55 System identification Using Kalman Filters 573 B.1.3 Unscented Kalman Filter (UKF) Unscented Kalman Filter algorithm Description: 1: Initialization: State mean and covarianc at k D0: Ox0 DEŒx0 and P0 DE .x0 Ox0/.x0 Ox0/ T 2: Prediction phase (a) Generation of 2nC1 sigma-points f%i ;k 1;!i giD0:::2n DUT.Oxk 1;Pxk 1/ (b) Predicted state: % i ;k Dfk.%i;k 1; uk 1/ and Ox k D 2n PiD0 !i % i ;k (c) Predicted covariance: P xk D 2n PiD0 !i .% i ;k O x k /.% i ;k O x k /T CQ 3: Correction phase (a) Measurement update: Yi;k Dhk.% i;k / (b) Measurement prediction: Oyk D 2n PiD0 !i Yi ;k (c) Innovation (Residual term): Qyk DYi;k Oyk (d) Innovation covariance: Pyk D 2n PiD0 !i Qyk Qy T k CR (e) Cross covariance: Pxkyk D 2n PiD0 !i .% i;k O x k /.Yi;k Oyk/ T CR (f) Updated state mean and Covariance: Kalman Gain matrix: Kk DPxkyk P 1 y k State update: Oxk D Ox k CKk Qyk Covariance update: Pxk DP xk KkPyk KT k References 1. Andrews HC, Patterson CL (1976) Singular value decompositions and digital image processing. IEEE Trans Acoust Speech Signal Process 24(1):26–53 2. Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25(1):539–575 3. Boyce WE, DiPrima RC (1977) Elementary differential equations and boundary value problems. John Wiley & Sons, New York 4. Craig R, Bampton MCC (1968) Coupling of substructures for dynamic analyses. AIAA J 6(7):1313–1319 5. Dormand JR, Prince PJ(1980) A family of embedded runge-kutta formulae. J Comput Appl Math 6(1):19–26 6. Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA J 3(2):380–380 7. Julier SJ, Uhlmann JK (1996) A general method for approximating nonlinear transformations of probability distributions. Technical report, University of Oxford, Departement of Engineering Science 8. Julier SJ, Uhlmann JK (1997) A new extension of the kalman filter to nonlinear systems. In: The proceedings of aeroSense: the 11th international symposium on aerospace/defence sensing, simulation and controls, Orlando, Florida, pp 182–193 9. Joseph J, LaViola Jr (2003) A comparison of unscented and extended kalman filtering for estimating quaternion motion. In the proceedings of the 2003 american control conference, Denver, Colorado, pp 2435–2440 10. Mathews JH, Fink KD (2004) Numerical methods using MATLAB. Prentice Hall, Upper Saddle River, New Jersey 11. Moore BC (1981) Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans Autom Control 26(1):17–32 12. Sorenson HW (1970) Least-squares estimation: from gauss to kalman. IEEE Spectr 7(7):63–68
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