Kontrollerbare

Kontrollerbare is the Swedish term for controllable, used in control theory to describe a dynamical system whose state can be driven from any initial state to any desired final state in finite time using an appropriate input. In English-language texts, the term is simply controllable. The concept applies to linear time-invariant state-space models, typically written as x' = Ax + Bu for continuous time or x_{k+1} = Ax_k + Bu_k for discrete time.

A system is kontrollerbar if the controllability matrix formed from B and A, namely [B, AB, A^2B, ..., A^{n-1}B], has full rank n, where n is the dimension of the state vector. Equivalently, the controllability Gramian is positive definite over a finite horizon. This is the Kalman controllability criterion. Kontrollerbar systems permit complete maneuvering of state variables through admissible inputs, enabling tasks such as reference tracking, stabilization, and setpoint changes.

The dual concept is observability, which concerns the ability to infer the internal state from outputs. Together, controllability and observability inform the possibility of state estimation and control design, often via Kalman decomposition or design in a controllable canonical form. A system’s controllability can depend on the number and placement of actuators; adding or removing inputs can change whether a system is kontrollerbar.

Limitations include model accuracy, actuator constraints, and energy or effort costs. In practice, engineers assess controllability to determine whether a proposed controller can achieve the desired performance and to identify necessary model or actuator modifications. See also observability, controllable canonical form, and Kalman decomposition.