Stateinput

Stateinput is a term used in dynamical systems and control theory to denote external signals that directly influence a system’s state. It is distinguished from internal state variables and from outputs.

In state-space form, the state evolves as ẋ = Ax + Bu for linear systems, with x the state vector and u the state input (also called the control or external input). The output is y = Cx + Du. The matrix B maps the input to state changes, while A, C, and D define the system dynamics and measurement relations. The input can include deliberate control actions as well as disturbances, depending on modeling choices.

The concept is central to controller design, where a control law u = u(x, t) or a dynamic controller uses the state input to drive the state toward a desired trajectory. Whether an element of u is treated as a known actuation, a disturbance, or both affects analyses of controllability, observability, and robustness.

Example: a mass–spring–damper system with external force u(t). Let x = [position; velocity]. The dynamics can be written as ẋ = [0 1; -k/m -c/m] x + [0; 1/m] u. Here the force u(t) directly influences the velocity component, acting as the state input.

In practice, practitioners decide how to model the input—known control signals, uncertain disturbances, or a combination—which shapes design choices and performance assessments. The term stateinput may be used interchangeably with control input in some texts, though others reserve state input for signals that directly actuate the state.