SC · Control

The Root Locus

As the gain K grows from zero to infinity, the closed-loop poles trace a path across the s-plane. That path is the root locus. The poles start at the open-loop poles and head toward the zeros (or infinity); following the path shows at a glance where the system goes unstable and at what K you reach the damping you want. Move K to land the poles on a target damping line.

Raise K to land poles on the target damping line

This is the same plant K/(s(s+1)(s+2)) as F1. The blue curves are the root locus the closed-loop poles trace, and the gold × marks are the poles at the current gain K. The amber dashed line is the target damping ratio ζ = 0.5. When K is small the poles are spread along the real axis, over-damped. As K grows, two poles meet on the real axis and split (the breakaway), curving up and down into a complex pair. Land that complex pole exactly on the amber ζ = 0.5 line. Raise it further and at K = 6 the poles cross the imaginary axis into instability.

s-plane (blue = root locus, × = current pole)

gain K0.30

Gain and dominant pole

K = 0.30 · s = -0.21 (real)

ζ = 1.00 → 0.50

The gain is small, so the poles are still on the real axis or barely damped. This is the over-damped region, slow with no oscillation. Raise K and two poles meet at the breakaway and split into a complex pair, getting faster.

Far from target damping

The path the poles trace

The closed-loop poles of a unit-feedback system are the roots of 1 + K·G(s) = 0, where G(s) = N(s)/D(s) is the open-loop transfer function. Rearranged, that is D(s) + K·N(s) = 0, so the roots change as the gain K varies. At K = 0 it is D(s) = 0, so the closed-loop poles start at the open-loop poles. As K goes to infinity, N(s) = 0 dominates and the poles converge to the open-loop zeros. When there are not enough zeros, the leftover branches run off to infinity. The whole trail the roots trace as K sweeps from zero to infinity is the root locus.

Observeas K: 0 → ∞, poles go to zeros

The locus starts at poles and ends at zeros.

Chooseclosed-loop poles solve 1 + K G(s) = ?

Closed-loop poles solve 1+KG=0.

Fill inlocus crosses jω at K = ?

The jω crossing is Routh’s K=6.

On your ownchoose K where locus meets the ?

Pick K where the target ζ line meets the locus.

How to read the path

The root locus can be sketched by hand from a few rules. On the real axis, the locus exists only where the number of open-loop poles and zeros to the right of a point is odd; in this example that is between 0 and −1, and to the left of −2. The two branches starting at 0 and −1 approach each other, meet at the breakaway point, leave the real axis, and become a complex pair. With no zeros, all three branches go to infinity, along asymptotes that make ±60° and 180° with the real axis. And at some gain the complex branch crosses the imaginary axis; from there it is unstable. That crossing gain and frequency were exactly K = 6, s = ±j√2 from the Routh test of F1. The root locus and Routh point to the same boundary.

Placing poles to design the response

The real use of the root locus is design. As we saw in D5, a second-order transient is set by the dominant pole position, that is the damping ratio ζ and natural frequency ωn. A straight line through the origin in the s-plane means a constant ζ (the cosine of its angle from the real axis is ζ). So you draw the line for the ζ you want, find where that line meets the root locus, and the gain K that produces that point is your design value. That is exactly what the first screen had you do. But a third, unseen pole moves along too, so for the dominant complex pair to lead the response you also check that the third pole is far enough to the left. When proportional gain alone is not enough, the compensators to come bend the locus itself into the shape you want.

Back to the first screen

We followed the path that the closed-loop poles of the same plant as F1 trace on the s-plane as the gain K grows. When K was small the poles were spread along the real axis, over-damped; raising K made two poles meet at the breakaway and split into a complex pair, curving up and down. Landing that complex pole on the amber ζ = 0.5 line gave the design gain that balances speed and overshoot. Raising it further took the poles across the imaginary axis at K = 6 into instability, exactly the boundary Routh gave in F1. Where Routh handed us a single boundary point as a number, the root locus showed the whole path the gain traces, letting us design not just stability but the shape of the response.

The root locus is the path the closed-loop poles (the roots of 1 + K·G(s) = 0) trace on the s-plane as the gain K runs 0→∞. They start at the open-loop poles at K=0 and go to the open-loop zeros at K→∞ (or to infinity along ±60°,180° asymptotes). The real-axis locus is where the count of poles and zeros to the right is odd; two branches meet and split at a breakaway point into a complex pair. The imaginary-axis crossing = the stability boundary, here the same K = 6, s = ±j√2 as F1’s Routh. A line through the origin is a constant damping ratio ζ (the cosine of its angle), so you read the design gain K where the target ζ line meets the locus.

On to the next unit

The root locus is a time-domain design tool that places poles directly on the s-plane. But real systems often have an unknown exact transfer function, and instead you probe them with sine waves at many frequencies and measure the response. The next unit, the Bode plot, draws the gain and phase versus frequency as a sum of straight lines on logarithmic axes. Each pole and zero shows up as one bend in the broken line, so you can sketch the frequency response of a complicated transfer function by hand and also read it back out of measured data. Where the root locus saw the response through pole positions, the Bode plot sees the same system through frequency.