# State-space representation

In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variablesâ€™ values depend on the values of the state variables.

The "state space" is the Euclidean space[citation needed] in which the variables on the axes are the state variables. The state of the system can be represented as a state vector within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors.

If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.[1][2] The state-space method is characterized by significant algebraization of general system theory, which makes it possible to use Kronecker vector-matrix structures. The capacity of these structures can be efficiently applied to research systems with modulation or without it.[3] The state-space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With ${\displaystyle p}$ inputs and ${\displaystyle q}$ outputs, we would otherwise have to write down ${\displaystyle q\times p}$ Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions.

The state-space model can be applied in subjects such as economics,[4] statistics,[5] computer science and electrical engineering,[6] and neuroscience.[7] In econometrics, for example, state-space models can be used to decompose a time series into trend and cycle, compose individual indicators into a composite index,[8] identify turning points of the business cycle, and estimate GDP using latent and unobserved time series.[9][10] Many applications rely on the Kalman Filter to produce estimates of the current unknown state variables using their previous observations.[11][12]