Group delay induced instability and its suppression for a controlled-double pendulum

The group delay of digital filters induced instability of a controlled-double pendulum was discussed, and then a discrete delayed linear quadratic controller was used to suppress the instability. First, the dynamic model of the double-pendulum on a moving cart was established and the free-decaying vibration of the system was measured. It was found that the settling time of the free system was very long, and hence, an active control was needed to shorten the settling time. Then, the effects of digital filters on the stability of the system under the linear quadratic control were discussed, where the Butterworth low pass digital filters were used to remove the heavy noise embedded in the control input signal. The conclusion of this research is that if the filter parameters are properly selected, the digital filters can not only enhance the performance of control system, but also reduce the cost of energy. However, when the specifications of the filters are set too high, the group delay will exceed the critical value and cause the instability of the closed loop system. The delay controls based on state transform are suitable for the dynamic systems involving digital filters because the group delay of a digital filter can be completely determined by its frequency response of phase and can be estimated accurately beforehand. At the last section of the paper, therefore, a newly proposed discrete delayed linear quadratic controller is represented to suppress the group delay induced instability of the double pendulum.