State-space representation

See also: quantum state space and configuration space (physics)

In control engineering and system identification, a state-space representation is a mathematical model of a physical system specified as a set of input, output, and variables related by first-order differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables. Output variables’ values depend on the state variable values and may also depend on the input variable values.

The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector, the state vector.

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 the 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 or without modulation.[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 inputs and outputs, we would otherwise have to write down 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 or a state observer to produce estimates of the current unknown state variables using their previous observations.[11][12]

  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. ^ Vasilyev A.S.; Ushakov A.V. (2015). "Modeling of dynamic systems with modulation by means of Kronecker vector-matrix representation". Scientific and Technical Journal of Information Technologies, Mechanics and Optics. 15 (5): 839–848. doi:10.17586/2226-1494-2015-15-5-839-848.
  4. ^ 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
  5. ^ Durbin, James; Koopman, Siem Jan (2012). Time series analysis by state space methods. Oxford University Press. ISBN 978-0-19-964117-8. OCLC 794591362.
  6. ^ 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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.
  10. ^ "State-Space Models with Markov Switching and Gibbs-Sampling", State-Space Models with Regime Switching, The MIT Press, 2017, doi:10.7551/mitpress/6444.003.0013, ISBN 978-0-262-27711-2
  11. ^ 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.
  12. ^ Harvey, Andrew C. (1990). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press. doi:10.1017/CBO9781107049994