Before you can read the parent note the parent topic, you must own every letter it uses. This page builds them from absolute zero, in the order they depend on each other. Nothing is used before it is drawn.
The picture: a signal is a curve. The horizontal position tells you when; the height tells you how much.
Figure s01 — A single wiggly curve. Look at the magenta curve (the signal), then the violet dashed drop-line marking one instant t, and the orange dashed line reading off its height x(t). Takeaway: one dot on the curve = one number at one moment.
WHY we need this notation. Control is about quantities that evolve: the rocket's angle, the motor's position, the error. A single number like "5°" is a snapshot; x(t) is the whole movie. We must talk about the movie, so we need the (t).
The picture: stack the target line r and the measured curve ym; the vertical distance between them, at each instant, is e. Above the target, e is negative (overshoot); below, positive.
Figure s02 — Two curves plus vertical bars. Look at the flat violet line (target r), the rising magenta curve (measurement ym), and the orange vertical bars between them — each bar's length is the error e at that instant. Takeaway: error is the shrinking gap you are trying to close.
WHY subtraction and not, say, a ratio? Because a difference keeps its sign. A ratio of 2 can't tell you whether you're above or below — but e=−3 says "you overshot by 3, push back the other way." The sign is the steering direction. That is why the loop is built around a subtraction.
Each block takes a signal in and gives a signal out. Think of each as a machine: a signal enters the left, a changed signal leaves the right.
WHY split hardware from software? The controller C is just arithmetic — it can output any number. The actuator A is metal — it can only push so hard (this limit is called saturation, explored in Actuator Saturation and Anti-Windup). Confusing the two hides real failures, so the parent keeps them as separate letters.
Here is the tool question. Each block delays, smooths, or amplifies its input — described by a differential equation (rates of change). Chaining boxes means chaining derivatives, which is horrible algebra in time.
The picture:s lives on a 2-D map (the s-plane). Left half = decaying = safe. Right half = exploding = unstable. Exactly on the vertical middle line = neither — sustained oscillation. A signal's behaviour is decided by where its special points sit on this map.
Figure s03 — The s-plane map. Look at the violet-shaded left half (poles here = decaying, STABLE, drawn as violet ×'s), the magenta-shaded right half (poles here = growing, UNSTABLE), and the orange dots sitting exactly on the vertical axis (σ=0) — poles here neither grow nor decay, giving forever-ringing oscillation (marginal stability). Takeaway: horizontal position = stability, height = wobble speed.
WHY this tool and not just solving the ODEs? Because once we are in s, "signal goes through box C then box A then box P" is simply C(s)A(s)P(s) — plain multiplication. That is the only reason the parent can write G=CAP so casually. Full details live in Transfer Functions and Laplace Domain.
The top (numerator) roots are called zeros — values of s that make the output vanish.
The bottom (denominator) roots are called poles — values of s that make the box "blow up." Poles decide stability (see Poles Zeros and Stability).
WHY a fraction? Because a physical system responds strongly to some frequencies and weakly to others. A ratio of two polynomials is exactly the shape that captures "amplify here, ignore there." Example: the parent's motor P(s)=s(s+2)1 has poles at s=0 and s=−2.
Chaining the forward boxes gives G=CAP. Going all the way around the loop (forward through G, back through the sensor H(s)) multiplies them: this round-trip factor GH is the loop gain.
The picture: the signal loops around, each lap scaled by −GH, the laps alternately adding and cancelling until they settle. If ∣GH∣ is small, they die fast; if too big, they grow — instability.
Figure s04 — Bars plus a running total. Look at the orange bars (each lap's contribution (−GH)n — notice they flip sign and shrink), the magenta dotted line (the running sum as laps pile up), and the violet dashed line it settles onto, 1/(1+GH). Takeaway: the feedback subtraction, repeated, is exactly the geometric series that produces 1+GH.
When the loop's denominator is a quadratic s2+2ζωns+ωn2, two numbers describe everything about how it settles (see Second-order System Response).
WHY these two and not the raw coefficients? Because ζ and ωn map directly to what you feel: speed and wobble. They let the parent say "bigger K raises ωn but lowers ζ" — the eternal speed-vs-stability trade-off — in plain human terms. Note ζ=0 (no damping) sits exactly on the imaginary axis — the marginal case from Figure s03: forever ringing.
WHY this tool? We often want the steady-state error — how far off we are after everything settles — without simulating forever. Multiplying by s and letting s→0 (the "DC" corner of the s-plane) reads that final value straight off the transfer function. That is how the parent gets ess=1/(1+G(0))=3/8 without any time-plot.
