Modal synthesis of controllers by a polynomial method for automatic control systems with excessive dimension of the control vector

Abstract

When using polynomial methods of synthesis of multichannel regulators, there is a need for polynomial matrix calculus. However, when using this method, objects with the number of output channels equal to the number of input channels are mainly considered. This is necessary for the convenience of solving a system of linear algebraic equations in matrix polynomial computation. At the same time, a fairly large number of real technical systems have an unequal number of input and output channels. At the same time, the issue of the synthesis of generators by the polynomial method for multichannel objects with an unequal number of input and output impacts has not been worked out sufficiently deeply. One of the special cases of such systems can be considered systems with excessive dimension of the control vector. Within the framework of this work, examples of such systems and the purposes of their use are given. An illustrative example of a linear model of an unstable object with three channels for input action and two channels for output action is given. It is necessary to achieve certain quality indicators of the output vector quantity, while the control is carried out in the feedback of the system and is summed up with the input effect. The simplicity of the system under consideration is connected with the convenience of demonstrating on this example a modal synthesis method using a polynomial matrix decomposition of the transfer functions of an object and a controller for such a class of objects. In accordance with the recommendations presented in the algorithm for synthesizing regulators for objects with a non-square matrix transfer function, to solve the problem of synthesizing a regulator for systems with an excessive dimension of the control vector, the transfer function of the control object is represented as a right polynomial matrix decomposition, and the regulator is represented as a left one. During the demonstration of the example of the algorithm, some clarifications and edits are proposed in it.