Abstract
The problem of the optimal estimation of continuous processes by discrete measurements in the presence of time lag (delay) is considered. On the basis of the theory of parametric transfer functions, an optimal, periodically nonstationary filter is developed, which affords a minimum of the estimation error variance at any instant of time. The comparison is performed of the obtained solution with the optimal stationary filter, which ensures a minimum of the mean (by continuous time) error variance. It is shown that in the problem of estimation of the Markov process of the first order, a simpler stationary filter with the fixer of order zero is insignificantly inferior to the optimal filter.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yarlykov, M.S., Primenenie markovskoi teorii nelineinoi fil’tratsii v radiotekhnike (Use of Markov Theory of Nonlinear Filtering in Radio Engineering), Moscow: Sovetskoe Radio, 1980.
Yarlykov, M.S., Statisticheskaya teoriya radionavigatsii (Statistical Theory of Radio Navigation), Moscow: Radio i Sviaz’, 1985.
Yarlykov, M.S. and Mironov, M.A., Markovskaya teoriya otsenivaniya sluchainykh protsessov (Markov Theory of Estimation of Random Processes), Moscow: Radio i Svyaz’, 1993.
Rosenwasser, E.N., Lineinaya teoriya tsifrovogo upravleniya v nepreryvnom vremeni (Linear Theory of Digital Control in Continuous Time), Moscow: Nauka, 1994.
Milocco, R. and DeDoná, J., Optimal Parallel Model Solutions Using the Polynomial Approach, Int. J. Control, 1996, vol. 5, no. 4, pp. 681–697.
Milocco, R.H. and Muravchik, C.H., \( \mathcal{H}_2 \)-Optimal Linear Robust Sampled-Data Filtering Design Using Polynomial Approach, IEEE Trans. Signal Proc., 2003, vol. SP-51, no. 7, pp. 1816–1824.
Wang, Z., Huang, B., and Huo, P., Sampled-Data Filtering with Error Covariance Assignment, IEEE Trans. Signal Proc., 2001, vol. SP-49, no. 3, pp. 666–670.
Sun, W., Nagpal, K.M., and Khargonekar, P.P., \( \mathcal{H}_\infty \) Control and Filtering for Sampled-Data Systems, IEEE Trans. Automat. Control, 1993, vol. AC-38, no. 8, pp. 1162–1175.
Mirkin, L. and Palmor, Z., On Sampled-Data \( \mathcal{H}_\infty \) Filtering Problem, Automatica, 1999, vol. 35, no. 5, pp. 895–905.
Rosenwasser, E.N., Polyakov, K.Y., and Lampe, B.P., Optimal Discrete Filtering for Time-Delayed Systems with Respect to Mean-Square Continuous-Time Error Criterion, Int. J. Adapt. Control Signal Proc., 1998, vol. 12, no. 5, pp. 389–406.
Rosenwasser, E.N. and Lampe, B.P., Multivariable Computer Controlled Systems: A Transfer Function Approach, London: Springer-Verlag, 2006.
Astrom, K.J., Introduction to Stochastic Control Theory, New York: Academic, 1970.
Rosenwasser, E.N., Periodicheski nestatsionarnye sistemy upravleniya (Periodically Nonstationary Control Systems), Moscow: Nauka, 1973.
Fihtengolts, G.M., Kurs differentsial’nogo i integral’nogo ischisleniya (Course of Differential and Integral Calculus), Moscow: Fizmatgiz, 1970.
Rosenwasser, E.N. and Lampe, B.P., Computer Controlled Systems: Analysis and Design with Process Orientated Models, London: Springer-Verlag, 2000.
Feuer, A. and Goodwin, G.C., Generalized Sample and Hold Functions—Frequency Domain Analysis of Robustness, Sensitivity and Intersampling Difficulties, IEEE Trans. Automat. Control, 1994, vol. AC-39, no. 5, pp. 1042–1047.
Barabanov, A.E., Optimal Control of the Linear Object with Stationary Noise and the Quadratic Quality Criterion, Available from VINITI, 1979, no. 3478-79.
Polyakov, K.Yu., Singular Problems of \( \mathcal{H}_2 \)-Optimization of Discrete Systems, Avtom. Telemekh., 2005, no. 3, pp. 20–33.
Polyakov, K.Yu. and Rybinskii, V.O., Optimal Digital Filtering of Continuous Signals by the Criterion of Minimum of Mean Variance, Proc. 6th Conf. Young Scientists “Navigation and Motion Control,” St. Petersburg: GNTS “Elektropribor,” 2005, pp. 91–98.
Polyakov, K.Yu. and Rosenwasser, E.N., Direct SD2.0—Package for Analysis and Direct Synthesis of Digital Control Systems, Proc. All-Russ. Scien. Conf. “Design of Scientific and Engineering Applications in MATLAB Medium,” Moscow, 2002, pp. 74–78.
Zhou, K., Doyle, J.C., and Glover, K., Robust and Optimal Control, New Jersey: Prentice Hall, 1996.
Van Loan, C.F., Computing Integrals Involving Matrix Exponential, IEEE Trans. Automat. Control, 1978, vol. AC-23, pp. 395–404.
Chang, Sh.S.L., Sintez optimal’nykh system avtomaticheskogo upravleniya (Synthesis of Optimal Automatic Control Systems), Moscow: Mashinostroenie, 1964.
Lampe, B.P. and Rosenwasser, E.N., Modified Wiener-Hopf Method in the Problem of \( \mathcal{H}_2 \)-Optimization of Pulse Systems, Avtom. Telemekh., 1999, no. 3, pp. 156–169.
Author information
Authors and Affiliations
Additional information
Original Russian Text © K.Yu. Polyakov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 5, pp. 135–150.
Rights and permissions
About this article
Cite this article
Polyakov, K.Y. Optimal digital filtering of continuous signals in time lag systems. Autom Remote Control 69, 858–873 (2008). https://doi.org/10.1134/S0005117908050111
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117908050111