2.3 Videotype Sampling in the Feedback Stabilization of Electromechanical Equilibria

Abstract

The feedback control of hydromagnetically contained plasmas with dimensions large compared to potentially unstable wavelengths requires a large number of spatially distributed feedback sensors and drivers. The multiplicity of signals to be amplified and processed suggests the use of computers or other discrete‐time devices which handle signals on a ‘time‐sharing’ basis. Typically, scanning techniques are envisioned to sense and drive, thus introducing to an analytical representation discreteness in both time and space. A general method, based on the Fourier superposition of wavetrains, is developed to describe infinite continuum systems with discrete spatial and temporal feedback. Dynamics are represented by a generalization of the dispersion equation, with Z transforms used to provide closed‐form expressions if the discreteness is in space or in time only. The Bers‐Briggs criterion is generalized to differentiate between absolute instabilities and amplifying waves with the discrete feedback. A quasi‐one‐dimensional model for the $\mathrm{m}\text{\hspace{0.17em}=\hspace{0.17em}1}$ mode of the z‐θ pinch is studied to delineate effects of spatial and temporal sampling rates on stability regimes.