A neighboring optimal feedback control scheme for systems using discontinuous control

The calculation and implementation of the neighboring optimal feedback control law for multiinput, nonlinear dynamical systems, using discontinuous control, is the subject of this paper. The concept of neighboring optimal feedback control of systems with continuous, unbounded control functions has been investigated by others. The differentiating features between this class of problems and that considered here are the control discontinuities and the inherent system uncontrollability during the latter stages of the control-law operating time.The neighboring control law is determined by minimizing the second-order terms in the expansion of the performance index about an optimal nominal path. The resulting gains are a function of the states associated with the nominal trajectory. The development of a feedback control scheme utilizing these gains requires a technique for choosing the gains appropriate for each neighboring state. Such a technique is described in this paper. The technique combines abootstrap algorithm for determining the number of neighboring switch times and the initial and final controls with a scheme based ontime-to-go along the nominal and neighboring paths until the next predicted switch time or the predicted final time. This scheme requires that the nominal state, which is used to specify the feedback gains, be chosen such that the predicted time-to-go from the neighboring state be identical to the time-to-go from the nominal state. This technique for choosing feedback gains possesses minimal storage requirements and readily leads to a real-time feedback implementation of the neighboring control law.The optimal feedback control scheme described in this paper is utilized to solve the minimum-time satellite attitude-acquisition problem. The action of the neighboring control scheme when applied to states which do not lie in an immediate neighborhood of the nominal path is investigated. For this particular problem, the neighboring control scheme performs quite well despite the fact that, when the state perturbations are finite, the terminal constraints can never be satisfied exactly.This research was sponsored by the National Aeronautics and Space Administration under Research Grant No. NGL-05-020-007 and is a condensed version of the investigation described in Ref. 1. The authors are indebted to Professor Arthur E. Bryson, Jr., for suggesting the topic and providing stimulating discussions.