S4-3.3 Stability & Root Locus

What this is and why it exists

A feedback loop can correct — or it can chase its own tail and oscillate to destruction. This unit holds the two classical answers to "will it?". Routh-Hurwitz answers yes-or-no from the polynomial's coefficients, no root-solving needed. Root locus answers the better question — how stability changes as you turn the gain knob — by drawing the paths every closed-loop pole follows. The lasting possession is the map from pole position to behaviour; the sketching rules are how you read that map for any proposed gain.

The vocabulary

• Characteristic equation — one plus the loop transfer function set to zero; its roots are the closed-loop poles.
• Routh-Hurwitz criterion — a tabular test on the characteristic polynomial's coefficients; sign changes in the first column count right-half-plane roots.
• Root locus — the set of paths the closed-loop poles trace as loop gain runs from zero to infinity.
• Asymptotes, breakaway points, departure angles — the sketching vocabulary: where locus paths head at high gain, where they leave the real axis, how they exit complex poles.
• jω-axis crossing — the gain at which a locus path crosses into the right half-plane: the stability boundary, findable by Routh or by substitution.

The mental model

The s-plane is a landscape of behaviours. A pole deep in the left half-plane is a fast, dying transient. A complex pair near the jω-axis is a slow ring — the closer, the longer it rings. Anything in the right half-plane grows: instability. Every analysis in this unit is locating poles on that landscape.

Routh-Hurwitz is the border guard. Build the array from the polynomial's coefficients; each sign change down the first column is one root across the border. The special cases are mechanical once seen — a lone zero in the first column takes a small-epsilon substitution; an all-zero row signals roots placed symmetrically about the origin, often a pure oscillation, handled through the auxiliary polynomial. Beyond yes-or-no, Routh with a symbolic gain K yields the exact range of K for stability — its most useful trick.

Root locus animates the picture. Closing a loop moves the poles, and as gain rises they move continuously: the locus is every position visited. The sketching rules are consequences of one angle condition, and a handful of them draw the whole figure: paths start at open-loop poles and end at open-loop zeros or along asymptotes; the real axis belongs to the locus wherever an odd count of poles and zeros lies to the right; asymptote angles and centroid come from small formulas; breakaway points and jω crossings complete the sketch. A minute's drawing then answers the design questions at sight — what gain rings, what gain destabilises, whether added damping is possible at all. Software plots the locus exactly; the hand sketch is what lets you argue with the software.

What you should now be able to explain or do

Run a Routh array, both special cases included, and count unstable roots. Extract the stabilising range of K with Routh. Sketch a root locus from the rules and find its jω crossing. Translate any pole position into expected time-domain behaviour in one sentence.

Check yourself