Foundations — Transfer function — Laplace domain, poles and zeros

This page assumes nothing. Before you can read the parent note Transfer function, every symbol it throws at you must first be earned. We build them one at a time, each one leaning on the one before.

0. What a "signal" and a "system" even are

Before any symbol, two words.

• A signal is just a number that changes over time. The angle of a rocket, the voltage on a wire, the height of a swing — each is a value you could plot as a wiggly line against a clock.
• A system (or machine) is a rule that turns one signal into another. You feed in an input signal, and out comes an output signal.

The whole topic is one question: given $u(t)$, what is $y(t)$?

1. The derivative $\frac{d}{dt}$ — the "rate of change" symbol

The picture: stand on the graph of $y(t)$ at one instant and lay a tiny straight ruler along the curve. Its tilt is $\dot{y}$. A steep upward line → big positive $\dot{y}$; a flat line → $\dot{y}=0$; a downhill line → negative $\dot{y}$.

Why the topic needs it: real machines are described by how their quantities change, not by their raw value. "The harder you push, the faster the velocity changes" is a sentence about $\ddot{}$ and $\dot{}$. These sentences are differential equations, next.

2. A differential equation — the machine's rule book

The picture: a mass on a spring with a damper (a shock absorber). $\ddot{y}$ is its acceleration, $3\dot{y}$ is the drag resisting motion, $2y$ is the spring pulling back, and $u(t)$ is the hand pushing it.

Why the topic needs it: every GNC loop — autopilot, attitude control — is one of these. Solving them directly for each new command $u(t)$ is slow and painful. The rest of this topic is a shortcut that turns this calculus into algebra.

3. The exponential $e^{something}$ — the shape everything is built from

The picture:

• If $a>0$: a curve that grows faster and faster (runaway).
• If $a<0$: a curve that decays smoothly toward zero (calming down).
• If $a=0$: a flat line (nothing happens).

Why the topic needs it: the parent note claims each pole gives a response "$\propto e^{pt}$". You now know exactly what that curve looks like: growing if the number in the exponent is positive, decaying if negative. Stability will just be "which way does the exponential go".

4. Imaginary and complex numbers — $j$, and $\sigma +j\omega$

The picture: don't think "spooky number", think point on a flat map. The horizontal axis is the real part $\sigma$; the vertical axis is the imaginary part $\omega$. So $\sigma +j\omega$ is just an address on a 2-D plane called the complex plane.

Why the topic needs it: the parent's pole $s=\sigma +j\omega$ is this address. "Left half-plane" simply means $\sigma <0$ — the point sits to the left of the vertical axis — and thanks to $e^{\sigma t}$ with $\sigma <0$, the machine calms down. Points sitting exactly on the vertical axis are the marginal edge. Now that sentence is literal, not magic.

5. The operator $\mathcal{L}$, the Laplace variable $s$, and the integral $\int$

Why the topic needs it: the parent defines $\mathcal{L}\{f(t)\}={\int }_0^{\infty }f(t)e^{-st}dt.$ You can now read every piece: the operator $\mathcal{L}$ (the machine), applied to the signal $f(t)$, is an integral (add up over all time) of that signal times the decaying-oscillating exponential $e^{-st}$ (built from the complex frequency $s$). The kernel $e^{-st}$ is chosen precisely because — from §3 — it is the eigenfunction of the derivative, which is the trick that converts $\frac{d}{dt}$ into "multiply by $s$". This is Laplace transform in full.

6. Polynomials, roots, and factoring — $N(s)$, $D(s)$

The picture: each factor $(s-r)$ is a "switch" that flips the whole product to $0$ when $s=r$. Roots are the special addresses on the complex plane where the polynomial vanishes.

Why the topic needs it: the transfer function is $G(s)=\frac{N(s)}{D(s)}$, a ratio of two polynomials. Zeros are roots of the top $N(s)$; poles are roots of the bottom $D(s)$. Finding them = finding roots. Everything the parent says about behaviour is read off these root addresses on the plane from §4 — including the edge case where a root lands on the imaginary axis (marginal ringing).

7. The ratio $\frac{Y(s)}{U(s)}$ — putting it together

Once §1–§6 are in hand, the transfer function is nothing new:

• Take the machine's differential equation (§2).
• Laplace-transform it with the operator $\mathcal{L}$ (§5) so every $\frac{d}{dt}$ becomes a multiply-by-$s$ (§3's eigenfunction trick), the initial-condition terms set to zero.
• The equation becomes a polynomial equation (§6).
• Rearrange to the ratio $G(s)=\frac{Y(s)}{U(s)}$ = output-over-input.

$Y(s)$ and $U(s)$ are the $s$-domain twins of $y(t)$ and $u(t)$ (the capital-letter convention from §5). That is the parent's block-diagram currency.

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