Control problem of the asynchronous spectrum of linear periodic systems with degenerate right lower diagonal block of averaging of coefficient matrix

Abstract

The present study considers a linear control system with a periodic matrix of coefficients and program control. The matrix under control is constant, rectangular, and its rank is not maximum. It is assumed that the control is periodic, and that the modulus of its frequencies, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency modulus of the coefficient matrix. The following problem is posed: to select such a control from an admissible set that the system would have periodic solutions, the frequency spectrum (the set of Fourier exponents) of which contains a predetermined subset, and the intersection of the modules of the frequencies of the solution and the matrix of coefficients is trivial. The posed problem can thus be termed the ‘problem of control of the asynchronous spectrum with the target set of frequencies’. The solution to the posed problem essentially depends on the structure of the average value of the matrix of coefficients. To date, this problem has been solved for systems with zero mean. In addition, the case is studied when the matrix under control has zero rows, the averaging of the matrix of coefficients has a degenerate left upper diagonal block, and the rest of its blocks are zero. The question for a system with a nontrivial right lower averaging block remained open. In the present work, we study the problem of control of the asynchronous spectrum for the indicated class of systems. It has been established, in particular, that for the solvability of this problem it is necessary that the block formed by the rows of the matrix of coefficients has an incomplete column rank.