kontrolligrammian

Kontrolligrammian is a term used in linear control theory to denote the controllability Gramian, a matrix that encapsulates how readily a linear system can be driven from one state to another by its input. The term appears in some languages as a direct translation of “controllability gramian.”

Definition for continuous-time systems: given x' = A x + B u, the controllability Gramian Wc is Wc = ∫_0^∞ e^{A t} B B^T e^{A^T t} dt, provided A is Hurwitz (all eigenvalues have negative real parts). For a finite horizon T, Wc(T) = ∫_0^T e^{A t} B B^T e^{A^T t} dt.

Discrete-time: Wc = ∑_{k=0}^{∞} A^k B B^T (A^T)^k, convergent when the spectral radius of A is less than one.

Properties: Wc is symmetric and positive semidefinite. Its rank equals the dimension of the controllable subspace of (A,B). The pair is controllable if and only if Wc is full rank (invertible in the square case).

Uses: Wc provides the minimum energy required to steer the state from an initial to a final state under certain boundary conditions; it is used in minimum-energy control problems and in balancing-based model reduction methods to identify states that are easily controllable.

Transformation: under state-space transformations, Wc changes by Wc' = T Wc T^T when A' = T A T^{-1}, B' = T B. It also relates to the observability Gramian in dual formulations.

See also: controllability, observability Gramian, Kalman decomposition, balanced truncation.