Regions of stability with unequal saturation limits and non-zero set point

Constraints on the magnitudes of control variables limit the region where open-loop unstable systems can be stabilized using feedback control. Variations in regions of stability with unequal control saturation limits and non-zero set points are illustrated for single-input unstable linear systems which have one or two unstable eigenvalues. The regions of stability for saddle-point- and unstable-node-type singularities increase with the increase in one of the saturation limits, but they become invariant when the larger control limit exceeds a certain value; the stability regions vanish for non-zero set-points that saturate the controls. The unstable-focus-type singularity exhibits strikingly different characteristics. These results suggest guidelines for obtaining desired stability regions for different types of singularities.