Control of Large Actuator Arrays Using Pattern-Forming Systems

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Pattern-forming systems are used to model many diverse phenomena from biology,chemistry and physics. These systems of differential equations havethe property that as a bifurcation (or control) parameter passes through acritical value, a stable spatially uniform equilibrium state gives way to astable pattern state, which may have spatial variation, time variation, orboth. There is a large body of experimental and mathematical work on pattern-forming systems.

However, these ideas have not yet been adequately exploited inengineering, particularly in the control of smart systems; i.e.,feedback systems having large numbers of actuators and sensors. With dramatic recent improvements in micro-actuator and micro-sensortechnology, there is a need for control schemes betterthan the conventional approach of reading out all of the sensor informationto a computer, performing all the necessary computations in a centralizedfashion, and then sending out commands to each individual actuator.Potential applications for large arrays of micro-actuators includeadaptive optics (in particular, micromirror arrays), suppressingturbulence and vortices in fluid boundary-layers, micro-positioning smallparts, and manipulating small quantities of chemical reactants.

The main theoretical result presented is a Lyapunov functional for thecubic nonlinearity activator-inhibitor model pattern-forming system.Analogous Lyapunov functionals then follow for certain generalizations ofthe basic cubic nonlinearity model. One such generalization is a complex activator-inhibitor equation which, under suitable hypotheses,models the amplitude and phase evolution in the continuum limitof a network of coupled van der Pol oscillators, coupled to a network of resonant circuits, with an external oscillating input. Potentialapplications for such coupled van der Pol oscillator networks includequasi-optical power combining and phased-array antennas.

In addition to the Lyapunov functional, a Lyapunov function for the truncated modal dynamics is derived, and the Lyapunov functional isalso used to analyze the stability of certain equilibria. Basic existence, uniqueness, regularity, and dissipativity properties ofsolutions are also verified, engineering realizations of the dynamicsare discussed, and finally, some of the potential applications areexplored.