Control of solitons in mems actuator arrays
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Signal transmission in large arrays MEMS and NEMS devices will be a major issue due to the sheer complexity, and it is likely that solutions to ensuing problems will have much in common with complex networks in the biological world. Motivated by this we specifically address the problem of how one may generate a compendium of stable travelling pulse patterns in a linear array of microactuators by controlling their own inherent dynamics appropriately. The approach we take is two fold. First we pick a known system that produces the desired patterns and derive controls that ensure smooth and quick transition from one pattern to another. Then we expect to embed these controlled dynamics into the linear array of actuators using standard available model based control strategies. The problem of transition from one pattern to another in a smooth and quick fashion has received only scant attention. We focus on a class of systems that possess special type of solutions known as solitons. All of the hitherto known soliton producing equations are completely integrable and can be expressed in a special form known as the Lax form. In the Lax formulation the problem of complete integrability is transformed into an analyticity problem on a Riemann surface referred to as the spectral curve. This allows one to relate the properties of the solution to the properties of the spectral curve. It can be shown that given a solution there exists an associated unique spectral curve and that given a spectral curve there exists a corresponding class of solutions that evolve on a unique g dimensional torus of the state space where g is the genus of the spectral curve. We classify solutions by their respective spectral curves and reduce the problem of smooth and quick transition from one class to another to that of a setpoint control problem on the space of spectral curves. It is the first time such results have been shown. The results are general and are applicable in a very wide sense. Namely, to any Liouville integrable system.