An inverted pendulum is a pendulum which has its center of mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart and pole. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downwards, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies.
First of all, kinematic and dynamic of this system modeled. Then, it implements on Matlab by the transformation matrix and state space method. After that, respond to this system analysis which is needed in order to control that one.
Second, linear controller especially PID method used to control this system. Then, another controller which is called LQR used to control this one better according to error and actuator force.
Finally, the last controller, LQR, turned to digital or discreet one.