Abstract
A nonlinear time-varying adaptive filter is introduced, and its derivation using optimal control concepts is given in detail. The filter, which is called the discrete Pontryagin filter, is basically an extension to Sridhar filtering theory. The proposed approach can easily replace the conventional methods of autoregressive (AR) and autoregressive moving average (ARMA) models in their many applications. Instead of using a large number of time-invariant parameters to describe the signal or the time series, a single time-varying function is enough. This function is estimated using optimization techniques. Many features are gained using this approach, such as simpler and compact filter equations and better overall accuracy. The statistical properties of the filter are given, and it is shown that the signal estimate will converge in thepth mean to the true value.
Similar content being viewed by others
References
Kalman, R. E.,A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering, Vol. 82, pp. 35–45, 1960.
Bellman, R., Kagiwada, H., Kalaba, R., andSridhar, R.,Invariant Imbedding and Nonlinear Filtering Theory, Journal of Astronautical Sciences, Vol. 13, pp. 110–115, 1966.
Detchmendy, D., andSridhar, R.,Sequential Estimation of States and Parameters in Noisy Nonlinear Dynamical Systems, Journal of Basic Engineering, Ser. D, Vol. 88, pp. 362–368, 1966.
Kagiwada, H., Kalaba, R. A., Schumitsky, A., andSridhar, R.,Invariant Imbedding and Sequential Interpolating Filters for Nonlinear Processes, Journal of Basic Engineering, Vol. 91, pp. 195–201, 1969.
Sage, A. P., andMelsa, J. L.,Estimation Theory with Applications to Communications and Control, McGraw-Hill, New York, New York, 1971.
Tesfatsion, L.,A New Approach to Filtering and Adaptive Control, Journal of Optimization Theory and Applications, Vol. 25, pp. 247–261, 1978.
Tesfatsion, L.,Direct Updating of Intertemporal Criterion Functions for a Class of Adaptive Control Problems, IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-9, pp. 143–151, 1979.
Kalaba, R., andTesfatsion, L.,An Exact Sequential Solution Procedure for a Class of Discrete-Time Nonlinear Estimation Problems, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 1144–1149, 1981.
Widrow, B., Mantley, P., Griffiths, L., andGoode, B.,Adaptive Antenna Systems, Proceedings of the IEEE, Vol. 55, pp. 2143–2159, 1967.
Cooley, T. F., andPrescott, E. C.,Estimation in the Presence of Stochastic Parameter Variations, Econometrica, Vol. 44, pp. 167–183, 1976.
Pagan, A. R.,Some Identification and Estimation Results for Regression Models with Stochastically Varying Coefficients, Journal of Econometrics, Vol. 13, pp. 341–364, 1980.
Chow, G. C.,Random and Changing Coefficient Models, Handbook of Econometrics, Edited by Z. Griliches and M. Intriligator, North-Holland, Amsterdam, Holland, Vol. 2, 1984.
Widrow, B., andStearns, S. D.,Adaptive Signal Processing, Prentice-Hall, Englewood Cliffs, New Jersey, 1985.
Friedlander, B.,System Identification for Adaptive Noise Cancelling, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-30, pp. 699–708, 1982.
Ljung, L., andSoderstrom, T.,Theory and Practice of Recursive Identification, MIT Press, Cambridge, Massachusetts, 1982.
Narayan, S. S., Peterson, A. M., andNarashima, J. J.,Transform Domain LMS Algorithm, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-31, pp. 609–615, 1983.
Mikhael, W. B., Wu, F., Kang, G., andFransen, L.,Optimum Adaptive Algorithms with Applications to Noise Cancellation, IEEE Transactions on Circuits and Systems, Vol. CAS-31, pp. 312–315, 1984.
Dembo, A., andZeitouni, O.,On the Parameter Estimation of Continuous-Time ARMA Processes from Noisy Observations, IEEE Transactions on Automatic Control, Vol. AC-32, pp. 361–364, 1987.
Dembo, A., andZeitouni, O.,Parameter Estimation of Partially Observed Continuous-Time Processes via the EM Algorithm, Stochastic Processes and Their Applications, Vol. 23, pp. 91–113, 1987.
Zeitouni, O., andDembo, A.,A Maximum A-Posteriori Estimator for the Trajectory of Diffusion Processes, Stochastics, Vol. 20, pp. 211–246, 1987.
Shumway, R., andStoffer, D.,An Approach to Time Series Smoothing and Forecasting Using the EM Algorithm, Journal of Time Series Analysis, Vol. 3, pp. 253–264, 1982.
Abutaleb, A.,New Results in Sridhar Filtering Theory, Journal of the Franklin Institute, Vol. 322, pp. 229–240, 1986.
Goodwin, G. C., andSin, K. S.,Adaptive Filtering, Prediction, and Control, Prentice-Hall, Englewood Cliffs, New Jersey, Chapter 8, 1984.
Graupe, D.,Time Series Analysis, Identification, and Adaptive Filtering, Krieger Publishing Company, Malabar, Florida, 1984.
Author information
Authors and Affiliations
Additional information
Communicated by R. Kalaba
Rights and permissions
About this article
Cite this article
Abutaleb, A.S., Papaiouannou, M. New results in Sridhar filtering theory: The discrete case. J Optim Theory Appl 64, 5–14 (1990). https://doi.org/10.1007/BF00940018
Issue Date:
DOI: https://doi.org/10.1007/BF00940018