Controllability of Unstable Objects

The application of unstable objects in engineering has increased interest to the problems of stabilizing control. Magnetic bearings are one of examples of such objects since the Earnshaw-Braunbeck theorem prohibits a stable suspension of ferromagnetic body in static magnetic and gravity fields. Restriction in control resources for an unstable object gives rise to restriction of attracting region of stabilizable equilibrium in the phase space for a system with feedback.This specificity of stabilizing unstable objects is reflected in the choice of a control-optimality criterion needing for its application the maximal attracting region at the given control restrictions. It is shown that to use this optimizing criterion it is sufficient to obtain the object's controlability in the partial variables that are unstable. This requirement is called stabilizability.The optimal control satisfying the criterion for maximal attracting region is synthesized for the objects having a given number of different positive roots of the characteristic equation coinciding with dimensionality the vector of elementary piece-linear admissable controls has.There has been studied the structure of phase space of an optimal system; on the basis of this study there has been made some conclusion pertaining to geometry of the unstable object cortrolability region. The results obtained have been applied to the synthesis of systems for stabilizing an elementary magnetic suspension, a magnetic suspension with elastic elements of design and with contour of eddy currents, and to stabilizing the shaft's magnetic bearings as well.