styrbare

Styrbare is a term used in control theory describing a dynamical system whose internal state can be driven to any desired value in finite time by applying suitable inputs. The adjective styrbar is used to describe such systems, with styrbarhet referring to the property itself. The concept is common in Norwegian and Swedish technical language, with English equivalent controllable or steerable.

For a continuous-time linear time-invariant system ẋ = A x + B u, the system is styrbar if the controllability matrix C = [B, AB, A^2B, ..., A^{n−1}B] has full rank n, where n is the dimension of the state vector x. This is the Kalman rank condition. An equivalent criterion is the PBH (Popov–Belevitch–Hautus) test: the pair (A, B) is controllable if rank [λI − A, B] = n for all eigenvalues λ of A.

Styrbarhet has practical significance in engineering. It determines whether state feedback can place closed-loop poles to achieve desired dynamics; without styrbarhet, some states cannot be influenced by inputs, constraining performance and controller design. In system validation, engineers assess controllability to decide whether actuators and state variables can be effectively controlled, and they may modify the system or add actuation to achieve styrbarhet.

See also controllability, Kalman criterion, PBH test, state feedback.