*3.2.1. Design of proportional feedback controller*

Proportional feedback control is a very simple and easily implementable controller methodology. Since only a sensor signal is passed through a negative amplifier and fed back to the system, the total computational and implementation cost is very low. Thus, it is possible to apply this control scheme using analog circuits since it does not need online calculation of parameters such as controller gains and current states' values. In addition to these, by using collocated sensor and actuator pair, proportional feedback controller provides active damping at the resonance frequencies of the structure. In fact, this active damping can be employed effectively in the lower-frequency modes of the structure because of stability issues. At the high-frequencies, the phase lag may deteriorate the stability of the collocation and the closed-loop system can be unstable. This high-frequency stability problem is eliminated usually by decreasing the proportional gain or including a low-pass filter in the closed-loop. In this paper, we consider a proportional velocity feedback controller along with a low-pass filter to eliminate high-frequency dynamics.

The undesired vibrations of the thin separation plate are originated by external disturbances and the controller is designed for the expected disturbances including frequencies up to the third resonance mode. By only aiming low-frequency vibration suppression of the plate, high frequency performance of the controller is not considered. For this purpose, a low pass filter which has a cut-off frequency around third resonance mode is considered. In order to have decay at the frequencies higher than the cut-off frequency, the filter is included as an input low-pass filter and is placed between the sensor signal and the controller.

Then, the amplifier gain is adjusted by utilizing the tools of Ziegler-Nichols method. In Ziegler-Nichols method, the important step is the determination of critical gain (gain margin of the closed-loop system). This critical gain is defined as the amplifier ratio at which response of the controlled plant has sustained oscillations and closed-loop system is at the stability limit. Proportional feedback criterion of Ziegler-Nichols method claims that 0.8 times the critical gain yields the best proportional controller performance.

Having determined the cut-off frequency of the filter and amplifier ratio, the proportional velocity feedback controller is finalized for the implementation and experiments.
