State Space
A representation of a dynamic system by a set of state variables evolving through first-order equations.
Definition
The state-space representation describes a dynamic system by a vector of state variables and a set of first-order differential (or difference) equations governing their evolution, together with equations relating states to outputs.
State-space design enables powerful methods such as the linear-quadratic regulator, which computes the optimal feedback gains to balance control effort against tracking error. Pairing such a controller with a state estimator yields a systematic route from model to working controller.
The state-space form generalizes classical control to systems with many coupled inputs and outputs, and it enables systematic design methods such as the linear-quadratic regulator, which computes optimal feedback gains by balancing control effort against tracking error. Paired with a state estimator like the Kalman filter, it yields a complete controller from a model. This model-based route is the foundation of modern control engineering.
It is typically written as dx/dt = Ax + Bu (state evolution) and y = Cx + Du (output), where u is the input and y the measurement.
Advantages
- Handles multiple inputs and outputs naturally.
- Extends to nonlinear and time-varying systems.
- Exposes internal structure like controllability and observability.
Why it matters
State-space is the modern foundation of control engineering, more general than the classical transfer-function view. It enables systematic design methods such as optimal and state-feedback control and connects directly to the Kalman filter.
Fusion connection
Plasma control models are cast in state-space form so multi-input, multi-output controllers can regulate several coupled quantities of the discharge at once.