SC · Control

The Root Locus

As the gain K grows from 0 to infinity, the closed-loop poles trace a path across the s-plane, and that path is the root locus. The poles start at the open-loop poles and head for the open-loop zeros (or infinity when there are no zeros), and following that path shows at a glance where the system becomes unstable and at what K you get the damping you want. In this lesson you move K to set the poles on a target damping line and follow that process yourself.

Raise K to land poles on the target damping line

This is the same plant as in the F1 unit, K/(s(s + 1)(s + 2)). In the upper plot, the three gray dots (0, −1, −2) are the open-loop poles, the starting points at K = 0; the blue line is the root locus traced by the closed-loop poles; and the gold × marks are the three poles at the current gain K. The orange dashed line is the line of damping ratio ζ = 0.5, which makes 60° with the negative real axis. At the starting K = 0.3, all three poles are on the real axis, overdamped; as you raise K, the two poles that started at 0 and −1 meet at a breakaway point near −0.42 (K ≈ 0.385) and then split into a complex pair that curves up and down. That complex pair reaches the orange line at K ≈ 1.04, and if you raise K further, it crosses the imaginary axis at K = 6 and the system becomes unstable.

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 even as a complex pair the damping ratio is large (ζ > 0.65) and they barely do more than decay. This is a region where the response is slow and has almost no oscillation. Raise K and the two poles meet at the breakaway point and split into a complex pair, making the response faster.

Far from target damping

The path the poles trace

The closed-loop poles of a unity-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, this is D(s) + K·N(s) = 0, so the roots change every time the gain K changes. At K = 0, 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 gather at the open-loop zeros. When there are not enough zeros, the leftover branches stretch out to infinity. The whole set of traces the roots draw as K sweeps from 0 to infinity is the root locus.

Observeas K: 0 → ∞, poles go to zeros

The locus starts at the open-loop poles and ends at the zeros (or infinity).

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

The closed-loop poles are the roots of 1 + K·G(s) = 0.

Fill inlocus crosses jω at K = ?

The gain where the locus meets the imaginary axis is the Routh test's K = 6.

On your ownchoose K where locus meets the ?

Choose K where the desired ζ line meets the locus.

How to read the path

With just a few rules, a root locus can be sketched by hand. On the real axis, the locus lies only on segments where the number of open-loop poles and zeros to the right of the point is odd; in this example that means 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 (about −0.42), and leave the real axis as a complex pair. Since there are no zeros, all three branches go to infinity, along asymptotes that start from −1 on the real axis and make ±60° and 180° with the real axis. If the complex branches cross the imaginary axis at some K, the system is unstable from there on, and that crossing gain and frequency are exactly the K = 6, s = ±j√2 found with the Routh test in the F1 unit. The root locus and the Routh test point to the same boundary.

Placing poles to design the response

The real use of the root locus is design. As the D5 unit showed, the transient response of a second-order system is set by the location of the dominant poles, that is, by the damping ratio ζ and the natural frequency ωn. A straight line through the origin of the s-plane stands for a constant ζ: the cosine of the angle that line makes with the negative real axis is ζ. So draw the line for the ζ you want and find where it meets the root locus; the gain K that puts the poles at that point is the design value, which is exactly what you did on the first screen. Because a third pole moves along in this system, you also check that it lies far enough to the left for the dominant complex poles to lead the response (at K ≈ 1.04, the third pole at −2.33 is about 7 times as far left as the complex poles' real part, −0.33). If proportional gain alone is not enough, the compensators that come later bend the locus itself into the shape you want.

Back to the first screen

On the first screen, you followed the path that the closed-loop poles of the same plant as in the F1 unit trace across the s-plane as the gain K grows. With small K the poles were scattered on the real axis, overdamped; raising K made two poles meet at the breakaway point and split into a complex pair curving up and down. Setting that complex pair on the orange ζ = 0.5 line gave a design gain that balances speed and overshoot. Raising K further made the poles cross the imaginary axis at K = 6 and become unstable, which was exactly the boundary the F1 unit found with the Routh test. If the Routh test pinned one point of the stability boundary in numbers, the root locus laid out the whole path the gain traces, so you could design not only stability but also the shape of the response.

The root locus is the path that the closed-loop poles (the roots of 1 + K·G(s) = 0) trace across the s-plane as the gain K grows from 0 → ∞. They start at the open-loop poles when K = 0 and go to the open-loop zeros as K → ∞ (to infinity when there are no zeros, here along ±60° and 180° asymptotes). On the real axis the locus lies on segments with an odd number of poles and zeros to the right, and it turns complex at the breakaway point where two branches meet and split. Crossing the imaginary axis = the instability boundary, which in this example is K = 6, s = ±j√2, the same as the Routh test of the F1 unit. A line through the origin is a constant damping ratio ζ (the cosine of its angle with the negative real axis), so you read the design gain K where the desired ζ line meets the locus.

On to the next unit

The root locus was a design tool that places poles directly on the s-plane. In real systems, however, the exact transfer function is often unknown, so instead you feed in sine waves of various frequencies and measure the frequency response. The next unit, Gain and Phase Margins, reads, on the open-loop Bode plot of the same plant, how far a stable system is from instability, in the form of two margins. At the frequency where the phase reaches −180°, √2 rad/s, the amount the gain still has before reaching 0 dB is the gain margin, and since that margin becomes 0 at K = 6, it is the same point where the root locus crosses the imaginary axis. The boundary the root locus showed through pole locations appears again, on the frequency axis, in the Bode plot.