Polynomial method for the synthesis of regulators for the special case of multichannel objects with one input variable and several output values

Abstract

Currently, an urgent task in control theory is the synthesis of regulators for objects with a smaller number of input values compared to output ones, such objects are described by matrix transfer functions of a non-square shape. A particular case of a multichannel object with one input variable and two / three / four output variables is considered; the matrix transfer function of such an object has not a square shape, but one column and two / three / four rows. To calculate the controllers, a polynomial synthesis technique is used, which consists in using a polynomial matrix description of a closed-loop control system. A feature of this approach is the ability to write the characteristic matrix of a closed multichannel system through the polynomial matrices of the object and the controller in the form of a matrix Diophantine equation. By solving the Diophantine equation, the desired poles of the matrix characteristic polynomial of the closed system are set. There are many options for solving the Diophantine equation and one of them is to represent the polynomial matrix Diophantine equation as a system of linear algebraic equations in matrix form, where the matrix of the system is the Sylvester matrix. The choice of the order of the polynomial matrix controller and the order of the characteristic matrix is carried out on the basis of the theorem given in the works of Chi-Tsong Chen, which always holds for controlled objects. If the minimum order of the controller is chosen in accordance with this theorem, and the Sylvester matrix has not full rank, then this means that there are more unknown elements in the system of linear algebraic equations than there are equations. In this case, the solution corresponding to the selected basic minor has free parameters, which are the parameters of the regulators. Free parameters of regulators can be set arbitrarily, which is used to set or exclude some zeros in a closed system. Thus, using various examples of objects with a non-square matrix transfer function, a polynomial synthesis technique is illustrated, which allows not only specifying the poles of a closed system, but also some zeros, which is a significant advantage, especially when synthesizing controllers for multichannel objects.