Terminal Pseudo-Optimal Control of a Nonlinear Dynamic Object

Abstract

Theoretically, this work belongs to a fairly broad class of articles and books devoted to solving control problems for dynamic objects with constraints on control actions and the Bolza functional. The necessary conditions for the existence of optimal controls for a terminal differential game are described by a two-point boundary value problem and the condition for choosing the control itself as a function dependent on the behavior of the Hamiltonian along the optimal trajectory. The main problem of finding optimal control is associated with finding a solution to the two-point boundary value problem. It should be noted that the existence of an optimal control is not necessary: the set of admissible controls may not even contain controls that transform the object from the initial state to a given set of goals. Typically, numerical methods are used to solve such problems. In this paper, an alternative to numerical methods for solving two-point boundary value problems, applied to the problem of synthesizing controls for nonlinear objects, is proposed. This approach is based on the assumption of the validity of R. Bellman’s inverse optimality principle, which maintains the functional relationship between the components of a two-point boundary value problem not only at the end of the transient process but throughout the entire control interval. Based on this, a new analytical method for constructing control for nonlinear objects, called the pseudo-optimal control synthesis method, is proposed. A condition is formulated for determining the set of initial conditions of the original nonlinear system that ensure the execution of the formulated control problem. Mathematical modeling of a quadcopter control system with synthesized control confirmed the theoretical results of the proposed method for synthesizing pseudo-optimal control for nonlinear dynamic objects.