Recursive Least-squares Algorithm Research Articles

Deep kernel recursive least-squares algorithm

We present a new kernel-based algorithm for modeling evenly distributed multidimensional datasets that does not rely on input space sparsification. The presented method reorganizes the typical single-layer kernel-based model into a deep hierarchical structure, such that the weights of a kernel model over each dimension are modeled over its adjacent dimension. We show that modeling weights in the suggested structure leads to significant computational speedup and improved modeling accuracy.

A new approach for designing and implementing ADF equalization for 5G frequency selective channel based on two operating phases of LS and RLS algorithms

In this paper, a new approach is proposed for implementing an adaptive decision feedback equalizer (ADFE) for the 5G channel. The proposed equalizer works in two phases. In the first phase, a least-squares (LS) algorithm with a variable-length training sequence is used to estimate the coefficients of the channel and the equalizer. In the second phase, the recursive least-squares algorithm estimates the channel and adapts the equalizer, jointly. According to the channel quality, a variable-length training sequence is used to estimate the channel vector and the coefficients of the equalizer. The feed-forward equalizer (FFE) compensates the effects of the transmitting filter and the channel filter. No matched filter is used in the receiver. The noise samples at the input of the proposed FFE are independent. The noise enhancement of the proposed FFE is less than the noise enhancement of its corresponding one in the conventional ADFE. The overall filtering response (OFR) from the input of the transmitting filter to the output of the FFE is calculated and used to estimate the coefficients of the feedback equalizer (FBE). The channel model, the FFE coefficients, the OFR vector, and the FBE coefficients are continuously updated every symbol period. Using a variable training sequence increases the bandwidth efficiency of the transmitted signal. Simulation results and real-time implementation measurements show that the convergence time and the steady-state error at the output of the proposed equalizer are smaller than their corresponding values in the conventional ADFE.

RANSAC과 순환최소자승 알고리즘을 이용한 고조파 등가 모델 파라미터 추정 연구

This paper presents a method for estimating the parameters of a harmonic equivalent model based on the RANSAC(random sample consensus) and RLS(recursive least square) algorithms. The harmonic contribution assessment basically requires the equivalent models of harmonic sources. This model consists of an equivalent impedance and an equivalent voltage source. Because the equivalent model parameters can not be measured directly, it is necessary to estimate the parameters of harmonic sources by measuring PCC voltage and current. In previous studies, several RLS-based methods have been proposed to obtain the equivalent parameters. However, if the measured data contains outliers, the methods lead to large errors in parameter estimation. Outlier is a data point that differs significantly from other measured data due to various causes such as measurement errors. In this paper, we propose a RANSAC based method to remove outlier from measured data set to improve parameter estimation performance. To verify the performance of the proposed method, a comparative analysis for the case of including data outlier, with the previous methods was performed.

Recursive least-squares algorithm for a characteristic model with coloured noise by means of the data filtering technique

This work investigates the identification problems for a characteristic model with autoregressive moving average noise, and the sum of all coefficients of the model is equal to one. A recursive least-squares (RLS) algorithm using the data filtering technique is derived for the model. The basic idea is to use a linear filter to filter the input–output data, to decompose a characteristic model into a system model and a noise model, but the sum of all coefficients remains equalling to one. Moreover, the traditional RLS algorithm is also presented and compared with the proposed algorithm in terms of computational complexity and effectiveness. Finally, three numerical examples show that the proposed algorithm outperforms the conventional RLS algorithm.

Efficient recursive least squares solver for rank-deficient matrices

Updating a linear least-squares solution can be critical for near real-time signal-processing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A∈Rn×m with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least-squares algorithm exploiting the rank-deficiency of A, achieving the update of the minimum-norm least-squares solution in O(mr) operations and, therefore, solving the linear least-squares problem from scratch in O(nmr) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity.

A Model-Based Sensor Fault Diagnosis Scheme for Batteries in Electric Vehicles

The implementation of each function of a battery management system (BMS) depends on sensor data. Efficient sensor fault diagnosis is essential to the durability and safety of battery systems. In this paper, a model-based sensor fault diagnosis scheme and fault-tolerant control strategy for a voltage sensor and a current sensor are proposed with recursive least-square (RLS) and unscented Kalman filter (UKF) algorithms. The fault diagnosis scheme uses an open-circuit voltage residual generator and a capacity residual generator to generate multiple residuals. In view of the different applicable state of charge (SOC) intervals of each residual, different residuals need to be selected according to the different SOC intervals to evaluate whether a sensor fault occurs during residual evaluation. The fault values of the voltage and current sensors are derived in detail based on the open-circuit voltage residual and the capacity residual, respectively, and applied to the fault-tolerant control of battery parameters and state estimations. The performance of the proposed approaches is demonstrated and evaluated by simulations with MATLAB and experimental studies with a commercial lithium-ion battery cell.

Adaptive method to predict and track unknown system behaviors using RLS and LMS algorithms

This study investigates the ability of recursive least squares (RLS) and least mean square (LMS) adaptive filtering algorithms to predict and quickly track unknown systems. Tracking unknown system behavior is important if there are other parallel systems that must follow exactly the same behavior at the same time. The adaptive algorithm can correct the filter coefficients according to changes in unknown system parameters to minimize errors between the filter output and the system output for the same input signal. The RLS and LMS algorithms were designed and then examined separately, giving them a similar input signal that was given to the unknown system. The difference between the system output signal and the adaptive filter output signal showed the performance of each filter when identifying an unknown system. The two adaptive filters were able to track the behavior of the system, but each showed certain advantages over the other. The RLS algorithm had the advantage of faster convergence and fewer steady-state errors than the LMS algorithm, but the LMS algorithm had the advantage of less computational complexity.

Recursive estimation of the stochastic model based on the Kalman filter formulation

Based on the batch expectation–maximization (EM) and recursive least-squares algorithms, we develop a new recursive variance components estimation (Recursive-VCE) algorithm that applies a Kalman filter and validates it by a simulated kinematic precise point positioning (PPP) experiment and a PPP test on real-world data. The Recursive-VCE algorithm processes the observations in an epoch-by-epoch or a group-by-group manner. Once new observations are obtained, it updates the estimates of the variance components in a recursive way or on the fly. Therefore, it does not require significant computing resources to store sufficiently large training datasets. The resulting algorithm is simple and able to be easily adapted to determine time-varying behaviours and is shown to converge faster than the batch EM algorithm because the EM algorithm updates the parameters only once after dealing with all the data. Hence, it is a good complement to other batch VCE methods, and its application in real-time data processing is promising.

Prediction of Time Series Empowered with a Novel SREKRLS Algorithm

For the unforced dynamical non-linear statespace model, a new Q1 and efficient square root extended kernel recursive least square estimation algorithm is developed in this article. The proposed algorithm lends itself towards the parallel implementation as in the FPGA systems. With the help of an ortho-normal triangularization method, which relies on numerically stable givens rotation, matrix inversion causes a computational burden, is reduced. Matrix computation possesses many excellent numerical properties such as singularity, symmetry, skew symmetry, and triangularity is achieved by using this algorithm. The proposed method is validated for the prediction of stationary and non-stationary MackeyGlass Time Series, along with that a component in the x-direction of the Lorenz Times Series is also predicted to illustrate its usefulness. By the learning curves regarding mean square error (MSE) are witnessed for demonstration with prediction performance of the proposed algorithm from where it’s concluded that the proposed algorithm performs better than EKRLS. This new SREKRLS based design positively offers an innovative era towards non-linear systolic arrays, which is efficient in developing very-large-scale integration (VLSI) applications with non-linear input data. Multiple experiments are carried out to validate the reliability, effectiveness, and applicability of the proposed algorithm and with different noise levels compared to the Extended kernel recursive least-squares (EKRLS) algorithm.

Калибровка цифровых антенных решёток с помощью алгоритмов адаптивной обработки сигналов

A method of antenna array calibration, based on the using of the recursive least squares (RLS) adaptive filtering algorithms is discussed. The matrix inversion lemma, the QR-decomposition and the Householder transform based RLS algorithms with quadratic computational complexity can be used for the method implementation in the narrowband adaptive arrays. The proposed calibration method is used in the antenna arrays with digital beam forming. The method demonstrates the ability to estimate and compensate the channel gain errors that ensures the average deviation of the calibrated array radiation pattern shape relatively the radiation pattern shape of the ideal array about –20 dB.

Analog-Hall-Sensor-Based Position Detection Method With Temperature Compensation for Permanent-Magnet Linear Motor

Analog Hall sensors are utilized to detect the mover position of tubular permanent-magnet synchronous linear motors because of their low cost and small size. Typical Hall-sensor-based position estimation methods rely on additional displacement sensors, such as optical grating rulers, to identify the measurement model of the Hall sensors, but this reliance increases the difficulty of implementation of these methods. Moreover, most position detection methods disregard temperature change in the motors, leading to a position estimation error. Thus, a measurement model that considers temperature change is established in this study to overcome these problems. An algorithm consisting of particle swarm optimization and a successive solving algorithm (SSA) is used to identify the measurement model using only Hall sensors. A new SSA with temperature compensation is proposed for real-time position estimation. In this method, the forgetting factor recursive least-squares algorithm is modified to update the measurement model at the present temperature to avoid model mismatch. The proposed method is validated using a self-designed experimental platform. The experimental results show that the model’s identification accuracy is equivalent to that of the fast Fourier transform algorithm using an optical grating ruler. The position estimation error is less than $120~\mu \text{m}$ under different operating conditions.

A Self-Adaptive Multikernel Machine Based on Recursive Least-Squares Applied to Very Short-Term Wind Power Forecasting

Wind power has contributed significantly to the increase in electricity generation, but a decision-making tool capable of dealing with its variability and limited predictability is necessary. For this purpose, a novel self-adaptive approach for kernel recursive least-squares machines named multiple challengers is introduced in this work, which is successfully used to produce very short-term wind power forecasts at eight wind farms in Australia. The proposed method is based on a competitive tracking method, and the algorithm deals with some common difficulties of kernel methods, e.g., the increasing kernel matrix size associated with time and memory complexities and the overfitting problem. The proposed method always considers the new information received by the model, thus identifying changes in the time series, avoiding abrupt loss of information and maintaining a controlled number of examples since there is an adaptive selection of the active kernel. It works with the smallest dictionary possible, reducing the probability of overfitting. Five minute-ahead wind power forecasts are produced and evaluated in terms of point forecast skill scores and calibration. The results of the proposed method are compared with those provided by other kernel-based versions of the recursive least-squares algorithm, an online version of the extreme learning machine method, and the persistence time series model. An increase in the number of kernels used in the ensemble system can lead to better results when compared with those of single-kernel models.

Channel compensation multipath mitigation technique for Kalman‐Based Least Mean Square based on Kalman estimation

AbstractIn the Global Navigation Satellite System, the traditional adaptive equalization algorithm re‐captures and tracks poor performance under the multipath channel for multipath errors caused by occlusion or reflection of buildings. A channel compensation multipath mitigation technique based on Kalman‐Based Least Mean Square based (KBLMS) on Kalman estimation. First, in the tracking loop of the receiver, a KBLMS‐based delay estimation module is designed to compensate for multipath distortion on the received signal. The coefficients of the filter are adaptively adjusted using the feedback signal. Second, a line of sight best estimate block is designed to generate a control error signal in the feedback loop for adaptively updating the filter coefficients. Finally, the performance of the proposed algorithm is verified by analyzing the performance of KBLMS, Recursive Least Squares (RLSs) and Least Mean Squares (LMS) algorithms through measured data and experimental simulation. The results show that KBLMS can converge quickly in multipath channels compared with RLS and LMS, and the code tracking error and carrier tracking error are reduced by 0.1chip, 0.2 cycle and final residual error is reduced by 0.015 compared with the RLS algorithm, which is 0.035 lower than the LMS algorithm, which shows that the KBLMS algorithm can make more accurate estimation.

Discrete-wavelet-transform recursive inverse algorithm using second-order estimation of the autocorrelation matrix

The recursive-least-squares (RLS) algorithm was introduced as an alternative to LMS algorithm with enhanced performance. Computational complexity and instability in updating the autocolleltion matrix are some of the drawbacks of the RLS algorithm that were among the reasons for the intrduction of the second-order recursive inverse (RI) adaptive algorithm. The 2nd order RI adaptive algorithm suffered from low convergence rate in certain scenarios that required a relatively small initial step-size. In this paper, we propose a newsecond-order RI algorithm that projects the input signal to a new domain namely discrete-wavelet-transform (DWT) as pre step before performing the algorithm. This transformation overcomes the low convergence rate of the second-order RI algorithm by reducing the self-correlation of the input signal in the mentioned scenatios. Expeirments are conducted using the noise cancellation setting. The performance of the proposed algorithm is compared to those of the RI, original second-order RI and RLS algorithms in different Gaussian and impulsive noise environments. Simulations demonstrate the superiority of the proposed algorithm in terms of convergence rate comparedto those algorithms.

An Air-to-Fuel ratio estimation strategy for turbocharged spark-ignition engines based on sparse binary HEGO sensor measures and hybrid linear observers

An effective Air-to-Fuel Ratio (AFR) control is paramount to ensure a good combustion and high catalyst efficiency. This work addresses the problem of determining continuous-time estimates of AFR in turbocharged Spark Ignition (SI) engines on the basis of binary sparse measurements of the exhaust gas Oxygen. The latter are provided by a HEGO (Heated Exhaust Gas Oxygen) sensor installed at the catalytic converter input in place of a more expensive linear UEGO (Universal Exhaust Gas Oxygen) sensor, as nowadays common in commercial cars. The HEGO sensor outputs two voltage values only, corresponding respectively to low or high concentration of the residual Oxygen in the exhaust gas (on/off behavior). In view of this, it can be classified as a binary sensor generating irregular and sparse measurements in that the useful information is only present at the instants of the on/off and off/on transitions. An estimation scheme based on the use of a recursive least-squares algorithm has been designed by resorting to the theory of linear hybrid observers with quantized inputs. A detailed convergence analysis of the state reconstruction error is also provided. The proposed hybrid observer scheme is employed in a PI control-loop designed to maintain the AFR close to a desired value. The effectiveness of the proposed method is demonstrated by several numerical simulations based on both synthetic and real data.

Estimation and Monitoring of COVID-19's Transmissibility From Publicly Available Data

The COVID-19 pandemic began in the city of Wuhan, China, at the end of 2019 and quickly spread worldwide. The disease is caused by contact with the SARS-CoV-2 virus, which probably jumped from an animal host to humans. SARS-CoV-2 infects various tissues in the body, notably the lungs, and patients usually die from respiratory complications. Mathematical models of the disease have been instrumental to guide the implementation of mitigation strategies aimed at slowing the spread of the disease. One of the key parameters of mathematical models is the basic reproduction ratio R0, which measures the degree of infectivity of affected individuals. The goal of mitigation is to reduce R0 as close or below 1 as possible, as it means that new infections are in decline. In the present work, we use the recursive least-squares algorithm to establish the stochastic variability of a time-varying R0(t) from eight different countries: Argentina, Belgium, Brazil, Germany, Italy, New Zealand, Spain, and the United States of America. The proposed system can be implemented as an online tracking application providing information about the dynamics of the pandemic to health officials and the public at large.

High compression rate, based on the RLS adaptive algorithm in progressive image transmission

The purpose of this paper is to develop a novel method based on recursive least squares (RLS) adaptive algorithm for progressive image transmission (PIT). The image is divided into non-overlapping blocks. Having an agreed vector sequence between the transmitter and the receiver, each block is related to a regressive model. Meanwhile, at the transmitter the blocks are estimated using the RLS algorithm. The high correlation between error vectors, regarding to the RLS execution, causes a very high compression rate in their transmission. The error vectors at the receiver are used to run the RLS algorithm and to estimate the image in a same manner. The method is easy to implement with a low computational complexity and achieves high quality, compared to other well-known methods. In comparison with its counterparts, simulation results show how efficient the proposed method is.

An orientation estimation strategy for low cost IMU using a nonlinear Luenberger observer

In this work a new orientation estimation strategy for a non-accelerated platform is presented. The proposed strategy is based on a low-cost inertial measurement unit (IMU). Orientation angles are obtained using a nonlinear Luenberger observer, while the common offset issues on the magnetometer are calibrated by a recursive least-square algorithm. The strategy is implemented using a TM4C123GXL microcontroller and a MPU9150 sensor (nine-axis: gyro, accelerometer and magnetometer), which makes it a low-cost, compact system. Experimental results for the calibration and estimation procedures are shown to validate the proposal.

Online identification of mechanistic milling force models

Mechanistic models are commonly used to compute the machining forces in milling operations. The constant coefficients of mechanistic models, also known as Specific Force Coefficients (SFC), are directly linked to the mechanics of shearing, friction, and ploughing actions during chip formation. Because of this direct linkage, online monitoring of them can provide informative metrics for process monitoring. However, SFC are usually determined offline by using the machining forces measured under various cutting conditions (e.g. feedrate and cutting speed). In this paper, the performance of Recursive Least Squares (RLS) and Kalman Filter (KF) algorithms in online monitoring of SFC is studied.In the RLS method, instantaneous milling forces are expressed as linear functions of unknown SFC. The RLS algorithm is then used to recursively identify the SFC from the milling forces measured at discrete time steps during the process. In the Kalman Filter method, the SFC are modelled as constant stochastic processes, and the dynamics of their variation is described by a state-space model in which the SFC are the state variables and the milling forces are outputs. A Kalman Filter state observer is then designed to recursively estimate the SFC from the forces measured at discrete time steps. Numerical simulations and experimental studies are presented to verify the effectiveness of the proposed methods in identifying SFC in various cutting conditions.

A Robust Online Parameter Identification Method for Lithium-Ion Battery Model Under Asynchronous Sampling and Noise Interference

Online identification of a battery model can capture the parameter variations in real time, thus, providing an accurate model under various conditions, such as a wide temperature range, different aging degree in electric vehicle application. However, the measurement noise and sampling asynchronization of voltage and current are usually inevitable in the battery management system (BMS), which can lead to a large identification error. In this article, the mechanism of sampling asynchronization and measurement noise is analyzed first, and then the identification sensitivity analyses on noise and sampling asynchronization are carried out. Simulation results indicate that even a small fluctuation as small as 5 mV or the sampling latency between voltage and current within 10 mS can cause relatively large identification errors, especially for polarization parameters. In order to guarantee the online identification accuracy and robustness in the BMS application, an improved robust recursive least-squares (RLS) algorithm with adaptive outlier boundary is proposed to eliminate the input outlier caused by sampling latency and suppress the effect of measurement noise utilizing the bias compensation method. The experimental results demonstrate that the proposed method can provide sufficient accuracy and robustness under noise interference and sampling latency in BMS application.