Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode

Kushner, Harold J.

doi:10.1109/tac.1967.1098582

Cited by 66 publications

(24 citation statements)

References 7 publications

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“…Early work on optimal nonlinear filtering can be traced back to Stratonovich [30]. The first rigorous derivation was given by Kushner in his seminal paper [19]; see also [20]. Such equations are referred to as Kushner's equations.…”

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confidence: 99%

State Observability and Observers of Linear-Time-Invariant Systems Under Irregular Sampling and Sensor Limitations

Wang

Yin

et al. 2011

IEEE Trans. Automat. Contr.

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Abstract-State observability and observer designs are investigated for linear-time-invariant systems in continuous time when the outputs are measured only at a set of irregular sampling time sequences. The problem is primarily motivated by systems with limited sensor information in which sensor switching generates irregular sampling sequences. State observability may be lost and the traditional observers may fail in general, even if the system has a full-rank observability matrix. It demonstrates that if the original system is observable, the irregularly sampled system will be observable if the sampling density is higher than some critical frequency, independent of the actual time sequences. This result extends Shannon's sampling theorem for signal reconstruction under periodic sampling to system observability under arbitrary sampling sequences. State observers and recursive algorithms are developed whose convergence properties are derived under potentially dependent measurement noises. Persistent excitation conditions are validated by designing sampling time sequences. By generating suitable switching time sequences, the designed state observers are shown to be convergent in mean square, with probability one, and with exponential convergence rates. Schemes for generating desired sampling sequences are summarized.

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confidence: 99%

State Observability and Observers of Linear-Time-Invariant Systems Under Irregular Sampling and Sensor Limitations

Wang

Yin

et al. 2011

IEEE Trans. Automat. Contr.

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“…If the system dynamics andlor the measurement function are nonlinear,a finite set of statistics sufficient to describe the conditional probability density function (CPDF) is not available (1,2,3). Even if the initial states and the process noise are assumed Gaussian,the nonlinear dynamical system results in a non-Gaussian CPDF.…”

Section: Approximations In Nonlinear Filtering Theorymentioning

confidence: 99%

Assumed density filter with application to homing missile guidance

Balakrishnan

Speyer²

1986

Astrodynamics Conference

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A maximum likelihood estimation method is developed for a class of problems where the dynamics are linear and the measurement function is nonlinear. In this method, called the assumed density filter(ADF), the form of the conditional probability density function(CPDF) is selected to be a function of a finite number of quantities. These quantities which describe the approximate shape of the CPDF around the mode are propagated through each measurement interval. At the measurement the CPDF is updated using Bayes theorem and its mode, computed numerically, is defined to be the best estimate of the state. The posteriori CPDF is then approximated by a Taylor series expansion about its mode to preserve the assumed functional form. The numerical results for a target-intercept problem indicate that the ADF is superior to the extended Kalman filter. Howe:er:fhe ADF has a negative range bias. It is analytically proved,with some approximations, that the maximum likelihood range estimates are smallerthan the mean range estimates. I . INTRODUCTIONTactical weapon systems require accurate tracking of maneuverable vehicles such as submarines and airplanes. During the last several years, there has been an active interest in the development of sophisticated filtering algorithms for tracking with bearings-only as the observations. Mathematically,this problem can be described in an inertial rectangular coordinate frame by a linear dynamical model and a nonlinear discrete observation model or in an inertial polar coordinate frame by a nonlinear dynamical model and a linear discrete observation model. Satisfactory results for this class of problems have been difficult to obtain using current mechanizable filters because of the nonlinearity and the passive nature of the observations. As a result, considerable research has been going on to improve the existing methods in order to obtain better estimates. APPROXIMATIONS IN NONLINEAR FILTERING THEORYThe target tracking problem is stochastic in nature. Analyses of stochastic problems are possible through statistical interpretations. In order to obtain mathematical expressions for the statistics, assumed to represent the best estimates of the states associated with a problem, knowledge of the underlying probability density function (PDF) is essential.If the system dynamics andlor the measurement function are nonlinear,a finite set of statistics sufficient to describe the conditional probability density function (CPDF) is not available (1,2,3). Even if the initial states and the process noise are assumed Gaussian,the nonlinear dynamical system results in a non-Gaussian CPDF. Second,the propagation equation for the conditional mean consists of expectations of nonlinear functions that are very difficult . -to evaluate. Third,since the CPDF is not Gaussian, the system of equations to describe the conditional moments and,hence,the filter as a whole becomes coupled and infinite-dimensional.To circumvent these difficulties,approximations have been attempted to realize estimation methods consisting o...

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“…In general, statistical nonlinear filtering techniques developed using minimumvariance [4] as well as maximum-likelihood [19,20] criteria are infinite-dimensional and too complicated to solve the filter differential equations. On the other hand, the Kalman-filter is a minimum-variance estimator and can be derived from a deterministic approach using H 2 -optimization theory.…”

Section: Introductionmentioning

confidence: 99%

$\mathcal{H}_{2}$ Filtering for Discrete-Time Affine Nonlinear Descriptor Systems

Aliyu

Perrier

2011

Circuits Syst Signal Process

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In this paper, we consider the Kalman (or H 2 )-filtering problem for discrete-time affine nonlinear descriptor systems. Two types of filter are discussed, namely, (i) singular; and (ii) normal. Sufficient conditions for the solvability of the problem in terms of discrete-time Hamilton-Jacobi-Bellman equations (DHJBEs) are presented. The results are also specialized to linear systems in which case the DHJBEs reduce to a system of linear matrix-inequalities (LMIs). Examples are also presented to illustrate the results.

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Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode

Abstract: The s i g n a l i s a s t o c h a s t i c process s a t i s f y i n g t h e

Cited by 66 publications

References 7 publications

State Observability and Observers of Linear-Time-Invariant Systems Under Irregular Sampling and Sensor Limitations

State Observability and Observers of Linear-Time-Invariant Systems Under Irregular Sampling and Sensor Limitations

Assumed density filter with application to homing missile guidance

$\mathcal{H}_{2}$ Filtering for Discrete-Time Affine Nonlinear Descriptor Systems

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