Planar steering of a single ferrofluid drop by optimal minimum power dynamic feedback control of four electromagnets at a distance

Any single permanent magnet or electromagnet will always attract a magnetic fluid. For this reason it is difficult to precisely position and manipulate ferrofluid at a distance from magnets. We develop and experimentally demonstrate optimal (minimum electrical power) 2-dimensional manipulation of a single droplet of ferrofluid by feedback control of 4 external electromagnets. The control algorithm we have developed takes into account, and is explicitly designed for, the nonlinear (fast decay in space, quadratic in magnet strength) nature of how the magnets actuate the ferrofluid, and it also corrects for electromagnet charging time delays. With this control, we show that dynamic actuation of electromagnets held outside a domain can be used to position a droplet of ferrofluid to any desired location and steer it along any desired path within that domain-an example of precision control of a ferrofluid by magnets acting at a distance.