Computer Models of an Optimally Controlled Inverted Physical Pendulum

A physical pendulum controlled by a motor located at a fixed point of suspension is considered in the work. The task is to bring and stabilize this pendulum in the upper unstable equilibrium position. Such a pendulum can be considered as a model of a person standing on an individual Segway-type vehicle. Two engine control algorithms based on the readings of the rotation angle and angular velocity sensors of the pendulum are considered. One of these algorithms maximizes the area of attraction of the upper equilibrium position, the second algorithm allows you to bring the pendulum to this position with minimal energy costs. Computer graphic illustrations help to understand the essence of these control algorithms. The construction of such drawings and animations based on the results of integrating differential equations of the controlled movement of a pendulum under various control laws is a meaningful educational task. This problem is solved by master’s students of the Department of Robotics, Mechatronics, Dynamics and Strength of Machines of the National Research University Moscow Power Engineering Institute when they study the course “Stability and stabilization of motion of dynamic systems.” The article discusses an example of performing such computational and graphical work using the Mathematica computer algebra system. Students present the result of the assignment in the form of a Mathematica program, a file with the nb extension. Such a program, as is known, contains text and executable cells, which allows you to ask not only questions involving symbolic transformations, numerical and analytical solutions, graphic constructions. It is also possible to formulate theoretical problems, the answer to which must be given in the form of text. The package Mathematica is available freely online on WolframCloud. To complete the task, the freely distributed computer algebra system Maxima can be used.