Polynomial matrix decomposition for the synthesis of non-square control systems

The application of advanced synthesis methods is due to the increasing complexity of control objects. Relatively simple objects are represented as a single-channel system or as a combination of such systems and are calculated separately. More complex systems must be viewed as multi-input and multi-output systems. There are several approaches to this. Within the framework of this paper we will consider the synthesis of a system presented in the form of a polynomial matrix decomposition. It allows us to write a closed loop system in such a way that, by analogy with single-channel systems, it is possible to single out the "numerator" and "denominator" not only of the object and the controller, but of the entire system. For multichannel objects, they will be written in a matrix form allowing you to select the characteristic matrix whose determinant is the characteristic polynomial. In this paper, an emphasis is placed on the derivation of four variants of the polynomial matrix description (PMD) of a closed system. Such a variety of representation of a closed-loop system follows from the equivalent writing of the transfer matrix in the form of left and right PMD of an object or controller. Of the four options for recording the system, two options – left and right – for the characteristic matrix are distinguished. When they are reduced to a diagonal form, the elements on the main diagonal contain the poles of a closed system along the corresponding channel. From the example given at the end of the paper, it can be seen that it is more convenient to use the left characteristic matrix because it has a lower dimension for a non-square object (the number of input and output quantities is not equal), with the number of input actions exceeding the number of output quantities, The right characteristic matrix can also be used to synthesize such a control object, but the resulting solution is more complicated and not obvious. The situation is reversed if we consider an object with fewer inputs than outputs. In this case, the right characteristic matrix will be smaller and more suitable for synthesis. It follows from this that the procedure for synthesizing a control system for non-square objects differs depending on the number of inputs and outputs.