Foundations — Root locus — Evans' method, rules for sketching

This page builds every symbol the parent note used, from absolute zero, in the order you need them. Nothing later depends on something not yet built.

1. The complex plane — the map everything lives on

Before we talk about poles moving, we need the place they move in.

Why do we need this? A real number is a point on a line — a 1D thing. But the answers to the equations in this topic (the poles) can be off the line: they oscillate. To store both "how fast it decays" ($\sigma$) and "how fast it wiggles" ($\omega$) you need two numbers, so you need a 2D map. That map is the complex plane.

A complex number also has two other descriptions we will need constantly:

Why these two? Because the parent's whole method — the "angle condition" and "magnitude condition" — is nothing but reading off these two numbers for a special quantity. We meet them again in §6.

2. Functions of $s$: $N(s)$, $D(s)$, and the ratio $G(s)H(s)$

Why a ratio and not just one polynomial? Because two different things break a system: the recipe blowing up ($D=0$) and the recipe vanishing ($N=0$). Keeping numerator and denominator separate lets us talk about each. Those two events have names — poles and zeros — coming next.

3. Poles and zeros — the pins the locus is stretched between

Why do we care so much about poles? A pole's location is the system's behaviour:

• Pole far left → response dies out fast (stable, snappy).
• Pole near the middle line → response dies slowly.
• Pole with a nonzero height $\omega$ → response oscillates at that frequency.
• Pole on the right → response grows → machine shakes apart.

So "where are the poles?" answers "is my drone stable and how does it fly?" — the entire job of GNC control loops.

4. Feedback and the gain knob $K$

Why a single knob and not many? Because for one knob the pole-paths are curves (1D), which you can draw by hand. Evans' whole method is built around this one-parameter sweep. See PID / PD controller design for how extra knobs add zeros to reshape those curves.

5. The characteristic equation — the master equation

Everything the parent does flows from one line. Here is where it comes from, built slowly.

The closed-loop transfer function (output over input, with feedback wired up) is

The right-hand form comes from multiplying through by $D(s)$: since $GH=N/D$,

This is the polynomial whose roots are the closed-loop poles — the thing Characteristic equation & closed-loop poles is all about. As $K$ varies, its roots trace the locus.

6. Reading $L(s)=-1$ as two conditions

The characteristic equation rearranges to $KG(s)H(s)=-1$. Now $-1$ is itself a complex number — go back to §1's picture and find it: it sits one unit to the left of centre. That means:

• its magnitude is $1$ (distance from centre),
• its angle is $180°$ (pointing straight left) — or $180°$ plus any full turn, i.e. $\pm 180°(2k+1)$.

For a point $s$ to be a pole, $L(s)$ must equal $-1$, so its magnitude and angle must match:

How do you compute an angle to a point? Draw an arrow from each pole and each zero to your test point $s$. Measure each arrow's angle. Add the zero-angles, subtract the pole-angles — that sum is $\angle G(s)H(s)$. Every sketching rule (real-axis test, departure angle, asymptotes) is just this arrow-arithmetic in disguise.

7. Two tools the parent borrows

8. How it all feeds the topic

Read it top-down: the map (§1) hosts poles/zeros (§3), which build the transfer function (§2). Feedback (§4) plus that function give the master equation (§5). Splitting $-1$ into size and direction (§6) yields the two conditions, and those — helped by the derivative and Routh (§7) — power every sketching rule in the parent note.

Equipment checklist

Connections

• Root locus — Evans' method, rules for sketching — the parent this page equips you for
• Characteristic equation & closed-loop poles
• Routh–Hurwitz stability criterion
• PID / PD controller design
• Bode plot & frequency response
• Nyquist criterion
• Damping ratio and settling time
• GNC control loops