Controllabilitymatriisi

Controllabilitymatriisi, or the controllability matrix, is a concept in linear control theory used to determine whether a linear system can be steered from any initial state to any final state using suitable inputs. For a continuous-time system x' = Ax + Bu, where x is the state and u is the input, the controllability matrix is defined as C = [B AB A^2B ... A^{n-1}B], with n the number of states. The discrete-time analogue uses the same construction for x_{k+1} = Ax_k + Bu_k. The pair (A, B) is controllable if rank(C) = n, a condition known as the Kalman rank condition. If the rank is less than n, the system has uncontrollable modes that cannot be influenced by the inputs.

The matrix C aggregates the immediate input influence (B) and the influence after one, two, and subsequent state transitions (AB, A^2B, etc.), illustrating how inputs propagate through the state dynamics. In practice, controllability is tested by computing the rank of C or by performing a Kalman decomposition to identify controllable and uncontrollable subspaces. Controllability is also related to the finite-horizon controllability Gramian Wc, defined by Wc = ∫_0^T e^{At} B B^T e^{A^T t} dt in continuous time (or the corresponding discrete sum); a full-rank Wc over a given interval implies controllability on that interval.

Applications of the controllabilitymatriisi include informing state-feedback design that places closed-loop eigenvalues, guiding observer and estimator construction, and aiding actuator placement and system analysis. Related concepts include controllability, observability, and the Kalman decomposition.