kontrollbara

Kontrollbara is a Swedish term used in control theory to denote that a system is controllable. In this sense, a controllable (kontrollbar) system is one for which any initial state can be driven to any desired final state in finite time by an appropriate input signal. The concept is central to the design of state-feedback controllers and observers, as controllability determines whether the system’s internal dynamics can be fully influenced from the inputs.

For linear time-invariant (LTI) systems described in continuous time by dx/dt = Ax + Bu, or in discrete time by x_{k+1} = Ax_k + Bu_k, controllability is captured by the controllability matrix. In continuous time, it is [B, AB, A^2B, ..., A^{n-1}B], and the system is controllable if this matrix has full rank equal to the number of states n. An equivalent condition is the Kalman rank criterion. In discrete time, the same shift applies with the corresponding powers of A. If the rank is deficient, some states cannot be steered to arbitrary values, no matter how inputs are chosen.

Practically, controllability underpins the possibility of placing closed-loop poles via state feedback and of achieving optimal control. If a system is not controllable, certain state directions are uncontrollable and cannot be moved by inputs alone. Related concepts include observability, stabilizability (a weaker property), and structural controllability, which examines controllability from the network structure rather than exact numerical parameters. Extensions to nonlinear systems involve local controllability and conditions such as the Lie-algebra rank criterion.

Kontrollbarhet is thus a key property in system modeling, analysis, and controller design, influencing what can be achieved through control actions.