Filtering for a class of nonlinear discrete-time stochastic systems with state delays

In this paper, the filtering problem is investigated for a class of nonlinear discrete-time stochastic systems with state delays. We aim at designing a full-order filter such that the dynamics of the estimation error is guaranteed to be stochastically, exponentially, ultimately bounded in the mean square, for all admissible nonlinearities and time delays. First, an algebraic matrix inequality approach is developed to deal with the filter analysis problem, and sufficient conditions are derived for the existence of the desired filters. Then, based on the generalized inverse theory, the filter design problem is tackled and a set of the desired filters is explicitly characterized. A simulation example is provided to demonstrate the usefulness of the proposed design method.     

On a class of SGS filters for LES with shape preserving concatenation properties

We develop a class of filter functions for large-eddy simulation which have the key property that multiple successive application even with different filter widths is equal to a single filtering employing filters from the same class but at an extended or equal filter width. In the context of the filter class development we obtain a functional delay equation which for special cases may be solved completely general. The presently developed class of filters may be used in conjunction with certain sub-grid scale models such as the approximate deconvolution model [3] where explicit multiple filtering is needed. Hence utilizing filters from the present class computational cost of filter evaluation may be considerably reduced.     

Generalized H ∞-optimal filtering under external and initial perturbations

We consider the problem of optimal filtering of unmeasured variables of a linear dynamical system by linear stationary filters. The filtering performance functional to be minimized is given by the maximum relative integral filtering error over all external perturbations as well as initial perturbations caused by the unknown initial conditions of the system. We show that the optimal filter implements a trade-off between an H _∞-optimal filter and a minimax observer.     

Projection–difference method for controlled Fourier filtering

A projection–difference method is developed for approximating controlled Fourier filtering for quasilinear parabolic functional-differential equations. The method relies on a projection–difference scheme (PDS) for the approximation of the differential problem and derives a O(τ^1/2 + h) bound on the rate of convergence of PDS in the weighted energy norm without prior assumptions of additional smoothness of the generalized solutions. The PDS leads to a natural approximation of the objective functional in the optimal Fourier filtering problem. A bound of the same order is obtained for the rate of convergence in the functional of the problems approximating the Fourier filter control problem.     

Fast algorithms for designing variable FIR notch filters

The paper presents fast algorithms for designing finite impulse response (FIR) notch filters. The aim is to design a digital FIR notch filter so that the magnitude of the filter has a deep notch at a specified frequency, and as the notch frequency changes, the filter coefficients should be able to track the notch fast in real time. The filter design problem is first converted into a convex optimization problem in the autocorrelation domain. The frequency response of the autocorrelation of the filter impulse response is compared with the desired filter response and the integral square error is minimized with respect to the unknown autocorrelation coefficients. Spectral factorization is used to calculate the coefficients of the filter. In the optimization process, the computational advantage is obtained by exploiting the structure of the Hessian matrix which consists of a Toeplitz plus a Hankel matrix. Two methods have been used for solving the Toeplitz‐plus‐Hankel system of equations. In the first method, the computational time is reduced by using Block–Levinson's recursion for solving the Toeplitz‐plus‐Hankel system of matrices. In the second method, the conjugate gradient method with different preconditioners is used to solve the system. Comparative studies demonstrate the computational advantages of the latter. Both these algorithms have been used to obtain the autocorrelation coefficients of notch filters with different orders. The original filter coefficients are found by spectral factorization and each of these filters have been tested for filtering synthetic as well as real‐life signals. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite-dimensional filter for a class of nonlinear systems with correlated noises

The purpose of this article is to compute an explicit formula for the unnormalized conditional density for the filter associated with a nonlinear filtering problem with correlated noises and a signal process with nonlinear terms in the drift. This article extends the result of Daum to nonlinear filtering systems with correlated noises and incorporates both the Kalman–Bucy and Bene? filters as particular cases.     

Finite-time filtering for switched linear systems with a mode-dependent average dwell time

This study considers the problem of finite-time filtering for switched linear systems with a mode-dependent average dwell time. By introducing a newly augmented Lyapunov–Krasovskii functional and considering the relationship between time-varying delays and their upper delay bounds, sufficient conditions are derived in terms of linear matrix inequalities such that the filtering error system is finite-time bounded and a prescribed noise attenuation level is guaranteed for all non-zero noises. Thus, a finite-time filter is designed for switched linear systems with a mode-dependent average dwell time. Finally, an example is given to illustrate the efficiency of the proposed methods.     

Self-triggered filter design for a class of nonlinear stochastic systems with Markovian jumping parameters

This paper is concerned with the self-triggered filtering problem for a class of Markovian jumping nonlinear stochastic systems. The event-triggered mechanism (ETM) is employed between the sensor and the filter to reduce unnecessary measurement transmission. Governed by the ETM, the measurement is transmitted to the filter as long as a predefined condition is satisfied. The purpose of the addressed problem is to synthesize a filter such that the dynamics of the filtering error is bounded in probability (BIP). A sufficient condition is first given to ensure the boundedness in probability of the filtering error dynamics, and the characterization of the desired filter gains is then realized by means of the feasibility of certain matrix inequalities. Furthermore, a self-triggered mechanism is designed to guarantee the filtering error dynamics to be BSP with excluded Zeno phenomenon. In the end, numerical simulation is carried out to illustrate the usefulness of the proposed self-triggered filtering algorithm.     

Filtering-based robust fault detection of fuzzy jump systems

This paper studies the robust fault detection filter (RFDF) design problems for uncertain nonlinear Markov jump systems with state delays and parameter uncertainties. By means of Takagi-Sugeno fuzzy models, the dynamics of filtering error generator and the fuzzy RFDF system are constructed. With the aid of the selected weighting matrix function, the design objective is to find an optimal RFDF which results in a minimal difference between the reference model (ideal solution) and the RFDF (real solution) to be designed. A sufficient condition is firstly established on the stochastic stability by using stochastic Lyapunov-Krasovskii functional approach. Then in terms of linear matrix inequalities techniques, sufficient conditions on the existence of fuzzy RFDF are presented and proved. Finally, the design problem is formulated as an optimization algorithm. Simulation results illustrate that the proposed RFDF can detect the faults shortly after the occurrences.     

On the best approximation matrix problem and matrix Fourier series

In this paper the concept of positive definite bilinear matrix moment functional, acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b] is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Riemann—Lebesgue, matrix property and the bessel—parseval matrix inequality are proved. The concept of total set with respect to a positive definite matrix functional is introduced, and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional is established.     

Explicit solution of a Kolmogorov equation

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 been the major tool in studying the Duncan-Mortensen-Zakai equation. Recently the second author has constructed general finite-dimensional filters which contain both Kalman-Bucy filters and Benes filter as special cases. In this paper we consider a filtering system with arbitrary nonlinear driftf(x) which satisfies some regularity assumption at infinity. This is a natural assumption in view of Theorem 10 of [DTWY] in a special case. Under the assumption on the observation h(x)=constant, we propose writing down the solution of the Duncan-Mortensen-Zakai equation explicitly.This research was supported by Army Grant DAAH-04-93G-0006.     

Optimization approach to the design of frequency sampling filters

Many digital signal processing applications require linear phase filtering. For applications that require narrow-band linear phase filtering, frequency sampling filters can implement linear phase filters more efficiently than the commonly used direct convolution filter. In this paper, a technique is developed for designing linear phase frequency sampling filters. A frequency sampling filter approximates a desired frequency response by interpolating a frequency response through a set of frequency samples taken from the desired frequency response. Although the frequency response of a frequency sampling filter passes through the frequency samples, the frequency response may not be well behaved between the specific samples. Linear programming is commonly used to control the interpolation errors between frequency samples. The design method developed in this paper controls the interpolation errors between frequency samples by minimizing the mean square error between the desired and actual frequency responses in the stopband and passband. This design method describes the frequency sampling filter design problem as a constrained optimization problem which is solved using the Lagrange multiplier optimization method. This results in a set of linear equations which when solved determine the filter's coefficients.This work was partially funded by The National Supercomputing Center for Energy and the Environment, University of Nevada Las Vegas, Las Vegas, Nevada and by NSF Grant MIP-9200581.     

The construction of wavelets from generalized conjugate mirror filters in

The classical constructions of wavelets and scaling functions from conjugate mirror filters are extended to settings that lack multiresolution analyses. Using analogues of the classical filter conditions, generalized mirror filters are defined in the context of a generalized notion of multiresolution analysis. Scaling functions are constructed from these filters using an infinite matrix product. From these scaling functions, non-MRA wavelets are built, including one whose Fourier transform is infinitely differentiable on an arbitrarily large interval.     

Limiting optimal adaptive filtering with unknown disturbance covariance

The accuracy of estimating the variance of the Kalman-Bucy filter depends essentially on disturbance covariance matrices and measurement noise. The main difficulty in filter design is the lack of necessary statistical information about the useful signal and the disturbance. Filters whose parameters are tuned during active estimation are classified with adaptive filters. The problem of adaptive filtering under parametric uncertainty conditions is studied. A method for designing limiting optimal Kalman-Bucy filters in the case of unknown disturbance covariance is presented. An adaptive algorithm for estimating disturbance covariance matrices based on stochastic approximation is described. Convergence conditions for this algorithm are investigated. The operation of a limiting adaptive filter is exemplified.     

基于仿酉矩阵的紧支撑二元正交小波滤波器组的构造

高维小波是处理多维信号的有力工具,张量积和栅格结构的小波有其自身的特点,但在实际应用中,我们仍需要构造小波滤波器来满足特定情形下的需要以提高滤波的效果,而构造正交滤波器,在多相域里就等价于构造仿酉阵,在本文中,我们通过对仿酉矩阵的研究,证明二元一次对称的仿酉阵一定能够块对角化,利用这种性质,给出了不可分离的二元正交小波滤波器组及线性相位小波滤波器的构造,并给出了相应的例子．     

The Influence of Rainflow Filtering on the Distributions of Characteristic Wave Parameters

A problem when filtering measurements of the sea-surface level is that the highest values, which are also the most interesting, might get lost. The rainflow filter is an alternative to the band-pass filters traditionally used. In this paper we investigate the properties of rainflow filtering by examining the distributions of some characteristic wave parameters. These distributions are calculated by means of the regression-approximation method which requires as input a spectral density. An approximation of the rainflow filter in the frequency domain, obtained by spectral simulation, is therefore used. The rainflow filter yields results which are closer to the distributions obtained from the original spectrum with high-frequency contents.     

A dynamical systems framework for intermittent data assimilation

We consider the problem of discrete time filtering (intermittent data assimilation) for differential equation models and discuss methods for its numerical approximation. The focus is on methods based on ensemble/particle techniques and on the ensemble Kalman filter technique in particular. We summarize as well as extend recent work on continuous ensemble Kalman filter formulations, which provide a concise dynamical systems formulation of the combined dynamics-assimilation problem. Possible extensions to fully nonlinear ensemble/particle based filters are also outlined using the framework of optimal transportation theory.     

Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper, the authors address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.     

Spline filter for optimum phase demodulation †

An optimum nonlinear filter is realized by sequentially updating the spline coefficients of the relevant conditional distribution. The nonlinear filtering problem considered is that of phase demodulation with a two-dimensional phase process model. Speed and accuracy comparison of spline realization with other realizations, such as Fourier filter and point mass, will be provided     

Signal Processing in Relaxation Experiments

Signal processing problems arising in the study of the linearly viscoelastic behavior of polymers and composites are considered. It is shown that the great amount of data conversions is associated with integral transforms using kernels which depend on the ratio or product of arguments for monotonic long-time-interval and wide-frequency-band functions (signals). A unified method of carrying out these integral transforms is developed by combining a logarithmic transformation of the signal time scale with digital filtering. For integral transforms leading to ill-conditioned inverse problems, a method of regularization is proposed based on choosing a sampling rate which ensures an acceptable error variance of the output signal. The specific features of the functional filters used for performing the functional (integral) transforms are discussed. Examples of performing the Heaviside-Carson sine transform and an inherently ill-conditioned problem of inverting the integral transform for determining the relaxation spectrum are represented by digital functional filters.