Root locus formulas

Negative feedback, with $G(s)H(s)=N(s)/D(s)$ in monic form. The ordinary root locus is for $K\ge 0$.

A point $s$ that is not itself an open-loop pole or zero is on the root locus ($K\ge 0$) if and only if the net angle is an odd multiple of $180^{\circ }$. $K$ does not appear, so this condition alone decides the shape of the locus.

Use this only at a point that already satisfies the angle condition. It is the product of the distances to all poles divided by the product of the distances to all zeros. If $m=0$ the denominator is $1$.

For $K\ge 0$. Angles are measured counter-clockwise from the positive real axis. There are exactly $n-m$ asymptotes.

Use the signed values: a pole at $-3$ contributes $-3$. The imaginary parts of conjugate pairs cancel, so ${\sigma }_A$ is always real. You only need it when $n-m\ge 2$. When $n-m=1$, the single asymptote is the negative real axis.

The roots are only candidates. Keep a root only if it lies on the locus: a real root must sit on a valid real-axis segment, and $K$ must come out real and positive.

For a pole $p_k$ that is not repeated. Use the angles of the vectors drawn from every other pole and from every zero to $p_k$. Add or subtract $360^{\circ }$ to bring the result into $(-180^{\circ },180^{\circ }]$. The conjugate pole departs at $-{\varphi }_d$.

For a zero $z_k$ that is not repeated. Use the angles of the vectors drawn from every other zero and from every pole to $z_k$. The conjugate zero has arrival angle $-{\varphi }_a$.

$A(s)$ is the auxiliary polynomial from the row just above the zero row (usually the $s^2$ row), evaluated at $K_{mar}>0$. $b$ usually depends on $K$, so substitute $K_{mar}$ before taking the square root. This is a crossing only if $b/a>0$ at $K_{mar}$. If $b/a<0$, the zero row signals a pair of real roots $\pm \sqrt{-b/a}$ instead.

When $n-m\ge 2$, $K$ does not appear in the $s^{n-1}$ coefficient of $D(s)+KN(s)$, so the closed-loop poles $r_i$ always add up to the sum of the open-loop poles. Use it to find the remaining real pole at the $j\omega$ crossing without dividing polynomials.

For $0<\zeta <1$. Closed-loop poles with damping ratio $\zeta$ lie on the two radial lines from the origin at angle $\theta$ above and below the negative real axis. The gain for a required $\zeta$ is $K$, from the magnitude condition, at the point where this line cuts the locus. If the line cuts the locus more than once, each crossing gives a valid gain.

Valid when the characteristic polynomial is linear in the parameter $\alpha \ge 0$. Apply every rule to $P(s)/Q(s)$ with $\alpha$ in place of $K$. The poles of this equivalent loop are the roots of $Q(s)$, i.e. the closed-loop poles at $\alpha =0$. They are generally not the open-loop poles of the physical system. If $P(s)$ or $Q(s)$ is not monic, move the leading coefficients into the gain first.

Applies to $K\le 0$ with negative feedback, or to $K\ge 0$ with positive feedback.

The centroid formula does not change. When $n>m$ there is always an asymptote at $0^{\circ }$, i.e. along the positive real axis.