The root locus is a graphical method for showing how the closed-loop poles of a feedback control system move in the s-plane as the gain K changes. It matters because pole locations determine whether a system is stable and how it responds to inputs or disturbances. Engineers use the plot to connect algebraic transfer functions with visible design choices.
A well drawn root locus quickly shows which gains give fast, slow, oscillatory, or unstable behavior.
For a unity feedback system with open-loop transfer function G(s), the closed-loop poles satisfy 1 + K G(s) = 0. As K increases from 0 to infinity, each branch of the root locus starts at an open-loop pole and moves toward an open-loop zero or toward infinity along an asymptote. Points in the left half of the s-plane usually represent stable decaying motion, while points in the right half represent growing unstable motion.
By adding poles, zeros, or choosing K, a controller designer can reshape the locus to meet stability, damping, and settling time goals.
Key Facts
- Closed-loop characteristic equation for unity feedback: 1 + K G(s) = 0.
- Root locus branches show the locations of closed-loop poles as K varies from 0 to infinity.
- Each branch starts at an open-loop pole when K = 0.
- Each branch ends at an open-loop zero or goes to infinity if there are more poles than zeros.
- A continuous-time system is stable if all closed-loop poles have negative real parts.
- For a complex pole s = σ + jω, the decay rate is set by σ and the oscillation frequency is set by ω.
Vocabulary
- Root locus
- A plot of the paths followed by closed-loop poles in the complex s-plane as a system gain changes.
- Closed-loop pole
- A root of the closed-loop characteristic equation that determines a mode of the feedback system response.
- s-plane
- The complex plane used in control engineering, with real part σ on the horizontal axis and imaginary part jω on the vertical axis.
- Gain K
- A multiplier in the loop transfer function that changes pole locations and therefore changes the system response.
- Stability boundary
- The imaginary axis in the continuous-time s-plane, separating stable left-half-plane poles from unstable right-half-plane poles.
Common Mistakes to Avoid
- Thinking the root locus is a time graph, which is wrong because it plots pole locations in the complex s-plane rather than output versus time.
- Ignoring the imaginary axis stability boundary, which is wrong because crossing into the right half-plane means the continuous-time closed-loop system becomes unstable.
- Assuming larger K always improves performance, which is wrong because increasing gain can move poles toward low damping or instability.
- Confusing open-loop poles with closed-loop poles, which is wrong because the root locus starts at open-loop poles but represents closed-loop pole locations for different gains.
Practice Questions
- 1 A unity feedback system has G(s) = 1/(s(s + 4)). Write the closed-loop characteristic equation for gain K and find the range of K for stable closed-loop poles.
- 2 For a closed-loop pole at s = -3 + j4, find the damping ratio ζ and natural frequency ωn. Use ωn = sqrt(σ^2 + ω^2) and ζ = -σ/ωn.
- 3 A root locus branch moves closer to the imaginary axis as K increases. Explain what this suggests about settling time, oscillation, and stability margin.