Kalman-Bucy Filter Research Articles

Filtrage lin�aire par morceaux avec petit bruit d'observation

We consider a piecewise linear filtering problem with small observation noise. In two different situations we construct an approximate finite-dimensional filter based on several Kalman-Bucy filters running in parallel and a procedure of tests. In the first case our work generalizes some results of Fleminget al. to more general piecewise linear dynamics.

Active noise cancellation by using the linear quadratic Gaussian independent modal space control

The objective of this study is to develop an active noise controller based on optimal control approaches for enclosed Gaussian noise fields. By independent modal space control, the individual mode in the acoustic field is suppressed by the corresponding modal control exerted by loudspeakers. The formulation of the modal controller is considerably simplified because modal coupling is neglected. For the Gaussian noises considered in this study, the linear quadratic Gaussian algorithm in conjunction with the Kalman–Bucy filter is employed to perform state feedback and estimation. The developed controller is validated by simulations for a duct and a rectangular room. The results indicate that the technique will yield significant noise reduction if one uses the same number of controlled modes, microphones, and loudspeakers. Satisfactory performance is possible if one carefully avoids placing microphones and loudspeakers at the nodal points.

Discrete-time piecewise linear filtering with small observation noise

A problem of discrete-time piecewise linear filtering with small observation noise is considered. The case in which the observation function is not one-to-one and has two intervals of linearity is studied, in particular when this function is symmetric. Under some detectability assumption, a procedure is presented to detect the intervals of linearity of the observation function. During such time intervals, one can then approximate the optimal filter by the corresponding Kalman-Bucy filter.

Robust estimation and compensation for actuator and sensor failures in linear systems

A computationally feasible technique for the robust detection and estimation for actuator and sensor failures is presented. Model errors and component failures are represented by a bias vector called the failure state in system and measurement equations. A Kalman-Bucy filter is implemented to estimate the system state, and to generate the corresponding residuals. These residuals are then processed by using an adaptive fading Kalman filter to give the failure state estimate. The final state estimate is obtained by compensating model errors and component failures in the filter based on no failure assumption. The divergency of the filter based on no failure assumption is avoided by stepwise compensation of the failure state. The technique is applicable to the detection, estimation and compensation of slowly varying model errors and suddenly occurring component failures, and to the discrimination between them

Efficiency of the Kalman-Bucy Filter in the Presence of Uncertain Disturbances

Stochastic guaranteeing estimation problem with uncertain bounded disturbances is considered. The Kalman-Bucy filter is used instead of the optimal one for approximate solving of this problem. By means of the duality theory the estimate of the nonoptimality degree is constructed. It is calculated without solving the optimal problem. Thus, the performance of the Kalman-Bucy filter can be evaluated by a simple tool. Numerical example shows a high efficiency of the proposed approach.

The Kalman-Bucy filter in the guaranteed estimation problem

The optimal guaranteed a priori estimation problem is considered. The Kalman-Bucy filter is used for the approximate solving of this problem. The analytical estimate for the nonoptimality degree of the Kalman-Bucy filter is obtained. This estimate is determined by the Kalman-Bucy filter characteristics only. Thus the Kalman-Bucy filter efficiency can be established without accurate solving of the difficult optimal guaranteed estimation problem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fixed memory filter for real-time estimation of noise-corrupted signals

Introduction I N many engineering applications, such as guidance and navigation of aeronautical vehicles, the signals need to be estimated from noisy measurements in real time. The (minimum-variance) Kalman-Bucy filter' that relies on a finite-dimensional state-space model of the plant dynamics and the measurement history is extensively used for this purpose. Techniques of adaptive filtering' that are based on the principle of recursive least squares algorithm and rely on an inputoutput relationship of the plant dynamics are also used. Given a linear (or linearized) time-invariant finite-dimensional model of plant dynamics in the autoregressive moving average (ARMA) setting, we propose a fixed-memory, nonrecursive filter for real-time signal estimation. The key idea is to construct a simple, nonrecursive filter based on weighted averaging of a finite array of past values of measured inputs and outputs of the plant. A fixed memory filter is very useful in the applications of smart sensors and/or active sensors where memory constraints are tight and computationally efficient algorithms are required for real-time implementation on a microchip collocated with the sensor. The filter memory length depends on the plant dynamics and the allowable bound of estimation error and, for real-time applications, is often selected as a tradeoff between execution time and accuracy of the filter. This Engineering Note presents the concept and formulation of a fixed-memory filter based on an ARMA model of plant dynamics. The criteria for selection of the filter memory length are not included in this Note. Although the filter algorithm is derived for single-input single-output (SISO) systems, the proposed filter can be extended to multi-input multioutput (MIMO) systems.

Adaptive model architecture and extended Kalman-Bucy filters

In radar systems, extended Kalman-Bucy filters (EKBFs) are used to estimate state vectors of objects in track. Filter models accounting for fundamental aerodynamic forces on reentry vehicles are well known. A general model structure accommodating the dynamics of reentry vehicles in both exoatmospheric and endoatmospheric flight is presented. The associated EKBFs for these various models are described and the resulting associated parameter estimation and identification problems are discussed. The effects of position, velocity, drag, and aerodynamic lift are described within a nested set of EKBF models. >

The Kalman-Bucy filter accuracy in the guaranteed parameter estimation problem with uncertain statistics

The guaranteed parameter estimation problem for a linear dynamical system with uncertain disturbance statistics is considered. The estimate of the Kalman-Bucy filter nonoptimality degree is constructed. It is based on the duality theory. This estimate is determined by the Kalman-Bucy filter characteristics only and can be calculated without knowledge of the accurate solution of the optimal guaranteed estimation problem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parameter estimation for kalman-bucy filter with small noise

The asymptotic behavior of the MLE of a parameter of partially observed linear system is studied. Using asymptotic expansion of this estimate by the powers of diffusion coefficient the consistency and asymptotic normality of MLE are proved. The time of observation is supposed to be fixed and the asymptotics corresponds to small noises in observed and nonobserved equations.

Numerical Estimation of States in a Diffusion Process using the Finite Element Method

The problem of state estimation in a distributed parameter system using discretization at the end and discretization at the beginning using finite element method has been considered. The distributed Kalman—Bucy filter algorithm of a diffusion process is solved numerically by the finite element method. The state-estimation problem of the same process is also solved by first approximating the diffusion process by a lumped model and then using the Kalman filter algorithm. This method is seen to be simpler in terms of computational effort and yields fairly accurate results.

Estimating the solutions of linear stochastic equations by the information criterion

The information criterion is applied to estimate the state of linear dynamic systems in the presence of random disturbances on the input and the output. The problem of finding the best information estimator is reduced to an optimal control problem. An expression for the maximum mean information is derived. Recursive Kalman-Bucy filters are constructed.

Recursive structure of noncausal Gauss-Markov random fields

An approach is developed for noncausal Gauss-Markov random fields (GMRFs) that enables the use of recursive procedures while retaining the noncausality of the field. Recursive representations are established that are equivalent to the original field. This is achieved by first presenting a canonical representation for GMRFs that is based on the inverse of the covariance matrix, which is called the potential matrix. It is this matrix rather than the field covariance that reflects in a natural way the MRF structure. From its properties, two equivalent one-sided representations are derived, each of which is obtained as the successive iterates of a Riccati-type equation. For homogeneous fields, these unilateral descriptions are symmetrized versions of each other, the study of only one Riccati equation being required. It is proven that this Riccati equation converges at a geometric rate, therefore the one-sided representations are asymptotically invariant. These unilateral representations make it possible to process the fields with well-known recursive techniques such as Kalman-Bucy filters and two-point smoothers.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Reduced order controller design for discrete time systems

Robust discrete control system design techniques and model reduction are discussed. A new linear quadratic guussian/loop transfer recovery procedure for discrete time systems is presented. In this technique, a full-state feedback or an output injection feedback is designed which has the desired loop shape, and then recovered by a realizable linear quadratic gaussian controller. To do this, results that show the effects of the weighting matrices (noise intensities) on linear quadratic regulator (Kalman-Bucy filter) return difference and inverse-return difference arc derived and a procedure to recover the linear quadratic regulator loop transfer function is described. The complexity of the resulting controller is then reduced without causing closed-loop instability. Two methods for model reduction are considered. The first is the discrete balanced realization and the second is P frequency weighting technique where it is possible to vary the approximation accuracy with frequency. The controller design and re...

On diffusion approximations for filtering

We consider a family of processes ( X ε , Y ε ) where X ε = ( X ε t ) is unobservable, while Y ε = ( Y ε t ) is observable. The family is given by a model that is nonlinear in the observations, has coefficients that may be rapidly oscillating, and additive disturbances that may be wide-band and non-Gaussian. Using results of diffusion approximation for semimartingales, we show the convergence in distribution (for ε → 0) of ( X ε , Y ε ) to a process ( X, Y) that satisfies a linear-Gaussian model. Applying the Kalman-Bucy filter for ( X, Y) to ( X ε , Y ε ), we obtain a linear filter estimate for X ε t , given the observations { Y ε s , 0 ⩽ s ⩽ t}. Such filter estimate is shown to possess the property of asymptotic (for ε → 0) optimality of its variance. The results are also applied to show the effects that a limiter in the observation equation may have on the signal-to-noise ratio and thus on the filter variance.

Spillover stabilization of large space structures

Active control of large flexible space structures is typically implemented to control only a few known elastic modes. This paper considers the stabilization of the neglected dynamics of the higher modes of vibration. An attempt is made to design modal controllers with improved spillover stability properties. Two formulations for designing the observer to improve spillover stability with minimum performance loss are proposed. One optimizes the noise statistics used in the design of the Kalman-Bucy Filter (KBF), while the other directly optimizes the the gain matrix of the KBF.

On a Necessary and Sufficient Condition for Finite Dimensionality of Estimation Algebras

Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It has proven to be an invaluable tool in the study of nonlinear filtering problems. In this paper, a simple algebraic necessary and sufficient condition is established for an estimation algebra of a special class of filtering systems to be finite-dimensional. Also presented is a rigorous proof of the Wei–Norman program which allows one to construct finite-dimensional recursive filters from finite dimensional estimation algebras.

Filtering then-dimensional logistic growth model

Following a detailed discussion of the linear Kalman–Bucy filter for the l–dimensional case a description of Zakai nonlinear filtering of the n-dimensional logistic is presented. Explicit formulas for the c ∞-densities for least squares estimation of growth variables conditioned on continually updated community production or consumption records, are obtained. Application is made to a model of the myxomatosis disease in European wild rabbits and to a coral growth model with predation. Extensive use is made of the techniques of linear stochastic partial differential equations developed by H. Kunita, especially backward stochastic calculus and decomposition techniques

Estimating integrated higher-order continuous time autoregressions with an application to money-income causality

An algorithm is presented for computing the Gaussian maximum likelihood estimator of the parameters of a multivariate continuous time autoregressive process with multiple roots that equal zero. Some of the variables might be observed at a point in time (‘stocks’) and some might be observed as an integral over a unit interval (‘flows’). This algorithm, based on the Kalman-Bucy filter, is used to investigate the possibility that previous findings that money Granger-causes industrial production spuriously arose because time-averaged variables were analyzed using discrete time methods.

New Kalman filter algorithms based on orthogonal transformations for serial and vector computers

A new numerically stable Kalman filter algorithm based on a special Givens transformation is developed theoretically. The advantage of the new Kalman filter algorithm vs. conventional formulations is examined for the support of inertial navigation systems (simulation data). The conventional Kalman-Bucy filter, the Bierman formulation and the new Kalman filter algorithms are implemented on IBM 4381 and 3090 computers and vector processors CRAY-1S and CRAY-X-MP. A comparison of CPU-times of the algorithms on the different computers is shown.