Statefeedback

Statefeedback, also referred to as state feedback, is a feedback control strategy in which the actuator input is a function of the system state. For a continuous-time linear system described by x_dot = Ax + Bu, the state feedback law u = -Kx yields the closed-loop dynamics x_dot = (A - BK)x. In discrete time, with x_{k+1} = Ax_k + Bu_k and u_k = -Kx_k, the closed-loop map is x_{k+1} = (A - BK)x_k. The matrices A, B (and optionally C for outputs) define the system, while K is the feedback gain chosen to achieve desired behavior.

Key objectives include stabilizing the system and shaping response by placing the closed-loop poles (eigenvalues of A - BK). This requires controllability of the pair (A, B); if it is controllable, poles can be assigned arbitrarily within the complex plane (subject to physical constraints). Statefeedback forms the basis for pole placement and many optimal control approaches.

In practice, full state information may not be available. When only outputs y = Cx are measurable, a state estimator such as a Luenberger observer can reconstruct x from y. The combined scheme uses a dynamic controller with u = -K x_hat, where x_hat is the estimated state. Optimal state feedback is often obtained via the Linear Quadratic Regulator (LQR), which minimizes a quadratic cost J and yields K from a Riccati equation.

Limitations include sensitivity to model errors, measurement noise, and actuator saturation. Extensions address these issues with integral action, robust or adaptive state feedback, and integration with observers for partial-state feedback in nonlinear or time-varying systems.