Controllability and Observability in Plasma Control
Two structural properties that decide whether a plasma quantity can be driven by the actuators and inferred from the sensors.
Two fundamental questions
Before designing any loop, two questions must be answered. Controllability: can the available actuators drive the plasma from any state to any desired state? Observability: can the available sensors determine the full state from their measurements? If a quantity is uncontrollable or unobservable, no controller, however clever, can regulate it.
The formal test
For a linear model with state matrix A, input matrix B, and output matrix C, controllability is tested by the rank of the controllability matrix [B, AB, A^2 B, ...] and observability by the rank of the observability matrix [C; CA; CA^2; ...]. Full rank means full controllability or observability. In practice, the useful question is not the binary rank but how strongly each mode is controllable or observable.
Degrees, not absolutes
Real systems are rarely perfectly controllable or hopelessly uncontrollable; they have modes that are strongly, weakly, or barely reachable. Singular-value analysis of the response matrices ranks the shape and profile combinations by how much actuator effort they need. Nearly uncontrollable directions demand huge coil currents for small effect and are best left un-targeted.
Consequences for design
- Weakly controllable shape modes are dropped from the target to avoid saturating coils
- Unobservable internal profiles require added diagnostics or must be assumed
- Actuator and sensor placement is chosen to make the important modes strongly controllable and observable
- Control authority limits, not just stability, set what shapes are achievable
Why it comes first
Controllability and observability are properties of the machine's design - its coils and sensors - not of the controller. They must be checked when the device is designed, because a control problem that is structurally impossible cannot be fixed in software. Good actuator and diagnostic layout is control design done in hardware.