Polynomial method for the synthesis of multichannel systems by transition to matrix polynomial representation

Polynomial methods for synthesizing linear regulators for automatic control systems with linear objects, proposed by a number of authors, including Chen, Kailath, Gaiduk, and others, along with methods of synthesis in the state space, are becoming increasingly widespread. The synthesis of multichannel regulators caused by the need to use the matrix polynomial calculus is of a special difficulty, which is aggravated by a significant increase in the dimension of the matrices during the transition from polynomial matrices to numeric ones, in which Sylvester matrices are used. Herewith, it is necessary to take into account the requirements of controllability and observability, leading to the need to check for the presence of identical roots in polynomial matrices corresponding to the numerator and denominator of the object. This leads to the requirement of a relatively prime matrix polynomial fraction, which can be significantly weakened if it is possible to include in the desired characteristic matrix of the system some zeros and poles of the object located far to the left of the imaginary axis. In calculations using numerical matrices and, consequently, using Sylvester matrices, the latter degenerate due to the lowering of the rank, which complicates the calculations. The research continues to study polynomial synthesis of multichannel regulators based on the results obtained by Chen and other researchers and presents an algorithm for the synthesis of regulators, the feature of which is the possibility of introducing additional so-called free parameters that allow additional requirements for the automatic control system. The free parameters allow to obtain strictly proper regulators, along with the proper regulators.