Can you move it? Can you see it?
Controllability concerns the actuator; observability concerns the sensors. Both matter before state feedback and estimation.
Distinguish a state you can move from one you can infer from measurements over time.
PID used measured error. We now return to the state model and ask whether the available actuator and sensors can reach and reveal every state.
Review the previous lesson: Thermostats, PI and heating limitsFirst, 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.
Build the idea step by step
Controllability asks whether inputs can move all the states in the ideal linear model. Observability asks whether known inputs and a history of measured outputs reveal the starting state. Measuring one quantity can be enough: successive positions can reveal velocity. For two states, the matrices below test whether there are two independent directions. Rank is the number of independent columns or rows. Full rank means the complete property holds; it does not promise small control forces or accurate estimates in noisy data.
Read the equations
For a two-state linear model, each matrix must have two independent columns for its respective test to pass. Independent means one column cannot be made by scaling the other.
What the symbols mean
- state, input, and output matrices from the state equation
- controllability matrix: paths from input to state
- observability matrix: paths from state to measurements
These tests belong to the stated linear model and sensor arrangement. A change of operating point can change the result. Force limits and measurement noise need separate checks.
This ideal double-integrator is controllable and observable with a position sensor. Noise changes estimation quality, not the algebraic rank.
Can a position sensor reveal velocity?
Use a unit-mass cart with position and velocity as states. Force changes velocity; the sensor reads position. Use consistent SI units.
The input reaches velocity directly and position through motion. These two columns are independent.
Position is measured directly; its rate gives velocity. Both states can be inferred from ideal measurements over time, although noise makes estimation harder.
More detail and model assumptions
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.
Connect the example to the lab
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.Why the next lesson follows
When the state is controllable, feedback gains can move its poles. Next we calculate those gains for a chosen response.