🌀 Root Locus: How Poles Move as Gain Changes

The root locus is a plot, in the complex s-plane, of every root of the closed-loop characteristic equation 1 + K·G(s)H(s) = 0 as the real, non-negative gain K varies continuously from 0 to infinity. It was introduced by Walter R. Evans in 1948 specifically so control engineers could see how feedback gain affects closed-loop pole locations — and therefore stability and transient response — without having to factor a new polynomial by hand for every candidate gain. This simulation performs that same exact root-solving numerically at every K instead of by hand.

At K = 0, the characteristic equation 1 + K·G(s)H(s) = 0 reduces to just the denominator of G(s)H(s) equal to zero — so the closed-loop poles are exactly the open-loop poles. As K → ∞, the equation is dominated by the numerator, so roots approach the open-loop zeros. If there are fewer zeros than poles, the "extra" branches have nowhere finite to go and instead diverge to infinity along straight-line asymptotes, whose angles are (2k+1)·180°/(n−m) and whose common origin (the centroid) is (Σpoles − Σzeros)/(n−m), where n is the pole count and m the zero count.

A linear system is stable if and only if every closed-loop pole has strictly negative real part — i.e. sits in the left half of the s-plane, shaded white here versus the red-tinted right half. As K increases along a branch, the moment that branch's real part crosses zero into the shaded region is the critical gain K_crit: for K below it the system is stable, for K above it that branch has gone unstable. This simulation finds K_crit by scanning the computed locus for a sign change in a branch's real part, then refining it with bisection on the exact numerically tracked root.

Yes. From your placed poles and zeros it builds the monic denominator polynomial Den(s) and numerator polynomial Num(s), forms the exact characteristic polynomial Den(s) + K·Num(s) for a swept sequence of K values, and finds all of that polynomial's complex roots simultaneously using the Durand–Kerner iteration — a genuine numerical polynomial root-finding method. Consecutive K samples are warm-started from the previous roots and matched by nearest distance so each branch stays continuous, exactly the same continuity a hand-drawn locus assumes.

If any open-loop pole already sits in the right half-plane (positive real part) and no zero or nearby pole reshapes the locus fast enough, the corresponding branch can remain in the right half-plane for small K, or a branch can cross into it very early. The "Complex pair" preset in this simulation loads poles at 1 ± 2j — already unstable in open loop — so you can see how much (or how little) feedback gain is needed, if any, to pull that branch back into the stable region.

Control engineers use the root locus to choose a gain K that places closed-loop poles where they want them — for example, far enough left for a fast response, or with a damping ratio that limits overshoot. It also motivates compensator design: adding a zero (as in a lead compensator) bends branches toward the left half-plane and can stabilize a marginally stable plant, while adding a pole (lag compensation) does the opposite locally but can improve steady-state accuracy. This is why nearly every classical control-systems course builds the root locus by hand before ever touching a computer.