Control Systems · 96 h · Topic 6 of 8

The root locus

Three poles with no zeros leave two branches heading into the right half-plane at 60°. A well-placed zero turns them vertical, and the loop is stable for any gain.

Angle condition Asymptotes Breakaway point Critical gain Lead network

01The loop poles as a function of the gain

The Routh criterion tells whether the loop is stable for one gain. The root locus, proposed by Walter Evans in 1948, shows how all the closed-loop poles move as the gain \( K \) goes from zero to infinity. They are the points of the s-plane that satisfy the characteristic equation \( 1 + K\,G(s) = 0 \), that is, two conditions:

\[ |K\,G(s)| = 1 \] \[ \angle G(s) = \pm 180° (2k + 1) \] The angle condition defines the locus; the magnitude condition says which gain corresponds to each point. With the locus in view, design consists of choosing K, or modifying G with a compensator, so that the dominant poles fall in the region required by the transient response specifications.

02Construction rules

Evans gave rules for drawing it by hand, which remain the best way to understand its shape:

It starts at the open-loop poles (K = 0) and ends at the zeros (K → ∞); if there are more poles than zeros, the remaining branches go to infinity.
On the real axis, the segments that have an odd number of poles and zeros to their right belong to the locus.
The branches that go to infinity follow asymptotes that leave the centroid at fixed angles:

\[ \sigma_a = \frac{\sum p_i - \sum z_j}{n - m} \] \[ \theta_a = \frac{180° (2k + 1)}{n - m} \] \( n \) poles and \( m \) zeros. With three poles and no zeros, the asymptotes make 60°, 180° and 300°: two branches cross into the right half-plane, and that is why the loop becomes unstable at high gain.

Two branches that meet on the real axis separate at the breakaway point, where \( dK/ds = 0 \).
The crossing of the imaginary axis is obtained with Routh: it is the critical gain.

Lab · the root locus

The locus of \( K\,G(s) \) with \( G(s) = 1/(s(s+a)(s+b)) \), with the option of adding a zero at \( -z \). The light-blue dots are the locus for K between 0.01 and 1000; the large ones, the closed-loop poles for the chosen K; the crosses, the open-loop poles, and the circle, the zero.

a =

b =

Gain K =

03Shaping the locus with poles and zeros

A zero attracts the branches: added to the left of the dominant poles, it bends the locus to the left, moves the asymptotes away from the imaginary axis and allows more gain with more damping. It is the effect of derivative action and of the lead network, which adds a zero and a farther pole. A pole added near the origin, like integral action or the lag network, pushes the branches to the right: it improves the steady-state error at the cost of stability, and so it is accompanied by a very close zero, so that the locus barely changes.

04In the lab

Activity 1 · Locus with software

Plot the root locus of the lab loop with the Python control library and verify the breakaway point and the critical gain calculated by hand.

Activity 2 · Design by pole placement

Choose the gain that gives a damping ratio of 0.5 in the dominant poles and verify the overshoot with a simulation.

Activity 3 · Lead zero

Add a zero to the loop built with op amps and measure how the critical gain and the overshoot change.

05Common mistakes

Forgetting that the locus is of the closed loop. The poles that move are those of \( 1 + KG = 0 \), not those of G.
Counting the real-axis segments wrongly. Poles and zeros are counted to the right of the point, not to the left.
Using the locus for a parameter that does not multiply the whole gain. The equation has to be rewritten so that it appears as a factor.
Ignoring the imperfect cancellation of a pole with a zero. It leaves a slow response of low amplitude.

06Self-assessment

Where is the centroid of the asymptotes of K/(s(s+1)(s+5))?

\( (0 - 1 - 5)/3 = -2 \).

What segments of the real axis belong to that locus?

From 0 to −1 and from −5 to −∞.

Where do the branches of the segment from 0 to −1 separate?

At s ≈ −0.47, with K ≈ 1.13.

What angles do the asymptotes have if a zero is added?

±90°: two branches go to infinity.

07Further reading

Katsuhiko Ogata. Modern Control Engineering. 5th ed., Prentice Hall, 2010 (Spanish edition: “Ingeniería de control moderna”, Pearson). The root locus rules and the design of lead and lag compensators.
Walter R. Evans. Control-System Dynamics. McGraw-Hill, 1954. The book by the creator of the method.
Gene F. Franklin, J. David Powell and Abbas Emami-Naeini. Feedback Control of Dynamic Systems. 8th ed., Pearson, 2019. Root locus design explained with real examples.

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