Microchaotic motion of digitally controlled machines

Without control, the desired motions of machines do not occur, and the desired equilibria and stationary motions are often unstable. Human operator or computer control may be needed to control and stabilize these machines. An important common feature of both analog and digital controllers is the time delay that is introduced into the system. Even when these delayed systems should be stable, the experiments show small stochastic oscillations around the desired motion, as are often experienced in robotics. In case of the stabilization of an inverted pendulum, the analysis of the equation of motion shows that chaotic vibrations occur around the equilibrium even when stochastic effects related to human control are not present. In advanced design work of digitally controlled machines, it is vital to know the characteristics of this chaotic behavior. The estimation of the distribution of vibration amplitudes and the frequency range should be available at the design stage. This initiates the analysis of the so-called microchaos or mu-chaos.