Equation Error Formulation Research Articles

A recursive system identification inertia estimator for traditional and converter-interfaced generators

This paper proposes a method to estimate the inertia constant of synchronous generators (SGs) and the virtual inertia provided by converter-interfaced generators (CIGs). It can be applied when the system is under normal operating conditions or after power imbalance events. The algorithm is based on a recursive system identification approach and uses the active power and bus frequency measurements proceeding from phasor measurement units. The measurements are preprocessed to improve the estimation accuracy. Subsequently, a model-independent rotor speed estimator is used to compute rotor speed variations (or internal frequency variations, in the case of CIGs). Afterwards, a recursive least-square equation error formulation is implemented to compute the SG and CIG parameters in the z-domain, which are later converted into their respective counterparts in the s-domain. The method is applied to several generation devices in the IEEE 39-bus and IEEE 118-bus test systems, including traditional SGs and CIGs providing virtual inertia. Numerical results show the accuracy and the low computational burden of the proposed method.

Recursive myriad-mean filters: Adaptive algorithms and applications

In this paper, a new class of recursive hybrid filtering structures is proposed for impulsive noise removal; the so-called recursive myriad-mean (RMyM) filters. More precisely, the output of the RMyM filter can be thought of as the sum of two independent weighted M-filters: the nonlinear weighted myriad acting on a subset of input samples and the linear weighted mean acting on a subset of filter’s previous outputs. The uncoupled structure of the proposed filters takes into account the benefits of both weighted M-estimators: the robustness against impulsive noise of the myriad operator and the desired spectral response induced by the linear feedback. Least mean absolute (LMA) based adaptive algorithms are developed for designing these filtering structures under the equation error formulation framework. The results of extensive simulations are shown to evaluate both the behavior of the adaptive algorithms as well as the performance of the proposed recursive filters against impulsive noise. Additionally, taking into account the uncoupled structure of the proposed recursive filters, a decision feedback equalizer (DFE) based on the RMyM filter is proposed, where its performance is compared to those yielded by various conventional DFE structures, under different conditions of impulsive noise.

Recursive Weighted Myriad Based Filters and Their Optimizations

This paper proposes two new recursive filtering structures based on the nonlinear myriad operator. First, we develop the general class of recursive weighted myriad (RWMy) filters as a robust filtering structure against impulsive noise that includes, as particular cases, a normalized version of the linear infinite impulse response (IIR) filter, the recursive mode-type filter, and the non-recursive weighted myriad filter. Secondly, considering the fact that the additive noise that contaminates the previous filter's outputs is no longer impulsive, we introduce a novel class of recursive hybrid myriad (RHMy) filters whose structure gathers the advantages of both the weighted myriad and the weighted mean in a single cost function to be minimized. Some properties of the RHMy filter are derived and a fast algorithm to compute the RHMy filter output is proposed. Furthermore, adaptive algorithms for designing the proposed recursive structures, under the equation error formulation framework, are developed. Finally, extensive numerical simulations are shown to evaluate both the iterative update of the adaptive algorithms and the performance of the proposed recursive filters against impulsive noise.

Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach

Abstract The objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework.

Asymptotically efficient estimators for the fittings of coupled circles and ellipses

Fitting a pair of coupled geometric objects to a number of coordinate points is a challenging and important problem in many applications including coordinate metrology, petroleum engineering and image processing. This paper derives two asymptotically efficient estimators, one for concentric circles fitting and the other for concentric ellipses fitting, based on the weighted equation error formulation and non-linear parameter transformation. The Kanatani–Cramér–Rao (KCR) lower bounds for the parameter estimates of the concentric circles and concentric ellipses under zero-mean Gaussian noise are provided to serve as the performance benchmark. Small-noise analysis shows that the proposed estimators reach the KCR lower bound performance asymptotically. The accuracy of the proposed estimators is corroborated by experiments with synthetic data and realistic images.

Effective bias reduction methods for passive source localization using TDOA and GROA

For passive source localization based on both TDOA and GROA, this paper proposes two bias reduction methods for the well-known Weighted-Least-Squares (WLS) estimator. We first derive the passive source localization bias from the two-step algebraic closed-form solution. This bias is found to be considerably larger than the Maximum Likelihood Estimator (MLE) and limits the WLS estimator’s practical applications. In this paper, We develop two methods to reduce the bias. The first one called Bias-Subtraction-Method (BSM) directly subtracts the expected bias from the solution of the WLS estimator, and the second one called Bias-Reduction-Method (BRM) imposes a constraint to the equation error formulation to improve the source location estimate. The noise covariance matrix must be known exactly in calculating the expected bias in BSM, and we only need to know the structure of it in BRM. For far-field sources localization when the noise is Gaussian and not too large, both of the two proposed methods can reduce the localization bias effectively and achieve the Cramer-Rao Lower Bound (CRLB) performance very well, and the BRM almost has the same performance as the MLE estimator. Simulations corroborate the performance of the two proposed methods.

Bias Reduction for an Explicit Solution of Source Localization Using TDOA

<?Pub Dtl=""?> This paper proposes two methods to reduce the bias of the well-known algebraic explicit solution (Chan and Ho, “A simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Process., vol. 42, pp. 1905–1915, Aug. 1994) for source localization using TDOA. Bias of a source location estimate is significant when the measurement noise is large and the geolocation geometry is poor. Bias also dominates performance when multiple times of independent measurements are available such as in UWB localization or in target tracking. The paper starts by deriving the bias of the source location estimate from Chan and Ho. The bias is found to be considerably larger than that of the Maximum Likelihood Estimator. Two methods, called BiasSub and BiasRed, are developed to reduce the bias. The BiasSub method subtracts the expected bias from the solution of Chan and Ho's work, where the expected bias is approximated by the theoretical bias using the estimated source location and noisy data measurements. The BiasRed method augments the equation error formulation and imposes a constraint to improve the source location estimate. The BiasSub method requires the exact knowledge of the noise covariance matrix and BiasRed only needs the structure of it. Analysis shows that both methods reduce the bias considerably and achieve the CRLB performance for distant source when the noise is Gaussian and small. The BiasSub method can nearly eliminate the bias and the BiasRed method is able to lower the bias to the same level as the Maximum Likelihood Estimator. The BiasRed method is extended for TDOA and FDOA positioning. Simulations corroborate the performance of the proposed methods.

Combined Regressor Methods and Adaptive IIR Filtering

An open issue in adaptive infinite-impulse response (IIR) filtering is that of convergence to a global minimum in the presence of observation noise when the system is insufficiently modeled . It is well known , that algorithms based on equation error (EE) formulation contain a single minimum that may be biased whereas, algorithms based on output error (OE) ensure the existence of an unbiased global minimum in presence of local minima. Recently, there have been several attempts to combine these formulations in order to ensure the existence and uniqueness of an unbiased minimum. Works presented here, EEOE and modified EEOE (MEEOE), are such attempts in the context of system identification. We will show, analytically and through simulations, the convergence properties of the MEEOE approach, in the context of system identification.

Recursive LMS L-filters for noise removal in images

The problem of designing the weights for recursive L-filters optimized by the least mean square (LMS) algorithm is addressed. The coefficients derived for nonrecursive filtering are not optimal for recursive implementation, where the estimate of current pixel depends on the past outputs of the filter. To combat this, analogous to the design of adaptive IIR filters, the optimization scheme referred to as equation-error formulation is employed. The recursive filter performs better in suppressing noise than its nonrecursive counterpart.

Design of analog equalizers for partial response detection in magnetic recording

Analog equalizers offer several advantages, such as high speed, small size, low power, and minimum delay, over sampled digital equalizers. (In this paper, the phrase analog equalizer refers to continuous-time equalizers having poles and zeros.) However, because of their feedback nature, their design often involves extensive search procedures. Recently, an analytical design technique based on equation-error formulation was proposed. In this paper, we study this approach for partial response (PR) equalization and show that it is quite sensitive to the choice of timing phase and it could result in quite suboptimal solution. We improve on it by using an iterative approach similar to that of Steiglitz and McBride. We also present a thorough simulation study to assess the performance loss incurred by low-order analog equalizers and produce performance charts that can be used for selecting the PR target for different channel characteristics. Application of this approach to multi-level decision feedback equalization detection is also discussed.

A general consistent equation-error algorithm for adaptive IIR filtering

This paper proposes a new adaptive IIR filtering algorithm based on equation error (EE) formulation. The algorithm allows for the compensation of the bias inherent to the equation error formulation. A study of the properties of the new algorithm shows that the asymptotic stability associated with the EE algorithm can be maintained. In addition, for sufficiently general conditions, the parameter estimates are consistent. Some methods recently proposed to remove the bias problem of the EE algorithm are particular cases of the general algorithm presented. Computer simulations are included to illustrate and compare the convergence properties of the proposed algorithm.

Bias removal in equation-error adaptive IIR filters

In the equation-error formulation of adaptive IIR filters, the estimated parameters contain bias when there is noise in the desired response. A method that can eliminate this bias is investigated. The idea is to maintain a quadratic constraint on the feedback coefficients so that the noise contributes only a constant term to the mean-square error. This term does not affect minimization and thus the bias is eliminated. A quadratically constrained stochastic gradient search method is applied for optimization and convergence behavior, when the noise is white, is analyzed. Adaptation of the feedback FIR filter in second-order cascade form, useful for stability monitoring, is also considered. When the noise is nonwhite, the technique requires an adaptive whitening filter. Simulation results are included to demonstrate the bias removal capability of the method, corroborate the theoretical developments, and compare with existing techniques.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An adaptive backpropagation cascade IIR filter

Among the various linear infinite impulse response (IIR) structures for adaptive filters, cascade IIR filters are notable for easy stability monitoring and good sensitivity performance. However, their high cost for computing gradients is a problem. A novel technique of backpropagating the desired signal is proposed, and a cascade structure satisfying the requirements of this technique is developed, resulting in an efficient adaptive IIR filter. It is shown that the equation-error formulation is just a special case of this backpropagation idea.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Linear, Frequency Domain Filter for Parameter Identification of Vibrating Structures

In this paper the authors present a linear, least-squares, frequency domain filter for the identification of structural vibration parameters, such as mass, stiffness, and damping, from observed forces and responses. The method is based on an equation error formulation which produces a measurement equation which is linear in the unknown parameters. Convergence of the parameter estimates is achieved rapidly and the period required to obtain convergence is indicated by the CPU time measured on an IBM 3083 computer. Results obtained using simulated data are used to compare the linear filter with alternative filter formulations. Experimental results from a portal frame rig are given to demonstrate the application of the method to physical measurements.

A method for linear model reduction using the input-independent equation error

The paper deals with the order-reduction problem for single-input single-output linear dynamical systems. Combining the uniform approximation approach with the equation error formulation, a simplification method is proposed which leads to input-independent-optimal models by a relatively simple calculation procedure.

Transition formulas for zero-pole modeling

Using the equation error formulation, transitional formulas are developed that provide solutions to the ARMA modeling problem when given solutions to either the AR or MA models. These formulas are useful in estimating ARMA models, and they also provide insight into how the more general ARMA modeling problem is related to the special cases of AR and MA modeling.

Iterative inverse filtering approach to the estimation of frequencies of noisy sinusoids

A signal modeling approach is proposed to reduce the problem of estimating frequencies of noisy sinusoids to the parameter estimation problem. Both white and colored noise are considered. The maximum likelihood approach leads to an algorithm computationally inefficient for low values of signal-to-noise ratio. Based on the equation-error formulation, an iterative inverse filtering algorithm is derived in the case of colored noise, and a generalized least-squares algorithm in the case of white noise. It is shown, on the basis of numerous experimental results, that the iterative inverse filtering algorithm provides highly accurate estimates of unknown frequencies for low values of signal-to-noise ratio, even in the case of a small number of sampling points.

Adaptive implementation of one-step-ahead optimal control via input matching

Directly adapting feedback control parameters to match an optimal input avoids the current limitations of indirect adaptive control and direct adaptive model matching. This original concept, termed input matching, allows combination of stable parameter adjustment algorithms and optimal performance considerations. Furthermore, the emphasis on input matching permits consistent control despite inconsistent parameter identification obviating efforts such as test or probing inputs required to ensure consistent identification. The control signal is reduced to a constant weighted sum of the measurable information-state vector components by the use of a one-step-ahead quadratic cost function to govern the behavior of a linear, time-invariant multivariable plant. The control effort from this linear combination proves globally estimable by a vector equation error formulation since the one-step-ahead cost function permits simple a posteriori input error calculation. Several simulations demonstrate the behavior of this new multivariable adaptive input matching control method.

An adaptive analogue tracker for automatic measurement of lung parameters

This paper describes the design and results obtained using simple electronic circuits specially designed for implementing a simple lung-parameter tracking algorithm for identification of the mechanical properties of the lung. Unlike the commonly used loop-flattening technique, the adaptive electronic tracker is able to monitor continuously the mechanical properties of the respiratory sysem. It is capable of tracking the rapid changes in lung parameters as the frequency of breathing changes. The design of the adaptive tracker is based on equation-error formulation and the global asymptotic stability of the adaptive tracking equations is guaranteed. The cheapness and simplicity of the tracker makes it suitable for clinical applications.

Identification of linear systems with time-delay operating in a closed loop in the presence of noise

The subject of this short paper is on-line identification of the parameter vector defining a linear dynamical system which operates in a closed loop in the presence of noise, and incorporates a time delay. The method is based on the equation error. Other known subsystems in the closed-loop system increase the dimension of the closed-loop parameter vector which tends to degrade the estimation convergence process. By means of "composite state variables," introduced in this short paper, this increase is prevented and the open-loop parameters are directly identified from closed-loop input-output data. The pure time delay in the closed loop causes a representation problem in the equation error formulation. This is overcome by "composite delayed state variables." The value of the time delay is determined by means of excess parameters provided by a "higher order model" and a simple on-line search procedure. The method is illustrated by simulated examples.