Swing up of an inverted pendulum by simulator-based foresight control

One of the typical nonlinear system is a pendulum. Cause of difficulties of controlling a pendulum is mismatching of number between inputs and outputs, that is to say, single input and double outputs. The feature of a pendulum is nonlinear as to angle and linear as to cart position. Another point is that state equations do not have terms of position of a cart. In addition, the angle of a pendulum must satisfy the dynamic equation at horizontal line. In order to swing up a pendulum of these properties, first, we define a reference angle of a pendulum which satisfies the dynamic equation at horizontal line. After the pendulum goes up around upright position, we change the curve of reference angle so that the pendulum standstill at the origin. In other word, when the pendulum system becomes a linear time-invariant system, we define the ideal curves of both angle and position which will converge to the origin. The angle of the pendulum is controlled by SFC (simulator-based foresight control) all the way which can be applicable for a nonaffine nonlinear system