitative effect of the torque shaping parameters. EAR-MINIMUM time attitude control of flexible spacecraft with active vibration suppression is a topic of current research. Although minimum time/fuel maneuvers have been previously examined,1 such bang-bang controls cause spillover effects that may induce high residual flexural energy. The source of the energy spillover into the flexural modes is the instantaneous switching that results from the bang-bang controls, a potential problem for the control hard- ware (reaction wheels, control moment gyros, etc.) as well. Consequently, minimum-time optimal control problems are often of academic interest only for producing theoretical lower bounds for the maneuver time. We have also found that near bang-bang controls are usually very sensitive to model er- rors; therefore, control shaping is an important issue in ob- taining robust controls. Upon modifying the control profile with a smoothing func- tion and transforming the independent variable (time), a near- minimum-time problem is generated with the mathematical form of a fixed-time nonlinear optimal control problem. The resulting boundary-value problem yields to a number of established methods of numerical solution. The controls are attenuated in such a way that the magnitude of the control rises smoothly from zero to the bounded maximum at the in- itiation of the maneuver and typically has an identically shaped reduction to zero at the final time. In addition, any in- stantaneous switch during the maneuver is shaped using a smooth, continuous function. The sharpness of the control trajectory is determined by a set of arbitrary parameters such that we can produce profiles with low control rates as well as controls that approach, to any desired degree, the bang-bang minimum time solution. The independent variable is transformed such that a linear free-final-time optimal control problem is converted to a nonlinear fixed-time problem, where the maneuver time is contained explicitly as a parameter in the transformed system. This augmented fixed-time problem can then be solved numerically for the controls and the corresponding minimum time required to complete the maneuver.