Glossary term
Root Locus
A control-design plot showing how closed-loop poles move as a gain or parameter changes.
Definition
methodRoot locus is a graphical control-design method that shows how closed-loop poles move in the complex plane as a gain or parameter changes.
Root locus links the open-loop transfer function to closed-loop stability and transient response. By tracking pole locations as controller gain varies, engineers can estimate damping, natural frequency, overshoot, settling time, instability boundaries, and the effect of adding compensator poles or zeros.
For a feedback system with open-loop transfer function G(s)H(s) and scalar gain K, the closed-loop characteristic equation is often written as:
The root locus is the set of roots of this equation as K varies. These roots are the closed-loop poles, so their positions determine stability and much of the transient response. Poles in the left half of the continuous-time complex plane are stable; poles near the imaginary axis produce slow decay or oscillation; poles in the right half plane indicate instability.
Engineering use
Root locus is used to choose controller gain, add lead or lag compensation, place dominant poles, check damping ratio, and understand why a system becomes unstable as feedback is increased. It is especially useful early in design because it shows the tradeoff between speed, damping, and robustness without requiring a full time-domain simulation for every gain value.
Zeros matter as much as poles. Adding a compensator zero can attract branches of the locus and improve damping. Adding a pole can slow the response or reduce phase margin. Time delay, actuator dynamics, filters, non-minimum-phase zeros, sampling, saturation, and sensor noise can all change whether a root-locus design works in the real system.
Design interpretation
The root locus is not only a drawing technique. It is a way to reason about how feedback changes the system dynamics. Desired pole regions may be drawn from overshoot, settling time, or bandwidth targets, but those targets assume a model structure and dominant-pole approximation. Final design still needs frequency-domain robustness checks, time-domain simulation, and validation against the physical plant.
Common mistakes
A common mistake is selecting a gain from the plot and ignoring actuator saturation, dead time, sample rate, unmodelled flexible modes, or measurement filtering. Another is assuming that moving dominant poles to a desirable location guarantees acceptable disturbance rejection and noise sensitivity. A strong root-locus review states the transfer function, feedback sign, gain range, pole-zero map, performance target, model limitations, and how the design was validated.