Method of Multiobjective Optimization of Controlled Systems with Distributed Parameters

The constructive method for multicriteria optimization of control processes of deterministic and not fully defined controlled systems with distributed parameters, described by linear multidimensional parabolic partial differential equations with internal and boundary control actions, is proposed. The optimization problem is considered when an uniform approximation accuracy of object’s final state to the required spatial distribution of controlled function is given. The suggested approach is based on the one-criterion option in the form of minimax convolution of the normalized quality criteria and the subsequent transition to the equivalent form of a typical variational problem with constraints. It is applied to the deterministic model of an object described by an infinite system of differential equations with respect to time-dependent modes of the controlled quantity expansion in a series of eigenfunctions of the initial-boundary value problem. Further procedures for the preliminary parametrization of control actions, based on analytical optimum conditions and reduction to semi-infinite programming problems, allow one to find the desired extremals using their Chebyshev properties and fundamental laws of the domain in typical application conditions of estimating the accuracy of approaching the object’s final state to the required one in a uniform metric. The obtained results are extended to the tasks of program control on the principle of guaranteed result by ensembles of object trajectories under conditions of interval uncertainty of the parametric characteristics of the distributed system and multiple external disturbances. A demonstrated example of a multicriteria optimization of an innovation technology of metal induction heating prior a hot forming is of special interest. The typical optimization criteria such as energy consumption, metal loss due to scale formation, and heating accuracy are considered as components of vector optimization criterion.