Filter Estimation Error Research Articles

Practical Finite-Time Command-Filtered Adaptive Backstepping With Its Applications to Quadrotor Hovers.

In this article, a practical finite-time command-filtered adaptive backstepping (PFTCFAB) control method is presented for a class of uncertain nonlinear systems with nonparametric unknown nonlinearities and external disturbances. Unlike PFTCFAB control techniques that use neural networks (NNs) or fuzzy-logic systems (FLSs) to deal with system uncertainties, the proposed method is capable of handling such uncertainties without the need for NNs or FLSs, thus reducing complexity and increasing reliability. In the proposed approach, novel function adaptive laws are designed to directly estimate unknown nonparametric nonlinearities and external disturbances by means of command filter techniques, and a type of practical finite-time command filters is proposed to obtain such laws. Moreover, the PFTCFAB controllers and finite-time command filters are designed with practical finite-time Lyapunov stability, which ensures finite-time stability of system tracking and filter estimation errors. Experimental results with a quadrotor hover system are presented and discussed to demonstrate the advantages and effectiveness of the proposed control strategy.

An Effective Integrity Monitoring Scheme for GNSS/INS/Vision Integration Based on Error State EKF Model

It is well-known that the Global Navigation Satellite System/Inertial Navigation System (GNSS/INS) /Vision integration has been increasingly used for safety-critical applications like self-driving cars. However, the performance of the GNSS/INS/Vision integration degrades due to unknown faults. Therefore, it is essential to develop integrity monitoring algorithms for the integrated navigation system. A new protection level (PL) formula to ensure the integrity of the state estimation of the GNSS/INS/Vision integration is proposed, which can be calculated by the relation between the filter estimation error and faults. An integrity risk allocation tree is developed for each sensor fault hypothesis including the nominal hypothesis, and the specific PL is given in the case of GNSS, INS and visual measurement faults. In addition, the performance of the proposed PL is comprehensively evaluated. A field vehicular test was conducted to evaluate the performance of the integrity monitoring algorithm for the GNSS/INS/Vision integration. The results demonstrate that the proposed PL for the GNSS/INS/Vision integration can fit the position error against six different fault modes.

Firms’ response to macroeconomic estimation errors

Initial Gross Domestic Product (GDP) announcements are important economic signals that convey information on the state of the economy but contain substantial estimation error. We investigate how GDP estimation errors affect firms' real decisions and profitability. We find that GDP estimation errors are positively associated with one-quarter-ahead changes in firms’ capital investments, production, inventory, and profitability. However, we observe long-run profitability reversals, which is consistent with initial over (under) production eventually being met with declines (increases) in future profitability. Furthermore, managerial responses to the estimation errors mimic the response to the true component of the GDP signal, suggesting that managers do not filter estimation errors and overreact to both GDP signal components. Our firm-level findings translate to the macroeconomic level, where we find that a long-run reversal follows a positive short-run aggregate investment response to GDP signal components.

On the Enhancement of Generalized Integrator-Based Adaptive Filter Dynamic Tuning Range

Many phase-locked loop (PLL) or frequency-locked loop (FLL) use quadrature signals (directly or indirectly) as the input variables. Generalized integrator (GI) is a popular quadrature signal generator (QSG) available in the literature. GI is also widely used in various industrial applications. In addition to being a QSG, GI also works as a kind of bandpass filter. However, due to structural limitations, the dynamic tuning range is limited for the standard GI. The limitation arises from using only one gain in the direct-phase estimation dynamics, while quadrature phase estimation dynamics does not use direct feedback of the filter estimation error. Some attempts have already been made to overcome this limitation by adding direct feedback of the filter estimation error to quadrature-phase dynamics as well. However, we have demonstrated in this article that this kind of implementation has some frequency domain limitations. In this article, we propose a novel GI-type adaptive filter using coordinate transformation. The resulting structure maintains the same kind of filtering property of the standard GI at the transformed coordinates level while at the same time enhancing the dynamic tuning range of standard GI. Details of the proposed technique, stability analysis, and discussion on gain tuning are provided in this article. Finally, comparative experimental results are provided with respect to GI-FLL to show the dynamic performance improvement. Experimental results demonstrate the suitability of the proposed technique.

Firmss Response to Macroeconomic Estimation Errors

Initial Gross Domestic Product (GDP) announcements are important economic signals that convey information on the state of the economy but contain substantial estimation error. We investigate how GDP estimation errors affect firms’ real decisions and profitability. We find that GDP estimation errors are positively associated with one-quarter-ahead changes in firms’ capital investments, production, inventory, and profitability. However, we observe long-run profitability reversals, which is consistent with initial over (under) production eventually being met with declines (increases) in future profitability. Furthermore, managerial responses to the estimation errors mimic the response to the true component of the GDP signal, suggesting that managers do not filter estimation errors and overreact to both GDP signal components. Our firm-level findings translate to the macroeconomic level, where we find that a long-run reversal follows a positive short-run aggregate investment response to GDP signal components.

Iterative Frequency-Weighted Filtering and Smoothing Procedures

Minimum-variance filters and smoothers exhibit performance degradations when they are designed with inexact models and noise statistics. Filter and smoother estimation errors are assumed herein to be generated by a first-order moving-average system. This assumed system is identified and used to design a frequency weighting function to improve mean square error performance. It is shown under prescribed conditions that the sequence of frequency-weighted estimation error variances are nonincreasing. An example is presented which demonstrates the efficacy of repeated frequency weighting iterations.

Error and Noise Analysis in an IMU using Kalman Filter

Kalman filtering is a well-established methodology used in various multi-sensor data fusion applications. In our experiment, we first obtain measurements from the accelerometer and gyroscope and fuse them using Kalman filter in an inertial measurement unit (IMU). We estimate Kalman filter output and estimation error. The affect of process noise and measurement noise on estimation error is tested. It is explored that the measurement noise has significant role to increase estimation error in the data fusion process.

Enhancement of Residual Echo for Robust Acoustic Echo Cancellation

This paper examines the technique of using a noise-suppressing nonlinearity in the adaptive filter error feedback-loop of an acoustic echo canceler (AEC) based on the least mean square (LMS) algorithm when there is an interference at the near end. The source of distortion may be linear, such as local speech or background noise, or nonlinear due to speech coding used in the telecommunication networks. Detailed derivation of the error recovery nonlinearity (ERN), which “enhances” the filter estimation error prior to the adaptation in order to assist the linear adaptation process, will be provided. Connections to other existing AEC and signal enhancement techniques will be revealed. In particular, the error enhancement technique is well-founded in the information-theoretic sense and has strong ties to independent component analysis (ICA), which is the basis for blind source separation (BSS) that permits unsupervised adaptation in the presence of multiple interfering signals. The single-channel AEC problem can be viewed as a special case of semi-blind source separation (SBSS) where one of the source signals is partially known, i.e., the far-end microphone signal that generates the near-end acoustic echo. The system approach to robust AEC will be motivated, where a proper integration of the LMS algorithm with the ERN into the AEC “system” allows for continuous and stable adaptation even during double talk without precise estimation of the signal statistics. The error enhancement paradigm encompasses many traditional signal enhancement techniques and opens up an entirely new avenue for solving the AEC problem in a real-world setting.

Blind Multidimensional Matched Filtering Techniques for Single Input Multiple Output Communications

In this paper, blind multidimensional matched filtering techniques for Single Input Multiple Output (SIMO) communications are examined. To improve the signal-to-noise ratio prior to the equalization process, three different techniques are proposed to blindly implement the multidimensional matched filtering. Different from the blind channel identification techniques for SIMO channels in the literature, the proposed techniques require neither the implementation inefficient matrix decomposition methods nor the higher-order statistics of the received data. Therefore, they are favorable over existing methods due to their simplicity in application. The matched filter equivalents are established through the use of an adaptive filter as well as the channel equalization being performed blindly. It has been shown that the equalization performance of the proposed methods are close to the matched filter bound, and to support this, a comparison between the matched filter estimation error performances of our proposed techniques is also provided. A short discussion to improve the convergence speed of the proposed approaches is also included in this paper.

Optimal Tuner Selection for Kalman Filter-Based Aircraft Engine Performance Estimation

A linear point design methodology for minimizing the error in on-line Kalman filter-based aircraft engine performance estimation applications is presented. This technique specifically addresses the underdetermined estimation problem, where there are more unknown parameters than available sensor measurements. A systematic approach is applied to produce a model tuning parameter vector of appropriate dimension to enable estimation by a Kalman filter, while minimizing the estimation error in the parameters of interest. Tuning parameter selection is performed using a multivariable iterative search routine that seeks to minimize the theoretical mean-squared estimation error. This paper derives theoretical Kalman filter estimation error bias and variance values at steady-state operating conditions, and presents the tuner selection routine applied to minimize these values. Results from the application of the technique to an aircraft engine simulation are presented and compared with the conventional approach of tuner selection. Experimental simulation results are found to be in agreement with theoretical predictions. The new methodology is shown to yield a significant improvement in on-line engine performance estimation accuracy.

Design and Analysis of Supervised and Decision-Directed Estimators of the MMSE/LCMV Filter in Data Limited Environments

We consider sample-matrix-inversion (SMI)-type estimates of the minimum-mean-square-error (MMSE) and the linearly constrained-minimum-variance (LCMV) linear filters obtained from data records of limited size. We quantify theoretically the (detrimental) effect of the desired-signal energy level on the mean square (MS) filter estimation error and the normalized output signal-to-interference-plus-noise ratio (SINR) by deriving a new exact analytical expression and a lower bound, respectively. For cases where accumulation of pure disturbance observations is not possible, we show theoretically how certain intuitive, pilot-assisted, and decision-directed adaptive filter implementations that utilize desired-signal-present data/observations perform close to their desired-signal-absent counterparts. Simulation studies illustrate our theoretical developments in the context of spread-spectrum communications over multipath fading channels under perfect and nonperfect synchronization.

GPS Composite Clock Analysis

The GPS composite clock defines GPS time, the timescale used today in GPS operations. GPS time is illuminated by examination of its role in the complete estimation and control problem relative to UTC/TAI. The phase of each GPS clock is unobservable from GPS pseudorange measurements, and the mean phase of the GPS clock ensemble (GPS time) is unobservable. A new and useful observability definition is presented, together with new observability theorems, to demonstrate explicitly that GPS time is unobservable. Simulated GPS clock phase and frequency deviations, and simulated GPS pseudorange measurements, are used to understand GPS time in terms of Kalman filter estimation errors.

The Smooth Variable Structure Filter

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, a new method for state estimation, referred to as the smooth variable structure filter (SVSF), is presented. The SVSF method is model based and applies to smooth nonlinear dynamic systems. It allows for the explicit definition of the source of uncertainty and can guarantee stability given an upper bound for uncertainties and noise levels. The performance of the SVSF improves with more refined definition of upper bounds on parameter variations or uncertainties. Furthermore, most filtering methods provide as their measure of performance the filter innovation vector or (output) estimation error. However in addition to the innovation vector, the SVSF has a secondary set of performance indicators that correlate to the modeling errors specific to each state or parameter that is being estimated. The combined robustness and multiple indicators of performance allow for dynamic refinement of internal models in the SVSF. Dynamic refinement and robustness are features that are particularly advantageous in fault diagnosis and prediction. In this paper, the applications of the SVSF to linear and nonlinear systems, including one pertaining to fault detection, are provided. The characteristics of this filter in terms of its accuracy and rate of convergence are discussed. </para>

Recursive Short-Data-Record Estimation of AV and MMSE/MVDR Linear Filters for DS-CDMA Antenna Array Systems

The presence of the desired signal during estimation of the minimum mean-square error (MMSE)/minimum-variance distortionless-response (MVDR) and auxiliary-vector (AV) filters under limited data support leads to significant signal-to-interference-plus-noise ratio (SINR) performance degradation. We quantify this observation in the context of direct-sequence code-division multiple-access (DS-CDMA) communications by deriving close approximations for the mean-square filter estimation error, the probability density function of the output SINR, and the probability density function of the symbol-error rate (SER) of the sample matrix inversion (SMI) receiver evaluated using both a desired-signal-"present" and desired-signal-"absent" input covariance matrix. To avoid such performance degradation, we propose a DS-CDMA receiver that utilizes a simple pilot-assisted algorithm that estimates and then subtracts the desired signal component from the received signal prior to filter estimation. Then, to accommodate decision-directed operation, we develop two recursive algorithms for the on-line estimation of the AV and MMSE/MVDR filter and we study their convergence properties. Finally, simulation studies illustrate the SER performance of the overall receiver structures.

Data record-based criteria for the selection of an auxiliary vector estimator of the mmse/mvdr filter

When the auxiliary vector (AV) filter generation algorithm utilizes sample average estimated input data statistics, it provides a sequence of estimates of the ideal minimum mean-square error or minimum-variance distortionless-response filter for the given signal processing/receiver design application. Evidently, early nonasymptotic elements of the sequence offer favorable bias/variance balance characteristics and outperform in mean-square filter estimation error the unbiased sample matrix inversion (SMI) estimator as well as the (constraint) least-mean square, recursive least-squares, "multistage nested Wiener filter", and diagonally-loaded SMI filter estimators. Selecting the most successful (in some appropriate sense) AV filter estimator in the sequence for a given data record is a critical problem that has not been addressed so far. We deal exactly with this problem and we propose two data-driven selection criteria. The first criterion minimizes the cross-validated sample average variance of the AV filter output and can be applied to general filter estimation problems; the second criterion maximizes the estimated J-divergence of the AV filter output conditional distributions and is tailored to binary phase-shift-keying-type detection problems.

Periodic measurement strategies for distributed parameter systems filtering

This paper deals with the optimization of the output matrix for a discrete time linear stochastic system. The output matrix varies as a periodic function of time, and its values are constrained to belong to a finite prescribed set. The aim is to minimize the average variance of the Kalman filter estimation error in the periodic steady state. The application regards the optimization both of the measurement points and of the scanning sequence for a distributed parameter system (DPS) of parabolic type. A modal approximation is used to reduce the DPS to finite dimension. The proposed solution algorithm makes use of heuristic rules that enable to overcome the difficulties arising from the cardinality of the admissible set, the possible slow convergence of the relevant Riccati equation and the high dimensionality of the lumped approximate model of the DPS. The numerical applications show that the periodic scanning policies, found by the optimization algorithm, cause a great improvement of the filter performance...

Second-order state estimation experiments using acceleration measurements

The estimation of dynamic states for feedback control of structural systems using second-order differential equations and acceleration measurements is described. The formulation of the observer model and the design of the observer gains are discussed in detail. It is shown that the second-order observer is highly stable because the stability constraints on the observer gains are model independent. The limitation of the proposed observer is the need for nearly collocated actuators and accelerometers. Experimental results using a control-struct ure interaction testbed are presented that show that the second-order observer provided more stability than a Kalman filter estimator without decreasing closed-loop performance. F LEXIBLE-BODY vibration control of large structural systems often requires estimation of dynamic states that cannot be measured directly. The state estimator or observer plays a key role in the overall stability of the control system. Reference 1 has shown that the linear quadratic regulator (LQR) technique provides 60 deg of phase margin and infinite gain margin when the full state is available. However, when an observer must be used in conjunction with LQR, no guaran- tees can be made on the robustness properties due to state reconstruction errors.2'3 Observer state reconstruction suffers from model errors, measurement noise, and unmodeled dy- namics.4 Although much work has been done on optimal state estimation using first-order Kalman filters, the closed-loop stability remains a strong function of the observer model accuracy. Although there is more design freedom when using a first- order model, second-order observers provide a great deal of physical insight when scientists attempt to perform state esti- mation of a second-order system. Second-order observer mod- els have received recent attention in the literature. Reference 5 utilized a Kalman filter for discretized (time and space) sec- ond-order structure models. In addition, robust computa- tional procedures for solution of the Kalman filter estimation error covariance matrices have been developed for second-or- der models in Ref. 6. A dissipative observer for use with velocity measurements that possessed the second-order form was introduced in Ref. 7. This observer was analogous to the collocated controller whereby symmetric positive definite feedback gains were designed to insure stability. Recently, the authors of Ref. 8 developed the conditions for optimal estima- tion using the second-order form assuming velocity measure- ments were available. Accelerometers are frequently used to measure the dynamic response of structural systems because of their low cost and their light weight and because they provide an inertial mea- surement. The first-order observer filters acceleration mea- surements in both space and time to derive the proper phase information to reconstruct the unmeasured states. Spatial fil- tering, which projects the measurement data onto the system coordinate basis, requires accurate dynamic models. The re- duced design freedom available with the second-order ob- server has usually required velocity measurements since nei- ther spatial nor temporal filtering of the measurement is performed. In this work, collocated acceleration measurements and control actuators are used for state estimation by augmenting the second-order observer with additional states. (The addi- tional states provide temporal filtering of the acceleration measurements to insure asymptotic convergence of the esti- mated state error.) By using collocated sensors and actuators, the observer does not require spatial filtering, which signifi- cantly reduces the sensitivity to model errors. Equations for the typical first-order observer and a logical form for the second-order observer are presented. Stability and performance considerations for the first- and second-or- der observers are discussed. A design method for second-order observer gains with acceleration measurements is then de- scribed. Practical aspects of the observers are examined, and both numerical and experimental results are presented.

Optimal Periodic Selection of the Measurement for the State Estimation of Linear Discrete Time Stochastic Systems

In this paper the optimization of the output matrix for a discrete time linear single output stochastic system is considered. The output matrix may vary as a periodic function of time, with fixed period, and its values are constrained to belong to a prescribed set. The optimization criterion is the average value of the trace of the covariance matrix of the Kalman filter estimation error in the periodic steady state. Two types of constraints are considered: the admissible set is either a closed sphere or a finite collection of matrices. In the first case a gradient based optimization is proposed, in the other a suboptimal solution is searched using heuristics.