controllability

Controllability describes whether the state of a dynamical system can be driven from any initial state to any final state in finite time using allowable inputs. In control theory, this concept is central to system design and analysis.

For a continuous-time linear time-invariant system x' = Ax + Bu with x ∈ R^n and u ∈ R^m, the system is controllable if, for any x0 and xT and any T > 0, there exists an input u(t) on [0, T] such that x(T) = xT. Equivalently, the controllability matrix C = [B AB A^2B ... A^{n−1}B] has full rank n. In discrete time, x_{k+1} = Ax_k + Bu_k, and the same type of matrix must have rank n.

The Kalman rank condition is often stated as rank C = n; an equivalent test is the Hautus criterion: rank [λI − A, B] = n for every eigenvalue λ of A. The controllability Gramian Wc(T) = ∫_0^T e^{At} B B^T e^{A^T t} dt is positive definite for some T if and only if the system is controllable.

If a system is not completely controllable, it decomposes into controllable and uncontrollable subspaces; states in the uncontrollable subspace cannot be driven by inputs. Controllability informs the feasibility of tasks such as pole placement, optimal control, and minimum-energy control. For nonlinear systems, local controllability can be studied via the Lie algebra rank condition, with Chow's theorem giving conditions for controllability on manifolds. Controllability is distinct from observability, although the two concepts are related in dual formulations.