Suboptimal Filter Research Articles

Suboptimal estimation of signals from uncertain observations using approximations of mixtures

In this paper, we propose a suboptimal filtering algorithm for the estimation of a Gaussian signal from uncertain observations, using covariance information. The suboptimal estimators are obtained as the expectation of the signal, given the observations, when a certain approximation is considered for the conditional distribution. The approximation is carried out via successive approximations of mixtures of Gaussian distributions by Gaussian distributions. On the other hand, by assuming that the uncertainty probability is unknown, a recursive estimation algorithm is proposed for that probability. This algorithm is obtained under a Bayesian viewpoint; specifically, by considering a Beta as the a priori distribution for the unknown parameter, the proposed estimators provide approximations for the mean of the a posteriori distributions.

A Single Parameter Tunable Quaternion Based Attitude Estimation Filter

: In human posture tracking applications, limb segment attitude can be estimated without the aid of a generated source using small inexpensive inertial/magnetic sensor modules. In the absence of an adequate process model and process noise characteristics or in an application in which the dynamic and measurement relations are non-linear, a simple complementary filter represents a computationally inexpensive solution that produces accurate attitude estimates superior to those of an improperly tuned Kalman filter. This paper presents the theory and experimental results for a single parameter tunable quaternion based attitude estimation filter. The described sub-optimal constant gain complementary filter is the first filter designed for inertial/magnetic sensor modules capable of estimating orientation in all attitudes without singularities while continuously correcting for drift without the need for still periods. Experimental results qualitatively and quantitatively demonstrate the accuracy and stability of the filtering algorithm.

Minimum entropy filtering for multivariate stochastic systems with non-Gaussian noises

In this note, a minimum entropy filtering algorithm is presented for a class of multivariate dynamic stochastic systems, which are represented by a set of time-varying difference equations and are subjected to the multivariate non-Gaussian stochastic inputs. Several new concepts including the hybrid random vector, hybrid probability and hybrid entropy are firstly established to describe the probabilistic property of the estimation errors. New relationships are provided between the probability density functions (PDFs) of the multivariate stochastic input and output for different mapping cases. Recursive algorithms are then proposed to design the real-time sub-optimal filter so that the hybrid entropy of the estimation error can be minimized. Finally, an improved algorithm is provided through the on-line tuning of the weighting matrices so as to guarantee the local stability of the error system.

Modified Riccati equation and its application to target tracking

The modified Riccati equation is studied here. This modified Riccati equation has already been associated with tracking a target under measurement uncertainty. The authors consider the special case of tracking a target without clutter but with a probability of detection of less than one. This special case has received some attention recently, especially in relationship with Cramer–Rao bounds. Here, some properties of the modified Riccati equation are derived and proved. Also the relationship of the modified Riccati equation with two special suboptimal filters is shown. Furthermore, the relationship with the Cramer–Rao bound and other bounds is studied.

SIGEST

Many important practical problems require a control system to be able to automatically, efficiently, and accurately detect changes in dynamical systems that the control system is monitoring. In some instances, public safely is involved. Well-known examples include analysis of seismic data to detect earthquakes, analysis of radar data in aircraft navigation, and monitoring of activities in chemical plants. The SIGEST paper in this issue, “A Nonlinear Filtering Approach to Changepoint Detection Problems: Direct and Differential-Geometric Methods” by M. H. Vellekoop and J. M. C. Clark, from the SIAM Journal on Control and Optimization, significantly advances the state of the art in this area. In particular, it considers the problem of online detection of parameter changes in dynamical systems, when the data are noisy, and when both the time of the change and the size of the change are unknown a priori. The problem of changepoint detection in dynamical systems on the basis of noisy observations has been well studied in cases when only one of these two factors is unknown. It can be solved by the Shiryayev–Wonham equation if the value after the change is known a priori, or by a Kalman filter if the time of the change is known a priori. But when both are unknown, the problem turns out to be much more difficult. Vellekoop and Clark use differential-geometric methods to derive an approximate, suboptimal filter for solving this challenging changepoint detection problem. The method they derive draws upon both of the approaches mentioned above. In their words, their new nonlinear filter can be interpreted as an adaptive version of the Shiryayev–Wonham equation for the detection of a priori known changes, combined with a modified Kalman filer structure to generate estimates of the unknown size of the change. The paper includes excellent mathematics, as well as nicely presented and readily understandable simulation results that compare the performance of the proposed new filters to more exact but computationally expensive approaches. These results indicate that the proposed new filters may be quite useful in practice. We hope that SIAM Review readers enjoy this fascinating glimpse into state-of-the-art research in optimal control.

Adapt the steady-state Kalman gain using the normalized autocorrelation of innovations

A discrete linear time-varying stochastic system with scalar measurement data is considered for which neither measurement noise variance nor power of process noise is known. A novel adaptive Kalman filter that gradually approaches the optimum steady-state gain is proposed in this letter. Different from previous adaptation schemes, our algorithm adjusts the Kalman gain depending on the normalized autocorrelation of the prediction error sequence of the suboptimal filter. We also present how to choose appropriate latency time and sample window length in the adaptation process. In our experiments, the filter shows advantages over several other methods under the same conditions.

State and Mode Estimation for Discrete-Time Jump Markov Systems

In this article we compute new state and mode estimation algorithms for discrete-time Gauss--Markov models whose parameter sets switch according to a known Markov law. An important feature of our algorithms is that they are based upon the exact filter dynamics computed in [R. J. Elliott, F. Dufour, and D. Sworder, IEEE Trans. Automat. Control, 41 (1996), pp. 1807--1810]. The fundamental and well-known obstacle in estimation of jump Markov systems is managing the geometrically growing history of candidate hypotheses. In our scheme, we address this issue by proposing an extension of an idea due to Viterbi. Our scheme maintains a fixed number of candidate paths in a history, each identified by an optimal subset of estimated mode probabilities. We compute finite-dimensional suboptimal filters and smoothers, which estimate the hidden state process and the mode probability. Our smoothers are based upon a duality between forward and backward dynamics. Further, our smoothing algorithms are general and can be configured into the standard forms of fixed point, fixed lag, and fixed interval smoothers. A computer simulation is included to demonstrate performance.

H∞ Sub-optimal Filter for Low-cost Integrated Navigation System

The integrated navigation system, specially the system based on GPS and INS, is a leading trend of navigation technology, and some of typical integrated navigation systems have been developed and tested in the past few years. On the other hand, because INS is very expensive in general, the application of GPS/INS system is also limited in some application fields. In this paper, a low-cost dynamic measurement unit (DMU) and GPS are integrated by using H∞ filter. At first, based on the mathematic description of H∞ filtering problem, a sub-optimal H∞ filter is discussed. Meanwhile, the relationship between H∞ filter and standard Kalman filter is studied. Aiming at the detail position/velocity integrated mode, the application of H∞ filter on low cost integrated navigation is researched through some simulations, including the performance of accuracy, robustness and stability.

A Wavelet-Based Reduced Rank Kalman Filter for Assimilation of Stratospheric Chemical Tracer Observations

A suboptimal Kalman filter system that evolves error covariances in terms of a truncated set of wavelet coefficients has been developed for the assimilation of chemical tracer observations of CH4. The truncation is carried out in such a way that the resolution of the error covariance is reduced only in the zonal direction, where gradients are smaller. Assimilation experiments, which lasted 24 days and used different degrees of truncation, were carried out. These experiments reduced the number of elements in the covariance matrix by 90%, 97%, and 99% and the computational cost of covariance propagation by 80%, 93%, and 96%, respectively. The difference in both error covariance and the tracer field between the truncated and full systems over this period was not found to be growing after about 5 days of assimilation. The largest errors in the tracer fields were found to occur in regions of largest zonal gradients at times when observations were made in the immediate vicinity.

Suboptimal filtering in a zero-guard, dicode PPM system operating over dispersive optical channels

Optimal filters for use in digital pulse position modulation (PPM) systems consist of a noise-whitened matched filter followed by a proportional-derivative-delay (PDD) network. Although the PDD network can be removed, with a small loss in sensitivity, the practical implementation of the remaining matched filter is complex. An alternative is to use suboptimal filtering and the use of a third-order Butterworth filter in a zero-guard interval dicode PPM system operating over a dispersive optical channel is examined. Gaussian-shape received pulses are assumed, and a bandwidth-limited PIN–bipolar receiver with both frequency invariant and variant noise is considered. This original analysis shows that the bandwidth of the Butterworth filter is relatively independent of the channel. It is also shown that the performance of such a filter is superior to that of a noise-whitened matched filter. It is concluded that a dicode PPM system can use a simple third-order Butterworth filter. The best predicted sensitivities when coding 1 Gbit/s PCM data are −37.48 dBm in a high-bandwidth link and −32.24 dBm with a link bandwidth equal to 1.2 times the data rate.

Extension of minimum variance estimation for systems with unknown inputs

In this paper, we address the problem of minimum variance estimation for discrete-time time-varying stochastic systems with unknown inputs. The objective is to construct an optimal filter in the general case where the unknown inputs affect both the stochastic model and the outputs. It extends the results of Darouach and Zasadzinski (Automatica 33 (1997) 717) where the unknown inputs are only present in the model. The main difficulty in treating this problem lies in the fact that the estimation error is correlated with the systems noises, this fact leads generally to suboptimal filters. Necessary and sufficient conditions for the unbiasedness of this filter are established. Then conditions under which the estimation error and the system noises are uncorrelated are presented, and an optimal estimator and a predictor filters are derived. Sufficient conditions for the existence of these filters are given and sufficient conditions for their stability are obtained for the time-invariant case. A numerical example is given in order to illustrate the proposed method.

Optimal and sub-optimal Loop Filter Responses for Continuous-Time ΔΣ Modulators

This paper describes a comprehensive analysis of the stability of continuous-time low-pass or band-pass ΔΣ modulators. This analysis is based on a fully S-domain approach which considers the modulator as a feedback control system with gain and phase margins. Theoretical responses for both the optimal filter and a sub-optimal one are derived using a mathematical formalism close to well-known boundary elements methods. A very accurate empirical model has also been developed and, for each case (optimal and sub-optimal), normalized sets of curves allowing to extract the maximum achievable loop gain for a given loop configuration. High achievable gains were found even in the case of ΔΣ modulators involving large delays (several clock times).

Discrete‐Time Risk‐Sensitive Filters with Non‐Gaussian Initial Conditions and Their Ergodic Properties

ABSTRACTIn this paper, we study asymptotic stability properties of risk‐sensitive filters with respect to their initial conditions. In particular, we consider a linear time‐invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk‐sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk‐sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non‐Gaussian initial conditions, we derive the expression for the risk‐sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon‐Nikodym derivative, we show that a suboptimal risk‐sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk‐sensitive filter for non‐Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims.

Suboptimal continuous-discrete filtering based on decomposition of observations

The continuous-discrete filtering problem for different types of observations is considered. In the work of Cho and Oh, the suboptimal continuous and discrete filters for this type of observations were developed. In this paper, we present a generalization of these filters to mixed continuous-discrete systems. The new suboptimal filter allows fully parallel processing of information and fits in with multisensor environment. The examples demonstrate the accuracy and efficiency of the proposed suboptimal linear and nonlinear continuous-discrete filters.

Suboptimal filtering and nonlinear time scale transformation for the analysis of multiexponential decays

Multiexponential decays may contain time-constants differing in several orders of magnitudes. In such cases, uniform sampling results in very long records featuring a high degree of oversampling at the final part of the transient. Here, we analyze a nonlinear time scale transformation to reduce the total number of samples with minimum signal distortion, achieving an important reduction of the computational cost of subsequent analyses. We propose a time-varying filter whose length is optimized for minimum mean square error.

Two-dimensional FlR filter design using matrix dilation approach

This paper develops an approach for the optimal design of two-dimensional (2-D) finite impulse response (FIR) filters based on the minimization of a new performance index. The approach gives analytical expressions for the optimal solution and offers a two-parameter family of suboptimal filters. It is shown that the conventional least square solutions is a member of this family. The approach is based on a result in linear algebra, and used in robust control theory, known as the dilation equation. An efficient numerical algorithm for solving the filter design problem using the dilation equation is proposed, and some techniques for choosing the design parameters are discussed. Finally, some examples are shown illustrating the flexibility of the design using the new approach.

Data analysis of gravitational-wave signals from spinning neutron stars. III. Detection statistics and computational requirements

We develop the analytic and numerical tools for data analysis of the continuous gravitational-wave signals from spinning neutron stars for ground-based laser interferometric detectors. The statistical data analysis method that we investigate is maximum likelihood detection which for the case of Gaussian noise reduces to matched filtering. We study in detail the statistical properties of the optimum functional that needs to be calculated in order to detect the gravitational-wave signal and estimate its parameters. We find it particularly useful to divide the parameter space into elementary cells such that the values of the optimal functional are statistically independent in different cells. We derive formulas for false alarm and detection probabilities both for the optimal and the suboptimal filters. We assess the computational requirements needed to do the signal search. We compare a number of criteria to build sufficiently accurate templates for our data analysis scheme. We verify the validity of our concepts and formulas by means of the Monte Carlo simulations. We present algorithms by which one can estimate the parameters of the continuous signals accurately. We find, confirming earlier work of other authors, that given a 100 Gflops computational power an all-sky search for observation time of 7 days and directed search for observation time of 120 days are possible whereas an all-sky search for 120 days of observation time is computationally prohibitive.

Lumpable hidden Markov models-model reduction and reduced complexity filtering

This paper is concerned with filtering of hidden Markov processes (HMP) which possess (or approximately possess) the property of lumpability. This property is a generalization of the property of lumpability of a Markov chain which has been previously addressed by others. In essence, the property of lumpability means that there is a partition of the (atomic) states of the Markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned. We prove necessary and sufficient conditions on the HMP for exact lumpability to hold. For a particular class of hidden Markov models (HMM), namely finite output alphabet models, conditions for lumpability of all HMP representable by a specified HMM are given. The corresponding optimal filter algorithms for the aggregated states are then derived. The paper also describes an approach to efficient suboptimal filtering for HMP which are approximately lumpable. By this we mean that the HMM generating the process may be approximated by a lumpable HMM. This approach involves directly finding a lumped HMM which approximates the original HMM well, in a matrix norm sense. An alternative approach for model reduction based on approximating a given HMM by an exactly lumpable HMM is also derived. This method is based on the alternating convex projections algorithm. Some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms.

Fault Detection Based on Wavearx Neural Network

A means to approximate the nonlinear dynamic system based on WaveARX neural network has been studied. After a nonparametric approximate model of the system was constructed from a priori experiment, a nonlinear suboptimal filter has been designed to generate the residuals for fault detection based on the analysis of the upper bound approximate error. The proposed method was applied to a batch process monitoring, the application results have shown the feasibility and effectiveness of the method.

On the sampled-data H∞ filtering problem

In this paper the H ∞ filtering problem of a continuous-time process is considered under the assumption that the measurement is sampled by a generalized sampling device. Both cases of known and unknown initial conditions are considered. The problem is treated via the lifting transformation, which converts the problem to a pure discrete-time setting. It is shown that if a suboptimal filter exists, it can always be presented in the form of a finite-dimensional discrete-time (possibly time-varying) filter followed by a stationary generalized hold function. This extends previous results in the literature, where a suboptimal filter of this structure (with an LTI discrete-time part of the filter) was obtained for the case of known initial conditions only. Also, the existence conditions of a suboptimal filter derived in this paper are simpler than those known in the literature.