Root locus analysis is a graphical way to see how a feedback system’s closed-loop poles move as a system parameter—usually gain K—changes. Because pole locations determine the growth, decay, and oscillation of system modes, the root locus helps engineers assess stability and anticipate system behavior.
A linear time-invariant (LTI) feedback system has open-loop poles and open-loop zeros. Its closed-loop poles are the roots of the closed-loop characteristic equation. When a parameter such as gain K varies, those roots generally move through the complex s-plane. The collection of their possible positions is the root locus.
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What is a root locus?
A graphical representation of the possible closed-loop pole locations as a selected system parameter, commonly gain K, varies.
What does the real part of a continuous-time pole tell you?
It determines the exponential growth or decay of the associated mode. A negative real part means decay, a positive real part means growth, and a zero real part means neither exponential growth nor decay.
What pole condition gives asymptotic stability in continuous time?
Every closed-loop pole must lie strictly in the left half-plane, so every pole has a negative real part.
How can root locus support damping and frequency design?
A designer can identify where a locus branch intersects a desired constant-damping-ratio line and natural-frequency circle, then find the gain for that pole location.
What is the dominant-pole approximation?
It approximates a system’s response using its slowest-decaying poles when other poles decay much faster. It is a heuristic and should be validated against the full model.
Quiz questions
Under the usual positive-gain convention, what phase must the loop transfer function have at a point on the root locus?Answer: An odd multiple of 180∘.
For a candidate point on a root locus, what does the magnitude condition specify?Answer: The gain that makes the loop-transfer-function magnitude equal to one.
If a system has P open-loop poles and Z open-loop zeros, with P>Z, how many root-locus branches extend toward infinity?Answer: P−Z branches.
In the s-plane, how are constant-damping-ratio lines oriented?Answer: They extend from the origin at an angle measured from the negative real axis.
Where must every closed-loop pole lie for a discrete-time system to be asymptotically stable?Answer: Inside the unit circle in the z-plane.