Parameterization of Some Control Problems by Linear Systems

Abstract

In the framework of control parameterization methods a number of optimization problems of linear phase systems with quadratic and bilinear functionals is considered. Approximation of the control is carried out in the class of piecewise linear functions and is formed as a linear combination of a special set of support functions with coefficients that are variables of the finite-dimensional problem. At the same time the interval control constraint in the variational problem automatically passes into similar constraints on the variables of the finite-dimensional problem. To characterize and effectively solve these problems, explicit expressions of the selected functionals are obtained with respect to parameters of approximations. As a result, a series of quadratic mathematical programming problems with the simplest restrictions on variables is formulated. The quadratic forms of the objective functions are determined by the Gram and Hessenberg matrices. It should be emphasized that the parametrization preserves the convexity property of the original optimal control problem. In addition, the simplest optimal control problem with a linear terminal functional after parameterization is solved without iterations. The connection between the coordinated problems at the level of optimality conditions is established. It consists in the fact that the differential condition of the extremum in a finite-dimensional problem is locally equivalent to the maximum principle for the variational problem at the points of set.

Keywords

• linear control system
• quadratic and bilinear functionals
• piecewise-linear approximation
• finite dimensional problems