Sequential Filtering Theory

Abstract

Given the following model for the d-dimensional system and observations:

$$ {x_{n + 1}} = {\Phi _{n + 1}}{x_n} + {G_{n + 1}}{u_n} $$

$$[{z_n} = {H_n}{x_n} + {v_n}$$

where x₁ ∈ N(0,Γ) and

$$[E{u_n}{u'_m} = {\delta _{n,m}}{Q_n}$$

$$ E{{v}_{n}}{{v'}_{m}} = {{\delta }_{{n,m}}}{{R}_{n}} $$

and Q _n 2265 0. We discovered last time that the sequential filter is described by the equations

$${\hat x_{n + 1|n}} = {\Phi _{n + 1}}{\hat x_{n|n - 1}} + {K_n}\left( {{I_n}} \right),$$

((4.1))

$${I_n} = {z_n} - {\hat x_{n|n - 1}},$$

((4.2))

Kn = Φ_n+1P_nH’_n(H_nP_nH’_n + R_n)^-1, (4.3) P_n+1=Φ_n+1(P_n-P_nH’_n(H_nP_nH’_n+R_n)^-1H_nP_n)Φ’_n+1+G_n+1Q_nG’_n+1(4.4) where ${\hat x_{1|0}} = 0$ and Pn is the error covariance matrix, defined by

$${P_n} = E{\tilde x_{n|n - 1}}{\tilde x'_{n|n - 1}}$$

$${\tilde x_{n|n - 1}} = {x_n} - {\hat x_{n|n - 1}}$$

so that ${P_0} = \Gamma .{\tilde x_{n|n - 1}}$ is called the filter error. Equation (4.4) is the matrix Riccati equation for the filter.