Linearisationdraft

Why Aircraft Need Derivatives

What you just saw

If you came here from the simulator, you watched two aircraft take off from the same trim, get the same elevator doublet, and — for a while — fly as one. The cyan aircraft is the linearised model; the amber one is the full nonlinear equations of motion. Fly politely and you cannot tell them apart. Do something rude with the stick and the trails part company.

If you haven’t seen it yet, go and fly it first — this page will still be here: Linear vs Nonlinear workspace →

That divergence raises the obvious question, and it deserves a proper answer.

The obvious question

The linear model is the one that’s wrong at the edges. So why is it the model that every flight dynamics course, every stability analysis, every autopilot, and every handling-qualities specification is built on?

Because the nonlinear equations, honest as they are, have almost no structure. The full six-degree-of-freedom equations are twelve coupled nonlinear ODEs. You can simulate them — that is literally what the amber aircraft is doing, step by step — but a simulation only ever answers one question: what happens this time, from this state, with these inputs. Change the gust, the mass, the entry speed, and you must fly it again. Simulation gives you a trajectory. It does not give you understanding.

Linearise about a trim condition, though, and the equations collapse into

\delta\dot{\vec{x}} = A\,\delta\vec{x} + B\,\delta\vec{u}

— linear, time-invariant, state-space form. And that form is a skeleton key:

The eigenvalues of $A$ are the aircraft’s modes — phugoid, short period, Dutch roll, spiral, roll subsidence. Stability stops being “run a thousand simulations and squint” and becomes a number you read off. (Fly the modes →, then see Aircraft Modes of Motion.)
Transfer functions, frequency response, and every classical and modern control design method follow — the autopilot and stability augmentation flying in the simulator are designed on exactly this model.
Handling-qualities requirements (MIL-STD-1797 and friends) are written in terms of the linear parameters: frequencies, damping ratios, time constants.
The answer holds for every sufficiently small perturbation at once — not one trajectory, but the local behaviour of all of them.

One model answers “what happened?”. The other answers “what happens, and why, for everything nearby”. That is a trade worth making — as long as you stay honest about nearby.

Why is linearisation even allowed?

Here is the part that usually gets rushed past. The right-hand side of the equations of motion is a function: $\dot{\vec{x}} = f(\vec{x}, \vec{u})$ . The entire licence to linearise is that this function has a first derivative in the proper, multivariable sense:

f(\vec{x}_0 + \delta\vec{x},\ \vec{u}_0 + \delta\vec{u}) \;=\; \underbrace{f(\vec{x}_0, \vec{u}_0)}_{=\,0\ \text{at trim}} \;+\; A\,\delta\vec{x} \;+\; B\,\delta\vec{u} \;+\; o\!\left(\lVert\delta\rVert\right)

$A$ and $B$ are not assumptions — they are Jacobian matrices, $A = \partial f/\partial \vec{x}$ and $B = \partial f/\partial \vec{u}$ , evaluated at the trim point. And the stability derivatives you have been collecting — $X_u$ , $M_q$ , $C_{L_\alpha}$ , $C_{n_\beta}$ — are nothing more or less than the entries of those matrices (see Stability Derivatives and Linearisation Theory).

Why does such a derivative exist at all? Because $f$ is built by composing smooth pieces — aerodynamic forces that depend on $\alpha$ , $\beta$ and $V$ ; kinematics that depend on the Euler angles; propulsion that depends on throttle — and compositions of differentiable functions are differentiable, with Jacobians that multiply (the chain rule, grown up). A function of a function with a first derivative still has a first derivative. That single fact is why the blocks assemble into $A$ and $B$ at all.

And why it stops working

The $o(\lVert\vec{h}\rVert)$ is a promise about small perturbations at that trim point — nothing more. The divergence you watched in the simulator is the discarded remainder growing: push the angle of attack toward the stall and the lift curve bends away from its tangent, the higher-order terms stop being negligible, and the amber aircraft goes somewhere the cyan one cannot follow.

The stability derivatives have not “failed” when that happens. They have gone back to being what they always were: one Jacobian, evaluated at a flight condition you have left. This is also why real aircraft data comes as tables of derivatives, one set per trim condition — a different point of linearisation, a different $A$ .

The next link in the chain

To recap the spine so far: one aircraft, two models — you watched them part company, and that is why motion and derivatives had to be understood: the linear twin is a derivative, a Jacobian frozen at trim. And we keep it, lies and all, because of what it carries: eigenvalues — the objects that turn “run another simulation” into “read the answer off a map.” What an eigenvalue actually is deserves its own page, and it’s the next one to read: What an Eigenvalue Actually Is →

Go deeper

The full story of the derivative itself — from the limit of rise-over-run, to why only the linear term survives, to the Jacobian, the chain rule as matrix multiplication, and the adjoint method: The Generalisation Ladder (interactive, scroll-driven).
The algebra for this aircraft: Comparison of the Linear and Nonlinear Equations of Motion.
What the payoff object actually is: What an Eigenvalue Actually Is — a growth rate, a direction the dynamics doesn’t rotate, and a map that replaces infinitely many simulations.
From linearisation to a working model: Worked Example, then Aircraft Modes of Motion.