(a) Block diagram of a typical feedback control system showing the control input , direct-acting disturbances , and sensor noise . The physical system is represented by and the control algorithm by . (b) Feedforward prefilter modifies the input signal from to , which is then substituted for the control signal in the closed-loop system in (a). The closed-loop system of (a) is represented in (b) by the block .
Measured frequency response (solid lines) of loaded and unloaded translation stage, overlaid with fits (dashed lines) to the loaded response by three different models (two, four, and six poles). (a) Magnitude response of the Bode plot. (b) Phase response.
Pole-zero plot of the system , showing how the prefilter “cancels out” poles of the system with zeros and then adds new poles at higher frequencies. The crosses indicate poles; the circles, zeros.
Signal flow in the design of an acausal feedforward filter, using the second-order model. The light-shaded boxes represent dynamics that are coded on computer. The dark-shaded box represents the physical system, including its analog closed-loop control.
Collage showing time series of measured stage responses using the two-pole dynamical model and its associated prefilter . Dotted-line triangular waveform represents the desired stage response . Dashed lines represent the signal fed to the stage. Solid lines represent the measured sensor signal. (a) Normal stage operation; (b) causal feedforward algorithm; (c) acausal feedforward algorithm. The phase shifts in the stage response observed in (a) and (b) are removed in (c). (Use the vertical dotted line as a reference.)
AFM images of a calibration grating. All images are taken at 40 Hz , scanning to the left. The white vertical dashed lines indicate the “turnaround” point in the image, with the distance to the right side of the image proportional to the phase lag. (a) Image taken by the scanner in its normal mode, without feedforward. (b), (d), (f): Images taken with causal feedforward filters of second, fourth, and sixth orders, respectively. (c), (e), (g): Same, with acausal filters.
AFM images of calibration grating. (a) “Standard” image, 1 Hz, without feedforward. (b)–(d) Images taken with a fourth-order feedforward filter, at indicated scan rates. All images are scanned over a area.
Phase shift (degrees) produced by different types of filters.
Average spacing of dots measured on images with different types of causal filters, compared to actual spacing and that expected without feedforward.
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