Dynamic Controller Approach to Algebraic Constraints in Linear Systems

This paper addresses the problem of rapidly driving a linear system's state near a manifold. These manifold or algebraic constraints represent desired surfaces in the state space whose neighborhood the system is required to move to. For practical reasons only linear type controls are allowed. To solve this, we propose to reformulate the original problem by extending the state space to include additional dynamics that will be part of the controller. The design involves a two-part control which make use of a linear dynamic as well as static feedback control. The feedback gains are chosen so that the extended system is stable. Furthermore they are adjusted so that the rate of the decay of the controller states will go to zero "faster" than the rate of the original states. For a scalar case we present a complete characterization on how to select the eigen values allocations to achieve the requirement. Examples are presented to illustrate the suggested approach.