Glossary term
State-Space Model
A mathematical representation of a dynamic system using state variables, inputs, outputs, and evolution equations.
Definition
modelA state-space model represents a dynamic system through state variables, inputs, outputs, and equations that describe how the state evolves over time.
State-space modelling is a general framework for linear, nonlinear, continuous-time, and discrete-time systems. It is used when internal dynamics, multiple inputs and outputs, estimation, modern control design, simulation, or numerical integration matter more than a single input-output transfer function.
A common continuous-time linear state-space model is:
where x is the state vector, u is the input vector, y is the output vector, and A, B, C, and D are matrices. The state contains the minimum internal information needed to predict future behaviour from the current time onward, given future inputs. Examples include position and velocity of a mechanical mass, capacitor voltages and inductor currents in a circuit, or concentrations in a process model.
Engineering use
State-space models support multi-input multi-output control, observer design, Kalman filtering, optimal control, simulation, digital control, and system identification. They can represent systems that are awkward or impossible to express cleanly as a single transfer function. Nonlinear state-space models use equations such as \dot{x}=f(x,u,t) and y=g(x,u,t), then may be linearized around an operating point for local control design.
Discrete-time models are used when the controller or estimator runs on sampled data:
The conversion from continuous to discrete form depends on the assumed input hold, sample time, and numerical method. The sample time must be chosen with bandwidth, delay, noise, computation, and actuator limits in mind.
Common mistakes
A common mistake is treating any chosen variables as good states. Good states should be physically meaningful or mathematically independent enough to predict the system without hidden history. Another is validating only output fit while ignoring whether the internal state, operating range, delay, saturation, and disturbance model are realistic. A strong state-space review states state definitions, inputs, outputs, units, operating point, linearization assumptions, sample time, disturbance model, and validation data.