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Research Papers

An Extended Kalman Filtering Mechanism Based on Generalized Interval Probability

[+] Author and Article Information
Jie Hu

The State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, China e-mail: hu_jie@hnu.edu.cn

Yan Wang

Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 e-mail: yan.wang@me.gatech.edu

Aiguo Cheng, Zhihua Zhong

The State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, China

Manuscript received October 1, 2014; final manuscript received March 11, 2015; published online July 1, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(3), 031002 (Jul 01, 2015) (11 pages) Paper No: RISK-14-1069; doi: 10.1115/1.4030465 History: Received October 01, 2014; Accepted April 27, 2015; Online July 01, 2015

Kalman filter has been widely applied for state identification in controllable systems. As a special case of the hidden Markov model, it is based on the assumption of linear dependency relationships and Gaussian noise. The classical Kalman filter does not differentiate systematic error from random error associated with observations. In this paper, we propose an extended Kalman filtering mechanism based on generalized interval probability, where state and observable variables are random intervals, and interval-valued Gaussian distributions model the noises. The prediction and update procedures in the new mechanism are derived. Two examples are used to illustrate the developed mechanism. It is shown that the method is an efficient alternative to sensitivity analysis for assessing the effect of systematic error.

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References

Kalman, R. E., 1960, “A New Approach to Linear Filtering and Prediction Problems,” ASME J. Basic Eng., 82(1), pp. 35–45. 10.1115/1.3662552
Julier, S. J., and Uhlmann, J. K., 1997, “A New Extension of the Kalman Filter to Nonlinear Systems,” International Symposium Aerospace/Defense Sensing, Simulation and Controls, Vol. 3, Orlando, FL, ISPE, Bellingham, pp. 3–2.
Wan, E. A., and Van Der Merwe, R., 2000, “The Unscented Kalman Filter for Nonlinear Estimation,” Proceedings of the 2000 IEEE Adaptive Systems for Signal Processing, Communications, and Control Symposium (AS-SPCC), IEEE, Piscataway, NJ, pp. 153–158.
Bucy, R. S., and Joseph, P. D., 1987, Filtering for Stochastic Processes With Applications to Guidance, Vol. 326, American Mathematical Society, Providence, RI.
Rauch, H. E., Striebel, C., and Tung, F., 1965, “Maximum Likelihood Estimates of Linear Dynamic Systems,” AIAA J., 3(8), pp. 1445–1450. 10.2514/3.3166
Einicke, G. A., 2006, “Optimal and Robust Noncausal Filter Formulations,” IEEE Trans. Signal Process., 54(3), pp. 1069–1077. 10.1109/TSP.2005.863042
Einicke, G. A., 2007, “Asymptotic Optimality of the Minimum-Variance Fixed-Interval Smoother,” IEEE Trans. Signal Process., 55(4), pp. 1543–1547. 10.1109/TSP.2006.889402
Morrell, D. R., and Stirling, W. C., 1991, “Set-Values Filtering and Smoothing,” IEEE Trans. Syst. Man Cybern., 21(1), pp. 184–193. 10.1109/21.101148
Morrell, D., and Stirling, W. C., 2003, “An Extended Set-Valued Kalman Filter,” Proceedings of the 3rd International Symposium on Imprecise Probabilities and their Applications, Carleton Scientific, Ontario, Canada, pp. 395–405.
Fung, P., and Grimble, M. J., 1983, “Dynamic Ship Positioning Using a Self-Tuning Kalman Filter,” IEEE Trans. Autom. Control, 28(3), pp. 339–350. [CrossRef]
Jetto, L., Longhi, S., and Venturini, G., 1999, “Development and Experimental Validation of an Adaptive Extended Kalman Filter for the Localization of Mobile Robots,” IEEE Trans. Rob. Autom., 15(2), pp. 219–229. 10.1109/70.760343
Rezaei, S., and Sengupta, R., 2007, “Kalman Filter-Based Integration of DGPS and Vehicle Sensors for Localization,” IEEE Trans. Control Syst. Technol., 15(6), pp. 1080–1088. 10.1109/TCST.2006.886439
Ito, K., Nguyen, B. M., Wang, Y., Odai, M., Ogawa, H., Takano, E., Inoue, T., Koyama, M., Fujimoto, H., and Hori, Y., 2013, “Fast and Accurate Vision-Based Positioning Control Employing Multi-Rate Kalman Filter,” Proceedings of the 39th IEEE Annual Conference on Industrial Electronics Society (IECON 2013), IEEE, Piscataway, NJ, pp. 6466–6471.
Westmore, D. B., and Wilson, W. J., 1991, “Direct Dynamic Control of a Robot Using an End-Point Mounted Camera and Kalman Filter Position Estimation,” Proceedings of 1991 IEEE International Conference on Robotics and Automation, IEEE, Piscataway, NJ, pp. 2376–2384.
Wang, J., and Wilson, W. J., 1992, “3D Relative Position and Orientation Estimation Using Kalman Filter for Robot Control,” Proceedings of 1992 IEEE International Conference on Robotics and Automation, IEEE, Piscataway, NJ, pp. 2638–2645.
Tang, Q., 2014, “Localization and Tracking Control for Mobile Welding Robot,” Ind. Rob. Int. J., 41(3), pp. 4–4.
Lee, J. H., and Ricker, N. L., 1994, “Extended Kalman Filter Based Nonlinear Model Predictive Control,” Ind. Eng. Chem. Res., 33(6), pp. 1530–1541. 10.1021/ie00030a013
Yang, S., and Liu, T., 1999, “State Estimation for Predictive Maintenance Using Kalman Filter,” Reliab. Eng. Syst. Saf., 66(1), pp. 29–39. 10.1016/S0951-8320(99)00015-0
Roshany-Yamchi, S., Cychowski, M., Negenborn, R. R., De Schutter, B., Delaney, K., and Connell, J., 2013, “Kalman Filter-Based Distributed Predictive Control of Large-Scale Multi-Rate Systems: Application to Power Networks,” IEEE Trans. Control Syst. Technol., 21(1), pp. 27–39. 10.1109/TCST.2011.2172444
Kanieski, J. M., Cardoso, R., Pinheiro, H., and Grundling, H. A., 2013, “Kalman Filter-Based Control System for Power Quality Conditioning Devices,” IEEE Trans. Ind. Electron., 60(11), pp. 5214–5227. [CrossRef]
Loebis, D., Sutton, R., Chudley, J., and Naeem, W., 2004, “Adaptive Tuning of a Kalman Filter via Fuzzy Logic for an Intelligent AUV Navigation System,” Control Eng. Pract., 12(12), pp. 1531–1539. 10.1016/j.conengprac.2003.11.008
Yun, X., and Bachmann, E. R., 2006, “Design, Implementation, and Experimental Results of a Quaternion-Based Kalman Filter for Human Body Motion Tracking,” IEEE Trans. Rob., 22(6), pp. 1216–1227. [CrossRef]
Volponi, A., Daguang, C., DePold, H., and Ganguli, R., 2003, “The Use of Kalman Filter and Neural Network Methodologies in Gas Turbine Performance Diagnostics: A Comparative Study,” ASME J. Eng. Gas Turbines Power, 125(4), pp. 917–924. 10.1115/1.1419016
Mneimneh, M., Yaz, E., Johnson, M., and Povinelli, R., 2006, “An Adaptive Kalman Filter for Removing Baseline Wandering in ECG Signals,” Proceedings of the Computers in Cardiology, IEEE, Piscataway, NJ, pp. 253–256.
An, L., and Sepehri, N., 2003, “Hydraulic Actuator Circuit Fault Detection Using Extended Kalman Filter,” Proceedings of the 2003 American Control Conference, Vol. 5, IEEE, pp. 4261–4266.
Harvey, A. C., 1985, “Trends and Cycles in Macroeconomic Time Series,” J. Bus. Econ. Stat., 3(3), pp. 216–227.
Stock, J. H., and Watson, M. W., 1996, “Evidence on Structural Instability in Macroeconomic Time Series Relations,” J. Bus. Econ. Stat., 14(1), pp. 11–30.
Wang, Y., 2013, “Generalized Fokker–Planck Equation with Generalized Interval Probability,” Mech. Syst. Signal Process., 37(1), pp . 92–104. [CrossRef]
Sainz, M. A., Armengol, J., Calm, R., Herrero, P., Jorba, L., and Vehi, J., 2014, Modal Interval Analysis, Springer, New York.
Dimitrova, N., Markov, S., and Popova, E., 1992, “Extended Interval Arithmetics: New Results and Applications,” Computer Arithmetics and Enclosure Methods, Elsevier, Amsterdam, pp. 225–232.
Shary, S. P., 1996, “Algebraic Approach to the Interval Linear Static Identification, Tolerance, and Control Problems, or One More Application of Kaucher Arithmetic,” Reliable Comput., 2(1), pp. 3–33. [CrossRef]
Kaucher, E., 1980, “Interval Analysis in the Extended Interval Space IR,” Fundamentals of Numerical Computation (Computer-Oriented Numerical Analysis), Springer, Austria, pp. 33–49.
Dempster, A. P., 1967, “Upper and Lower Probabilities Induced by a Multivalued Mapping,” Ann. Math. Stat., 38(2), pp. 325–339. [CrossRef]
Shafer, G., 1976, A Mathematical Theory of Evidence, Vol. 1. Princeton University Press, Princeton, NJ.
Walley, P., 1991, Statistical Reasoning With Imprecise Probabilities, Chapman and Hall, London.
Williamson, R. C., and Downs, T., 1990, “Probabilistic Arithmetic. I. Numerical Methods for Calculating Convolutions and Dependency Bounds,” Int. J. Approximate Reasoning, 4(2), pp. 89–158. 10.1016/0888-613X(90)90022-T
Ferson, S., Ginzburg, L., Kreinovich, V., Longpré, L., and Aviles, M., 2002, “Computing Variance for Interval Data is NP-Hard,” ACM SIGACT News, 33(2), pp. 108–118. [CrossRef]
Ferson, S., Kreinovich, V., Ginzburg, L., Myers, D. S., and Sentz, K., 2002, Constructing Probability Boxes and Dempster–Shafer Structures, Vol. 835, Sandia National Laboratories, Livermore, CA.
Weichselberger, K., 2000, “The Theory of Interval-Probability as a Unifying Concept for Uncertainty,” Int. J. Approximate Reasoning, 24(2), pp. 149–170. [CrossRef]
Neumaier, A., 2004, “Clouds, Fuzzy Sets, and Probability Intervals,” Reliable Comput., 10(4), pp. 249–272. [CrossRef]
Wang, Y., 2008, “Imprecise Probabilities With a Generalized Interval Form,” Proceedings of the 3rd International Workshop on Reliability Engineering Computing (REC’08), NSF Workshop on Imprecise Probability in Engineering Analysis and Design, Georgia Institute of Technology, Savannah, GA, pp. 45–59.
Wang, Y., 2010, “Imprecise Probabilities Based on Generalised Intervals for System Reliability Assessment,” Int. J. Reliability Saf., 4(4), pp. 319–342. [CrossRef]
Wang, Y., 2008, “Semantic Tolerance Modeling With Generalized Intervals,” J. Mech. Des., 130(8), p. 081701. 10.1115/1.2936900
Körner, R., 1997, “On the Variance of Fuzzy Random Variables,” Fuzzy Sets Syst., 92(1), pp. 83–93. 10.1016/S0165-0114(96)00169-8
Sun, Y., and Ralescu, D., 2015, “A Normal Hierarchical Model and Minimum Contrast Estimation for Random Intervals,” Ann. Inst. Stat. Math., 67(2), pp. 313–333.
Xiang, G., Ceberio, M., and Kreinovich, V., 2007, “Computing Population Variance and Entropy Under Interval Uncertainty: Linear-Time Algorithms,” Reliable Comput., 13(6), pp. 467–488. [CrossRef]
Gardeñes E, Jorba, L., Calm, R., Estela, R., Mielgo, H., and Trepat, A., 2001, “Modal Intervals,” Reliable Comput., 7, pp. 77–111. [CrossRef]
Moore, R. E., Cloud, M. J., and Kearfott, R. B., 2009, Introduction to Interval Analysis, SIAM, Philadelphia.
Wang, Y., 2011, “Independence in Generalized Interval Probability,” Proceedings of the 1st International Conference on Vulnerability and Risk Analysis and Management (ICVRAM 2011) and 5th International Symposium on Uncertainty Modeling and Analysis (ISUMA 2011), ASCE, Reston, VA, pp. 37–44.
Markov, S., 1979, “Calculus for Interval Functions of a Real Variable,” Computing, 22(4), pp. 325–337. [CrossRef]
Lewis, F. L., and Lewis, F., 1986, Optimal Estimation: With an Introduction to Stochastic Control Theory, Wiley, New York.
Greg, W., and Gary, B., 2006, An Introduction to Kalman Filter, UNC-Chapel Hill, NC, TR 95-041.
Hu, J., Aminzadeh, M., and Wang, Y., 2014, “Searching Feasible Design Space by Solving Quantified Constraint Satisfaction Problems,” ASME J. Mech. Des., 136(3), p. 031002. 10.1115/1.4026027
Ding, Y., Ceglarek, D., and Shi, J., 2002, “Fault Diagnosis of Multistage Manufacturing Processes by Using State Space Approach,” ASME J. Manuf. Sci. Eng., 124(2), pp. 313–322. 10.1115/1.1445155
Izquierdo, L. E., Shi, J., Hu, J., and Wampler, C., 2007, “Feedforward Control of Multistage Assembly Processes Using Programmable Tooling,” Trans. NAMRI/SME, 35, pp. 295–302.
Zhong, J., Liu, J., and Shi, J., 2010, “Predictive Control Considering Model Uncertainty for Variation Reduction in Multistage Assembly Processes,” IEEE Trans. Autom. Sci. Eng., 7(4), pp. 724–735. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Cumulative distribution functions of Gaussian distribution with precise and imprecise parameters: (a) x∼(0.5,1), (b) x∼([0.2,0.8],1), (c) x∼(0.5,[0.8,1.2]), and (d) x∼([0.2,0.8],[0.8,1.2])

Grahic Jump Location
Fig. 2

Comparison between the results from the classical Kalman filter and the proposed mechanism in Example 1: (a) estimated voltage values of the classical Kalman filter, (b) mean value comparison, and (c) variance comparison

Grahic Jump Location
Fig. 3

Assembly sequence, locating, and measurement points in Example 2

Grahic Jump Location
Fig. 4

Results comparison between classical and the proposed Kalman filter: (a) variances estimated at stage 1, (b) variances estimated at stage 2, (c) variances estimated at stage 3, and (d) variances estimated at stage 4

Grahic Jump Location
Fig. 5

Variances updated at stage 4: (a) variances comparison between the two methods and (b) variances computed by the proposed method

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