Controllability, Observability, and Canonical Forms
Stub. The two structural questions of a state-space system: controllability (which states can be reached by choosing an input, and the state-transfer problem of how) and observability (which states can be inferred from an output); the invariant subspaces that organize the uncontrollable/unobservable parts of a system; and the Kalman canonical decomposition that separates all four combinations.
Corresponds to lectures: Controllability and state transfer; Observability and state estimation; Invariant subspaces; Kalman canonical form. Asks the two questions that determine what you can and can’t do with a state-space system: controllability — can the input steer the state from anywhere to anywhere (and how, concretely, via the state-transfer problem) — and its dual, observability — can the state be reconstructed from the observed output alone. Invariant subspaces (under ) are the right language for the states a given input can never reach, or that leave no trace in the output, and the Kalman canonical form decomposes any system into four pieces — controllable/observable, controllable/unobservable, uncontrollable/observable, uncontrollable/unobservable — which is the structural prerequisite for both state feedback (state-feedback-and-linear-quadratic-regulation, which needs controllability) and state estimation (estimation-and-the-kalman-filter, which needs observability).
Outline (TODO — flesh out each)
- Controllability
- The state-transfer problem
- Observability
- State estimation from observability
- Invariant subspaces
- The Kalman canonical form