A new method for the nonlinear transformation of means and covariances in filters and estimators. (English) Zbl 0973.93053

Based on the intuition that it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation, the authors propose a new approach for applying the linear estimation theory to nonlinear systems. Instead of approximating the Taylor series to an arbitrary order, the paper considers the approximation of the first three moments of the prior distribution accurately, using a set of samples. The proposed algorithm predicts the mean and covariance accurately up to the third-order and, because the higher-order terms in the series are not truncated, it is possible to reduce the errors in the higher-order terms as well.
The new linear estimator is shown to yield a performance equivalent to the Kalman filters for linear systems, and generalizes elegantly to nonlinear systems without the linearization steps required by the extended Kalman filter (EKF). The authors prove analytically that the expected performance of the new approach is superior to that of the EKF method. Empirical evidence is provided to support the theoretical conclusions, demonstrating that the new filter is easier to implement; it does not involve any linearization steps, and eliminates the derivation and evaluation of the Jacobian matrices.
The proposed algorithm has been extended to capture the first four moments of a Gaussian distribution, and the first three moments of an arbitrary distribution. Various applications have been found to be suitable to represent the new algorithm; e.g. high-order nonlinear coupled systems, navigation systems for high-speed road vehicles, public transportation systems, underwater systems, etc.


93E11 Filtering in stochastic control theory
93C10 Nonlinear systems in control theory
Full Text: DOI Link


[1] Zhu F L. Research on observers of nonlinear control systems (in Chinese). PhD thesis. Shanghai: Shanghai Jiaotong University, 2001
[2] Xiong K, Zhang H Y. Application of particle filter in INS nonlinear alignment (in Chinese). J Chin Inertial Tech, 2003, 11: 20-26
[3] Julier S J, Uhlmann J K, Hugh F D-W. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Trans Automat Control, 2000, 45: 477-482 · Zbl 0973.93053
[4] He Q. Study on method and application of set-membership estimation theory (in Chinese). PhD thesis. Hunan: Hunan University, 2002
[5] He Q, Zhang J. A square root extended set membership algorithm with applications to nonlinear system estimation. In: IEEE Proceed Int Conf on Intelligent Computation Tech and Automation, Changsha, Hunan, China, 2008. 559-562
[6] He Q, Zhang J. Set membership state estimation for nonlinear systems in the presence of bounded disturbances. In: IEEE Proceedings The 26th Chin Control Conf, Zhangjiajie, Hunan, China, 2007. 196-201
[7] Scholte E, Campell M E. A nonlinear set-membership filter for on-line applications. Int J Robust Nonlinear Control, 2003, 13: 1337-1358 · Zbl 1046.93042
[8] Scholte E, Campell M E. Robust nonlinear model predictive control with partial state information. IEEE Trans Control Syst Tech, 2008, 16: 636-651
[9] Schlaepfer F M, Schweppe F C. Continuous-time state estimation under disturbances bounded by convex sets. IEEE Trans Automat Control, 1972, 17: 197-205 · Zbl 0258.93020
[10] Alamo T, Bravo J M, Camacho E F. Guaranteed state estimation by zonotopes. Automatica, 2005, 41: 1035-1043 · Zbl 1091.93038
[11] Zhou B, Han J D. A UD factorization-based adaptive extended set-membership filter (in Chinese). Acta Automat Sin, 2008, 34: 150-158 · Zbl 1174.93701
[12] Zhou B, Han J D. A UD factorization-based nonlinear adaptive set-membership filter for ellipsoidal estimation. Int J Robust Nonlin Control, 2008, 18: 1513-1531 · Zbl 1151.93434
[13] Zhou B, Han J D. An enhanced adaptive set-membership filter for nonlinear ellipsoidal estimation. In: Proceedings IEEE American Control Conference, New York, USA, 2007. 5135-5140
[14] Laurent E G, Giuseppe C. Robust filtering for discrete-time systems with bounded noise and parametric uncertainty. IEEE Trans Automat Control, 2001, 46: 1084-1089 · Zbl 1008.93065
[15] Cheng P. Theory of Linear System (in Chinese). Beijing: BUAA Press, 2004. 89-94
[16] Gao J Y. Computer Control System (in Chinese). Beijing: Higher Education Press, 2004. 169-174
[17] Jean-Jacques E S,
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.