Generalized PI controllability

Integral controllable systems can be stabilized by all sufficiently low-gain integral controllers. The concept of integral controllability is extended to two generalized stability regions and to proportional controllability and PI controllability with respect to those regions. Furthermore, closed eigenvalue formulae are derived for the maximal low integral gain and the maximal low proportional gain for generalized stability. Related eigenvalue formulae are then used in combination to find all low-gain PI controllers that place the closed-loop eigenvalues in the desired region.