PI-controller for inverted pendulum

animated, with derivation in jpg-image

http://simulations.narod.ru/
It is inverted pendulum. Down point is base can be controlled. It can be moved horizontally only. PI (not PID) controller used to set the pendulum in up direction. There are friction in base and frictiopn from air simulated. Random kick is applied periodically. The PI-controller for base velocityt is
dx/dt=p*phi+(q-g)*integral(phi*dt)
where phi is angle counted from vertical position. There was found theoretically that optimal p and q is defined by equation: p^2/R=-q, p<0,q<0. R is pendulum length.

See formula.jpg for derivation of equation of motion.

See formulas2_pd.jpg for derivation optimal gains. In this formulas phi=theta-pi is angle counted from vertical position. The PI-contoller:
d2x/dt2=p*dtheta/dt+(q-g)*(theta-pi)

See animation here:
http://www.youtube.com/watch?v=_2g8bzSk18I