Can you move it? Can you see it?
Controllability concerns the actuator; observability concerns the sensors. Both matter before state feedback and estimation.
First, picture it
A cart motor can push the cart but a broken force path cannot influence the pole. A camera measuring only position may show where the cart is, while its velocity is not directly displayed. The question is whether the model and a history of measurements let us infer that hidden velocity.
What the model says
For ẋ=Ax+Bu, controllability asks whether a suitable input history can move the state between states in finite time in the ideal linear model. Observability asks whether the initial state can be reconstructed from the output history y=Cx when the input is known. For n states, the controllability matrix [B AB … A^(n−1)B] and observability matrix formed from C, CA, …, CA^(n−1) must each have rank n for the full properties. Stabilizability and detectability are weaker conditions relevant to control and estimation, but the rank tests make the basic distinction visible.
rank[B AB … Aⁿ⁻¹B]=n; rank[C; CA; …; CAⁿ⁻¹]=nThe ranks are properties of the stated A, B and C at the chosen operating point; they do not remove force limits or noise.
This ideal double-integrator is controllable and observable with a position sensor. Noise changes estimation quality, not the algebraic rank.
A two-state example you can calculate
Let x=[position, velocity], A=[[0,1],[0,0]], B=[0,1]ᵀ and C=[1,0]. The input changes velocity first; velocity later changes position. [B AB]=[[0,1],[1,0]] has rank 2, so both states are controllable in this ideal model.
The sensor sees position only, but [C; CA]=[[1,0],[0,1]] also has rank 2. A position history reveals velocity in the ideal noise-free model. In real measurements, differentiating noisy positions is difficult; a Kalman filter deals with that practical estimation problem.
Why the cart-pole needs a stated sensor set
The current Lab's Kalman lesson measures cart position and pole angle, then estimates the velocities. It does not provide a sensor-selection switch. Showing a successful estimate under this fixed setup helps intuition, but it is not a general observability test for every possible sensor arrangement.
Make it concrete
Open the Kalman lesson and identify which cart-pole states are measured and which are estimated. Compare true and estimated velocity after the same kick.
Inspect measured and hidden statesMake a prediction before moving a control.
- PREDICTCan position and angle measurements help infer both velocities?
- CHANGE ONE THINGKeep the Kalman preset and seed; change only assumed sensor variance.
- OBSERVECompare true and estimated velocities, not only angle.
- EXPLAINDistinguish mathematical observability from noisy estimate quality.
Full-rank ideal-model tests do not guarantee good numerical conditioning, practical sensor accuracy, actuator authority or safe recovery near a rail end.
Check your understanding+
If you can measure a state, does that prove you can control it?
No. Observability depends on A and C; controllability depends on A and B. They are separate tests.