Vibration Control, Sensor Gain, and Servo Bandwidth

I have had many conversations through the years regarding sensor resolution and control. It is often incorrectly believed that increasing resolution, meaning higher sensitivity, results in a higher bandwidth servo. If we assume for this discussion that servo gains have been tuned so they are at their maximum values and the loop is meeting gain and phase margin requirements, increasing the sensor gain requires the control gain to come down, in order to preserve the overall loop gain. This misconception about resolution and bandwidth stems from an interesting truth that occurs when increasing sensor gain, that is performance does increase. One might ask, if performance increases, then how is it bandwidth does not increase? The answer lies in the definitions of bandwidth and controllability.

Bandwidth is a term defined by the gain of the Closed (or Open) Loop Magnitude plot (Bode Plot). Bandwidth is generally associated with frequencies that the controller can track or attenuate. One common definition, although one I do not use, is based on signal filter theory, and defines the bandwidth as the -3dB point in the Closed Loop Transfer Function. If we link “bandwidth” to improving following error, this definition is extremely optimistic, and generally not valid for frequencies slightly lower than the bandwidth frequency. I always use the 0 dB frequency in the Open Loop transfer function, and while it also can be challenged for its authenticity, it is true that frequencies close to, but lower than the bandwidth frequency are more attenuated than those in the -3dB definition.

With bandwidth established, meaning we agree that frequencies below the value are tracked or attenuated, let’s move on to “Controllability”. This is where sensor resolution plays a major role. Simply put, a control system cannot control what it cannot sense. As sensor resolution (gain) is increased, and bandwidth is conserved, meaning control gain is lowered to accommodate the sensor gain increase, the system can sense deeper into the signal’s actual magnitude, and attenuate it to the level set by the loop gain, or the sensor noise floor. This truth is where the misconception is conceived, meaning the convolution of higher bandwidth (and subsequent higher gain), with improved disturbance rejection, when if fact the improvement is all coming from resolution, not servo bandwidth or gain.

The figures below walk us through how this works. If there were a vibration that is “in band”, meaning well below than the bandwidth frequency, and the zero to peak amplitude was 1 nm, a sensor with only 4nm of resolution would not sense this motion whatsoever. This is “Controllability”, and in this case, the harmful signal is not controlled. If, however, the sensor has 0.1 nm resolution, the controller can drive the disturbance to the floor of the sensor (0.1 nm) or to limit of its gain at that frequency, which ever one happens first.

In the frequency domain, an external sensor measuring the outcome of control would look like something like the Figure 2 below. The family of solid color curves represent an auto spectrum plot, and the dashed curves represent the integrated area under the auto power spectrum curve. The Green curve is with the control off, and the low frequency peak is at full amplitude. With control on, and coarse resolution, the low frequency peak would be attenuated to the floor of the sensor (1 count=4 nm for example). The Red curve is with fine resolution and the low frequency curve is attenuated to its floor (1count=0.1 nm). Note that no attenuation is realized for any of the curves beyond the bandwidth frequency.

Figure 3 below is actual data from a focus servo loop showing vibration as a function of sensor resolution. The peak near 12 Hz was attenuated 10x better with finer resolution with no change in servo bandwidth.

If you would like to discuss this article topic in greater detail or have questions about how to control and reject problem frequencies within your motion system, please reach out and let’s get to work!