Tilstandsregulering

Tilstandsregulering, or state regulation, is a control technique in which the state of a dynamic system is driven to a desired value or trajectory by using feedback from the system’s state. The standard modelling framework is a state-space representation: ẋ = Ax + Bu, y = Cx + Du, where x ∈ R^n is the state vector, u ∈ R^m the input, and y the measured output.

The goal is to design a control law u = -Kx + r(t) (or u = -Kx for regulation to the origin) so that the closed-loop system ẋ = (A − BK)x + Br(t) is stable and meets performance criteria. Stability requires that the eigenvalues of (A − BK) lie in the left half-plane (continuous time) or inside the unit circle (discrete time).

Prerequisites include controllability and observability. If the state cannot be measured directly, observers such as a Luenberger observer or a Kalman filter provide estimates for feedback.

Common design methods include state feedback with pole placement, and optimal control via the Linear Quadratic Regulator (LQR), which minimizes a cost function J = ∫ (x^TQx + u^TRu) dt using appropriate weight matrices Q and R.

In digital implementations, the continuous model is discretized, yielding x[k+1] = Ad x[k] + Bd u[k]. The controller is executed at sampling instants, possibly with a state estimate ĥx[k].

Tilstandsregulering finds applications in robotics, aerospace, automotive, and process industries, where fast, robust regulation of states such as position, velocity, temperature, or pressure is required.