Different ways of finding the feedback matrix for a linear dynamical system

Abstract

Using of feedback in linear dynamical systems is an important task, because it allows to correct the control function by using the information about the state of the system. Using of the feedback matrix K, which makes possible to make a relationship between the state and control static and linear is particularly relevant. The complexity arises if the boundary condition is imposed on a state function not only in the initial, but also at the final point. We need to expand the defined parametrically matrix M of the closed system into a series and solve the necessary equations to find the feedback matrix K .First we need to answer the question: what are the properties of the matrix M in order for these equations to be solvable. Within the framework of this article, we consider types of matrices M for which the answer to the posed question is not difficult. The first type includes matrices in which all elements except for the main diagonal are equal to zero, the second type includes those in which all elements except for some column are zero, the third type includes the matrices where the zeros are outside of some row. The fourth type is a matrix, where non-zero elements are arranged diagonally starting with k + 1 elements of the first row. The matrices of the first three types allow us to find the connection between the components of the boundary conditions necessary for the existence of the feedback matrix K. For matrices of the fourth type, it is difficult to obtain such connection analytically. However, it will not be difficult to calculate the matrix exponent by numerical methods for matrices of thistype, which also facilitates the solution of the problem.