2016 , Volume 21, ¹ 4, p.64-98
Kulikova M.V., Kulikov G.Y.
Numerical methods for nonlinear filtering of signals and measurements
This paper studies numerical methods of contemporary nonlinear Kalman filtering for estimation of unknown vector of state in stochastic continuous-time systems presented by Ito-type stochastic differential equations with discrete measurements. The elaborated methods are analysed and compared in the case of severe conditions of tackling a seventh-dimensional radar tracking problem, where an aircraft executes a coordinated horizontal turn. The latter problem is considered to be a challenging example for testing nonlinear filtering algorithms. This paper explores such effective state estimation methods as the cubature and unscented Kalman filters, including their square-root versions. Implementation particulars and performances of the mentioned techniques are studied for various values of aircraft’s turn rate and sampling time. New variants of the extended and unscented Kalman filters are also presented for treating continuousdiscrete stochastic systems. It is shown that the new methods outperform the traditional extended Kalman filter in the considered air traffic control scenario.
Keywords: Ito-type stochastic differential equations with discrete measurements, optimal estimate of the state vector of stochastic system, extended Kalman filter, unscented Kalman filter, cubature Kalman filter
Kulikova Maria Vyacheslavovna
Office: CEMAT Instituto Superior Tecnico Universidade de Lisboa
Address: Portugal, Lissabone, 1049-001 Lisboa
Kulikov Gennadiy Yurievich
Dr. , Associate Professor
Position: Senior Research Scientist
Office: CEMAT, Instituto Superior Tecnico, Lissabone
Address: Portugal, Lissabone, Av. Rovisco Pais 1
Phone Office: (351) 21401 607
 Jazwinsky, A. Stochastic processes and filtering theory. N. Y.: Academic Press; 1970: 376.
 Kalman, R.E., Bucy, R.S. New results in linear filtering and prediction theory. Journal of Basic Engineering. 1961; 83(1):95–108.
 Arasaratnam, I., Haykin, S. Cubature Kalman filters. IEEE Transactions on Automatic Control. 2009; 54(6):1254–1269.
 Sinitsyn, I.N. Fil'try Kalmana i Pugacheva [Kalman and Pugachev filters]. Moscow: Universitetskaya Kniga, Logos; 2006: 643. (In Russ.)
 Maybeck, P.S. Stochastic models, estimation and control. London: Acad. Press; 1982: 291.
 Sarkka, S. On unscented Kalman filter for state estimation of continuous-time nonlinear systems. IEEE Transactions on Automatic Control. 2007; 52(9):1631–1641.
 Benzerrouk, H., Nebylov, A. Robust nonlinear filtering applied to integrated navigation system INS/GNSS under non Gaussian measurement noise effect. Proc. of the IEEE “Aerospace Conference”. 2012; USA,MT: Big Sky; 2012:1-8.
 Benzerrouk, H., Nebylov, A., Salhi, H. Contribution in information signal processing for solving state space nonlinear estimation problems. Journal of Signal and Information Processing. 2013; 4(4):375–384.
 Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H. A new approach for filtering nonlinear systems. Proc. of the “American Control Conference”. Seattle, WA; 1995: 1628–1632.
 Julier, S.J., Uhlmann, J.K. A new extension of the Kalman filter to nonlinear systems. Proc. of the Intern. Conf. “Signal Processing, Sensor Fusion, and Target Recognition”. Orlando FL; 1997: 182–193.
 Julier, S., Uhlmann, J., Durrant-Whyte, H. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control. 2000; 45(3):477–482.
 Wan, E.A., Van der Merwe, R. The unscented Kalman filter for nonlinear estimation. Proc. of the Adaptive Systems for Signal Processing, Communications, and Control Symposium. Lake Louise, Alta; 2000: 666–672.
 Ito, K., Xiong, K. Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic Control. 2000; 45(5):910–927.
 Nørgaard, M., Poulsen, N.K., Ravn, O. New developments in state estimation for nonlinear systems. Automatica. 2000; (36):1627–1638.
 Arasaratnam, I., Haykin, S., Elliott, R.J. Discrete-time nonlinear filtering algorithms using Gauss — Hermite quadrature. Proc. IEEE. 2007; 95(5):953–977.
 Van der Merwe, R., Wan, E.A. Efficient derivative-free Kalman filters for online learning. Proc. of the European Symposium on Artificial Neural Networks. Bruges, Belgium; 2001: 205–210.
 Julier, S., Uhlmann, J. Unscented filtering and nonlinear estimation. Proc. IEEE. 2004; 92(3):401–422.
 Van der Merwe, R., Wan, E.A. The square-root unscented Kalman filter for state and parameter-estimation. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing. Salt Lake City, UT. 2001; (6):3461–3464.
 Liu, G., Worgotter, F., Markelic, I. Square-root sigma-point information filtering. IEEE Transactions on Automatic Control. 2012; 57(11):2945–2951.
 Nørgaard, M., Poulsen, N.K., Ravn, O. Advances in derivative-free state estimation for nonlinear systems. Tech. Rep. IMM-REP-1998-15, Tech. Univ. of Denmark; 2000.
 Arasaratnam, I., Haykin, S. Square-root quadrature Kalman filtering. IEEE Transactions on Signal Processing. 2008; 56(6):2589–2593.
 Tang, X., Wei, J., Chen, K. Square-root adaptive cubature Kalman filter with application to spacecraft attitude estimation. Proc. of the Conf. “Information Fusion (FUSION)”. Singapore; 2012: 1406–1412.
 Chandra, K.P.B., Gu, D.-W., Postlethwaite, I. Square root cubature information filter. IEEE Sensors Journal. 2013; 13(2):750–758.
 Arasaratnam, I., Haykin, S. Cubature Kalman smoother. Automatica. 2011; (47):2245–2250.
 Wan, E.A., van der Merwe, R. The unscented Kalman filter. Chapter 7: Kalman filtering and neural networks. Ed. S. Haykin. N. Y.: John Wiley & Sons; 2001: 221–280.
 Arasaratnam, I., Haykin, S., Hurd, T.R. Cubature Kalman filtering for ñontinuousdiscrete systems: theory and simulations. IEEE Transactions Signal Processing. 2010; 58(10):4977–4993.
 Konakov, A.S., Shavrin, V.V., Tislenko, V.I., Savin, A.A. Comparative analysis of the mean square error determining the coordinates of an object in a strapdown inertial navigation system using a variety of nonlinear filtering algorithms. Proceedings of Tomsk State University of Control Systems and Radioelectronics. 2012; (1):5–9. (In Russ.)
 Sytnik, A.A., Rayevskiy, N.V., Klyuchka, K.N., Protasov, S.Yu. Comparison of methods of the filtration in problems of statistical regularization at estimation of parameters of radar-tracking systems. Proceedings of Voronezh State University. Series: System analysis and information technologies. 2013; (1):10–16. (In Russ.)
 Kailath, T., Sayed, A. H., Hassibi, B. Linear estimation. New Jersey: Prentice Hall; 2000: 854.
 Grewal, M.S., Andrews, A.P. Kalman filtering: theory and practice. New Jersey: Prentice Hall; 2001: 401.
 Kulikova, M.V., Semoushin, I.V. Score evaluation within the extended square-root information filter. Lecture Notes in Computer Science. 2006; (3991):473–481.
 Kulikova, M.V. Likelihood gradient evaluation using square-root covariance filters. IEEE Transactions on Automatic Control. 2009; 54(3):646–651.
 Tsyganova, Yu.V., Kulikova, M.V. On efficient parametric identification methods for linear discrete stochastic systems. Automation and Remote Control. 2012; 73(6):962–975.
 Kulikova, M.V., Pacheco, A. Kalman filter sensitivity evaluation with orthogonal and J-orthogonal transformations. IEEE Transactions on Automatic Control. 2013; 58(7):1798–1804.
 Tsyganova, J.V., Kulikova, M.V. State sensitivity evaluation within UD based array covariance filter. IEEE Transactions on Automatic Control. 2013; 58(11):2944–2950.
 Kulikova, M.V., Tsyganova, Yu.V. A general approach to constructing parameter identification algorithms in the class of square root filters with orthogonal and J-orthogonal tranformations. Automation and Remote Control. 2014; 75(8):1402–1419.
 Julier, S.J. The scaled unscented transformation. Proc. of the “American Control Conference”. Anchorage; 2002:4555–4559.
 Frogerais, P., Bellanger, J.-J., Senhadji, L. Various ways to compute the continuousdiscrete extended Kalman filter. IEEE Transactions on Automatic Control. 2012; 57(4):1000– 1004.
 Kloeden P.E., Platen E. Numerical solution of stochastic differential equations. Berlin: Springer; 1999: 856.
Kulikova M.V., Kulikov G.Y. Numerical methods for nonlinear filtering of signals and measurements // Computational technologies. 2016. V. 21. ¹ 4. P. 64-98