Solving the robustpeak-to-peak$\mathcal {L}_{1}$filtering problem requires a solution when persistent impulsive disturbances affect the model. In this letter, we solve the$\mathcal {L}_{1}$problem for a state-space model based on the backward Euler method with norm-bounded colored disturbances. Thepeak-to-peaklemma is modified and a new theorem is proved to numerically compute the bias correction gain$\bf {K}$for a recursive$\mathcal {L}_{1}$filter. The gain$\bf {K}$is computed numerically using a linear matrix inequality. Based on a numerical example of a quasi-periodic process subject to Gauss-Markov measurement disturbances with overshooting, it is shown that the gain$\bf {K}$of an$\mathcal {L}_{1}$filter satisfies the rule of thumb, i.e. it ranges between the Kalman gain and the unbiased finite impulse response (UFIR) filter gain. Accordingly, the$\mathcal {L}_{1}$filter occupies an intermediate place between the Kalman and UFIR filters. Herewith, having two additional tuning factors, it is more complex in tuning.

The$H_\infty$problem is reformulated for the bias correction gain of the recursive$H_\infty$-Kalman filter. For uncertain processes, the gain is computed using a linear matrix inequality, a bounded real lemma modified for non-predictive state space models based on Euler's backward method, and a new theorem. It is shown numerically that the gain of the$H_\infty$-Kalman filter is between the optimal Kalman gain and the gain of the robust unbiased finite impulse response filter. The filter performances are compared in terms of root mean square error, as well as the newly introduced robustness and estimation quality factors.

The Bayesian approach resulting in the Kalman filter (KF), often struggle with model uncertainties, particularly when noise statistics are inaccurate. Inspired by transfer learning, this letter presents a novel Bayesian transfer filtering framework that significantly enhances estimation accuracy by incorporating the unbiased finite impulse response (UFIR) structure for adaptive regularization. To adaptively adjust the UFIR filtering estimate, the statistical significance of the transfer-regularization is learned and the variational Bayesian method is applied to learn the regularization factor directly from the data. It is shown that this adaptive strategy not only improves the interpretability and transferability but also removes the need for heuristic selection, which is a common limitation in traditional regularization-based transfer methods. Numerical simulations and water tank experiments collectively confirm the effectiveness of the proposed framework under uncertain noise statistics.

Integrating the advantage of the unbiased finite impulse response (UFIR) filter into the Kalman filter (KF) is a practical yet challenging issue, where how to effectively borrow knowledge across domains is a core issue. Existing methods often fall short in addressing performance degradation arising from noise uncertainties. In this article, we delve into a Bayesian transfer filter (BTF) that seamlessly integrates the UFIR filter into the KF through a knowledge-constrained mechanism. Specifically, the pseudo marginal measurement likelihood of the UFIR filter is reused as a constraint to refine the Bayesian posterior distribution in the KF. To optimize this process, we exploit the Kullback-Leibler (KL) divergence to measure and reduce discrepancies between the proposal and target distributions. This approach overcomes the limitations of traditional weight-based fusion methods and eliminates the need for error covariance. Additionally, a necessary condition based on mean square error criteria is established to prevent negative transfer. Using a moving target tracking example and a quadruple water tank experiment, we demonstrate that the proposed BTF offers superior robustness against noise uncertainties compared to existing methods.

Ultrawideband (UWB)-based localization is widely used in environments inaccessible to global navigation satellite system signals. To improve the precision of UWB-based localization, a robust distributed adaptive extended unbiased finite impulse response (EFIR) filtering algorithm is developed. The algorithm is designed to reduce round-off errors and adaptively adjust noise covariances using the expectation-maximization (EM) approach. Based on extensive experimental testing, the EFIR algorithm is shown to outperform the distributed extended Kalman filter-based algorithm and distributed EFIR filter-based algorithm under harsh conditions.

Receding horizon (RH) finite impulse response (FIR) filters (observers) are used in model predictive and state feedback control. For such applications, we develop a robust RH$H_{2}$suboptimal FIR (RH$H_{2}$-FIR) filter under the conditions when an uncertain system is also disturbed, and the initial and data errors are not necessarily Gaussian. The RH$H_{2}$-FIR filter is obtained by minimizing the squared Frobenius norm of the weighted error-to-error transfer function, where the weights are related to the errors. The filter developed is tested by a quasi periodic disturbed process with an uncertain system matrix. Based on numerical examples and experimental data, it is shown that the RH$H_{2}$-FIR filter outperforms the optimal FIR, Kalman, maximum likelihood FIR, and unbiased FIR (UFIR) filters. It is also almost as robust as the UFIR filter. Note to Practitioners—Monitoring and control of modern automation systems require accurate discrete-time algorithms. The standard techniques such as Kalman filter are applicable under ideal conditions and have poor performance otherwise. The discrete convolution-based batch FIR filters are more accurate and robust. Under industrial uncertainties, disturbances, initial errors, and data errors, FIR filters are able to extract features of dynamic processes with high accuracy. They can also be combined with convolutional neural networks.

State estimation of industrial processes is often carried out under harsh conditions in uncertain environments. Therefore, robust state estimators are required. To address this demand, in this article, a batch$H_\infty$finite impulse response ($H_\infty$-FIR) filter is developed for disturbed systems under measurement and initial errors. The gain for the batch$H_\infty$-FIR filter is numerically computed using linear matrix inequality (LMI). Since LMI contains a quadratic form with respect to the filter gain, a theorem is proved and an iterative computation algorithm is developed. Based on numerical simulations of harsh conditions and an experimental example, it is shown that the$H_\infty$-FIR filter significantly outperforms the Kalman filter and is almost as robust as the unbiased FIR filter.

In indoor robot localization by using ultra-wideband (UWB), the extended Kalman filter (EKF)-based algorithms suffer from the colored measurement noise (CMN) that degrades the localization accuracy and causes the divergence. To overcome this issue, we develop a hybrid colored EKF and colored extended unbiased finite impulse response (EFIR) filter (cEKF/EFIR filter) employing measurement differences. We also develop this algorithm using a filter bank on merged averaging horizons to be adaptive to time-varying CMN and call it the adaptive EKF/EFIR (aEKF/EFIR) filter. Experimental testing is provided in UWB-based indoor mobile robot localization environments. It is shown that the end-to-end colored EKF/EFIR and aEKF/EFIR filtering algorithms have better performances than the EKF, EFIR filter, and their modifications for CMN.

Strictly nonwhite measurement noise (NMN) is observed in many industrial processes. Therefore, effective smoothing is often required to extract useful information about the process state with maximum accuracy. This article proposes a batch$q$-lag optimal finite impulse response (OFIR) smoother, operating under NMN with full block covariance matrices. It is shown that the OFIR smoother significantly outperforms the Rauch–Tung–Striebel (RTS) smoother and the unbiased FIR (UFIR) smoother. Testing is provided based on object tracking. The results are validated by a practical example of a three degree-of-freedom helicopter system, which confirms that OFIR smoothing provides better noise reduction than UFIR smoothing, RTS smoothing, and modified RTS smoothing using state augmentation and measurement differencing.

A new robust$q$-lag unbiased finite impulse response (UFIR) smoother is obtained in discrete-time state-space in batch and recursive forms for linear time-varying (LTV) systems under arbitrary zero mean noise and uncertainties. It is shown that recursive computation of the error covariance of the UFIR smoother is partly available for LTV systems and fully available for linear time-invariant systems. Examples of a polynomial model with colored noise and an uncertain harmonic process have proved that the$q$-lag UFIR smoother outperforms the Rauch-Tung-Striebel or Kalman smoother.

Accurate indoor localization information of the quadrotor plays an important role in many Internet-of-Things applications. To improve the estimation accuracy and robustness, a fixed-lag extended finite impulse response smoother (FEFIRS) algorithm is proposed for fusing the inertial navigation system (INS) and ultra wideband (UWB) data tightly, which employs a distance between the UWB reference nodes and a blind node measured by the INS and UWB. The FEFIRS algorithm consists of an extended unbiased finite impulse response (EFIR) filter and a fixed-lag unbiased FIR (UFIR) smoother. The EFIR filter is employed to improve the robustness, and the fix-lag UFIR smoother is capable of improving the accuracy. Based on extensive test investigations employing real data, the proposed FEFIRS has higher accuracy and robustness than the Kalman-based solutions in the tightly integrated INS/UWB-based indoor quadrotor localization.

A novel approach is proposed to H2finite impulse response (FIR) prediction in discrete-time state-space. The biased-constrained H2optimal unbiased FIR (H2-OUFIR) predictor derived under disturbances and measurement errors is shown to have the maximum likelihood form and be equivalent to the OUFIR predictor under Gaussian noise. The derivation is provided using the backward Euler method by minimizing the squared weighted Frobenius norm. A bias-constrained suboptimal H2FIR filtering algorithm using the linear matrix inequality is also designed. The H2-OUFIR predictor performance is investigated by simulations and experimentally in a comparison with the Kalman and unbiased FIR predictors.

In this article, the multipass optimal finite impulse response (OFIR) filtering approach is developed for industrial processes with unknown initial conditions under temporary model mismatches. The forward and backward OFIR filters are derived in batch and fast iterative forms using recursions. The double-pass OFIR (DOFIR) filter supported by the unbiased FIR (UFIR) filter and triple-pass OFIR (TOFIR) filter starting with some initial values are designed and extensively investigated using simulations and experimental data. It is shown that the DOFIR and TOFIR filters are able to essentially improve the performance close to the initial values and are more robust against temporary model mismatches than the Kalman, OFIR, and UFIR filters.

Under coloured noise, known modifications of the Kalman filter (KF) exist only for discrete‐time state‐space models produced by the forward Euler (FE) method, which fits with feedback control. In this study, the authors modify the KF and unbiased finite impulse response (UFIR) filter using the backward Euler (BE) method for models with coloured measurement noise (CMN), which better fits systems without feedback. The FE‐ and BE‐based models differ by time indexes in the system input and noise that is essential for time‐varying and Markov jump systems. Employing measurement differencing, two KF algorithms and a unique UFIR algorithm are derived for time‐correlated and de‐correlated noise. An equivalence of the KF algorithms is proved analytically and confirmed by simulations. Numerical examples are given for target tracking and experimental verification is provided for visual object tracking. The high efficiency of the designed algorithms in removing CMN is demonstrated experimentally.

The distributed wireless sensor network (WSN) technologies have experienced rapid developments in recent years owing to mobility, scalability, and flexibility of solutions with different kinds of consensus achieved using the Kalman filter (KF). In this article, we employ a robust linear technique known as unbiased finite impulse response (UFIR) filtering and develop it for distributed WSNs with optimal consensus on estimates. A batch of distributed UFIR (dUFIR) filter is designed to operate on individual and average data horizons and then represented with an iterative algorithm using recursions. Based on simulations of a circularly traveling and rapidly maneuvering object covered by a WSN with 50 nodes, the dUFIR filter is shown to have higher robustness against the distributed KF (dKF). An experimental verification is conducted for a known ground truth of a moving vehicle covered by a WSN with eight nodes. The trade-off between the dUFIR filter, dKF, and dH∞filter is investigated in detail under different operation scenarios and tuning modes.

In this paper, an iterative finite impulse response (FIR) filter is proposed for discrete time-varying state-space models, with the purpose of a new initialization strategy for the iterative FIR structure as well as consideration of possible unexpected state dynamics in a finite horizon. A compensation variable that satisfies the Gaussian property is introduced into the state equation, and its probability density function (pdf) is estimated analytically together with the pdf of state variable using the variational Bayesian inference technique. Different from the existing methods, the proposed filter exploits the FIR structure from the perspective of pdf propagation, which provides a new efficient way to use the iterative FIR filtering structure without any particular initialization scheme. Moreover, the effects of uncertainties (caused by initialization and/or possible unmodeled state dynamics) on the filtering output are loosened adaptively. Two examples of applications demonstrate that the proposed algorithm can not only provide optimal estimates when the model used perfectly matches the measurements, but can also exhibit better robustness than the Kalman filter, optimal FIR filter, maximum likelihood FIR filter, and some commonly used robust and/or adaptive Kalman filters when the underlying process suffers from unpredicted uncertainties.

Copy number alterations (CNAs) are hallmarks of cancer, which are now been routinely measured by different techniques and used for diagnostic and prognostic purpose. Efficient and accurate detection of the breakpoint positions in heterogeneous cancer sample measured with intrinsic random noise and subjected to technical and biological biases is a challenging practical and methodological problem. To improve the CNA estimates, the authors present the probabilistic approach for breakpoints detection that gives confidence masks (the system of local segmentation profiles with confidence probabilities) tuned using experts estimates. The authors show that the asymmetric exponential power distribution matches well the uncertainties (jitter) in the breakpoint locations. The confidence upper and lower boundary masks for the breakpoint location are built using this function. The confidence masks are then tuned based on the medical expert annotations of the training set of the breakpoints obtained by the standard circular binary segmentation (CBS) algorithm. Comparison of modified confidence masks and experts annotations on the testing set of CNA profiles of neuroblastoma showed improvement of the CNA estimates.

This letter develops the Kalman and unbiased finite impulse response filtering algorithms for linear discrete-time state-space models with Gauss-Markov colored process noise (CPN) employing state differencing. The approach avoids problems caused by matrix augmentation, but requires solving a nonsymmetric algebraic Riccati equation to specify the system matrix modified for CPN. Higher accuracy of the algorithms proposed is demonstrated by simulation. A comparative analysis of filtering estimates is provided based on navigation data of walking humans.

It is proved that the iterative computation form for the mean square error (MSE) matrix of the batch unbiased finite impulse response (UFIR) filter exactly equals that of the iterative UFIR filter form, unlike what was previously thought. Based on the iterative MSE matrix form, we suggest two strategies for defining the optimal horizon length for the UFIR filter. The results are verified using the two-state polynomial and harmonic models.

The particle filter (PF) is a popular filtering algorithm in various localization problems represented by nonlinear state-space models. Although the PF can provide accurate localization results, it often fails in localization because of the sample impoverishment phenomenon. In this paper, we propose a novel nonlinear filtering method that combines a PF with a robust filter, called a finite impulse response (FIR) filter, in order to accomplish accurate and reliable localization. The proposed filter is called the composite particle/FIR filter (CPFF). In the CPFF framework, the PF is the main filter used in normal situations. When PF failures occur, the FIR filter is used to recover the PF from failures. To detect PF failures, a new decision-making algorithm is proposed in this paper. The proposed CPFF is applied to indoor human localization using a wireless sensor network. The CPFF is accurate and reliable under conditions in which the pure PF typically exhibits degraded accuracy or failures in localization.

Industrial wireless sensor networks (WSNs) often operate under harsh conditions that require robustness from an estimator of a measured quantity. We propose a novel distributed unbiased finite-impulse response (UFIR) filter called micro-UFIR filter that, unlike the micro-Kalman filter (micro-KF), is robust against modeling errors in uncertain noise environments. The micro-UFIR filter is derived based on average consensus on measurements and, unlike the micro-KF, requires only one consensus filter. Better robustness of the micro-UFIR filter is shown analytically and confirmed by simulations of a WSN and a vehicle travelling along a circular trajectory under unpredictable impacts, impulsive noise, and errors in the noise statistics.

Industrial processes are often organized using mechanical systems with multiple degrees-of-freedom (DOF). For real-time operation of such systems in noise environments, fast, optimal, and robust estimators are required. In this paper, information gathering about multi-DOF system states is provided using the optimal finite impulse response (OFIR) filter. To use this filter in real time, a fast iterative algorithm is developed with a pseudocode available for immediate use. Although the iterative algorithm utilizes Kalman recursions, it is more robust against uncertainties and model errors owing to the transversal structure. We use this algorithm to estimate state in the 1-DOF torsion system and the 3-DOF helicopter system.

In this survey, the authors examine the trade‐off between the unbiased, optimal, and in‐between solutions in finite impulse response (FIR) filtering. Specifically, they refer to linear discrete real‐time invariant state‐space models with zero mean noise sources having arbitrary covariances (not obligatorily delta shaped) and distributions (not obligatorily Gaussian). They systematically analyse the following batch filtering algorithms: unbiased FIR (UFIR) subject to the unbiasedness condition, optimal FIR (OFIR) which minimises the mean square error (MSE), OFIR with embedded unbiasedness (EU) which minimises the MSE subject to the unbiasedness constraint, and optimal UFIR (OUFIR) which minimises the MSE in the UFIR estimate. Based on extensive investigations of the polynomial and harmonic models, the authors show that the OFIR‐EU and OUFIR filters have higher immunity against errors in the noise statistics and better robustness against temporary model uncertainties than the OFIR and Kalman filters.

In this letter, the maximum likelihood (ML) finite-impulse response (FIR) filter is proposed for discrete time-variant state-space models with nonsingular system matrix. The ML FIR filter has the deadbeat property and its form is universal for all known bias constrained FIR filters. By the identity weighting matrix, the ML FIR filter becomes the unbiased FIR filter, which ignores the noise statistics and the initial error statistics. Otherwise, the ML FIR filter is equivalent to the optimal FIR filter with embedded unbiasedness and to the minimum variance unbiased FIR filter. An example of a stochastic resonator demonstrates higher immunity of the ML FIR filter against errors in the imprecisely defined noise statistics than in the Kalman filter.

We propose a fast iterative algorithm for optimal finite impulse response (OFIR) filtering of linear discrete time-invariant state-space models in white Gaussian noise. The OFIR filter is known to have the BIBO stability and better robustness against the Kalman filter (KF). The iterative OFIR algorithm is KF-like; that is, its estimate appears much faster than in the batch OFIR filter. A dramatic reduction of computation time is demonstrated in the full-horizon iterative OFIR algorithm which operates as fast as KF. We also notice a considerable reduction of the computational resources allowed by iterations.

The need for accurate, fast, and reliable indoor localization using wireless sensor networks (WSNs) has recently grown in diverse areas of industry. Accurate localization in cluttered and noisy environments is commonly provided by means of a mathematical algorithm referred to as a state estimator or filter. The particle filter (PF), which is the most commonly used filter in localization, suffers from the sample impoverishment problem under typical conditions of real-time localization based on WSNs. This paper proposes a novel hybrid particle/finite impulse response (FIR) filtering algorithm for improving reliability of PF-based localization schemes under harsh conditions causing sample impoverishment. The hybrid particle/FIR filter detects the PF failures and recovers the failed PF by resetting the PF using the output of an auxiliary FIR filter. Combining the regularized particle filter (RPF) and the extended unbiased FIR (EFIR) filter, the hybrid RP/EFIR filter is constructed in this paper. Through simulations, the hybrid RP/EFIR filter demonstrates its improved reliability and ability to recover the RPF from failures.

Grid navigation spaces nested with the radio-frequency identification (RFID) tags are promising for industrial and other needs, because each tag can deliver information about a local two-dimensional or three-dimensional surrounding. The approach, however, requires high accuracy in vehicle self-localization. Otherwise, errors may lead to collisions; possibly even fatal. We propose a new extended finite impulse response (EFIR) filtering algorithm and show that it meets this need. The EFIR filter requires an optimal averaging interval, but does not involve the noise statistics which are often not well known to the engineer. It is more accurate than the extended Kalman filter (EKF) under real operation conditions and its iterative algorithm has the Kalman form. Better performance of the proposed EFIR filter is demonstrated based on extensive simulations in a comparison to EKF, which is widely used in RFID tag grids. We also show that errors in noise covariances may provoke divergence in EKF, whereas the EFIR filter remains stable and is thus more robust.

We discuss the code reading error probability (EP) in the radio frequency identification (RFID) surface acoustic wave tags with pulse position coding and peak-pulse detection. EP is found in the most general form assuming M groups of codes with N slots each and allowing individual signal-to-noise ratios (SNRs) in each slot. We show that if the RFID tag is designed such that the spurious responses are attenuated on more than 20 dB below On-pulses, then EP can be achieved at the level of 10−8(one false per 108readings) with SNR > 17 dB for any reasonable M and N.

An optimal finite impulse response (FIR) estimator is adapted for discrete filtering, smoothing, and prediction of hybrid (continuous/discrete) models over N nearest past measurement points. Its unbiased FIR (UFIR) version ignoring noise and initial errors is also discussed for near optimal estimation when N >;>; 1. The UFIR estimator is represented with an iterative Kalman-like algorithm efficient for highly oversampled data. An example of applications is given for the Global Positioning System-based measurement of time errors in a crystal clock.

Two finite impulse response (FIR) estimators (optimal and unbiased) are addressed for filtering, smoothing and predicting linear time-invariant state-space signal models perturbed by white Gaussian noise in receiver channels with imbedded digital signal processing units. The FIR estimators are efficient in estimating oversampled and highly oversampled signals, respectively. Special attention is paid to the unbiased FIR (UFIR), owing to its ability of becoming optimal when the processing memory is large. An iterative UFIR algorithm is discussed in detail and compared with the Kalman filter. The optimal memory and errors are also discussed for such kind of estimators. Examples of applications are given for one-dimensional tracking of a two-state polynomial model and state estimation in a harmonic one. Based on this study, the authors show that the UFIR estimator is more efficient than the Kalman filter in blindly estimating receiver channels under the model temporary uncertainties.

A smoothing finite impulse response (FIR) filter is addressed for discrete time-invariant state-space polynomial models commonly used to model signals over finite data. A general gain is derived for the relevant p-lag unbiased smoothing FIR filter. An application is given for the time interval errors of local slave clocks of digital communications networks. An excellent performance of the best unbiased fit is demonstrated along with its ability to extrapolate linearly the clock behaviors for holdover required by the IEEE Standards.

We address a Kalman-like estimator for solving universally the problems of filtering (p = 0), prediction (p > 0), and smoothing (p < 0) of discrete time-varying state-space models with no requirements for noise and initial conditions. The estimator proposed overperforms the Kalman one when 1) noise covariances and initial conditions are not known exactly, 2) noise constituents are not white sequences, and 3) both the system and measurement noise components need to be filtered out and the deterministic state estimated. Otherwise, the Kalman-like and Kalman filters produce similar errors. A numerical comparison of the Kalman and Kalman-like estimators is provided.

The noise power gain (NPG) is addressed and the error bound (EB) is specified via NPG for a general p-shift linear time-variant finite impulse response (FIR) transversal estimator intended for filtering (p = 0), prediction (p > 0), and smoothing (p < 0) of discrete-time K-state space signal models with M states measured. A fast iterative algorithm for NPG of the unbiased FIR estimator is provided along. An example is given for the two-state space model.

The noise power gain (NPG) matrix is specialized in state space for transversal finite impulse response (FIR) estimators intended for filtering, prediction, and smoothing of discrete time-variantK-state models withMstates measured. A computationally efficient iterative algorithm for NPG associated with unbiased estimation is provided along. Based on a numerical example, we show that the estimates are well bounded with the error bound (EB) specified in the three-sigma sense by the main components of the NPG matrix and measurement noise variance. In turn, the cross-components in the NPG matrix represent interactions in the estimator channels. It is concluded that EB can serve as an efficient measure of errors in optimal and suboptimal FIR and Kalman structures.

The gain of an optimal finite impulse response (FIR) filter of the system k-state is derived and analyzed for discrete real-time state-space models with jointly independent zero-mean noises of arbitrary distributions and covariances. Several special cases are considered, and the gain is modified accordingly. It is stated that if the model is deterministic or slowly changing with time and such that an averaging horizon is large, N Gt 1, then a relatively simple unbiased FIR filter produced by the deadbeat constraint yields virtually an optimal estimate. An application is given to the crystal clock time error estimation via GPS-based measurements with temporary uncertainty in the presence of sawtooth noise.

This correspondence examines the joint conditional probability density function (PDF) of the main variables (envelope, phase, and their eta-order time derivatives) of a time-varying random signal in the presence of additive Gaussian noise. The main variables are conditioned with respect to the given variables, which are the amplitude, phase, and their derivatives of the signal alone. We prove a theorem stating that some of the conditional PDFs of the main variables do not depend on some of the given variables. This theorem, together with Bayes's theorem, can substantially simplify the derivations of conditional PDFs and give alternative forms of them. Both theorems can also help in finding reasonable approximations, as we demonstrate for the phase and first time derivative of the envelope.