Selection of Weight Coefficients of Quadratic Quality Functional in Solving ADOC Problem in the Letov—Kalman Formulation

Abstract

For linear stationary one-dimensional control objects, the inverse problem of analytical design of optimal controller (ADOC) is considered, which consists in determining the weight coefficients of the quadratic functional of the optimality of the control process, providing a closed control system with the set values of primary quality indicators (static error, transient time and overshoot). It is analyzed in relation to both the ADOC problem in the Letov-Kalman formulation. A method of its solution is proposed based on the transformation of the ADOR problem to a canonical form in which the control object is described by a matrix differential equation in the Frobenius form, and the quality functional is defined as an integral of the sum of the products of the canonical phase coordinates of the object with the corresponding weight coefficients, as well as the square of the control signal. It is shown that the solution of the inverse canonical ADOC Letov-Kalman problem is determined by the values of only three non-zero weighting coefficients of the criterion, and one of them has a single value. The values of the other two coefficients are proposed to be found in the process of modeling the synthesized optimal control system from the conditions of ensuring for it the values of primary quality indicators no more than the specified ones. The results obtained, presented in the form of Theorems 1 and 2, are extended to the synthesis of astatic control systems, in which an additional integrator is connected to the plant output to obtain astaticism. Since such an "extended" control object is described using a state vector that has the first two phase coordinates of the canonical form, the synthesis of the optimal system is carried out without converting the object description to the canonical form of the phase variable and vice versa. The construction of an astatic control system is illustrated by an example.