Abstract
We consider an application of the auxiliary quality functional (AQF) method for identifying the parameters of a linear dynamical system in case when the filtering is done with a square root covariance filter. We construct a new algorithm for computing the gradient of the auxiliary quality functional. The advantages of this algorithm are that it is stable to computer rounding errors and does not require the user to write down the “differentiated” Kalman filter in the standard form for every unknown system parameter. All values necessary to compute the values of the AQF gradient are computed in terms of the square root covariance filter with orthogonal transformations.
Similar content being viewed by others
References
Semushin, I.V., Using the Active Filtering Principle for Nonstationary Stochastic Processes, in Proc. of III NTK, Novgorod Branch of Ulyanov (Lenin) LETI, Novgorod: LETI Branch, 1968, p. 64.
Hampton, R.L.T., On Unknown State-Dependent Noise, Modeling Errors, and Adaptive Filtering, Comput. Elect. Engng., 1975, vol. 2, pp. 195–201.
Semoushin, I.V. and Tsyganova, J.V., Auxiliary Performance Functional Approach to Adaptive and Learning Filtering and Control, Conf. Proc. Eur. Control Conf. ECC’99, Karlsruhe, 1999.
Semoushin, I.V. and Tsyganova, J.V., Indirect Error Control for Adaptive Filtering, Proc. Third Eur. Conf. Numerical Math. Advanced Appl. ENUMATH’99, Jyvaskyla, July 26–30, 1999, Singapore: World Scientific, 2000, pp. 333–340.
Bierman, G.J., Belzer, M.R., Vandercraft, J.S., and Porter, D.W., Maximum Likelihood Estimation Using Square Root Information Filters, IEEE Trans. Automat. Control, 1990, vol. 35, no. 12, pp. 1293–1299.
Kulikova, M.V. and Semoushin, I.V., On the Evaluation of Log Likelihood Gradient for Gaussian Signals, Int. J. Appl. Math. Statist., 2005, vol. 3, no. 5, pp. 1–14.
Kulikova, M.V., New Square-Root Algorithms for Log-Likelihood Gradient Evaluation, IEEE Trans. Automat. Control, 2009, vol. 54, no. 3, pp. 646–651.
Kulikova, M.V., Maximum Likelihood Estimation via the Extended Covariance and Combined Square-Root Filters, Math. Comput. Simulat., 2009, no. 79, pp. 1641–1657.
Grewal, M.S. and Andrews, A.P., Kalman Filtering: Theory and Practice Using MATLAB, New York: Wiley, 2001, 2nd ed.
Kaminski, P.G., Bryson, A.E., and Schmidt, S.F., Discrete Square Root Filtering: A Survey of Current Techniques, IEEE Trans. Automat. Control, 1971, vol. AC-16, no. 6, pp. 727–735.
Mosca, E., Optimal, Predictive and Adaptive Control, Upper Saddle River: Prentice-Hall, 1995.
Aström, K.J., Maximum Likelihood and Prediction Error Methods, Automatica, 1980, vol. 16, pp. 551–574.
Semushin, I.V., Identification of Linear Stochastic Objects from Incomplete Noisy Measurements of the State Vector, Autom. Remote Control, 1985, vol. 46, no. 8, part 1, pp. 975–985.
Semushin, I.V., Adaptive Control for a Stochastic Linear Object under Uncertainty, in Nelineinye dinamicheskie sistemy: kachestvennyi analiz i upravlenie (Nonlinear Dynamical Systems: Qualitative Analysis and Control), Moscow: Inst. Sist. Anal., 1994, no. 2, pp. 104–110.
Ljung, L., System Identification: Theory for the User, Upper Saddle River: Prentice Hall, 1999. Translated under the title Identifikatsiya sistem. Teoriya dlya pol’zovatelya, Tsypkin, Ya.Z., Ed., Moscow: Nauka, 1991.
Saridis, G.N., Self-organizing Control of Stochastic Systems, New York: Marcel Dekker, 1977. Translated under the title Samoorganizuyushchiesya stokhasticheskie sistemy upravleniya, Tsypkin, Ya.Z., Ed., Moscow: Nauka, 1980.
Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, New York: Academic, 1977.
Verhaegen, M. and Van Dooren, P., Numerical Aspects of Different Kalman Filter Implementations, IEEE Trans. Automat. Control, 1986, vol. AC-31, no. 10, pp. 907–917.
Faddeev, D.K. and Faddeeva, V.N., Vychislitel’nye metody lineinoi algebry (Computational Methods in Linear Algebra), Moscow: Nauka, 1963.
Schmidt, S.F., Computational Techniques in Kalman Filtering, in Theory Appl. Kalman Filtering, London: NATO Advisory Group for Aerospace Research and Development, 1970, no. 139.
Battin, R., Astronautical Guidance, New York: McGraw-Hill, 1964. Translated under the title Navedenie v kosmose, Moscow: Mashinostroenie, 1966.
Bellantoni, J.F. and Dodge, K.W., A Square Root Formulation of the Kalman-Schmidt Filter, AIAA J., 1967, vol. 5, pp. 1309–1314.
Andrews, A., A Square Root Formulation of the Kalman Covariance Equations, AISS J., 1968, vol. 6, pp. 1165–1166.
Park, P. and Kailath, T., New Square-root Algorithms for Kalman Filtering, IEEE Trans. Automat. Control, 1995, vol. 40, no. 5, pp. 895–899.
Author information
Authors and Affiliations
Additional information
Original Russian Text © Yu.V. Tsyganova, 2011, published in Avtomatika i Telemekhanika, 2011, No. 9, pp. 142–160.
Rights and permissions
About this article
Cite this article
Tsyganova, Y.V. Computing the gradient of the auxiliary quality functional in the parametric identification problem for stochastic systems. Autom Remote Control 72, 1925–1940 (2011). https://doi.org/10.1134/S0005117911090141
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117911090141