Exact finite-dimensional filters for certain diffusions with nonlinear drift

Beneš, V. E.

doi:10.1080/17442508108833174

Cited by 386 publications

(182 citation statements)

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“…'Generally', it seems, such filters will not exist and though there remains the tantalizing possibility of whole new classes of useful models for which they do exist, there are at the moment no clear ideas as to how and where to look for them. All the same a number of new filters, both 'model cases' and filters of importance in practice, have been discovered using these Lie-algebra ideas [4,13,18,19,[47][48][49][50][51]56]. Since exact finite dimensional filters can not exist in many cases it is natural to look for approximate ones.…”

Section: Y(t) = Y(o)+ Jh(x(s))ds J(t)=h(x(t))mentioning

confidence: 99%

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Lectures on Linear and Nonlinear Filtering

Hazewinkel

1988

Analysis and Estimation of Stochastic Mechanical Systems

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Section: Y(t) = Y(o)+ Jh(x(s))ds J(t)=h(x(t))mentioning

confidence: 99%

“…4) 0 are corrupted by further (measurement) noise v(t). Technically speaking, equations (1.1), (1.2) are to be regarded as Ito stochastic differential equations; cf section 5 below for more remarks.…”

Section: Y(t) = Y(o)+ Jh(x(s))ds J(t)=h(x(t))mentioning

confidence: 99%

Lectures on Linear and Nonlinear Filtering

Hazewinkel

1988

Analysis and Estimation of Stochastic Mechanical Systems

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“…Although the general solution of the mean-square filtering problem for nonlinear state and observation equations confused with white Gaussian noises is given by the Kushner equation for the conditional density of an unobserved state with respect to observations Kushner (1964), there are a very few known examples of nonlinear systems where the Kushner equation can be reduced to a finite-dimensional closed system of filtering equations for a certain number of lower conditional moments (see Kalman and Bucy (1961), Wonham (1965) and Benes (1981) for more details). The complete classification of the "general situation" cases (this means that there are no special assumptions on the structure of state and observation equations and the initial conditions), where the nonlinear finite-dimensional filter exists, is given in Yau (1994).…”

Section: Introductionmentioning

confidence: 99%

ean-Square Filtering for Polynomial System States Confused with Poisson Noises over Polynomial Observations

Maldonado¹,

Karimi

2011

MIC

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In this paper, the mean-square filtering problem for polynomial system states confused with white Poisson noises over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations with white Poisson noises. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third order sensor filtering problem for a quadratic state, assuming a conditionally Poisson initial condition for the extended third order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.

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“…Recursive least squares (which may be considered as a special case of Kalman filtering [2], [3]), continues to be a key algorithm in digital signal processing. However, the assumptions of linearity and of quadratic costs (or, equivalently, Gaussian noise) are not suitable for some applications, which has motivated nonlinear filters such as the extended Kalman filter (EKF), the unscented Kalman filter (UKF) [4], particle filters, and exact recursive filters [5], [6].…”

Section: Introductionmentioning

confidence: 99%

Autonomous state space models for recursive signal estimation beyond least squares

Zalmai

Wildhaber

Loeliger

2017

2017 25th European Signal Processing Conference (EUSIPCO)

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Abstract-The paper addresses the problem of fitting, at any given time, a parameterized signal generated by an autonomous linear state space model (LSSM) to discrete-time observations. When the cost function is the squared error, the fitting can be accomplished based on efficient recursions. In this paper, the squared error cost is generalized to more advanced cost functions while preserving recursive computations: first, the standard sample-wise squared error is augmented with a sampledependent polynomial error; second, the sample-wise errors are localized by a window function that is itself described by an autonomous LSSM. It is further demonstrated how such a signal estimation can be extended to handle unknown additive and/or multiplicative interference. All these results rely on two facts: first, the correlation function between a given discrete-time signal and a LSSM signal can be computed by efficient recursions; second, the set of LSSM signals is a ring.

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Exact finite-dimensional filters for certain diffusions with nonlinear drift

Cited by 386 publications

References 3 publications

Lectures on Linear and Nonlinear Filtering

Lectures on Linear and Nonlinear Filtering

ean-Square Filtering for Polynomial System States Confused with Poisson Noises over Polynomial Observations

Autonomous state space models for recursive signal estimation beyond least squares

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