On controllability and observability of multichannel automatic control systems in the synthesis by the modal method using the polynomial matrix decomposition of the object and controller

When considering the issue of synthesis of an automatic control system by a modular method using a polynomial decomposition of the transfer functions of an object and a controller, synthesis algorithms for fully controlled systems are proposed. However, the question arises about the possibilities of using this algorithm if this condition is not met. The consideration of this is-sue turned out to be especially relevant for multichannel models of objects with a non-square transfer function (having an unequal number of input and output channels). It is shown that for some fundamental terms of the theory of automatic control, such as controllability, reachability, observability, stability and some others, there are special definitions of them in the case of considering this type of objects. The term non‒square object is proposed for use, which is used mainly in foreign literature. Some restrictions on the modal synthesis of regulators by a method using a polynomial matrix separation of the object and the regulator are considered. Examples of internally and asymptotically unstable systems are given. A hypothesis is put forward about the stability of the controlled system. An example of a multichannel system "inverted pendulum on a cart" is considered, which is an object with a non-square matrix transfer function (in this example, the number of input actions is less than the number of output parameters). Using the static characteristics of this object, it is demonstrated that not always controlled systems can be stabilized in a given position. For example, in the case of setting the desired angle of an inverted pendulum other than zero, it is impossible to hold the position of the cart in a given coordinate. At the same time, if you set the angle at the equilibrium point as the desired angle of the inverted pendulum, then stabilization of the cart at a given coordinate becomes possible.