Editing State-space representation (section)

{{Short description|Mathematical model of a system in control engineering}}
{{distinguish|quantum state space|configuration space (physics)}}In [[control engineering]] and [[system identification]], a '''state-space representation''' is a [[mathematical model]] of a [[physical system]] that uses [[State variable|state variables]] to track how inputs shape system behavior over time through [[First-order differential equation|first-order differential equations]] or [[Difference equation|difference equations]]. 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 [[phase space]] in certain [[Dynamical system|dynamical systems]]) is a geometric space where the axes are these state variables, and the system’s state is represented by a state [[vector (mathematics)|vector]].

For [[Linear system|linear]], [[Time-invariant system|time-invariant]], and finite-dimensional systems, the equations can be written in [[Matrix (mathematics)|matrix]] form,<ref>
{{cite book |author1=Katalin M. Hangos |author1-link=Katalin Hangos |url=https://books.google.com/books?id=-w7rbl1QKVsC&pg=PA254 |title=Intelligent Control Systems: An Introduction with Examples |author2=R. Lakner |author3=M. Gerzson |publisher=Springer |year=2001 |isbn=978-1-4020-0134-5 |page=254 |name-list-style=amp}}</ref><ref>
{{cite book |author1=Katalin M. Hangos |url=https://books.google.com/books?id=Jgtwi3o-mmUC&pg=PA25 |title=Analysis and Control of Nonlinear Process Systems |author2=József Bokor |author3=Gábor Szederkényi |publisher=Springer |year=2004 |isbn=978-1-85233-600-4 |page=25 |name-list-style=amp}}</ref> offering a compact alternative to the [[frequency domain]]’s [[Laplace transform|Laplace transforms]] for [[multiple-input and multiple-output]] (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial conditions. This approach turns [[systems theory]] into an algebraic framework, making it possible to use [[Kronecker product|Kronecker]] structures for efficient analysis. 

State-space models are applied in fields such as economics,<ref>{{Citation |last1=Stock |first1=J.H. |title=Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics |date=2016 |work=Handbook of Macroeconomics |volume=2 |pages=415–525 |publisher=Elsevier |language=en |doi=10.1016/bs.hesmac.2016.04.002 |isbn=978-0-444-59487-7 |last2=Watson |first2=M.W.}}</ref> statistics,<ref>{{Cite book |last1=Durbin |first1=James |title=Time series analysis by state space methods |last2=Koopman |first2=Siem Jan |date=2012 |publisher=Oxford University Press |isbn=978-0-19-964117-8 |oclc=794591362}}</ref> computer science, electrical engineering,<ref>{{Cite journal |last=Roesser |first=R. |date=1975 |title=A discrete state-space model for linear image processing |journal=IEEE Transactions on Automatic Control |volume=20 |issue=1 |pages=1–10 |doi=10.1109/tac.1975.1100844 |issn=0018-9286}}</ref> and neuroscience.<ref>{{Cite journal |last1=Smith |first1=Anne C. |last2=Brown |first2=Emery N. |date=2003 |title=Estimating a State-Space Model from Point Process Observations |journal=Neural Computation |volume=15 |issue=5 |pages=965–991 |doi=10.1162/089976603765202622 |issn=0899-7667 |pmid=12803953 |s2cid=10020032}}</ref> 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,<ref>James H. Stock & Mark W. Watson, 1989. [https://www.nber.org/chapters/c10968.pdf "New Indexes of Coincident and Leading Economic Indicators]," NBER Chapters, in: NBER Macroeconomics Annual 1989, Volume 4, pages 351-409, National Bureau of Economic Research, Inc.</ref> identify turning points of the business cycle, and estimate GDP using latent and unobserved time series.<ref>{{Cite journal |last1=Bańbura |first1=Marta |last2=Modugno |first2=Michele |date=2012-11-12 |title=Maximum Likelihood Estimation of Factor Models on Datasets with Arbitrary Pattern of Missing Data |journal=Journal of Applied Econometrics |volume=29 |issue=1 |pages=133–160 |doi=10.1002/jae.2306 |issn=0883-7252 |s2cid=14231301 |hdl-access=free |hdl=10419/153623}}</ref><ref>{{Citation |title=State-Space Models with Markov Switching and Gibbs-Sampling |date=2017 |work=State-Space Models with Regime Switching |pages=237–274 |publisher=The MIT Press |doi=10.7551/mitpress/6444.003.0013 |isbn=978-0-262-27711-2}}</ref> Many applications rely on the [[Kalman filter|Kalman Filter]] or a state observer to produce estimates of the current unknown state variables using their previous observations.<ref>{{Cite journal |last=Kalman |first=R. E. |date=1960-03-01 |title=A New Approach to Linear Filtering and Prediction Problems |url=https://asmedigitalcollection.asme.org/fluidsengineering/article/82/1/35/397706/A-New-Approach-to-Linear-Filtering-and-Prediction |journal=Journal of Basic Engineering |language=en |volume=82 |issue=1 |pages=35–45 |doi=10.1115/1.3662552 |issn=0021-9223 |s2cid=259115248}}</ref><ref>Harvey, Andrew C. (1990). ''Forecasting, Structural Time Series Models and the Kalman Filter''. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107049994</ref>