Foundations — Impulse response and transfer function (GNC connection)

This page assumes you know nothing about the notation in the parent note. We build every symbol from the ground up, each one leaning only on the ones before it. Read top to bottom.

1. A function of time: $y(t)$, $u(t)$

Picture a strip chart: the horizontal axis is time $t$ (seconds), the vertical axis is the quantity. As the pen moves right, it traces a curve — that curve is $y(t)$.

• $u(t)$ = the input: what we do to the system (a thruster command, a push).
• $y(t)$ = the output: what the system does back (an angle, a tank level).

Why the topic needs this: the whole subject is "input goes in, output comes out." Both are functions of time, so we must first agree what a function-of-time looks like.

2. The derivative: $y^{\prime }$, $\dot{y}$, $y^{(n)}$

We need a way to say "how fast is the output changing right now?"

Why a slope and not something else? Rate of change is the natural language of physics: velocity is the slope of position, and Newton's laws relate forces to rates of change. To describe a spacecraft turning, we must talk about how fast its angle moves.

Three notations that all mean the same thing:

Look at the figure: at the peak the slope is zero (flat top); going up the slope is positive; coming down it is negative. That sign-changing slope is $y^{\prime }$.

3. An ODE and its coefficients: $a_n{y}^{(n)}+\cdots +a_{0}y=u$

The parent's central object is $a_n{y}^{(n)}+\cdots +a_1{y}^{\prime }+a_{0}y=u(t).$

• $a_0,a_1,\dots ,a_n$ are constant numbers (they don't change with time).
• Each term mixes some derivative of $y$ with a weight $a_k$.

Why this shape? Real machines obey balance laws (mass × acceleration = force, etc.), and those laws naturally relate a quantity to its rates of change. A Linear Constant-Coefficient ODE is the simplest family rich enough to model springs, tanks, and spacecraft — see Linear constant-coefficient ODEs.

4. The Dirac delta: $\delta (t)$ — the perfect tap

We want to "kick" the system as sharply as possible. What does an idealised tap look like?

Build it as a limit: take a rectangle of width $w$ and height $1/w$. Its area is always $1$. Now shrink $w\to 0$: it gets taller and thinner but keeps area $1$. That limiting spike is $\delta (t)$. See the shrinking rectangles in the figure.

Why the topic needs it: the delta is the mathematical "single perfect tap." The response to it is the system's fingerprint. And sifting is the tool that lets us later chop any input into a row of taps. More at Dirac delta function.

5. Zero-state (at rest before $t=0$)

Picture a still swing hanging straight down before you push it. We insist on this so that the response we measure comes purely from the input, not from some leftover motion. This is the "zero-state" assumption behind every formula in the parent.

6. The impulse response: $h(t)$

Picture the still swing, one sharp shove, then it swings and slowly dies down. That whole dying-down curve is $h(t)$ — the system's fingerprint or personality.

Why "$h$"? It's the response handed back to a single tap. Because the system is LTI, this one curve secretly contains the response to everything — that's proved next.

7. Convolution: the $\ast$ symbol

Now the punchline machinery. We cut an arbitrary input into taps and add up the responses.

Read the symbols slowly:

• $\tau$ (Greek "tau") is a dummy time — the instant a particular tap was applied.
• $u(\tau )$ = how strong the tap at time $\tau$ was.
• $h(t-\tau )$ = the fingerprint, shifted to start at $\tau$, evaluated at the present time $t$ (time-invariance lets us just shift it).
• Integrating over all $\tau$ = adding up the after-effects of every past tap. See Convolution.

Why an integral and not a plain sum? The taps are packed infinitely densely in time, so "add them up" becomes "integrate." For a causal system ($h=0$ for $t<0$) with input starting at $0$, the limits shrink to $0$ to $t$: only past taps can affect the present.

8. The Laplace transform: $\mathcal{L}$, $s$, $F(s)$

Convolutions are painful integrals. We want a world where they become plain multiplication.

• $s$ is a (possibly complex) number; think of it as a knob.
• $e^{-st}$ is a weighting that emphasises different time-behaviours as $s$ changes.
• Capital letters ($F$, $Y$, $U$, $H$) mean "the Laplace transform of" the lowercase function. See Laplace transform.

Three facts the topic leans on: $\mathcal{L}\{f^{\prime }\}=sF(s)-f(0),\mathcal{L}\{\delta \}=1,\mathcal{L}\{u\ast h\}=U(s)H(s).$ The last one is the whole reason we bother: convolution in time becomes multiplication in $s$.

9. The transfer function $H(s)$ and poles

• The bottom polynomial $a_n{s}^n+\cdots +a_0$ is the characteristic polynomial — see Characteristic polynomial and roots.
• Poles = the values of $s$ that make the denominator zero (where $H$ blows up).

This is the crux of Guidance Navigation and Control (GNC) and Feedback control systems: keep the poles left, keep the spacecraft calm.

Prerequisite map

Equipment checklist

Test yourself — can you answer each before revealing?

Connections

• Parent topic
• Dirac delta function
• Laplace transform
• Convolution
• Characteristic polynomial and roots
• Linear constant-coefficient ODEs
• Stability and poles (left-half plane)
• Feedback control systems
• Guidance Navigation and Control (GNC)