Itô-Taylor-based square-root unscented Kalman filtering methods for state estimation in nonlinear continuous-discrete stochastic systems. (English) Zbl 1458.93251
Summary: This paper addresses the problem of square-rooting in the unscented Kalman filtering (UKF) methods rooted in the Itô-Taylor approximation of the strong order 1.5. Since its discovery the UKF has become one of the most powerful state estimation means because of its outstanding performance in numerous stochastic systems of practical value, including continuous-discrete ones. Besides, the main shortcoming of this technique is the need for the Cholesky decomposition of covariance matrices derived in its time and measurement update steps. Such a factorization is time-consuming and highly sensitive to round-off and other errors committed in the course of computation, which can result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually overcome by means of square-root filter implementations, which propagate not the covariance itself but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of the UKF to high-dimensional stochastic systems preclude from designing conventional square-root UKF methods. We resolve it with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding \(J\)-orthogonal square roots. Our novel square-root filters are justified theoretically and examined and compared numerically to the existing UKF in a flight control scenario.
MSC:
| 93E11 | Filtering in stochastic control theory |
| 93C55 | Discrete-time control/observation systems |
| 93C10 | Nonlinear systems in control theory |
Keywords:
continuous-discrete nonlinear stochastic model; discrete-discrete unscented Kalman filter; orthogonal and \(J\)-orthogonal square-root implementations; radar tracking; turning aircraft scenario with ill-conditioned measurementsReferences:
| [1] | Al-Tayie, J. K.; Acarnley, P. P., Estimation of speed, stator temperature and rotor temperature in cage induction motor drive using the extended Kalman filter, IEE Proc. — Electr. Power Appl., 144, 5, 301-309 (1997) |
| [2] | Andrews, A., A square root formulation of the Kalman covariance equations, AIAA J., 6, 6, 1165-1166 (1968) · Zbl 0159.26403 |
| [3] | Arasaratnam, I.; Haykin, S., Cubature Kalman filters, IEEE Trans. Autom. Control, 54, 6, 1254-1269 (2009) · Zbl 1367.93637 |
| [4] | Arasaratnam, I.; Haykin, S.; Hurd, T. R., Cubature Kalman filtering for continuous-discrete systems: theory and simulations, IEEE Trans. Signal Process., 58, 10, 4977-4993 (2010) · Zbl 1391.93223 |
| [5] | Bellantoni, J. F.; Dodge, K. W., A square root formulation of the Kalman-Schmidt filter, AIAA J., 5, 7, 1309-1314 (1967) |
| [6] | Bhar, R.; Chiarella, C., Interest rate futures: estimation of volatility parameters in an arbitrage-free framework, Proceedings of the IEEE/IAFE 1996 Computational Intelligence for Financial Engineering, 168-182 (1996) |
| [7] | Björck, A., Numerical Methods in Matrix Computations (2015), Springer: Springer Cham · Zbl 1322.65047 |
| [8] | Bojanczyk, A.; Higham, N. J.; Patel, H., Solving the indefinite least squares problem by hyperbolic QR factorization, SIAM J. Matrix Anal. Appl., 24, 4, 914-931 (2003) · Zbl 1036.65035 |
| [9] | Boje, E.; Petrick, M., Application of the extended Kalman filter to a lysine hydrochlorination process, Control Eng. Pract., 8, 3, 291-297 (2000) |
| [10] | Caccia, M.; Bruzzone, G.; Veruggio, G., Active sonar-based bottom-following for unmanned underwater vehicles, Control Eng. Pract., 7, 4, 459-468 (1999) |
| [11] | Campbell, J. K.; Synnott, S. P.; Bierman, G. J., Voyager orbit determination at Jupiter, IEEE Trans. Autom. Control, 28, 3, 256-269 (1983) |
| [12] | Castro, G. J.; Nieto, J.; Gallego, L. M.; Pastor, L.; Cabello, E., An effective camera calibration method, Proceedings of the 5th International Workshop on Advanced Motion Control, 171-174 (1998) |
| [13] | Chen, H.; Kremling, A.; Allgöwer, F., Nonlinear predictive control of a benchmark CSTR, Proceedings of the 3rd European Control Conference ECC’95, 3247-3252 (1995) |
| [14] | Crowley, T. J.; Choi, K. Y., On-line monitoring and control of a batch polymetrization reactor, J. Process Control, 6, 2-3, 119-127 (1996) |
| [15] | Y. Diab, F. Auger, E. Schaeffer, M. Wahbeh, Estimating lithium-ion battery state of charge and parameters using a continuous-discrete extended Kalman filter, Energies 10(8) 1075. |
| [16] | Dyer, P.; McReynolds, S., Extensions of square root filtering to include process noise, J. Opt. Theory Appl., 3, 6, 444-458 (1969) · Zbl 0165.10602 |
| [17] | Ennola, K.; Sarvala, J.; Dévai, G., Modelling zooplankton population dynamics with the extended Kalman filter technique, Ecol. Model., 110, 2, 135-149 (1998) |
| [18] | Evenson, G., Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model, J. Geophys. Res., 97, C11, 17905-17924 (1992) |
| [19] | Fossen, T. I.; Sagatun, S. I.; Sorensen, A. J., Identification of dynamically positioned ships, Control Eng. Pract., 4, 3, 369-376 (1996) · Zbl 0875.93430 |
| [20] | Fung, P. T.-K.; Grimble, M. J., Dynamic ship positioning using a self-turned Kalman filter, IEEE Trans. Autom. Control, 28, 3, 339-350 (1983) · Zbl 0508.93052 |
| [21] | Gao, B.; Gao, S.; Hu, G.; Zhong, Y.; Gu, C., Maximum likelihood principle and moving horizon estimation based adaptive unscented Kalman filter, Aerosp. Sci. Technol., 73, 184-196 (2018) |
| [22] | Gao, B.; Hu, G.; Gao, S.; Zhong, Y.; Gu, C., Multi-sensor optimal data fusion for INS/GNSS/CNS integration based on unscented Kalman filter, Int. J. Control Autom. Syst., 16, 129-140 (2018) |
| [23] | Ghil, M.; P., M. R., Data assimilation in meteorology and oceanography, Adv. Geophys., 33, 141-266 (1991) |
| [24] | Goodwin, G. C.; Sin, K. S., Adaptive Filtering Prediction and Control (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0653.93001 |
| [25] | Grewal, M. S.; Andrews, A. P., Kalman Filtering: Theory and Practice (2001), Prentice Hall: Prentice Hall New Jersey |
| [26] | Henrion, D.; Hippe, P., Hyperbolic QR factorization for J-spectral factorization of polynomial matrices, Proceedings of the 42nd IEEE Conference on Decision and Control, 4, 3479-3484 (2003) |
| [27] | Higham, N. J., J-orthogonal matrices: properties and generalization, SIAM Rev., 45, 3, 504-519 (2003) · Zbl 1034.65026 |
| [28] | Hilaly, A. K.; Karim, M. N.; Linden, J. C., A study on real-time optimization of a fedbatch recombinant Escherichia coli fermentation, Control Eng. Pract., 3, 4, 485-493 (1995) |
| [29] | Hohman, D.; Murdock, T.; Westerfield, E.; Hattox, T.; Kusterer, T., GPS roadside integrated presition positioning system, Proceedings of the IEEE Position Location and Navigation Symposium, 221-230 (2000) |
| [30] | Hu, G.; Gao, S.; Zhong, Y., A derivative UKF for tightly coupled INS/GPS integrated navigation, ISA Trans., 56, 135-144 (2015) |
| [31] | Hu, G.; Ni, L.; Gao, B.; Zhu, X.; Wang, W.; Zhong, Y., Model predictive based unscented Kalman filter for hypersonic vehicle navigation with INS/GNSS integration, IEEE Access, 8, 4814-4823 (2019) |
| [32] | Jazwinski, A. H., Stochastic Processes and Filtering Theory (1970), Academic Press: Academic Press New York · Zbl 0203.50101 |
| [33] | Julier, S. J.; Uhlmann, J. K., Reduced sigma point filters for the propagation of means and covariances through nonlinear transformations, Proceedings of the American Control Conference, 887-892 (2002) |
| [34] | Julier, S. J.; Uhlmann, J. K., Unscented filtering and nonlinear estimation, Proc. IEEE, 92, 3, 401-422 (2004) |
| [35] | Julier, S. J.; Uhlmann, J. K.; Durrant-Whyte, H. F., A new approach for filtering nonlinear systems, Proceedings of the American Control Conference, 1628-1632 (1995) |
| [36] | Julier, S. J.; Uhlmann, J. K.; Durrant-Whyte, H. F., A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Autom. Control, 45, 3, 477-482 (2000) · Zbl 0973.93053 |
| [37] | Kailath, T.; Sayed, A. H.; Hassibi, B., Linear Estimation (2000), Prentice Hall: Prentice Hall New Jersey |
| [38] | Kalman, R. E., A new approach to linear filtering and prediction problem, ASME J. Basic Eng., 82, 1, 35-45 (1960) |
| [39] | Kalman, R. E.; Bucy, R. S., New results in linear filtering and prediction theory, ASME J. Basic Eng., 83, 1, 95-108 (1961) · Zbl 07915244 |
| [40] | Kaminski, P. G.; Bryson, A. E.; Schmidt, S. F., Discrete square-root filtering: a survey of current techniques, IEEE Trans. Autom. Control AC-16, 727-735 (1971) |
| [41] | Kulikov, G. Yu.; Kulikova, M. V., Estimating the state in stiff continuous-time stochastic systems within extended Kalman filtering, SIAM J. Sci. Comput., 38, 6, A3565-A3588 (2016) · Zbl 1353.65009 |
| [42] | Kulikov, G. Yu.; Kulikova, M. V., Accurate continuous-discrete unscented Kalman filtering for estimation of nonlinear continuous-time stochastic models in radar tracking, Signal Process., 139, 25-35 (2017) |
| [43] | Kulikov, G. Yu.; Kulikova, M. V., Accurate state estimation of stiff continuous-time stochastic models in chemical and other engineering, Math. Comput. Simul., 142, 62-81 (2017) · Zbl 1540.93109 |
| [44] | Kulikov, G. Yu.; Kulikova, M. V., Do the cubature and unscented Kalman filtering methods outperform always the extended Kalman filter?, IFAC-PapersOnLine, 50, 1, 3762-3767 (2017) |
| [45] | Kulikov, G. Yu.; Kulikova, M. V., Square-root Kalman-like filters for estimation of stiff continuous-time stochastic systems with ill-conditioned measurements, IET Control Theory Appl., 11, 9, 1420-1425 (2017) |
| [46] | Kulikov, G. Yu.; Kulikova, M. V., Accuracy issues in Kalman filtering state estimation of stiff continuous-discrete stochastic models arisen in engineering research, Proceedings of 2018 22nd International Conference on System Theory, Control and Computing (ICSTCC), 800-805 (2018) |
| [47] | Kulikov, G. Yu.; Kulikova, M. V., Estimation of maneuvering target in the presence of non-Gaussian noise: a coordinated turn case study, Signal Process., 145, 241-257 (2018) |
| [48] | Kulikov, G. Yu.; Kulikova, M. V., Numerical stability of EKF-based software sensors in chemical engineering: a Van der Vusse reaction case study, Proceedings of 2018 22nd International Conference on System Theory, Control and Computing (ICSTCC), 268-291 (2018) |
| [49] | Kulikov, G. Yu.; Kulikova, M. V., Stability analysis of Extended, Cubature and Unscented Kalman Filters for estimating stiff continuous-discrete stochastic systems, Automatica, 90, 91-97 (2018) · Zbl 1387.93161 |
| [50] | Kulikov, G. Yu.; Kulikova, M. V., Moore-Penrose-pseudo-inverse-based Kalman-like filtering methods for estimation of stiff continuous-discrete stochastic systems with ill-conditioned measurements, IET Control Theory Appl., 12, 16, 2205-2212 (2018) |
| [51] | Kulikov, G. Yu.; Kulikova, M. V., Numerical robustness of extended Kalman filtering based state estimation in ill-conditioned continuous-discrete nonlinear stochastic chemical systems, Int. J. Robust Nonlinear Control, 29, 5, 1377-1395 (2019) · Zbl 1410.93123 |
| [52] | (in press) |
| [53] | Kulikov, G. Yu.; Kulikova, M. V., Hyperbolic-singular-value-decomposition-based square-root accurate continuous-discrete extended-unscented Kalman filters for estimating continuous-time stochastic models with discrete measurements, Int. J. Robust Nonlinear Control, 30, 5, 2033-2058 (2020) · Zbl 1465.93213 |
| [54] | Kulikova, M. V., Square-root algorithms for maximum correntropy estimation of linear discrete-time systems in presence of non-Gaussian noise, Syst. Control Lett., 108, 8-15 (2017) · Zbl 1375.93122 |
| [55] | Kulikova, M. V.; Tsyganova, J. V., Improved discrete-time Kalman filtering within singular value decomposition, IET Control Theory Appl., 11, 15, 2412-2418 (2017) |
| [56] | Lancaster, P., Theory of Matrices (1970), Academic Press: Academic Press New York · Zbl 0212.05201 |
| [57] | Ledsham, W. H.; Staelin, D. H., An extended Kalman-Bucy filter for atmospheric temperature profile retrieval with a passive microwave sounder, J. Appl. Meteorol., 17, 7, 1023-1033 (1978) |
| [58] | Lefferts, E. J.; Markley, F. L.; Shuster, M. D., Kalman filtering for spacecraft attitude estimation, J. Guid., 5, 5, 417-429 (1982) |
| [59] | Leonard, J. J.; Durrant-Whyte, H. F., Mobile robot localization by tracking geometric beacons, IEEE Trans. Robot. Autom., 7, 3, 376-382 (1991) |
| [60] | Lewis, F. L., Optimal Estimation: With an Introduction to Stochastic Control Theory (1986), John Wiley & Sons: John Wiley & Sons New York · Zbl 0665.93065 |
| [61] | Menegaz, H. M.; Ishihara, J. Y.; Borges, G. A., New minimum sigma set for unscented filtering, Int. J. Robust Nonlinear Control, 25, 17, 3286-3298 (2015) · Zbl 1338.93375 |
| [62] | Menegaz, H. M.; Ishihara, J. Y.; Borges, G. A.; Vargas, A. N., A systematization of the unscented Kalman filter theory, IEEE Trans. Autom. Control, 60, 10, 2583-2598 (2015) · Zbl 1360.93705 |
| [63] | Merwe, R. V.d.; Wan, E. A., The square-root unscented Kalman filter for state and parameter-estimation, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing Proceedings, 6, 3461-3464 (2001) |
| [64] | Misu, T.; Hashimoto, T.; Ninomiya, K., Optimal guidance for autonomous landing of spacecraft, IEEE Trans. Aerosp. Electron. Syst., 35, 2, 459-473 (1999) |
| [65] | Mun-Li, H.; Kleeman, L., Ultrasonic classification and localization of 3D room features using maximum likelihood estimation – part I, Robotica, 15, 483-491 (1997) |
| [66] | Munack, A.; Buning, E.; Speckmann, H., A high-performance control system for spreading liquid manure, Control Eng. Pract., 9, 4, 387-391 (2001) |
| [67] | Nilsson, B.; Nygards, J.; Larsson, U.; Wernersson, A., Control of flexible mobile manipulators: positioning and vibration reduction using an eye-in-hand range camera, Control Eng. Pract., 7, 741-751 (1999) |
| [68] | Pai, W.-C.; Doerschuk, P. C., Statistical AM-FM models, extended Kalman filter demodulation, Cramér-Rao bounds, and speech analysis, IEEE Trans. Signal Process., 48, 8, 2300-2313 (2000) |
| [69] | Park, P.; Kailath, T., New square-root algorithms for Kalman filtering, IEEE Trans. Autom. Control, 40, 5, 895-899 (1995) · Zbl 0829.93075 |
| [70] | Paynter, S. J.; Bishop, R. H., Adaptive nonlinear attitude control and momentum management of spacecraft, J. Guid. Control Dyn., 20, 5, 1025-1032 (1997) · Zbl 0882.93062 |
| [71] | Potter, J. E.; Stern, R. G., Statistical filtering of space navigation measurements, Proceedings of the AIAA Guidance Control Conference (1963) |
| [72] | Prasad, G.; Irwin, G. W.; Swidenbank, E.; Hogg, B. W., Plant-wide predictive control for thermal power plant based on a physical plant model, IEE Proc. — Control Theory Appl., 147, 5, 523-537 (2000) |
| [73] | Psiaki, M. L.; Huang, L.; Fox, S. M., Ground tests of mangnetometer-based autonomous navigation (MAGNAV) for low-earth-orbiting spacecraft, J. Guid. Control Dyn., 16, 1, 206-214 (1993) |
| [74] | Quine, B. M., A derivative-free implementation of the extended Kalman filter, Automatica, 42, 1927-1934 (2006) · Zbl 1130.93426 |
| [75] | Ray, L. R., Nonlinear tire force estimation and road friction identification: simulation and experiments, Automatica, 33, 10, 1819-1833 (1997) · Zbl 0900.93213 |
| [76] | Rivals, I.; Personnaz, L., A recursive algorithm based on the extended Kalman filter for the training of feedforward neural models, Neurocomputing, 20, 1-3, 279-294 (1998) |
| [77] | Rocadenbosch, F.; Vazquez, G.; Comeron, A., Adaptive filter solution for processing lidar returns: Optical parameter estimation, Appl. Opt., 37, 30, 7019-7034 (1998) |
| [78] | Särkkä, S., On unscented Kalman filter for state estimation of continuous-time nonlinear systems, IEEE Trans. Autom. Control, 52, 9, 1631-1641 (2007) · Zbl 1366.93660 |
| [79] | Särkkä, S., Bayesian Filtering and Smoothing (2013), Cambridge University Press: Cambridge University Press Cambridge, U.K. · Zbl 1274.62021 |
| [80] | Simon, D., Optimal State Estimation: Kalman, H Infinity and Nonlinear Approaches (2006), Wiley: Wiley Hoboken, New Jersey |
| [81] | Spirito, M. A., Further results on GSM mobile station location, Electron. Lett., 35, 11, 867-869 (1999) |
| [82] | Teixeira, B. O.S.; Santillo, M. A.; Erwin, R. S.; Bernstein, D. S., Spacecraft tracking using sampled-data Kalman filters, IEEE Control Syst. Mag., 28, 4, 78-94 (2008) · Zbl 1395.93548 |
| [83] | Tham, J. L.; Wang, H.; Teoh, E. K., Multi-sensor fusion for steerable four-wheeled industrial vehicles, Control Eng. Pract., 7, 1233-1248 (1999) |
| [84] | Voorrips, A. C.; Heemink, A. W.; Komen, G. J., Wave data assimilation with the Kalman filter, J. Mar. Syst., 19, 4, 267-291 (1999) |
| [85] | Wan, E. A.; Merwe, R. V.d., The unscented Kalman filter for nonlinear estimation, Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium, 153-158 (2000) |
| [86] | Wan, E. A.; Merwe, R. V.d., The unscented Kalman filter, (Haykin, S., Kalman Filtering and Neural Networks (2001), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York), 221-280 |
| [87] | Wilson, D. I.; Agarwal, M.; Rippin, D. W.T., Experiences implementing the extended Kalman filter on an industrial batch reactor, Comput. Chem. Eng., 22, 1653-1672 (1998) |
| [88] | Kulikov, G. Yu.; Kulikova, M. V., The accurate continuous-discrete extended Kalman filter for radar tracking, IEEE Trans. Signal Process., 64, 4, 948-958 (2016) · Zbl 1412.94052 |
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