Robust two-stage Kalman filters for systems with unknown inputs

A method is developed for the state estimation of linear time-varying discrete systems with unknown inputs. By making use of the two-stage Kalman filtering technique and a proposed unknown inputs filtering technique, a robust two-stage Kalman filter which is unaffected by the unknown inputs can be readily derived and serves as an alternative to the Kitanidis' (1987) unbiased minimum-variance filter. The application of this new filter is illustrated by optimal filtering for systems with unknown inputs     

Kalman filters for non-linear systems: a comparison of performance

The Kalman filter is a well-known recursive state estimator for linear systems. In practice, the algorithm is often used for non-linear systems by linearizing the system's process and measurement models. Different ways of linearizing the models lead to different filters. In some applications, these?‘Kalman filter variants’?seem to perform well, while for other applications they are useless. When choosing a filter for a new application, the literature gives us little to rely on. This paper tries to bridge the gap between the theoretical derivation of a Kalman filter variant and its performance in practice when applied to a non-linear system, by providing an application-independent analysis of the performances of the common Kalman filter variants.

This paper separates performance evaluation of Kalman filters into (i) consistency, and (ii) information content of the estimates; and it separates the filter structure into (i) the process update step, and (ii) the measurement update step. This decomposition provides the insights supporting an objective and systematic evaluation of the appropriateness of a particular Kalman filter variant in a particular application.     

Multiscale systems, Kalman filters, and Riccati equations

An algorithm analogous to the Rauch-Tung-Striebel algorithm-consisting of a fine-to-coarse Kalman filter-like sweep followed by a coarse-to-fine smoothing step-was developed previously by the authors (ibid. vol.39, no.3, p.464-78 (1994)). In this paper they present a detailed system-theoretic analysis of this filter and of the new scale-recursive Riccati equation associated with it. While this analysis is similar in spirit to that for standard Kalman filters, the structure of the dyadic tree leads to several significant differences. In particular, the structure of the Kalman filter error dynamics leads to the formulation of an ML version of the filtering equation and to a corresponding smoothing algorithm based on triangularizing the Hamiltonian for the smoothing problem. In addition, the notion of stability for dynamics requires some care as do the concepts of reachability and observability. Using these system-theoretic constructs, the stability and steady-state behavior of the fine-to-coarse Kalman filter and its Riccati equation are analysed     

Kalman filtering for continuous-time uncertain systems withMarkovian jumping parameters

Studies the problem of Kalman filtering for a class of uncertain linear continuous-time systems with Markovian jumping parameters. The system under consideration is subjected to time-varying norm-bounded parameter uncertainties in the state and measurement equations. Stochastic quadratic stability of the above system is analyzed. A state estimator is designed such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties, which is in terms of solutions of two sets of coupled algebraic Riccati equations     

A modified extended Kalman filter for linear discrete-time systems with unknown parameters

This technical communique presents a modified extended Kalman filter for estimating the states and unknown parameters in discrete-time, multi-input multi-output linear systems. The hyperstability of the filter is guaranteed by introducing a compensator into the estimation mechanism. It is proved that the estimates for the states and unknown parameters converge to the exact values if some conditions are assumed to the estimation mechanism. A numerical example shows that the proposed filter is much more effective than the extended Kalman filter in the estimation of unknown parameters.     

Separate-bias estimation with reduced-order Kalman filters

This paper presents the optimal two-stage Kalman filter for systems that involve noise-free observations and constant but unknown bias. Like the full-order separate-bias Kalman filter, this new filter provides an alternative to state vector augmentation and offers the same potential for improved numerical accuracy and reduced computational burden. When dealing with systems involving accurate, essentially noise-free measurements, this new filter offers an additional advantage, a reduction in filter order. The optimal separate-bias reduced order estimator involves a reduced order filter for estimating the state, the order equalling the number of states less the number of observations     

A parallel decomposition for Kalman filters

A decomposition is given for the implementation of the Kalman filter as a collection of parallel processors. This decomposition is based on the representation of the system as a direct sum of observability subspaces     

Statistical properties of response for nonlinear structural systems with random parameters

A probabilistic algorithm for calculating the statistical properties of response due to uncertainties in geometry and material properties for geometrically nonlinear structural dynamics is presented. It is shown here that this technique is readily applicable to geometrically nonlinear small strain dynamic problems in which the equations of motion of structures are derived using the finite element method. A general formulation for problems of this type is presented. The efficiency of the method is illustrated by application to nonlinear vibration of a cantilever beam with random parameters.     

Statistics of eigensolutions for systems with random shape parameters using BEM

This paper presents a probabilistic boundary element method for analysis of the statistics of structural eigenvalues and eigenvectors, when the shape parameters of the structures are considered as random variables. Using this method, engineers are able to estimate the errors of the structural eigenvalues and eigenvectors resulting from manufacturing errors, and evaluate the differences between the experimental results and numerical results, which are given by the finite element method or boundary element method, etc. This method can be used to design and analyse the components of engineering structures because of its simplicity and effectiveness.     

Learning algorithms for neural networks with the Kalman filters

Based on various approaches, several different learing algorithms have been given in the literature for neural networks. Almost all algorithms have constant learning rates or constant accelerative parameters, though they have been shown to be effective for some practical applications. The learning procedure of neural networks can be regarded as a problem of estimating (or identifying) constant parameters (i.e. connection weights of network) with a nonlinear or linear observation equation. Making use of the Kalman filtering, we derive a new back-propagation algorithm whose learning rate is computed by a time-varying Riccati difference equation. Perceptron-like and correlational learning algorithms are also obtained as special cases. Furthermore, a self-organising algorithm of feature maps is constructed within a similar framework.     

Stability of linear dynamic systems with random time-dependent parameters

We present conditions of asymptotic stability for the solution of a system of linear differential equations that depend on stochastic processes.Translated from Kibernetika, No. 3, pp. 70–72, 75, May–June 1990.     

Kalman filters in non-uniformly sampled multirate systems: For FDI and beyond

The first part of the paper is the development of a data-driven Kalman filter for a non-uniformly sampled multirate (NUSM) system. Algorithms for both one-step predictor and filtering are developed and analysis of stability and convergence is conducted in the NUSM framework. The second part of the paper investigates a Kalman filter-based methodology for unified detection and isolation of sensor, actuator, and process faults in the NUSM system with analysis on fault detectability and isolability. Case studies using data respectively collected from a pilot experimental plant and a simulated system are conducted to justify the practicality of the proposed theory.     

Robust Kalman filtering for uncertain discrete-time systems with probabilistic parameters bounded within a polytope

We consider the problem of finite horizon discrete-time Kalman filtering for systems with parametric uncertainties. Specifically, we consider unknown but deterministic uncertainties where the uncertain parameters are assumed to lie in a convex polyhedron with uniform probability density. The condition and a procedure for the construction of a suboptimal filter that minimizes an expected error covariance over-bound are derived.     

Sensitivity analysis of linear control systems with random plan parameters

Abstract-The effects of random variations of the plant parameters in multivariable linear control systems are analyzed in terms of stochastic sensitivity operators. Both tie-varying and time-invariant control systems are analyzed. Conditions are derived for the stochastic sensitivity operator which are sufficient to guarantee that the closed-loop realization of the control system is less sensitive to the random parameter variations than the nominally equivalent open-loop system.     

A stability criterion for a class of discrete nonlinear systems with random parameters

In this paper, discrete nonlinear models with random parameters are studied. A new frame to classify and analyze discrete stochastic nonlinear systems has been developed from deterministic nonlinear systems to stochastic nonlinear systems. This frame is broad and includes a large class of stochastic nonlinear systems. A stability criterion developed for this frame is a non-Lyapunov method of stability analysis and is easily applied. In addition, this derived sufficient condition of stability is obtained without the assumption of stationarity for the random noise as frequently assumed in the literature.     

General two-stage extended Kalman filters

A general two-stage extended Kalman filter (GTSEKF), which extends the linear general two-stage Kalman filter to nonlinear systems, is further proposed. A new nonlinear two-stage transformation is introduced to facilitate achieving this extension. As in the linear one, the GTSEKF is derived mainly by applying the nonlinear two-stage transformation to the well-known extended Kalman filter (EKF), and is shown to be equivalent to the EKF with a decoupled computing structure. A nonlinear filter for estimating constant parameters in dynamic systems is presented to illustrate one application of the proposed GTSEKF. A literature example is also given to demonstrate the correctness and usefulness of the proposed results.     

Decoupled Kalman filters for phased array radar tracking

Kalman filters have been used in numerous phased array radars to track satellites, reentry vehicles, and missiles. This paper considers the design of these filters to reduce computational requirements, ill-conditioning, and the effects of nonlinearities. Several special coordinate systems used to represent the Kalman filter error covariance matrix are described. These covariance coordinates facilitate the approximate decoupling required for practical filter design. A tutorial discussion and analysis of ill-conditioning in Kalman filters is used to motivate these design considerations. This analysis also explains several well-known phenomena reported in the literature. In addition, a discussion of nonlinearities and methods to mitigate their ill effects is included.     

Frequency-domain properties of Kalman filters

This paper explores the frequency-domain characteristics of the steady-state Kalman filter. It gives insight into the properties of the filter that are useful for analysis and design. The results are illustrated by a number of industrial applications.     

A Chi-square test for fault-detection in Kalman filters

A test for real-time detection of soft failures in navigation systems using Kalman filters has been proposed by Kerr. The test is based on the overlap between the confidence regions associated with two estimates, one obtained from a Kalman filter using on-line measurements, and the other based solely on a priori information. An alternate computational technique is presented which is based on constructing a chi-square test statistic from the difference between the two estimates and comparing it to a precomputed threshold. The chi-square test avoids the iterative computations required by the two-ellipsoid method for dimensions of two and higher.     

Distributed control of multi-agent systems with random parameters and a major agent

Distributed control of the multi-agent systems involving a major agent and a large number of minor agents is investigated in this paper. There exist Markov jump parameters in the dynamic equation and random parameters in the index functions. The major agent has salient impact on others. Each minor agent merely has tiny influence, while the average effect of all the minor agents is not negligible, which plays a significant role in the evolution and performance index of each agent. Besides the state of the major agent, each minor agent can only access to the information of its state and parameters. Based on the mean field (MF) theory, a set of distributed control laws is designed. By the probability limit theory, the uniform stability of the closed-loop system and the upper bound of the corresponding index values are obtained. Via a numerical example, the consistency of the MF estimation and the influence of the initial state values and parameters on the index values are demonstrated.