We developed a novel sensor positioning method: sensor averaging. By placing 2 sensors apart by 2d, and use the average of the two outputs to control the middle point. The averaged output has less observability on the vibration modes with wavelengths close to 4d. Therefore, we can attenuate the undesired vibration modes robustly. The Bode plots of the resonance modes are shown in the following figure. The broad notch filters undesired resonance modes. The advantage of this averaging methods are:
• It eliminate sensor/actuator noncollocation problem.
• It robustly attenuates undesired resonance modes without adversely affecting phase
• The attenuation is a function of wavelengths, and is independent of structure boundary conditions, structure lengths, and structure positions.

    Actuator Averaging

The dual of sensor averaging is actuator averaging. It is not as straight forward as sensor averaging, but the effect is very similar. We place two actuators set apart by 2d, and apply the same force to control the middle point. The resulting force has less controllability on the vibration modes with wavelengths close to 4d. Therefore, we can again attenuate the undesired vibration modes robustly. The bode plots of actuator averaging is the same as the previous figure for sensor averaging.

We can place 2 sensors and 2 actuators, and together we can create a even broader notch to filter the undesired resonance modes. The experiment of 10 ft beam suspension with free-free boundaries were not successful until we use both sensor averaging and actuator averaging. The resulting closed-loop system is robustly stable for varying boundary conditions.