Edinburgh Napier University

  The two dimensional inverted pendulum is unstable in that it may fall over at any time in either direction unless a suitable control force is applied. This paper designs a controller numerically and symbolically by using Ackermann's formula of the Pole placement method. The controller will balance an inverted pendulum on a moving cart whilst both controlling its position and keeping the pendulum upright in its state of unstable equilibrium. The relative stability is assessed using the Nyquist diagram since the pole placement methods can yield designs but cannot guarantee to have good relative stability. Simulation and experiments are carried out for the single link inverted pendulum system by using different sets of desired closed-loop poles. A comparison of the symbolic elements of a discrete control matrix and a continuous control matrix is undertaken.