Finite Impulse Response Model Research Articles

Asymptotic Theory for Regularized System Identification Part I: Empirical Bayes Hyperparameter Estimator

Regularized techniques, also named as kernel-based techniques, are the major advances in system identification in the last decade. Although many promising results have been achieved, their theoretical analysis is far from complete and there are still many key problems to be solved. One of them is the asymptotic theory, which is about convergence properties of the model estimators as the sample size goes to infinity. The existing related results for regularized system identification are about the almost sure convergence of various hyper-parameter estimators. A common problem of those results is that they do not contain information on the factors that affect the convergence properties of those hyper-parameter estimators, e.g., the regression matrix. In this paper, we tackle problems of this kind for the regularized finite impulse response model estimation with the empirical Bayes (EB) hyper-parameter estimator and filtered white noise input. In order to expose and find those factors, we study the convergence in distribution of the EB hyper-parameter estimator, and the asymptotic distribution of its corresponding model estimator. For illustration, we run Monte Carlo simulations to show the efficacy of our obtained theoretical results.

Handling of time delays in system identification with regularized FIR models

Abstract System identification with finite impulse response (FIR) models has been recently revised by introducing specific forms of regularization in the estimation. By utilizing prior knowledge of the dynamic process under investigation different kernels for regularization can be derived. However, the dynamical processes considered are mainly restricted to time-delay free systems. Therefore, in this paper we propose two novel methods to handle the time-delay case. Both methods incorporate the time-delay estimation in the hyperparameter optimization of the regularized FIR models. The first method is built on a single kernel (SK) and the second utilizes a multiple kernel (MK) approach. Simulations on different dynamical systems show the superior behavior of both methods in comparison to standard regularization methods, whereupon the MK outperforms the SK approach.

A mechanism-data hybrid-driven framework for identifying dynamic characteristic of actual chemical processes

As a prerequisite of accurate process analysis, prediction and optimization, the precise identification of process dynamic characteristics is of great significance. Traditional identification methods and data-driven technologies suffer from low accuracy and noise interference when facing the identification of actual chemical process. In this contribution, a mechanism-data hybrid-driven framework is developed to simultaneously capture the delay time and dynamic response of process variables. Combining non-parametric identification method, theoretical model and big data analysis technology, the proposed framework is able to derive more accurate and reliable identification results from the distributed control system (DCS) data directly, avoiding drawbacks brought by single method. The presented framework successfully identifies the flowrate dynamic properties of an actual deethanization process, and outperforms traditional approaches including finite impulse response model and Welch's averaged periodogram method. The contribution is significant for dynamic characteristic identification of actual chemical process, facilitating the developments of high-accuracy models for process prediction, control and intelligent optimization.

A diffusion bias-compensated LMS algorithm for distributed estimation with ARMAX models

In this paper, we study the distributed estimation problem with colored noise over adaptive networks. The nodes estimate and track an identical parameter vector in the network. The colored noise is described by a finite impulse response (FIR) model with an unknown variance, and thus leads to a bias in the diffusion least squares algorithm. Accordingly, we propose a diffusion bias-compensated LMS algorithm to deal with this situation. The performance of the proposed algorithm is analyzed, which shows that it can achieve the mean-square stability on the condition that the step size is sufficiently small. Furthermore, we obtain the corresponding closed mean-square deviation containing the white noise variance. Finally, the given simulations show the effectiveness of our proposed algorithm.

Displacement reconstruction based on Kalman smoothing in multi-rate suspension health monitoring systems

Suspension displacements and wheel center accelerations are important signals for suspension health monitoring systems to improve vehicle reliability and safety. The current way to obtain these signals is to install sensors on vehicles to conduct direct measurements. Usually, displacements are sampled at a slower rate than accelerations due to technical or economic limitations in real scenarios. This paper introduces a method for displacement reconstruction with low-sampling-rate displacement and high-sampling-rate acceleration measurements by formulating the reconstruction problem as a state estimation problem. A state-space model is established by identifying two data-driven models: a time-series Auto-Regressive model and a Finite Impulse Response model. Then, Kalman smoothing is used to estimate the displacement. A series of experiments have been done to show that the estimates from Kalman smoother coincide with the measurements.

Transmissibility-Based Fault Detection for Robotic Applications With Time-Varying Parameters

This article investigates transmissibility operators for time-variant systems with bounded nonlinearities. Transmissibility operators are mathematical objects that characterize the relationship between two subsets of responses of an underlying system. The underlying system parameters and nonlinearities are assumed to be unknown. Time-domain transmissibility operators are independent of the system inputs and initial conditions. In this article, we propose an algorithm to identify time-varying transmissibilities using recursive least-squares and noncausal finite impulse response models. The identified transmissibilities are then used for fault detection in three different systems. The first system is an autonomous multirobotic system formulated to emulate connected autonomous vehicle platoons, where the varying parameters are the robot mass and the ground friction coefficient. The second system is a flexible structure with unknown excitation, where the variant parameter is the location of the excitation. Finally, the third system is a robotic manipulator that picks objects with different mass values.

Evaluating increases in sensitivity from NORDIC for diverse fMRI acquisition strategies

As the neuroimaging field moves towards detecting smaller effects at higher spatial resolutions, and faster sampling rates, there is increased attention given to the deleterious contribution of unstructured, thermal noise. Here, we critically evaluate the performance of a recently developed reconstruction method, termed NORDIC, for suppressing thermal noise using datasets acquired with various field strengths, voxel sizes, sampling rates, and task designs.Following minimal preprocessing, statistical activation (t-values) of NORDIC processed data was compared to the results obtained with alternative denoising methods. Additionally, we examined the consistency of the estimates of task responses at the single-voxel, single run level, using a finite impulse response (FIR) model. To examine the potential impact on effective image resolution, the overall smoothness of the data processed with different methods was estimated. Finally, to determine if NORDIC alters or removes temporal information important for modeling responses, we employed an exhaustive leave-p-out cross validation approach, using FIR task responses to predict held out timeseries, quantified using R2.After NORDIC, the t-values are increased, an improvement comparable to what could be achieved by 1.5 voxels smoothing, and task events are clearly visible and have less cross-run error. These advantages are achieved with smoothness estimates increasing by less than 4%, while 1.5 voxel smoothing is associated with increases of over 140%. Cross-validated R2s based on the FIR models show that NORDIC is not measurably distorting the temporal structure of the data under this approach and is the best predictor of non-denoised time courses. The results demonstrate that analyzing 1 run of data after NORDIC produces results equivalent to using 2 to 3 original runs and that NORDIC performs equally well across a diverse array of functional imaging protocols.Significance Statement: For functional neuroimaging, the increasing availability of higher field strengths and ever higher spatiotemporal resolutions has led to concomitant increase in concerns about the deleterious effects of thermal noise. Historically this noise source was suppressed using methods that reduce spatial precision such as image blurring or averaging over a large number of trials or sessions, which necessitates large data collection efforts. Here, we critically evaluate the performance of a recently developed reconstruction method, termed NORDIC, which suppresses thermal noise. Across datasets varying in field strength, voxel sizes, sampling rates, and task designs, NORDIC produces substantial gains in data quality. Both conventional t-statistics derived from general linear models and coefficients of determination for predicting unseen data are improved. These gains match or even exceed those associated with 1 voxel Full Width Half Max image smoothing, however, even such small amounts of smoothing are associated with a 52% reduction in estimates of spatial precision, whereas the measurable difference in spatial precision is less than 4% following NORDIC.

Identification of MISO Hammerstein system using sparse multiple kernel-based hierarchical mixture prior and variational Bayesian inference

The Hammerstein model is a cascade composition of a static memoryless nonlinear function followed by a linear time-invariant dynamical subsystem, which is capable of modeling a wide range of nonlinear dynamical systems. Model structural parameter selection (including the model order and the nonlinearity order) and sparse representation of the static nonlinear function are two issues that receive increasing interests in Hammerstein system identification. In this paper, a novel Bayesian sparse multiple kernel-based identification method (BSMKM) for multiple-input single-output (MISO) Hammerstein system is proposed to handle those issues, where the basis-function model and the finite impulse response model are used to represent the nonlinear part and the linear part respectively. Firstly, in order to jointly realize the model parameter estimation, the sparse representation of static nonlinear function (the nonlinearity order selection can also be realized indirectly) and the model order selection of linear dynamical system, a hierarchical prior distribution is constructed based on Gaussian scale mixture model and sparse multiple kernel, which can characterize both inter-group sparsity and intra-group correlation structure. Then, a full Bayesian method based on variational Bayesian inference is proposed to estimate all unknown model parameters, including finite impulse response coefficients, hyperparameters and noise variance. Finally, the performance of the proposed BSMKM identification method is evaluated by numerical experiments using both simulation and real data.

Edge-computing based soft sensors with local Finite Impulse Response models for vehicle wheel center loads estimation under multiple working conditions

Wheel center loads are usually required to be measured for the durability assessment of vehicle key components, and thus play an important role in the reliability design of vehicles. The current method of obtaining these loads is to install sophisticated and expensive sensors on prototype vehicles to conduct direct measurement. However, it is economically infeasible to equip many prototype vehicles simultaneously for road testing. The paper aims to design and implement edge-computing based soft sensors to estimate these wheel center loads in real time. A set of Finite Impulse Response (FIR) models are trained offline under multiple working conditions, and a Maximum Likelihood Estimation (MLE) classifier is used to determine which working condition the vehicle currently is under so that the wheel center loads can be estimated accurately. The soft sensors are implemented with an edge computing paradigm, and the whole process of data collection, pre-processing, calculation, and upload runs automatically on edge devices. Experimental results show that the estimate from the proposed method fits the measurement well, and the edge-computing based soft sensors meet the real-time requirements for wheel center load estimation.

Perspective: Disentangling the effects of tES on neurovascular unit.

Transcranial electrical stimulation (tES) can modulate the neurovascular unit, including the perivascular space morphology, but the mechanisms are unclear. In this perspective article, we used an open-source "rsHRF toolbox" and an open-source functional magnetic resonance imaging (fMRI) transcranial direct current stimulation (tDCS) data set to show the effects of tDCS on the temporal profile of the haemodynamic response function (HRF). We investigated the effects of tDCS in the gray matter and at three regions of interest in the gray matter, namely, the anodal electrode (FC5), cathodal electrode (FP2), and an independent site remote from the electrodes (PZ). A "canonical HRF" with time and dispersion derivatives and a finite impulse response (FIR) model with three parameters captured the effects of anodal tDCS on the temporal profile of the HRF. The FIR model showed tDCS onset effects on the temporal profile of HRF for verum and sham tDCS conditions that were different from the no tDCS condition, which questions the validity of the sham tDCS (placebo). Here, we postulated that the effects of tDCS onset on the temporal profile of HRF are subserved by the effects on neurovascular coupling. We provide our perspective based on previous work on tES effects on the neurovascular unit, including mechanistic grey-box modeling of the effects of tES on the vasculature that can facilitate model predictive control (MPC). Future studies need to investigate grey-box modeling of online effects of tES on the neurovascular unit, including perivascular space, neurometabolic coupling, and neurovascular coupling, that can facilitate MPC of the tES dose-response to address the momentary ("state") and phenotypic ("trait") factors.

System identification with piecewise-constant finite impulse response model and its statistical property

Finite impulse response (FIR) models have attracted much attention in system identification in recent years. However, finite impulse response models have many parameters, and it is sometimes difficult to identify them under conditions with limited computational resources. This paper considers a finite impulse response model with a reduced number of parameters. In particular, supposing the purpose of the model is to estimate the unit step response at a given time, it is shown that one of the desirable models for this purpose is a model in which the impulse response values are piecewise constants. The statistical properties of such a piecewise-constant FIR model obtained by system identification are clarified. The usefulness of the proposed method is confirmed through numerical examples.

Maximum Likelihood Estimation in Data-Driven Modeling and Control

Recently, various algorithms for data-driven simulation and control have been proposed based on the Willems’ fundamental lemma. However, when collected data are noisy, these methods lead to ill-conditioned data-driven model structures. In this article, we present a maximum likelihood framework to obtain an optimal data-driven model, the signal matrix model, in the presence of output noise. Data compression and noise-level estimation schemes are also proposed to apply the algorithm efficiently to large datasets and unknown noise-level scenarios. Two approaches in system identification and receding horizon control are developed based on the derived optimal estimator. The first one identifies a finite impulse response model. This approach improves the least-squares estimator with less restrictive assumptions. The second one applies the signal matrix model as the predictor in predictive control. The control performance is shown to be better than existing data-driven predictive control algorithms, especially under high noise levels. Both approaches demonstrate that the derived estimator provides a promising framework to apply data-driven algorithms to noisy data.

Primary–auxiliary model scheduling based estimation of the vertical wheel force in a full vehicle system

In this work we study the estimation problem of the vertical wheel force in a vehicle system subject to non-stationary and unknown excitation, which is a common and critical problem in vehicle durability design. The vehicle system can be approximated by a linear time invariant system under each working condition. However, the working conditions are unknown during the estimation process. A primary–auxiliary model scheduling procedure based on time-domain transmissibilities is proposed and performed to identify the unknown working conditions and estimate the unknown vertical wheel force: In addition to constructing a primary transmissibility family from the pseudo-inputs to the output during the offline stage, an auxiliary transmissibility family is constructed by further decomposing the pseudo-input vector into two parts. The auxiliary family enables identifying the unknown working condition under which the system is currently running, and then an appropriate transmissibility from the primary transmissibility family for estimating the unknown output can be selected during the online estimation stage. Moreover, Finite Impulse Response (FIR) models, ridge regression, and Bayes classifiers are applied to realize the model scheduling procedure. A numerical example and a real-world application to the estimation of the vertical wheel force in a full vehicle system are, respectively, conducted to demonstrate the effectiveness of the proposed method. The results show that the estimates are in good agreement with reference measurements.

Complexity curtailed MLSE equalizer based on a compact look-up-table for C-band DSB IM/DD transmission.

Chromatic dispersion appears to be a major performance limiting problem in optical intensity modulation direct detection (IM/DD) transmission systems, especially for a double-sideband (DSB) signal. We propose a complexity-reduced look-up-table based maximum likelihood sequence estimation (LUT-MLSE) for DSB C-band IM/DD transmission based on pre-decision-assisted trellis compression and a path-decision-assisted Viterbi algorithm. To further compress the size of the LUT and reduce the length of the training sequence, we proposed a finite impulse response (FIR) and LUT hybrid channel model for the LUT-MLSE. For PAM-6 and PAM-4, the proposed methods can compress the size of the LUT into 1/6 and 1/4, and reduce the number of multipliers by 98.1% and 86.6% with slight performance degradation. We successfully demonstrate a 20-km 100-Gb/s PAM-6 and a 30-km 80-Gb/s PAM-4 C-band transmission over dispersion-uncompensated links.

A kernel-based identification approach for a class of nonlinear systems with quantized output data

In this paper, we consider the problem of identification for a class of nonlinear systems with Hammerstein structure, which is a combination of some basis functions for the static nonlinearity and finite impulse response models for the linear block under quantized output data. Following the empirical Bayes method, we model the impulse response of the system using a Gaussian process whose covariance matrix depends on some suitable kernels. Then, we compute the minimum mean-square error estimation of the impulse response via a special case of Markov Chain Monte Carlo sampling approach, namely, Gibbs sampler. The estimation of the impulse response depends on the kernel hyperparameters, the noise variance and the coefficients of the static nonlinearity. These unknowns are estimated by the marginal likelihood maximization, which is solved via the expectation-conditional maximization algorithm combining the Gibbs sampler. The expectation-conditional maximization method is a variation of the standard expectation-maximization method for solving maximum-likelihood problems. Numerical simulations show the effectiveness of the proposed scheme.

Signal Estimation for Vehicle Body Accelerations Using Piecewise Linear System Identification in the Frequency Domain

Abstract In this work, we investigate a signal estimation problem which is common and critical for durability design of vehicle bodies. The relation between the frequency responses of accelerometers is the target to model so that the ones of easy-to-measure accelerometers can estimate the responses of hard-to-measure accelerometers. A piecewise linear frequency-domain identification method relying on finite impulse response (FIR) models is proposed and performed to tackle the nonlinearity issue in the signal estimation problems: first, the interesting frequency range is segmented into three subranges which are clearly identified by peak histograms of frequency signals. Then, FIR models which provide a satisfactory description of the system are constructed to estimate the frequency responses of the interesting signals at subranges, one for each. The performance of the proposed approach is validated by using real-world data under multiple working conditions. The results show that the proposed method has a good estimation accuracy, and it brings the benefit that the number of accelerometers can be significantly reduced during the durability design of vehicle bodies.

Data-driven aerodynamic analysis of structures using Gaussian Processes

An abundant amount of data gathered during wind tunnel testing and health monitoring of structures inspires the use of machine learning methods to replicate the wind forces. This paper presents a data-driven Gaussian Process-Nonlinear Finite Impulse Response (GP-NFIR) model of the nonlinear self-excited forces acting on structures. Constructed in a nondimensional form, the model takes the effective wind angle of attack as lagged exogenous input and outputs a probability distribution of the forces. The nonlinear input/output function is modeled by a GP regression. Consequently, the model is nonparametric, thereby circumventing to set up the function’s structure a priori. The training input is designed as random harmonic motion consisting of vertical and rotational displacements. Once trained, the model can predict the aerodynamic forces for both prescribed input motion and aeroelastic analysis. The concept is first verified for a flat plate’s analytical solution by predicting the self-excited forces and flutter velocity. Finally, the framework is applied to a streamlined and bluff bridge deck based on Computational Fluid Dynamics (CFD) data. The model’s ability to predict nonlinear aerodynamic forces, flutter velocity, and post-flutter behavior are highlighted. Applications of the framework are foreseen in the structural analysis during the design and monitoring of slender line-like structures.

Recursive identification of IIR system using binary input/output measurements

The identification of system using binary measurements of both the input and output is a problem relevant to a number of applications. Although the identification of finite impulse response (FIR) models has been thorough, more complex model structures still need to be investigated. In this paper, the problem of identifying an infinite impulse response (IIR) system is studied. A three-step real-time algorithm is proposed. The proposed algorithm is based on the use of least mean squares (LMS) algorithm to estimate the correlation function, then estimate the parameters of the system. A convergence analysis of the proposed algorithm based on those of LMS algorithm is presented. Some simulation results are included to illustrate the performance of the proposed algorithm.

Fouling Detection in Industrial Heat Exchanger Using Number of Transfer Units Method, Neural Network, and Nonlinear Finite Impulse Response Models

Due to reduced energy efficiency and productivity loss caused by fouled heat exchangers in industrial plants there is an obvious need for detection of fouling formation. Based on continuously measured temperatures and flow rates collected from the refinery plant history database, a semi-empirical number of transfer units (NTU) model and neural network-based models are developed. In order to confirm the reliability of proposed fouling factor calculation, the entire procedure was performed by developing a dynamic nonlinear finite impulse response (NFIR) model. Developed models are intended for fouling detection for industrial shell and tube heat exchangers. The performance criteria of developed models together with residual monitoring indicate that not only neural networks but the NTU and NFIR models effectively detect fouling formation.

Estimating the SARS-CoV-2 infected population fraction and the infection-to-fatality ratio: a data-driven case study based on Swedish time series data

We demonstrate that finite impulse response (FIR) models can be applied to analyze the time evolution of an epidemic with its impact on deaths and healthcare strain. Using time series data for COVID-19-related cases, ICU admissions and deaths from Sweden, the FIR model gives a consistent epidemiological trajectory for a simple delta filter function. This results in a consistent scaling between the time series if appropriate time delays are applied and allows the reconstruction of cases for times before July 2020, when RT-PCR testing was not widely available. Combined with randomized RT-PCR study results, we utilize this approach to estimate the total number of infections in Sweden, and the corresponding infection-to-fatality ratio (IFR), infection-to-case ratio (ICR), and infection-to-ICU admission ratio (IIAR). Our values for IFR, ICR and IIAR are essentially constant over large parts of 2020 in contrast with claims of healthcare adaptation or mutated virus variants importantly affecting these ratios. We observe a diminished IFR in late summer 2020 as well as a strong decline during 2021, following the launch of a nation-wide vaccination program. The total number of infections during 2020 is estimated to 1.3 million, indicating that Sweden was far from herd immunity.