rootlocus

Root locus is a graphical method in control theory used to analyze how the closed-loop poles of a system move in the complex plane as a single gain parameter varies. It helps assess stability, transient response, and design choices for controllers.

For a system with open-loop transfer function G(s)H(s) in unity feedback, the closed-loop transfer function is T(s) = G(s)H(s) / (1 + K G(s)H(s)), where K is a real nonnegative gain. The root locus plots the roots of 1 + K G(s)H(s) = 0 as K ranges from 0 to ∞. The locus begins at the open-loop poles (K = 0) and ends at the open-loop zeros (K → ∞) or at infinity. Real-axis segments of the locus occur where an odd number of poles and zeros lie to the right.

Key features include breakaway and break-in points on real-axis segments, where branches depart from or rejoin the axis, and asymptotes that appear for high-order systems (when the number of poles exceeds zeros). The asymptotes have centroid σ* = (sum of poles − sum of zeros) / (n − m) and angles θk = (2k+1)π / (n − m), where n is the number of poles and m the number of zeros. Complex branches often tend toward these asymptotes and may move toward zeros or infinity.

Analysis considerations: stability requires all closed-loop poles to lie in the left-half plane for the range of K of interest. Crossing into the right-half plane indicates potential instability as K varies. Root locus is commonly used to select gain and to inform compensator design, often in conjunction with Bode and Nyquist methods. It applies to rational G(s) and can accommodate non-unity feedback with appropriate formulation.