Kalmanov filter

The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate. The estimate is updated using a state transition model and measurements. ${\hat {x}}_{k\mid k-1}$ denotes the estimate of the system's state at time step k before the k-th measurement y_k has been taken into account; $P_{k\mid k-1}$ is the corresponding uncertainty.

For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich. In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before the summer of 1961, when Kalman met with Stratonovich during a conference in Moscow.

Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically. Furthermore, Kalman filtering is a concept much applied in time series analysis used for topics such as signal processing and econometrics. Kalman filtering is also one of the main topics of robotic motion planning and control and can be used for trajectory optimization. Kalman filtering also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands.

The algorithm works by a two-phase process having a prediction phase and an update phase. For the prediction phase, the Kalman filter produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with greater certainty. The algorithm is recursive. It can operate in real time, using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required.

Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán: "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense, although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian.

Extensions and generalizations of the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion, and distributed sensor networks to develop distributed or consensus Kalman filtering.

Literatura

• Einicke, G.A. (2019). Smoothing, Filtering and Prediction: Estimating the Past, Present and Future (2nd ed.). Amazon Prime Publishing. ISBN 978-0-6485115-0-2. 
• Jinya Su; Baibing Li; Wen-Hua Chen (2015). „On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs”. Automatica. 53: 149—154. doi:10.1016/j.automatica.2014.12.044 . 
• Gelb, A. (1974). Applied Optimal Estimation. MIT Press. 
• Kalman, R.E. (1960). „A new approach to linear filtering and prediction problems” (PDF). Journal of Basic Engineering. 82 (1): 35—45. S2CID 1242324. doi:10.1115/1.3662552. Архивирано из оригинала (PDF) 2008-05-29. г. Приступљено 2008-05-03. 
• Kalman, R.E.; Bucy, R.S. (1961). „New Results in Linear Filtering and Prediction Theory”. Journal of Basic Engineering. 83: 95—108. CiteSeerX 10.1.1.361.6851 . S2CID 8141345. doi:10.1115/1.3658902. 
• Harvey, A.C. (1990). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press. ISBN 978-0-521-40573-7. 
• Roweis, S.; Ghahramani, Z. (1999). „A Unifying Review of Linear Gaussian Models” (PDF). Neural Computation. 11 (2): 305—345. PMID 9950734. S2CID 2590898. doi:10.1162/089976699300016674. 
• Simon, D. (2006). Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley-Interscience. Архивирано из оригинала 2010-12-30. г. Приступљено 2006-07-05. 
• Warwick, K. (1987). „Optimal observers for ARMA models”. International Journal of Control. 46 (5): 1493—1503. doi:10.1080/00207178708933989. 
• Bierman, G.J. (1977). Factorization Methods for Discrete Sequential Estimation. Mathematics in Science and Engineering. 128. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-44981-4. 
• Bozic, S.M. (1994). Digital and Kalman filtering. Butterworth–Heinemann. 
• Haykin, S. (2002). Adaptive Filter Theory. Prentice Hall. 
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• Manolakis, D.G. (1999). Statistical and Adaptive signal processing. Artech House. 
• Welch, Greg; Bishop, Gary (1997). „SCAAT: incremental tracking with incomplete information” (PDF). SIGGRAPH '97 Proceedings of the 24th annual conference on Computer graphics and interactive techniques. ACM Press/Addison-Wesley Publishing Co. стр. 333—344. ISBN 978-0-89791-896-1. S2CID 1512754. doi:10.1145/258734.258876. 
• Jazwinski, Andrew H. (1970). Stochastic Processes and Filtering. Mathematics in Science and Engineering. New York: Academic Press. стр. 376. ISBN 978-0-12-381550-7. 
• Maybeck, Peter S. (1979). „Chapter 1” (PDF). Stochastic Models, Estimation, and Control. Mathematics in Science and Engineering. 141-1. New York: Academic Press. ISBN 978-0-12-480701-3. 
• Moriya, N. (2011). Primer to Kalman Filtering: A Physicist Perspective. New York: Nova Science Publishers, Inc. ISBN 978-1-61668-311-5. 
• Dunik, J.; Simandl M.; Straka O. (2009). „Methods for Estimating State and Measurement Noise Covariance Matrices: Aspects and Comparison”. 15th IFAC Symposium on System Identification, 2009. France. стр. 372—377. ISBN 978-3-902661-47-0. doi:10.3182/20090706-3-FR-2004.00061. 
• Chui, Charles K.; Chen, Guanrong (2009). Kalman Filtering with Real-Time Applications. Springer Series in Information Sciences. 17 (4th изд.). New York: Springer. стр. 229. ISBN 978-3-540-87848-3. 
• Spivey, Ben; Hedengren, J. D.; Edgar, T. F. (2010). „Constrained Nonlinear Estimation for Industrial Process Fouling”. Industrial & Engineering Chemistry Research. 49 (17): 7824—7831. doi:10.1021/ie9018116. 
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Spoljašnje veze

• A New Approach to Linear Filtering and Prediction Problems, by R. E. Kalman, 1960
• Kalman and Bayesian Filters in Python. Open source Kalman filtering textbook.
• How a Kalman filter works, in pictures. Illuminates the Kalman filter with pictures and colors
• Kalman–Bucy Filter, a derivation of the Kalman–Bucy Filter
• MIT Video Lecture on the Kalman filter на сајту YouTube
• Kalman filter in Javascript. Open source Kalman filter library for node.js and the web browser.
• An Introduction to the Kalman Filter Архивирано 2021-02-24 на сајту Wayback Machine, SIGGRAPH 2001 Course, Greg Welch and Gary Bishop
• Kalman Filter webpage, with many links
• Kalman Filter Explained Simply, Step-by-Step Tutorial of the Kalman Filter with Equations
• „Kalman filters used in Weather models” (PDF). SIAM News. 36 (8). октобар 2003. Архивирано из оригинала (PDF) 2011-05-17. г. Приступљено 2007-01-27. CS1 одржавање: Формат датума (веза)
• Haseltine, Eric L.; Rawlings, James B. (2005). „Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation”. Industrial & Engineering Chemistry Research. 44 (8): 2451. doi:10.1021/ie034308l. 
• Gerald J. Bierman's Estimation Subroutine Library: Corresponds to the code in the research monograph "Factorization Methods for Discrete Sequential Estimation" originally published by Academic Press in 1977. Republished by Dover.
• Matlab Toolbox implementing parts of Gerald J. Bierman's Estimation Subroutine Library: UD / UDU' and LD / LDL' factorization with associated time and measurement updates making up the Kalman filter.
• Matlab Toolbox of Kalman Filtering applied to Simultaneous Localization and Mapping: Vehicle moving in 1D, 2D and 3D
• The Kalman Filter in Reproducing Kernel Hilbert Spaces A comprehensive introduction.
• Matlab code to estimate Cox–Ingersoll–Ross interest rate model with Kalman Filter Архивирано 2014-02-09 на сајту Wayback Machine: Corresponds to the paper "estimating and testing exponential-affine term structure models by kalman filter" published by Review of Quantitative Finance and Accounting in 1999.
• Online demo of the Kalman Filter. Demonstration of Kalman Filter (and other data assimilation methods) using twin experiments.
• Botella, Guillermo; Martín h., José Antonio; Santos, Matilde; Meyer-Baese, Uwe (2011). „FPGA-Based Multimodal Embedded Sensor System Integrating Low- and Mid-Level Vision”. Sensors. 11 (12): 1251—1259. Bibcode:2011Senso..11.8164B. PMC 3231703 . PMID 22164069. doi:10.3390/s110808164 . 
• Examples and how-to on using Kalman Filters with MATLAB A Tutorial on Filtering and Estimation
• Explaining Filtering (Estimation) in One Hour, Ten Minutes, One Minute, and One Sentence by Yu-Chi Ho
• Simo Särkkä (2013). "Bayesian Filtering and Smoothing". Cambridge University Press. Full text available on author's webpage https://users.aalto.fi/~ssarkka/.