Modal design of a polynomial control system for the inverted pendulum with two degrees of freedom

An example of calculating an automatic control system for a multi-channel linearized model of a control object is given ‒ a 2-degree of freedom pendulum on a cart with the ability to move, for a mobile board and a support rack, in two planes. This model is described by a non-square matrix transfer function with a large number of output parameters compared to the number of input parameters. A description of a linear model of an object is obtained through the state space and through a matrix transfer function. According to the obtained descriptions, it can be seen that for a linearized version of the description of the object, for different coordinate axes, the description of the system is symmetrical. In this regard, there is an approach to synthesizing the control system for each coordinate of the mobile platform separately. However, with the further transition to a nonlinear description of the system, the symmetry of the description will be violated and it will be problematic to apply the obtained regulators for a nonlinear object in some cases. In this article, the work of the algorithm for the synthesis of the control system of the mobile platform for stabilizing the support rack to a vertical position and bringing the mobile platform to a given position at both coordinates at the same time is illustrated. The calculation was carried out using the algorithm of synthesis of the controller by a modular method using a polynomial matrix decomposition of the matrix transfer functions of the object and the controller. The simulation of the operation of the resulting automatic control system shows that the controller copes with the task of stabilizing the angle of the support post in the vertical position and bringing the mobile platform to the desired position.