Linear controlled objects with one input and one output (single input - single output, SISO), and objects with multiple inputs and multiple outputs (multi-input – multi-output, MIMO) have different formalized controller synthesis algorithms. At the same time, objects with an unequal number of inputs and outputs, in many cases, are built by the developer intuitively, when changing the existing calculation algorithms for each control object, therefore, the development of a formalized calculation algorithm for this type of objects is relevant. Within the framework of this work, it is proposed to extend the synthesis technique for multichannel objects, which is the polynomial synthesis technique, to objects with a smaller number of inputs compared to the number of outputs, namely, to objects with one input and several outputs (single input – multi-output, SIMO). The reasoning developed in the work is an example of calculating an electromechanical tension control system in the material transportation zone of the production line, which has one input – the voltage supplied to the electric motor and four outputs–- the armature current, the rotation speed of the electric motor shaft, the rotation speed of the roll, the tension in the zone under consideration and the elastic moment. The tension in the considered zone is an adjustable coordinate. The use of the polynomial synthesis method for objects with a non-square matrix function made it possible to place the poles of a closed system in a given position, and the transfer function does not contain zeros according to the assignment. It was also possible to set the disturbance-stimulated zeros of the closed system in such a way that a second-order astatism is obtained.

Currently, an urgent task in control theory is the synthesis of regulators for objects with a smaller number of input values compared to output ones, such objects are described by matrix transfer functions of a non-square shape. A particular case of a multichannel object with one input variable and two / three / four output variables is considered; the matrix transfer function of such an object has not a square shape, but one column and two / three / four rows. To calculate the controllers, a polynomial synthesis technique is used, which consists in using a polynomial matrix description of a closed-loop control system. A feature of this approach is the ability to write the characteristic matrix of a closed multichannel system through the polynomial matrices of the object and the controller in the form of a matrix Diophantine equation. By solving the Diophantine equation, the desired poles of the matrix characteristic polynomial of the closed system are set. There are many options for solving the Diophantine equation and one of them is to represent the polynomial matrix Diophantine equation as a system of linear algebraic equations in matrix form, where the matrix of the system is the Sylvester matrix. The choice of the order of the polynomial matrix controller and the order of the characteristic matrix is carried out on the basis of the theorem given in the works of Chi-Tsong Chen, which always holds for controlled objects. If the minimum order of the controller is chosen in accordance with this theorem, and the Sylvester matrix has not full rank, then this means that there are more unknown elements in the system of linear algebraic equations than there are equations. In this case, the solution corresponding to the selected basic minor has free parameters, which are the parameters of the regulators. Free parameters of regulators can be set arbitrarily, which is used to set or exclude some zeros in a closed system. Thus, using various examples of objects with a non-square matrix transfer function, a polynomial synthesis technique is illustrated, which allows not only specifying the poles of a closed system, but also some zeros, which is a significant advantage, especially when synthesizing controllers for multichannel objects.

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Построить регулятор для какого-либо объекта возможно множеством способов. В рамках данной работы рассмотрим варианты построения регулятора в зависимости от представления модели нелинейного объекта. Управление, сформированное регуляторами, будет компенсирующее, т. е. исходная система, замкнутая найденным управлением, будет эквивалентна линейной. Такие компенсирующие воздействия будем искать, основываясь на линеаризации обратной связью, которая заключается в нахождении нелинейных обратных связей, компенсирующих нелинейные функции самого объекта. Для объекта, заданного нелинейным дифференциальным уравнением первого порядка, найдены четыре варианта его записи. Для каждого варианта сформировано компенсирующее управление. Приведены структурные схемы вариантов исходной модели объекта и замкнутых систем, а также их математическое описание.