Control of a continuum electromechanical instability

The stabilization of continuum mechanical and electro-mechanical systems by means of coupling to active distributed structures is an application of continuum feedback control that has promise. The stabilization of classic forms of Rayleigh-Taylor type instability with a continuum of feedback is presented. The effect of spacewise sampling on the dynamics is discussed in detail for a typical case where the instability occurs, because a thin, conducting, liquid or elastic film is stressed by a sufficiently large constant electric field. The stabilizing influence of feedback provided by an external electrostatic structure is presented as it depends on 1) the effect of longitudinal constraints on the film (fixed ends), 2) the degree of coupling between the film and the feedback structure (proximity), and 3) the effect of spacewise sampling to obtain the feedback signals. The stability conditions are found by a "spliced" solution that accounts for 1) and 3), a normal-mode solution that describes 1), 2), and 3) and a traveling-wave solution that includes the effects of 2) and 3). It is shown that the sampling interval places an upper bound on both the electric pressure and feedback gain consistent with stability. This bound is given, as it depends on the proximity of the feedback structure and the longitudinal boundaries.