Properties of Stabilizing PID Gain Set in Parameter Space

The properties of the PID gain set for which all the closed-loop poles lie in the left half of the complex plane are examined on the parameter plane. A new method of determining the increase in the number of unstable poles when the gain crosses the boundary of stability regions is proposed. This method is useful for determining the number of unstable poles in each region. These results are extended to Gamma-stability based on conformal mapping.