mathematics//dynamical systems//linearization
Linearization replaces a nonlinear dynamics locally by a matrix, so that the whole linear toolbox (eigenvalues, Kalman filters, LQR, frequency response) can be used on a system that is not linear; the work is in not confusing the two. The world is full of sines, square roots and saturations, and almost every powerful tool is linear: the bridge is that, close to an **operating point**, every smooth function looks like a line. It is the Taylor polynomial applied to \(f\). A linear map respects **superposition**, and \(Fx+b\) with \(b\ne0\) is affine; a curved trajectory can still belong to a linear system, because linear does not mean straight in time. Much of engineering is linear by this route: a sensor's calibration line and a neural layer before its activation are matrices, trusted where they were fitted.
Linearization replaces a nonlinear dynamics locally by a matrix, so that the whole linear toolbox (eigenvalues, Kalman filters, LQR, frequency response) can be used on a system that is not linear; the work is in not confusing the two. The world is full of sines, square roots and saturations, and almost every powerful tool is linear: the bridge is that, close to an operating point, every smooth function looks like a line. It is the Taylor polynomial applied to fff. A linear map respects superposition, and Fx+bFx+bFx+b with b≠0b\ne0b=0 is affine; a curved trajectory can still belong to a linear system, because linear does not mean straight in time. Much of engineering is linear by this route: a sensor's calibration line and a neural layer before its activation are matrices, trusted where they were fitted.
Linearizing around an equilibrium starts from f(x∗,u∗)=0f(x^*,u^*)=0f(x∗,u∗)=0 and works with deviation variables, δx=x−x∗\delta x=x-x^*δx=x−x∗ and δu=u−u∗\delta u=u-u^*δu=u−u∗. First order gives
δx˙≈Fδx+Bδu,F=∂f∂x∣∗,B=∂f∂u∣∗.\delta\dot x\approx F\delta x+B\delta u,\qquad F=\left.\frac{\partial f}{\partial x}\right|_*,\quad B=\left.\frac{\partial f}{\partial u}\right|_*.δx˙≈Fδx+Bδu,F=∂x∂f∗,B=∂u∂f∗.
FFF and BBB are Jacobians: entry (i,j)(i,j)(i,j) of FFF says how much the rate of state iii changes when state jjj moves a little. Diagonal entries are self-effects; the others are couplings. In a drone at hover, sinθ≈θ\sin\theta\approx\thetasinθ≈θ and cosθ≈1\cos\theta\approx1cosθ≈1 turn the planar equations into three decoupled lines, x¨≈gθ\ddot x\approx g\thetax¨≈gθ, z¨≈δT/m\ddot z\approx\delta T/mz¨≈δT/m, θ¨=τ/I\ddot\theta=\tau/Iθ¨=τ/I: to move sideways it must tilt, 5° giving about 0.86 m/s20.86\ \text{m/s}^20.86 m/s2 (small-angle approximation).
Linearizing is approximating with Jacobians near an equilibrium: useful, local, and dependent on the operating point.
A tank emptying through an orifice behaves as a first-order lag whose time constant is 2Ah∗/q∗2Ah^*/q^*2Ah∗/q∗, so the same tank answers twice as fast at a quarter of its level, and a controller tuned at one point needs its gains changed at another (gain scheduling).
The eigenvalues allow a local conclusion and no more. If every eigenvalue of FFF has negative real part there is local asymptotic stability; if one has positive real part, instability. With zero real parts the test is inconclusive: x˙=−x3\dot x=-x^3x˙=−x3 and x˙=x3\dot x=x^3x˙=x3 both have a zero Jacobian at the origin, one attracts and the other repels, and the discarded term decides (equilibrium and stability). The same equation can give opposite verdicts at two equilibria, as the hanging and inverted pendulum show.
If the point is not an equilibrium, f(x∗,u∗)f(x^*,u^*)f(x∗,u∗) appears as well; around a nominal trajectory the matrices depend on time. The matrix adds no forces: it reorganizes the sensitivity of mechanisms already modelled. Its modes are the same approximation read as collective patterns (linear field).
Linearizing again and again is one of the most profitable patterns in engineering. The extended Kalman filter relinearizes its model at every step around the current estimate, Newton's method linearizes the gradient to optimize, backpropagation multiplies Jacobians layer by layer (linearize-solve-repeat). The discipline is to know in which region the approximation holds and to relinearize on leaving it; a Jacobian that loses rank at a point can also declare uncontrollable a system that the nonlinear model can still steer.