Root locus: follow the poles
Root locus shows where closed-loop poles move as a gain changes.
Trace two poles as a positive gain changes, without assuming every gain increase improves the response.
System type predicts a class of tracking error. To choose gain while watching transient stability, we follow the closed-loop poles.
Review the previous lesson: Count integrators, then check the errorFirst, picture it
Instead of testing a thousand gain values in time simulations, trace every possible closed-loop pole position as K sweeps from zero upward. Crossing into the right half-plane warns of instability.
Build the idea step by step
A root locus is a map of closed-loop poles as one gain changes. Fix the plant and the feedback connection, then increase the positive gain K. For each K, solve the same characteristic equation. Plotting those solutions shows when poles split into an oscillating pair or cross into the unstable right half-plane. It also shows why increasing gain does not always mean faster settling. The worked quadratic lets us draw this idea without using the full construction rules.
Read the equations
For unity negative feedback, this characteristic equation gives closed-loop pole positions as K changes.
What the symbols mean
- adjustable positive loop gain
- fixed part of the open-loop transfer function
- complex frequency, in inverse seconds
This locus varies one scalar gain while keeping the rest of the linear loop fixed. A voltage limit is not included in the pole calculation. Compare the predicted motion with applied input in the lab.
As K rises from 0, poles at 0 and −2 meet at −1, then become −1 ± j√(K−1). This example never crosses into the right half-plane for K>0.
Two poles become an oscillating pair
Take a normalized open-loop plant with poles at zero and minus 2.
The closed-loop equation is a quadratic. At gain 1, it has a repeated pole at minus 1.
At gain 4, the poles have real part minus 1 and imaginary parts plus and minus square root of 3. Decay remains, but oscillation appears.
More detail and model assumptions
A curve of closed-loop poles
Write the loop as L(s)=KG₀(s). For every chosen K, solve 1+KG₀(s)=0 and mark its closed-loop poles. Connecting those locations as K varies makes the root locus. It is not a plot of motor speed against time, and each point belongs to a particular gain.
For G₀=1/[s(s+2)], the equation is s²+2s+K=0. The poles start at 0 and −2, meet at −1 when K=1, then become −1±j√(K−1). Their real part remains −1 for K>1, so this example grows more oscillatory without crossing into the right half-plane.
Use the right stability boundary
For a continuous-time model, a closed-loop pole crossing the imaginary axis into the right half-plane signals an unstable linear mode. A sampled controller instead has poles in the z-plane, where the boundary is the unit circle |z|=1. The Lab frequency workbench uses this discrete boundary; do not read it as an s-plane plot.
The Lab's root locus sweeps a proportional controller only. If I, D, lead or lag is active, the current time run has another characteristic equation. Compare the locus with a time trace only under the same controller and nominal model assumptions.
Poles: -2.52, -0.24 − 0.86j, -0.24 + 0.86j. stable: all poles in the left half-plane. Damping ratio of the complex pair ≈ 0.27. For this loop the pair crosses the imaginary axis at K = 6, ω = √2 rad/s. × marks the open-loop poles.
Connect the example to the lab
In the motor lab, raising Kp can first speed response, then make control effort and oscillation worse. Root locus explains the linear-model pole movement behind that trend.
Open the motor's discrete locusMake a prediction before moving a control.
- PREDICTCan larger K make a system more oscillatory without making it unstable?
- CHANGE ONE THINGSolve s²+2s+K=0 at K=.25, 1 and 4.
- OBSERVEPlot the real and imaginary parts, then compare with the chapter sketch.
- EXPLAINSeparate a claim about oscillation from a claim about stability.
A root-locus plot is not the time response itself; infer a trend from poles, then verify with the actual model and limits.
Check your understanding+
What happens when a closed-loop pole crosses into the right half-plane?
Its corresponding mode grows exponentially; the linear closed loop becomes unstable.Why the next lesson follows
A single gain moves all frequencies together. Lead and lag networks shape different frequency ranges differently.