Block diagonally dominant positive definite approximate filters and smoothers

✍ Scribed by Kurt S. Riedel

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 526 KB
• Volume: 29
• Category: Article
• ISSN: 0005-1098
• DOI: 10.1016/0005-1098(93)90074-4

✦ Synopsis

We examine stochastic dynamical systems where the transition matrix, 4, and the system noise, FQF r, covariance are nearly block diagonal. When HrR -IH is also nearly block diagonal, where R is the observation noise covariance and H is the observation matrix, our suboptimal filter/smoothers are always positive semidefinite, and have improved numerical properties. Applications for distributed dynamical systems with time dependent pixel imaging are discussed.

I. Introduction

IN THIS ARTICLE, we examine suboptimal filters and smoothers of stochastic systems when the dynamics and the measurements are nearly block diagonal (N.B.D.). We assume that the transition matrix, ~(i + 1, i), the system noise covariance, [FQrr]i, the initial state covariance, P(0 1 0), and the measurement information matrix, Ji-=HrRi-tHi, are all N.B.D~ We then derive estimation equations for the state vector, xi, and the covariance, P(i [ i), which approximate the optimal estimates to second order in e. When q~, FQF r, P(0, 0) and Ji are diagonal, Cohn and Parrish (1991) showed that P(ili) is diagonal. Our work generalizes these results from exactly diagonal systems to approximately block diagonal systems.

Our stochastic systems are similar to the widely studied weakly coupled system (Kokotovic et al. (1969); Sezer and Siljak (1986); Gajic et al. (1990); Shen and Gajic (1990)).

Our N.B.D. systems are not limited to two block systems, but apply to an arbitrary number of blocks. Furthermore, we require only that HrRi-tHi is N.B.D. This contrasts to the stronger hypothesis of weakly coupled systems that Hi and Ri are separately weakly coupled. The existing theory of weakly coupled systems concentrates on the convergence of approximations to the complete system as E tends to zero. Thus the existing analysis considers only the case where • is sufficiently small as to preclude the loss of positive definiteness in the approximate equations. Therefore previous analyses have not explicitly required positive definiteness.

Our emphasis is on well-conditioned approximation of xi and P(ili) for finite, but small values of the coupling parameter, •. Formally, our expansions require that the zeroth order N.B.D. matrices are all uniformly much larger than the remaining offdiagonal terms. In practice, the coupling parameter, •, is not vanishingly small, and there may be component directions where the first order terms almost cancel the zeroth order terms. To prevent the approximate covariance matrix, Pt'~(ili), from losing *