SYNTHESIS OF DESIGN PARAMETERS OF MULTI-PURPOSE DYNAMIC SYSTEMS

Abstract

Two problems related to the optimization of linear stationary dynamic systems are considered. A general formulation of the multi-purpose problem of optimal control with the choice of design parameters is given. As a special case, the problem of multi-objective optimization of a linear system according to an integral quadratic criterion with a given random distribution of initial deviations is considered. The solution is based on the method of simultaneously reducing two positive-definite quadratic forms to diagonal form. Analytical results have been obtained that make it possible to calculate the mathematical expectation of the criterion under the normal multidimensional distribution law of the vector of random initial perturbations. The inverse problem of stability theory is formulated: to find a vector of structural parameters that ensure the stability of the system and a given average value of the quadratic integral quality criterion on a set of initial perturbations. The solution of the problem is proposed to be carried out in two stages. The first stage involves deriving a general solution to the Lyapunov matrix equation in terms of the elements of the system matrix. To achieve this, the state space is mapped onto the eigen-subspace of the positive-definite matrix corresponding to the integral quadratic performance criterion. It has been established that this solution is determined by an arbitrary skew-symmetric matrix or by the corresponding set of arbitrary constants. In contrast, when the system matrix depends linearly on the vector of design parameters, a linear system of equations can be formulated with respect to the unknown parameters and arbitrary constants present in the general solution of the inverse stability problem. In general, such a system is consistent and admits an infinite number of solutions that satisfy the initial requirements for the elements of the symmetric matrices in the Lyapunov.