Optimal filtering for linear systems with state and observation delays. (English) Zbl 1086.93058

Summary: The optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman-Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman-Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable \((\tau =qh, q\) is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman-Bucy filter available for linear systems without delays and two versions of the extended Kalman-Bucy filter for time delay systems.


93E11 Filtering in stochastic control theory
Full Text: DOI


[1] Kalman, Journal of Basic Engineering, Part D 83 pp 95– (1961)
[2] Alexander, SPIE Data Structures and Target Classification 1470 (1991)
[3] . Discrete-time state estimation with two counters and measurement delay. Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996; 1472-1476.
[4] Hsiao, Journal of Dynamic Systems, Measurement, and Control 118 pp 803– (1996)
[5] , , . Incorporation of the time-delayed measurements in a discrete-time Kalman filter. Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, U.S.A., 1998; 3972-3977.
[6] Basin, International Journal of Robust and Nonlinear Control 14 pp 685– (2004)
[7] Shi, IEEE Transactions on Automatic Control 43 pp 1022– (1998)
[8] Basin, Asian Journal of Control 5 pp 557– (2003)
[9] Basin, Journal of The Franklin Institute 341 pp 267– (2004)
[10] Mahmoud, IEEE Transactions on Circuits and Systems 50 pp 98– (2003)
[11] . Control of Systems with Aftereffect. American Mathematical Society: Providence, 1996.
[12] . Introduction to the Theory and Applications of Functional Differential Equations. Kluwer: New York, 1999.
[13] . Time-Delay Systems: Analysis, Optimization and Applications. North-Holland: Amsterdam, 1987.
[14] Robust Control and Filtering for Time-Delay Systems. Marcel Dekker: New York, 2000.
[15] Delay Effects on Stability: A Robust Control Approach. Springer: Heidelberg, 2001.
[16] . Deterministic and Stochastic Time-Delayed Systems. Birkhäuser: Boston, 2002.
[17] Gu, Journal of Dynamic Systems, Measurement and Control 125 pp 158– (2003)
[18] Richard, Automatica 39 pp 1667– (2003)
[19] . Stochastic Systems: Theory and Applications. World Scientific: Singapore, 2001.
[20] Basin, IEEE Transactions on Automatic Control 50 pp 684– (2005)
[21] Stochastic Processes and Filtering Theory. Academic Press: New York, 1970. · Zbl 0203.50101
[22] Applied Optimal Estimation. MIT Press: Cambridge, MA, 1974.
[23] Introduction to Stochastic Control Theory. Academic Press: New York, 1970. · Zbl 0226.93027
[24] Feedback Control Systems. McGraw-Hill: New York, 1958.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.