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State-space representation

Not to be confused with quantum state space or configuration space (physics).

In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or 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 systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector.

For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form,[1][2] offering a compact alternative to the frequency domain’s 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 structures for efficient analysis.

State-space models are applied in fields such as economics,[3] statistics,[4] computer science, electrical engineering,[5] and neuroscience.[6] 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,[7] identify turning points of the business cycle, and estimate GDP using latent and unobserved time series.[8][9] Many applications rely on the Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations.[10][11]

  1. ^ Katalin M. Hangos; R. Lakner & M. Gerzson (2001). Intelligent Control Systems: An Introduction with Examples. Springer. p. 254. ISBN 978-1-4020-0134-5.
  2. ^ Katalin M. Hangos; József Bokor & Gábor Szederkényi (2004). Analysis and Control of Nonlinear Process Systems. Springer. p. 25. ISBN 978-1-85233-600-4.
  3. ^ Stock, J.H.; Watson, M.W. (2016), "Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics", Handbook of Macroeconomics, vol. 2, Elsevier, pp. 415–525, doi:10.1016/bs.hesmac.2016.04.002, ISBN 978-0-444-59487-7
  4. ^ Durbin, James; Koopman, Siem Jan (2012). Time series analysis by state space methods. Oxford University Press. ISBN 978-0-19-964117-8. OCLC 794591362.
  5. ^ Roesser, R. (1975). "A discrete state-space model for linear image processing". IEEE Transactions on Automatic Control. 20 (1): 1–10. doi:10.1109/tac.1975.1100844. ISSN 0018-9286.
  6. ^ Smith, Anne C.; Brown, Emery N. (2003). "Estimating a State-Space Model from Point Process Observations". Neural Computation. 15 (5): 965–991. doi:10.1162/089976603765202622. ISSN 0899-7667. PMID 12803953. S2CID 10020032.
  7. ^ James H. Stock & Mark W. Watson, 1989. "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.
  8. ^ Bańbura, Marta; Modugno, Michele (2012-11-12). "Maximum Likelihood Estimation of Factor Models on Datasets with Arbitrary Pattern of Missing Data". Journal of Applied Econometrics. 29 (1): 133–160. doi:10.1002/jae.2306. hdl:10419/153623. ISSN 0883-7252. S2CID 14231301.
  9. ^ "State-Space Models with Markov Switching and Gibbs-Sampling", State-Space Models with Regime Switching, The MIT Press, pp. 237–274, 2017, doi:10.7551/mitpress/6444.003.0013, ISBN 978-0-262-27711-2
  10. ^ Kalman, R. E. (1960-03-01). "A New Approach to Linear Filtering and Prediction Problems". Journal of Basic Engineering. 82 (1): 35–45. doi:10.1115/1.3662552. ISSN 0021-9223. S2CID 259115248.
  11. ^ Harvey, Andrew C. (1990). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107049994

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