Controllaw

Controllaw, also written as control law, is the rule or function that determines the control input applied to a dynamic system. It specifies how the actuator signal u(t) should be generated from available information about the system, such as its state x(t), measured outputs y(t), and time t. The goal of a control law is to drive the system toward a desired behavior, such as stabilizing an equilibrium, tracking a reference trajectory, or rejecting disturbances.

Common forms include linear state-feedback laws, where u = -Kx, and output feedback laws, where u = φ(y, t). Classical examples include proportional-integral-derivative (PID) control, which computes a scalar input from the error between a reference and a measured output. More general laws may be nonlinear, time-varying, or dynamic, using estimators to reconstruct the state (observer-based control) or incorporating internal models to achieve robust performance.

Design and analysis of controllaws involve criteria like stability, convergence, performance, and robustness. Methods include pole placement, linear-quadratic regulator (LQR) design, H-infinity techniques, and model predictive control, which supplies a control law by solving an optimization problem at each time step. Controls can be implemented in continuous time or discrete time, and often rely on state estimation when full state information is not available.

Applications span robotics, aerospace, automotive systems, process industries, and electrical networks. In modern practice, controllaws are embedded in digital controllers and may adapt online to changing conditions, relying on models, measurements, and possibly learning components.