Admissible Controls in a Nonlinear Time-Optimal Problem with Phase Constraints

Abstract

Abstract The paper is devoted to constructing admissible controls in a problem of optimal control by a nonlinear dynamic system under constraints on the current phase state. The dynamic system under consideration describes the controlled motion of a carrier rocket from the launching point to the time when the carrier rocket enters a given elliptic earth orbit. A problem consists in designing a program control for the carrier rocket that provides the maximal value of the payload mass led to the given orbit and the fulfillment of a number of additional restrictions on the current phase state of the dynamic system at the atmospheric part of the trajectory. The restrictions considered are due to the need to take into account the values of the dynamic velocity pressure, the attack angle and slip angle when the carrier moves in dense layers of the atmosphere. Such a problem is equivalent to a nonlinear time-optimal problem with phase constraints for carrier rockets of some classes. The algorithm for constructing admissible controls ensuring the fulfillment of additional phase constraints is suggested. The methodological basis of this algorithm is the application of some predictive control. This control is constructed in the problem without taking into account the constraints above. For a deterministic model of the atmosphere, such a predictive control is used to predict the values of a part of the phase state of the dynamic system at the next time. The prediction results are applied in the procedure of desired control construction. This procedure essentially takes into account specific features of the additional constraints. The results of numerical modeling are presented.