Holding a Drone's Altitude

A drone's altitude controller pushes back against height error and climb rate. Tuned exactly right, the system matrix has a repeated eigenvalue, and every recovery path bends to arrive along one single direction: a degenerate node.

Click or drag anywhere to start a new trajectory there.

How it works

Systems with repeated eigenvalues

In one sentence

If $A$ has a repeated eigenvalue $\lambda$ with only one independent eigenvector $\boldsymbol\eta$ , then $\mathbf x = c_1e^{\lambda t}\boldsymbol\eta + c_2e^{\lambda t}(t\boldsymbol\eta + \boldsymbol\rho)$ with $(A - \lambda I)\boldsymbol\rho = \boldsymbol\eta$ .

General formula

{\htmlData{sym=0}{z}}' = {\htmlData{sym=1}{v}}, \qquad {\htmlData{sym=1}{v}}' = -{\htmlData{sym=2}{K_p}}\,{\htmlData{sym=0}{z}} - {\htmlData{sym=3}{K_d}}\,{\htmlData{sym=1}{v}}

The factor $t$ in the second solution makes every path line up with the single eigenvector before it dies out: the phase-plane picture of $te^{\lambda t}$ .

What the symbols mean

$z$
Altitude error
How far the drone is above (+) or below (−) its target height.
$v$
Climb rate
How fast the altitude is changing.
$K_p$
Proportional gain
How hard the drone pushes back per meter of error.
$K_d$
Derivative gain
How hard it resists its own climb rate.

The controller corrects the height error $z$ with thrust $-K_pz - K_dv$ , where $v$ is the climb rate.
The matrix $\begin{bmatrix}0 & 1\\ -K_p & -K_d\end{bmatrix}$ has a repeated eigenvalue $\lambda = -K_d/2$ exactly when $K_d^2 = 4K_p$ .
A repeated eigenvalue with only one eigenvector gives a degenerate node: paths all bend to come in along that one direction.
It is critical damping seen in the phase plane: the fastest return to hover with no overshoot.

Learn the math

Repeated EigenvaluesDiff Eq · Lesson 35, Systems of Differential Equations