Controllability and Observability
Two structural properties defining whether a system can be fully steered and fully inferred.
Definition
Controllability is whether the available inputs can drive a system from any state to any other. Observability is whether the system's full internal state can be reconstructed from its measured outputs. Both are structural properties of a state-space model.
These properties are binary in the classical definition but graded in practice: a system can be technically controllable yet nearly uncontrollable if the required inputs are enormous. Measures like the controllability Gramian quantify this margin and guide sensor and actuator placement.
These properties determine what any controller or estimator can possibly achieve, so they are checked before design begins. They are effectively graded rather than strictly binary: a system can be technically controllable yet require impractically large inputs, a condition the controllability Gramian quantifies. Analyzing them guides where to place sensors and actuators, ensuring the chosen hardware can both steer the system and reconstruct its internal state from measurements.
Why they pair
- Controllability is required for state-feedback control to work.
- Observability is required for a state estimator such as the Kalman filter.
- Together they determine what control architecture is even possible.
Why it matters
Before designing a controller, an engineer must know whether the system can be controlled and observed with the chosen actuators and sensors. A lack of either points to a fundamental need for more or different hardware, not just better tuning.
Fusion connection
Analyzing controllability and observability guides the placement of coils and diagnostics on a fusion machine, ensuring the plasma can both be steered and be inferred from measurements.