Fuzzy Variable-Length Sliding Window Blockwise Least Square Algorithm with Application to Vehicle Heading Determination


1 Mechanical Engineering Department, Amirkabir University of Technology

2 University of Tehran, Campus #2, Control & Intelligent Processing Centre of Excellence, School of Electrical and Computer Engineering, Karegare Shomali st., Tehran, Iran, P.O. Box 1439-515

3 Corresponding Author, Mechanical Engineering Department, Amirkabir University of Technology


In ground vehicles, three-axis magnetometers may be corrupted by both soft- and hard-iron disturbances. Therefore, it may not be possible to achieve qualified headings without online calibration of this magnetic system. First contribution of this paper is focused on improving the order of persistent excitation of the squared signal matrix through incorporation of a direction cosine matrix in estimation model. As the main contribution, a fuzzy change detection scheme for adjusting the length of data sliding window of blockwise least square (BLS) algorithms is presented in the framework of on-line estimation of system parameters under both abrupt and gradual changes. This is called fuzzy variable-length sliding window (FVLSW) BLS. Two change detection indices including generalized likelihood ratio and averaged parameter estimation errors together with their changes are considered as inputs of the fuzzy system. The defuzzified outputs consists a forgetting factor in order to place more emphasis on the recent data, and two adjusted lengths of data history windows.Simulations and real experiments revealed that the proposed approach has superior performance with respect to the latest variable-length sliding window (VLSW) BLS estimation algorithm. The superiority is more significant when the measurement noise power is substantial.


[1] Deutschmann, J.K., and Izhtack, B., Evaluation of Attitude and Orbit Determination using Actual Earth Magnetic Field Data, J. Guidance, Vol. 24, pp. 616-623, (2001).
[2] Alonso, R., and Shuster, M.D., A New Algorithm for Attitude-Independent Magnetometer Calibration, J. Astronautical Sciences, pp. 513-527, (2003).
[3] Ma, G.F.,  and Jiang, X.Y., Unscented Kalman Filter for Spacecraft Attitude Estimation and Calibration using Magnetometer Measurements, Proc. of the Fourth Int. Conf. on Machine Learning and Cybernetics, Guangzhou, Korea,  pp. 18-21, August (2005).
[4] Gebre-Egziabher, D., and Elkaim, G.H., Calibration of Strapdown Magnetometers in Magnetic Field Domain, ASCE Journal of Aerospace Engineering, Vol. 19, pp. 87-102, (2006).
[5] Kao, W.W., and Tsai, C.L., Adaptive and Learning Calibration of Magnetic Compass, IOP J. Measurement  Science and Technology, Vol. 17, pp. 3073-3082, (2006).
[6] Ding, F., and Xiao, Y., A Finite-Data-Window Least Squares Algorithm with a Forgetting Factor for Dynamical Modeling, Applied Mathematics and Computation, Vol. 186, pp. 184–192, (2007).
[7] Jiang, J., and Zhang, Y., A Novel Variable-Length Sliding Window Blockwise Least-Squares Algorithm for On-Line Estimation of Time-Varying Parameters, Int. J. Adapt. Control Signal Process, Vol. 18, pp. 505–521, (2004).
[8] Basseville, M., and Nikiforov, I., Detection of Abrupt Changes: Theory and Application, Englewood Cliffs, New Jersey, Prentice Hall, 1993.
[9] Liu, H., and He, Z., A Sliding-Exponential Window RLS Adaptive Algorithm: Properties and Applications, Signal Processing, Vol. 45, pp. 357–368, (1995).
[10] Lee, Y.H., Kim, M., Chu, Y.H., and Han, C., Adaptive Multivariate Regression Modeling Based on Model Performance Assessment, Chemo Metrics and Intelligent Laboratory Systems, 78, pp. 63– 73, (2005).
[11] Rogers, R.M., Applied Mathematics in Integrated Navigation Systems, 2nd ed. Virginia, AIAA Series, 2003.
[12] Astrom, K.J., and Wittenmark,  B., Adaptive Control, 1st ed., Addison-Wesley, 1989.
[13] Keighobadi, J., Menhaj, M.B., and Kabganian, M., Nonlinear Control of a Wheeled Mobile Robot: a Fuzzy Approach, Amirkabir J.  Science and Technology Vol. 17, pp. 41-51(in Persian), (2007) .
[14] Malhotra, A., and Huang, B., Detection of Abrupt Change and Applications in Sensor Decalibration Monitoring, ISA Trans., Vol. 41, pp. 155–166, (2002).
[16] Http://www.ngdc.noaa.gov/seg/WMM/DoDWMM.shtml, accessed May 2006.