Modeling of Singularly Perturbed multi-criterion Optimal Control Workflows

Abstract

One-criterion problem of control the complex metallurgical process presented by variable parameters in vector notation was considered. In the implementation vector notation presentation transformed into canonical form by means of piecewise continuous functions optimizing control variables on a given interval [0, T]. Pontryagin's maximum principle is used to determine the optimal conditions for solving one-criterion problem, which is formulated as regular multi-criteria (maximum performance, minimum loss of ferrous and nonferrous metals and energy, etc.) optimization problem. Assumed that a piecewise continuous equations in the zone u ⊂ U, and their existence in space U ⊂ E as the main way to solve multi-criteria optimization problems of metallurgical processes is acceptable convolution of trajectory one-criterion optimum problem. In the modeling of singularly perturbed multi-criteria optimal control problems of metallurgical processes was used the method of Whitham or the search of controlled variable and control action carried out in the form of an asymptotic series in small parameter with the highest differential equations derivatives. It is shown that the expansion of control actions equations system in an asymptotic series in the small parameter allows a distributed optimal control problem for considered process to be transformed to recurrent sequence of simple regular tasks. Finding the coefficients of the asymptotic expansion of the optimal control actions for multi-criteria metallurgical process is the desired solution with given initial conditions and singular perturbations.