control//closed-loop stability//root locus

The root locus is the plot of the paths that the closed-loop poles of a feedback loop follow in the complex plane as one gain is raised from zero to infinity, and it is the graphical tool for choosing that gain: it shows at a glance where the loop becomes faster, where it starts to ring and where it goes unstable. Each point on the paths is a root of the characteristic polynomial \(1+K\,G(s)=0\) for some value of \(K\), with \(G\) the open-loop transfer function.

The root locus is the plot of the paths that the closed-loop poles of a feedback loop follow in the complex plane as one gain is raised from zero to infinity, and it is the graphical tool for choosing that gain: it shows at a glance where the loop becomes faster, where it starts to ring and where it goes unstable. Each point on the paths is a root of the characteristic polynomial 1+K G(s)=01+K,G(s)=01+KG(s)=0 for some value of KKK, with GGG the open-loop transfer function.

The paths obey simple rules. They start at the open-loop poles (where K=0K=0K=0) and end at the open-loop zeros or run off to infinity along fixed asymptotes; a stretch of the real axis belongs to the locus when an odd number of poles and zeros lies to its right. Take a motor position loop with an integrator and a motor lag, G(s)=1/(s(s+10))G(s)=1/\big(s(s+10)\big)G(s)=1/(s(s+10)). At low gain both poles are real and the loop creeps; at K=25K=25K=25 they meet at −5-5−5, the critically damped point; above it they leave the axis as a complex pair whose real part stays at −5-5−5 and whose imaginary part grows, so the loop rings faster and faster with the same decay. Add a third pole (a sensor filter at −20-20−20) and two branches bend to the right and cross the imaginary axis: there is now a gain beyond which the loop oscillates and then diverges.

The locus shows what one knob can and cannot do.

If no point on the paths satisfies the specification (a damping ratio, a settling time), no value of that gain will, and the controller needs another structure: a zero placed by derivative or lead action pulls the branches left, a pole close to the origin from integral action pulls them right.

It and the frequency methods answer the same question from two sides. The root locus works with the poles and needs a model in polynomial form; the Nyquist stability criterion and the stability margins work with the frequency response, can be measured on hardware and handle a pure delay, which has no finite polynomial. Where the locus crosses the imaginary axis is where the gain margin runs out.

It is drawn by software today (control.root_locus in python-control, rlocus in MATLAB), and its value is the reading: which pole pair dominates, which branch limits the gain, what one added filter does to the loop.

It varies only one parameter. For two gains at once, or for a mass that changes with payload, the same reasoning runs on the characteristic polynomial with the Routh-Hurwitz criterion or with a sweep; choosing every pole at once is pole placement.