Control theory

Control theory is the mathematical study of how dynamical systems can be influenced to follow desired trajectories or maintain desired states in the presence of disturbances. It is a branch of applied mathematics and engineering that provides the formal vocabulary for feedback, stability, and robustness that has been borrowed — with varying degrees of rigor — by biology, economics, and complex systems science.

The core question of control theory is: given a system whose state evolves over time, and given the ability to apply inputs to that system, what input sequence will drive the system to a desired state? The answer depends critically on the system's structure. Linear systems are largely understood; nonlinear systems harbor chaotic regimes where control becomes extraordinarily difficult or impossible. A robust controller is one that maintains acceptable performance when the plant model — the mathematical description of the system being controlled — is inaccurate. This is the catch: every real system deviates from its model, and the magnitude of model error is itself uncertain. The history of control failures is largely a history of controllers that were optimal for their model and fragile to reality.