High gain feedback,tolerance, and stabilization of linear systems

The problem of stabilizing linear continuous-time systems is revisited with the objective of investigating the tolerance allowed in the implementation of stabilizing feedback controllers. It is shown that, for high gain feedback, this tolerance can be described by a cone in the feedback parameter space, called the “tolerance cone”. The tolerance cone describes fractional (percentage) errors the feedback controller can tolerate without jeopardizing the internal stability of the controlled system. The larger the vertex angle of this cone, the more tolerance is available when implementing the controller. The vertex angle of the tolerance cone is determined by the proximity to singularity of a certain matrix derived from the controlled system.