Controllability and Observability

Can your actuators move every part of the system? Can your sensors see every part of it? These two questions have exact answers for linear systems, from a pair of matrix rank tests. A missing direction means no controller can ever fix it.

Actuator Sensor

The rank tests

For a system $\dot{\mathbf{x}} = A\mathbf{x} + B u$, $y = C\mathbf{x}$ with $n$ states:

$$\mathcal{C} = \begin{bmatrix} B & AB & \cdots & A^{n-1}B \end{bmatrix} \qquad \mathcal{O} = \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{bmatrix}$$

The system is controllable if $\mathcal{C}$ has rank $n$, and observable if $\mathcal{O}$ has rank $n$.

What to notice

Two carts joined by a spring have two ways to move: together (the sum mode) and toward and away from each other (the difference mode). Pushing only cart 1 reaches both modes, so the rank is full. Pushing both carts equally can only move them together, so the stretching mode can never be controlled: the rank drops to 2. Likewise, measuring only the midpoint never sees the stretching, and measuring only the gap never sees the carts drift together. Real designs check this before choosing where to put actuators and sensors.