Root Locus
The root locus plots how closed-loop poles move as a gain varies. Learn the sketching rules, asymptotes, breakaway points and jω crossings, and how to pick a gain for a desired damping ratio.
Changing the controller gain moves the closed-loop poles. The root locus is the path traced by every pole as K goes from 0 to ∞. From a single sketch you can see the gain range for stability, where the response becomes oscillatory, and which gain gives the damping you want.
Evans' rules let you sketch it by hand in minutes; software draws it exactly.
An analogy
The root locus is like a map of every route a car could take as you turn one steering knob — you pick the spot on the map that gives the ride you want, then read off the knob setting.
Start and end
Branches start (K = 0) at the open-loop poles and end (K → ∞) at the open-loop zeros or head to infinity along asymptotes. There are as many branches as poles, and the locus is symmetric about the real axis.
Real-axis segments and asymptotes
A real-axis point lies on the locus if the number of real poles and zeros to its right is odd. The (n − m) branches going to infinity follow straight asymptotes radiating from the centroid σ_a at angles (2q + 1)·180°/(n − m).
Breakaway and jω crossings
Where two branches meet on the real axis they break away into the complex plane; solve dK/ds = 0. Crossings of the imaginary axis mark the stability limit; find them from the Routh array (critical K and ω).
Design with the locus
Draw radial lines of constant ζ (angle arccos ζ from the negative real axis) and circles of constant ω_n. The intersection with the locus gives the gain for that damping. If the locus doesn't pass through the desired region, add a compensator: lead (zero–pole pair) pulls the locus left for speed and damping; lag improves steady-state error without moving the locus much.
Formulas
- Angle condition:
∠G(s)H(s) = ±180°(2q + 1) - Magnitude condition:
|K·G(s)H(s)| = 1 - Asymptotes:
θ_a = (2q + 1)·180° / (n − m), σ_a = (Σp − Σz)/(n − m) - Breakaway points:
dK/ds = 0
Worked example: Sketch rules for K/(s(s + 1)(s + 5))
Given: Poles 0, −1, −5; no zeros
Find: Asymptotes, real-axis segments, breakaway and jω crossing.
- n − m = 3 → asymptotes at 60°, 180°, 300°; centroid σ_a = (0 − 1 − 5)/3 = −2
- Real axis: [−1, 0] and (−∞, −5] are on the locus
- K = −(s³ + 6s² + 5s) → dK/ds = −(3s² + 12s + 5) = 0 → s = −0.47 (valid breakaway)
- jω crossing (Routh): K = 30, s = ±j2.24
Answer: Breakaway −0.47, asymptotes from −2 at ±60°/180°, crosses jω at K = 30
Worked example: Gain for ζ = 0.5
Given: G(s) = K/(s(s + 4)), unity feedback
Find: K for ζ = 0.5 and the resulting ω_n.
- Closed loop: s² + 4s + K → 2ζω_n = 4, ω_n² = K
- ζ = 2/√K = 0.5 → K = 16
- ω_n = 4 rad/s, poles −2 ± j3.46, overshoot 16 %
Answer: K = 16
Common mistakes
- Wrong: Forgetting asymptotes when n > m. — Right: n − m branches go to infinity along asymptotes; they often cross into the RHP.
- Wrong: Reporting every dK/ds = 0 root as a breakaway. — Right: Only roots lying on valid real-axis segments are breakaway points.
Applications
- Servo gain selection: Engineers read the gain for a target ζ directly off the locus.
- Compensator design: Lead/lag networks are placed to reshape the locus through desired pole locations.
- Sensitivity insight: The locus shows how poles move if a parameter (not only gain) drifts — useful for robustness checks.
Key takeaways
- The locus shows closed-loop poles for all gains.
- Starts at poles, ends at zeros or asymptotes.
- Breakaway from dK/ds = 0; jω crossing from Routh.
- Pick K where the locus meets the desired ζ line; reshape with lead/lag.
Going further
- Angle of departure: From complex poles, branches leave at an angle found from the angle condition, which matters for lightly damped plants such as flexible structures.
- Complementary locus: For positive feedback (or negative K) the angle condition becomes 0°, giving the complementary root locus — useful when a parameter can have either sign.
- Root contours: Varying two parameters produces families of loci (root contours), revealing how, say, both gain and a time constant affect pole positions.