The method of synthesizing linear regulators for linear objects using polynomial decomposition, along with classical synthesis methods, such as synthesis in the state space using full and reduced order observers, transfer functions and synthesis using logarithmic frequency characteristics (LATCH), optimal synthesis methods, and others, finds more and more widespread. With a polynomial method of synthesis, as a rule, a transition from polynomial representations to matrix numeric ones is used, which leads to equations with a non-square degenerate Sylvester matrix. When solving the problem of formalizing synthesis algorithms for multichannel systems, it is necessary to return to the synthesis algorithm for single-channel systems. In this paper, based on the results obtained by Chen, Kailath and other authors,  based on an analysis of the calculation of numerous examples of calculating single-channel systems, a formal algorithm for the synthesis of regulators is presented. When solving the problem of formalizing synthesis algorithms for multi-channel systems, rely on algorithms for the synthesis of single-channel systems. In this article, is use the results obtained by Chen, Kailath and other authors, which lists the requirements for the polynomial description of the object: the proper (strictly proper) of the transfer function of the object, the mutual simplicity of the polynomials numerator and denominator transfer function of the object. Of particular note is the requirement of mutual simplicity of numerator and denominator polynomials – failure to do so leads, first of all, to degeneration of the Sylvester matrix, and can also lead to violation of controllability, observability, etc. In addition, it is necessary to take into account the restriction imposed on the choice of the degree of the regulator, which is equivalent to limiting the desired characteristic polynomial of the closed system. Based on the analysis of calculations of numerous examples of synthesis of single-channel regulators, six of which are given in this article, a formalized algorithm for the synthesis of regulators is presented. In many studies, in solving the synthesis problem, linearly dependent rows / columns in the Sylvester matrix are zeroed together with the corresponding parameters of the regulator. In this article is propose linearly dependent rows with corresponding unknown parameters of the regulator to be transferred to the right side of the equation. This leads to the appearance of free regulator parameters, which can be set arbitrarily (in some cases additional limiting are imposed). This corresponds to the general solution of a system of linear equations, which when given by a free parameter of the regulator of specific values leads to different versions of the synthesized regulator.

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