Stability analysis for digital PD control of flexible systems including damping

In this work we extent our recent results on the stability of single-rigid/single-flexible mode system to cases involving damping. We present closed form analytical expressions that describe the boundaries of the stability regions for digital PD control systems. This is obtained using a newly adopted approach based on the critical stability constraints of Jury test. The considered system simulates many practical systems such as antenna, space shuttle, and robot arm. It is found that, the stability regions for damped flexible systems have three identifiable and distinguished topologies corresponding to three classes of damped system. The three classes are separated from each other by two surfaces in the three-dimension-space of the system-parameters. The stability region for the first class is almost a right triangle in the gain space where the third boundary, the hypotenuse, is described by the lowest root of a cubic equation. The stability region for the second class is clearly larger in size and the three roots of the cubic equation participate in defining the third boundary. In the third class, the stability region is further larger in size and the highest root of the cubic equation defines the third boundary. A peculiar situation is found where a stable system of the second class is possible with negative derivative gain. Numerical simulation is presented to verify this peculiar situation.