Approach to Numerical Solution of Nonlinear Optimal Feedback Control Problems

Abstract

In the paper, we consider optimal feedback control problems for dynamic, in the general case, nonlinear systems with lumped parameters based on continuous and discrete feedback on the object’s state. To calculate the values of the feedback control’s parameters, we propose to use the measured values of observable components of the phase vector or the object’s output at the current and some previous (past) moments of time in order to compensate for the inability to measure all the components of the object’s phase state. As a result of this kind of formation of the dependence of the parameters of the synthesized control on a part of the object’s state, the process under consideration will be described by ordinary differential equations with time-constant delay arguments in the phase state. The feedback control problem is solved numerically by reducing it to a finite-dimensional optimization problem. To this end, we derive formulas for the gradient of the objective functional of the reduced problem with respect to the optimizable parameters are zonal values of the feedback parameters. These formulas make it possible to formulate necessary first-order optimality conditions, as well as to use them for numerical solution to model problems using first-order optimization methods.