The complex range of gain falls mainly in the s plane

Abstract: "This technical report promotes graphically based methods for determining the gain margin and phase margin of linear time-invariant single-input, single-output control systems. The gain margin, or amount of gain that can be increased before the closed-loop system becomes unstable, can be determined from a graph showing the angle of each closed- loop system eigenvalue in the complex plane as a logarithmicfunction of real gain. By indentifying the gain interval for which all eigenvalues have angles within the stable region, the gain margin can be calculated. At any constant real gain, the phase margin, or range of phase angle corresponding to a stable closed-loop system, can be determined from a graph of the angle of each closed-loop system eigenvalue in the complex plane as a function of gain angle.The proposed methods do not require frequency calculations, and serve as alternate means for stability-robustness studies. Furthermore, thephase margin determination highlights the importance of root sensitivity, with the practical design guideline of not selecting control gainsnear break-points."