Inverted Pendulum with Horizontal and Vertical Motion: As a Benchmark of Bilinear System

An inverted pendulum is one of the most well known control problems: many textbooks on modern control theory treat it and many control courses in university use it as a laboratory experiment. But, usually, only the horizontal input to the cart is considered in those textbooks or benchmark problems. On the other hand, if we carefully analyze how we (human being) stabilize the inverted pendulum, we will find that we use not only the horizontal motion but also the vertical motion of the hand. In this note, we will consider the inverted pendulum on the cart with horizontal and vertical motion. We will show that the pendulum system can exactly be transformed into a bilinear system. We would like to propose to use this system as a benchmark problem of bilinear systems. We will also show one of our control strategy for bilinear systems based on an inverse optimal control, which efficiently uses the vertical motion of the cart.