Abstract
Cubature Kalman filter (CKF) discussed in the last chapter deals with nonlinear systems with single set of sensors and with Gaussian noise. In this chapter, variants of CKF, namely the cubature information filter (CIF), cubature \(\mathcal{H}_{\infty }\) filter (C\(\mathcal{H}_{\infty }\)F) and cubature \(\mathcal{H}_{\infty }\) information filter (C\(\mathcal{H}_{\infty }\)IF), and their square-root versions, will be explored. Each of these filters is suitable for particular applications. For example, the CIF is suitable for state estimation of nonlinear systems with multiple sensors in the presence of Gaussian noise; the C\(\mathcal{H}_{\infty }\)F is suitable for nonlinear systems with Gaussian or non-Gaussian noises; and finally, the C\(\mathcal{H}_{\infty }\)IF is useful for estimating the states of nonlinear systems with multiple sensors in the presence of Gaussian or non-Gaussian noise.
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Notes
- 1.
Square-root factors of the information matrix and information state
$$\begin{aligned} \mathbf P ^{-1}&=\mathbf P ^{-T/2}{} \mathbf P ^{-1/2}\\ \Rightarrow \mathbf Y&=\mathbf Y _s\mathbf Y _s^T\\ \mathbf P ^{-1}{} \mathbf x&=\mathbf P ^{-T/2}{} \mathbf P ^{-1/2}{} \mathbf x \\ \Rightarrow \mathbf y&=\mathbf Y _s\mathbf y _s. \end{aligned}$$ - 2.
The basic structure of the Householder matrix, \(\varTheta \), is
$$\begin{aligned} \varTheta = \mathbf I -\frac{2}{c^Tc}cc^T \end{aligned}$$where c is a column vector and \(\mathbf I \) is the identity matrix of the same dimension.
- 3.
If \(b=a\varTheta \), with \(\varTheta \) as J-unitary matrix, then Hassibi (2000)
$$\begin{aligned} bJb^T = a\varTheta J\varTheta ^T a^T = aJa^T. \end{aligned}$$ - 4.
QR is the orthogonal triangular decomposition and can be found in MATLAB using the command ‘qr’ .
References
Anderson BDO, Moore JB (1979) Optimal filtering, vol 21. Prentice-Hall, Englewood Cliffs, pp 22–95
Arasaratnam I, Haykin S (2009) Cubature kalman filters. IEEE Trans Autom control 54(6):1254–1269
Bar-Shalom Y, Li XR, Kirubarajan T (2004) Estimation with applications to tracking and navigation: theory algorithms and software. Wiley, New York
Chandra KB, Gu DW, Postlethwaite I (2013) Square root cubature information filter. IEEE SensJ 13(2):750–758
Chandra KPB, Gu D-W, Postlethwaite I (2014) A cubature \(H_{\infty }\) filter and its square-root version. Int J Control 87(4):764–776
Chandra KPB, Gu D-W, Postlethwaite I (2016) Cubature \(H_{\infty }\) information filter and its extensions. Eur J Control 29:17–32
Eustice RM, Singh H, Leonard JJ, Walter MR (2006) Visually mapping the rms titanic: conservative covariance estimates for slam information filters. Int J Robot Res 25(12):1223–1242
Grewal M, Andrews A (2001) Kalman filtering: theory and practice using matlab
Grewal MS, Andrews AP (2010) Applications of kalman filtering in aerospace 1960 to the present [historical perspectives]. IEEE Control Syst Mag 30(3):69–78
Hassibi B, Kailath T, Sayed AH (2000) Array algorithms for h/sup/spl infin//estimation. IEEE Trans Autom Control 45(4):702–706
Ishihara J, Macchiavello B, Terra M (2006) H8 estimation and array algorithms for discrete-time descriptor systems. In: 2006 45th IEEE conference on decision and control. IEEE, pp 4740–4745
Kailath T, Sayed AH, Hassibi B (2000) Linear estimation, vol 1. Prentice Hall, Upper Saddle River
Kurtz MJ, Henson MA (1995) Nonlinear output feedback control of chemical reactors. In: Proceedings of the 1995 American control conference, vol 4 IEEE, pp 2667–2671
Lee D-J (2008) Nonlinear estimation and multiple sensor fusion using unscented information filtering. IEEE Signal Process Lett 15:861–864
Li W, Jia Y (2010) H-infinity filtering for a class of nonlinear discrete-time systems based on unscented transform. Signal Process 90(12):3301–3307
Mutambara AG (1998) Decentralized estimation and control for multisensor systems. CRC Press, Boca Raton
Park P, Kailath T (1995) New square-root algorithms for kalman filtering. IEEE Trans Autom Control 40(5):895–899
Raol J, Girija G (2002) Sensor data fusion algorithms using square-root information filtering. IEE Proc-Radar Sonar Navig 149(2):89–96
Rigatos GG (2009) Particle filtering for state estimation in nonlinear industrial systems. IEEE Trans Instrum Meas. 58(11):3885–3900
Shen X-M, Deng L (1997) Game theory approach to discrete \(h_\infty \) filter design. IEEE Trans Signal Process 45(4):1092–1095
Sibley G, Sukhatme GS, Matthies LH (2006) The iterated sigma point kalman filter with applications to long range stereo. Robot: Sci Syst 8:235–244
Simon D (2006) Optimal state estimation: Kalman, H infinity, and nonlinear approaches. Wiley, New York
Terra MH, Ishihara JY, Jesus G (2009) Fast array algorithms for \(h_\infty \) information estimation of rectangular discrete-time descriptor systems. In: 2009 IEEE control applications, (CCA) & intelligent control, (ISIC). IEEE, pp 637–642
Tsyganova J, Kulikova M (2013) State sensitivity evaluation within ud based array covariance filters. IEEE Trans Autom Control 58(11):2944–2950
Uppal A, Ray W, Poore A (1974) On the dynamic behavior of continuous stirred tank reactors. Chem Eng Sci 29(4):967–985
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Chandra, K.P.B., Gu, DW. (2019). Variants of Cubature Kalman Filter. In: Nonlinear Filtering. Springer, Cham. https://doi.org/10.1007/978-3-030-01797-2_6
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