A dynamical system can be described by an internal state vector containing the information needed to propagate the model forward once the future input is known.

A continuous-time linear state-space model has the form

$$\dot{\mathbf x}(t)=A\mathbf x(t)+B\mathbf u(t),$$

$$\mathbf y(t)=C\mathbf x(t)+D\mathbf u(t).$$

A discrete-time model has the analogous form

$$\mathbf x_{k+1}=A\mathbf x_k+B\mathbf u_k,$$

$$\mathbf y_k=C\mathbf x_k+D\mathbf u_k.$$

The state need not be directly observable. Position and velocity, capacitor voltages, temperatures or stored inventories can all serve as state variables when they contain enough information for the chosen model to predict subsequent evolution.

A state-space model exposes internal variables that a purely input-output description can hide. Different choices of state can represent the same externally observed behavior.