State-space representation

In <a href="/facts/Control_engineering/5ckfXmB8">control engineering</a> and <a href="/facts/System_identification/v2xk1X0e">system identification</a>, a state-space representation is a <a href="/facts/Mathematical_model/CGhyuxgz">mathematical model</a> of a <a href="/facts/Physical_system/tdSfxvIB">physical system</a> that uses <a href="/facts/State_variable/HNF1dCXu">state variables</a> to track how inputs shape system behavior over time through <a href="/facts/First-order_differential_equation/ygKGQ8kD">first-order differential equations</a> or <a href="/facts/Difference_equation/JolXC71M">difference equations</a>. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to <a href="/facts/Phase_space/qPe4Fq57">phase space</a> in certain <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a>) is a geometric space where the axes are these state variables, and the system’s state is represented by a state <a href="/facts/Vector_(mathematics)/U8mM7A5F">vector</a>.
For <a href="/facts/Linear_system/uEl20Kc1">linear</a>, <a href="/facts/Time-invariant_system/NpWUk0YG">time-invariant</a>, and finite-dimensional systems, the equations can be written in <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> form, offering a compact alternative to the <a href="/facts/Frequency_domain/pyTVrUSq">frequency domain</a>’s <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transforms</a> for <a href="/facts/Multiple-input_and_multiple-output/1EXPB7Vl">multiple-input and multiple-output</a> (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial conditions. This approach turns <a href="/facts/Systems_theory/C56Ck7DQ">systems theory</a> into an algebraic framework, making it possible to use <a href="/facts/Kronecker_product/gABGx6I6">Kronecker</a> structures for efficient analysis. 
State-space models are applied in fields such as economics, statistics, computer science, electrical engineering, and neuroscience. In <a href="/facts/Econometrics/7U52gzHt">econometrics</a>, for example, state-space models can be used to decompose a <a href="/facts/Time_series/fSXPR817">time series</a> into trend and cycle, compose individual indicators into a composite index, identify turning points of the business cycle, and estimate GDP using latent and unobserved time series. Many applications rely on the <a href="/facts/Kalman_filter/wNK7rnbk">Kalman Filter</a> or a state observer to produce estimates of the current unknown state variables using their previous observations.

State-space representation open-in-new

State-space representation