10.26731/1813-9108.2017.4(56).84-99
In modal synthesis of multichannel linear systems using left / right polynomial expansions, the method is based on the solution of a matrix polynomial equation, called the diophantine equation. In its solution, one needs to pass from polynomial descriptions to the equivalent numerical matrices. In this case, the emerging matrix non-polynomial equivalent equation, as a rule, has a reduced-rank matrix (a degenerate matrix). To solve such a system of equations, it is necessary to transfer linearly dependent rows with the corresponding unknowns to the right-hand member. In addition, in order to proceed to a square matrix for unknowns, one needs to remove the linearly dependent columns of this matrix and the corresponding columns from the right side of the equation. After solving the "truncated" system of equations, it is necessary to return to the original system. In this case, we obtain a solution of the linear system with additional constraints. As a rule, these constraints are superimposed on the form of the desired characteristic matrix and on the polynomial matrices corresponding to the polynomial decomposition of the multichannel regulator. The article discusses various cases of the task object corresponding to singular and not singular polynomial matrix "denominator" of the object. An illustration of the calculations, in order to avoid cumbersome calculations, is carried out by the example of an object of a small order. The proposed technique is illustrated by the example of synthesis of a two-channel system.
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