Reglerproblemet

Reglerproblemet, or the regulation problem, is a fundamental task in control theory concerned with designing a controller that makes the output of a dynamic system follow a desired reference signal while rejecting disturbances, such that the closed-loop system is stable and the regulation error vanishes in steady state.

Typically modeled with a plant P(s) mapping control input u to output y, and with exogenous signals: a reference r(t) and disturbances d(t) generated by an external process (an exosystem). The objective is to find a controller K that, when connected to P, yields e(t) = y(t) − r(t) → 0 as t → ∞ (or keeps y close to r) for all admissible references and disturbances, while keeping the closed-loop stable.

A central result is the Internal Model Principle: to achieve perfect regulation for a class of exogenous signals, the controller must contain an internal copy of the dynamics that generate those signals. Practically this means embedding a model of the exosystem within the controller so that the combined system can anticipate and cancel the effect of the disturbances and reference dynamics. The problem can be formalized by regulator equations linking plant, controller, and exosystem.

Solvability depends on properties of the plant such as stabilizability and detectability and on the existence of an internal model that can be stabilized. Design methods include adding integral action for constant disturbances, incorporating an internal model for periodic or sinusoidal signals, and using standard control design tools (e.g., pole placement, LQR) within an inner–outer loop to ensure both regulation and stability.

Reglerproblemet is a cornerstone in both theoretical and applied control, underpinning the broader field of output regulation and robust control with applications in process control, robotics, and systems requiring precise tracking and disturbance rejection.