Fixed-Interval Kalman Smoothing Algorithms in Singular State–Space Systems

Abstract

Fixed-interval Bayesian smoothing in state–space systems has been addressed for a long time. However, as far as the measurement noise is concerned, only two cases have been addressed so far : the regular case, i.e., with positive definite covariance matrix; and the perfect measurement case, i.e., with zero measurement noise. In this paper we address the smoothing problem in the intermediate case where the measurement noise covariance is positive semi definite with arbitrary rank. We exploit the singularity of the model in order to transform the original state–space system into a pairwise Markov model with reduced state dimension. Finally, the a posteriori Markovianity of the reduced state enables us to propose a family of fixed-interval smoothing algorithms.

Appendix A

1.1 A.1 Some Useful Results for Gaussian Variables

The proof of Proposition 6 is based on the following two classical results on Gaussian variables.

Proposition 8

Let $p({\bf x})\! \sim \!{\cal N}\! (\widehat{\bf x} ,\! {\bf P}_{x})$ and $p({\bf y}|{\bf x})\! \sim \!{\cal N} \!({\bf A} {\bf x}\, +$ b, P _y|x ). Then

$$ p({\bf x} , {\bf y}) \sim {\cal N} \left( \left[ \begin{array}{c} \widehat{\bf x} \\ {\bf A} \widehat{\bf x} + {\bf b} \end{array} \right] , \left[ \begin{array}{cc} {\bf P}_x & \ \ {\bf P}_x {\bf A}^T \\ {\bf A} {\bf P}_x &\ \ {\bf A} {\bf P}_x {\bf A}^T + {\bf P}_{y|x} \end{array} \right]\right) . $$

Proposition 9

Let  $p({\bf x}, {\bf y})\, \sim \,{\cal N} \left(\, \left[ \begin{array}{c} \widehat{\bf x} \\ \widehat{\bf y} \end{array} \right] , \left[ \begin{array}{cc} {\bf P}_x \,\,{\bf P}_{x,y} \\ {\bf P}_{y,x} \,\,{\bf P}_y \end{array} \right]\, \right)$ . Then

$$ p({\bf x}|{\bf y}) \sim {\cal N} \left(\widehat{\bf x} + {\bf P}_{x,y} {\bf P}_{y}^{-1} ({\bf y} - \widehat{\bf y}\,) , {\bf P}_x - {\bf P}_{x,y} {\bf P}_{y}^{-1} {\bf P}_{y,x} \right). $$

1.2 A.2 Proof of Proposition 6

Let us address the calculation of Eqs. 22 and 23. First, from Eq. 14 we have

$$ \begin{array}{rll} &&{\kern-6pt}p(\underbrace{({\bf x}_{n+1} , {\bf y}_{n+1})}_{{\bf z}_{n+1}}|{\bf z}_n)\notag\\ &&{\kern6pt}\sim {\cal N} \left(\left[ \begin{array}{ll} \overline{{\cal F}}_{n}^{\overline{\bf x},\overline{\bf x}} &\,\, \overline{{\cal F}}_{n}^{\overline{\bf x},\overline{\bf y}} \\ \overline{{\cal F}}_{n}^{\overline{\bf y},\overline{\bf x}} &\,\, \overline{{\cal F}}_{n}^{\overline{\bf y},\overline{\bf y}} \cr \end{array} \right] \left[ \begin{array}{l} \overline{{\bf x}}_{n} \\ \overline{\bf y}_{n} \end{array} \right] , \left[ \begin{array}{ll} \overline{\bf Q}_n &\,\, \overline{\bf S}_n \\ (\overline{\bf S}_n)^T &\,\, \overline{\bf R}_n \end{array} \right] \right) , \end{array} $$

(62)

and by using Proposition 9 we get

$$ \begin{array}{lll} p({\bf x}_{n+1}|{\bf z}_n, {\bf y}_{n+1}) & = &p({\bf x}_{n+1}|{\bf x}_n, {\bf y}_{0:n+1}) \\ & \sim &{\cal N}\Bigg( \underbrace{\left[\overline{{\cal F}}_{n}^{\overline{\bf x},\overline{\bf x}} - \overline{\bf S}_n (\overline{\bf R}_n)^{-1} \overline{{\cal F}}_{n}^{\overline{\bf y},\overline{\bf x}} \right]}_{{\bf A}_n} {\bf x}_n \\ &&{\kern20pt}+\, \underbrace{\left[\overline{{\cal F}}_{n}^{\overline{\bf x},\overline{\bf y}} - \overline{\bf S}_n (\overline{\bf R}_n)^{-1} \overline{{\cal F}}_{n}^{\overline{\bf y},\overline{\bf y}} \right]}_{{\bf B}_n} {\bf y}_n \\ \label{transition-Gauss} && {\kern20pt}+\, \overline{\bf S}_n (\overline{\bf R}_n)^{-1} {\bf y}_{n+1} ,\\ &&{\kern20pt}\underbrace{\left[ \overline{\bf Q}_n - \overline{\bf S}_n (\overline{\bf R}_n)^-1 (\overline{\bf S}_n)^T \right]}_{{\bf C}_n} \Bigg) \end{array} $$

(63)

(the first equality holds because (x _n , y _n ) is an MC). On the other hand,

$$ \widetilde{\alpha}_n = p({\bf x}_{n}|{\bf y}_{0:n+1}) \sim {\cal N}(\,\widehat{\bf x}_{n|0:n+1} , {\bf P}_{n|0:n+1}) . $$

(64)

Applying Proposition 8 to Eqs. 63 and 64 we get

$$ \begin{array}{rll} &&{\kern-6pt} p({\bf x}_{n} , {\bf x}_{n+1}|{\bf y}_{0:n+1})\\ && \sim \,{\cal N} \left(\left[ \begin{array}{c} \widehat{\bf x}_{n|0:n+1} \\ {\bf A}_n \widehat{\bf x}_{n|0:n+1} + {\bf B}_n {\bf y}_n + \overline{\bf S}_n (\overline{\bf R}_n)^{-1} {\bf y}_{n+1} \end{array} \right]\right.,\\ \label{pxnxn} & &{\kern8pt} \left.\left[ \begin{array}{cc} {\bf P}_{n|0:n+1} &\,\, {\bf P}_{n|0:n+1} {\bf A}_n^T \\ {\bf A}_n {\bf P}_{n|0:n+1} &\,\, {\bf A}_n {\bf P}_{n|0:n+1} {\bf A}_n^T + {\bf C}_n \end{array} \right] \right) . \end{array} $$

(65)

Applying Proposition 9 in Eq. 65, and observing that conditionnally on (x _n + 1 , y _0:n + 1), x _n and y _n + 2:N are independent, we get

$$ \begin{array}{rll} \nonumber &&{\kern-6pt} p({\bf x}_n|{\bf x}_{n+1}, {\bf y}_{0:N}) \\ &&{\kern4pt}\sim \Bigg(\widehat{\bf x}_{n|0:n+1} + \underbrace{{\bf P}_{n|0:n+1} {\bf A}_n^T {\bf P}_{n+1|0:n+1}^{-1}}_{{\bf K}_{n|0:N}} ({\bf x}_{n+1} - \widehat{\bf x}_{n+1|0:n+1}), \\ \label{pxn|xn+1,y_{0:N}} &&{\kern28pt}{\bf P}_{n|0:n+1} - {\bf P}_{n|0:n+1} {\bf A}_n^{T} {\bf P}_{n+1|0:n+1}^{-1} {\bf A}_n {\bf P}_{n|0:n+1}\Bigg). \end{array} $$

(66)

from which we get formula 45 for the Kalman smoothing gain K _n|0:N . Injecting Eq. 66 in Eq. 22 and using Proposition 8 again we eventually get Eqs. 46 and 47.

1.3 A.3 Proof of Proposition 7

Injecting Eq. 45 in Eqs. 46 and 47 leads respectively to:

$$ \begin{array}{rll} \widehat{\bf x}_{n|0:N} & = & \widehat{\bf x}_{n|0:n+1} \nonumber\\ &&+\, {\bf P}_{n|0:n+1} \Bigg[ \underbrace{\overline{{\cal F}}_{n}^{\overline{\bf x},\overline{\bf x}} - \overline{\bf S}_n (\overline{\bf R}_n)^{-1} \overline{{\cal F}}_{n}^{\overline{\bf y},\overline{\bf x}}}_{{\bf A}_n} \Bigg]^T \lambda_{n+1} , \end{array} $$

(67)

$$ {\bf P}_{n|0:N} = {\bf P}_{n|0:n+1} - {\bf P}_{n|0:n+1} {\bf A}_n^T \Lambda_{n+1} {\bf A}_n {\bf P}_{n|0:n+1}. $$

(68)

On the other hand, from Eqs. 40 and 42 we get

$$ {\bf P}_{n|0:n+1} {\bf A}_n^T = {\bf P}_{n|0:n} \big({\bf K}_n^{\lambda}\big)^T. $$

(69)

Injecting Eq. 69 in Eqs. 67 and 68 leads respectively to Eqs. 51 and 52.

It remains to show Eqs. 55 and 56. Injecting Eq. 51 in Eqs. 52 and 53 in Eq. 54 leads respectively to:

$$ \lambda_n = \big({\bf K}_n^{\lambda}\big)^T \lambda_{n+1} + {\bf P}_{n|0:n}^{-1} \big[\,\widehat{\bf x}_{n|0:n+1} - {\widehat{\bf x}}_{n|0:n}\big] $$

(70)

$$ \label{Lambda_n-proof} \Lambda_n = \big({\bf K}_n^{\lambda}\big)^T\! \Lambda_{n+1} {\bf K}_n^{\lambda} + {\bf P}_{n|0:n}^{-1} \big[{\bf P}_{n|0:n}\! -\! {\bf P}_{n|0:n+1}\big] {\bf P}_{n|0:n}^{-1}. $$

(71)

Finally injecting Eqs. 40 and 41 in Eq. 70 we get Eq. 55, and injecting Eqs. 40 and 42 in Eq. 71 we get Eq. 56.