Recursive Least-squares Algorithm Research Articles

Online Processing of Vehicular Data on the Edge Through an Unsupervised TinyML Regression Technique

The Internet of Things (IoT) has made it possible to include everyday objects in a connected network, allowing them to intelligently process data and respond to their environment. Thus, it is expected that those objects will gain an intelligent understanding of their environment and be able to process data more efficiently than before. Particularly, such edge computing paradigm has allowed the execution of inference methods on resource-constrained devices such as microcontrollers, significantly changing the way IoT applications have evolved in recent years. However, although this scenario has supported the development of Tiny Machine Learning (TinyML) approaches on such devices, there are still some challenges that require further investigation when optimizing data streaming on the edge. Therefore, this article proposes a new unsupervised TinyML regression technique based on the typicality and eccentricity of the samples to be processed. Moreover, the proposed technique also exploits a Recursive Least Squares (RLS) filter approach. Combining all these features, the proposed method uses similarities between samples to identify patterns when processing data streams, predicting outcomes based on these patterns. The results obtained through the extensive experimentation utilizing vehicular data streams were highly encouraging. The proposed algorithm was meticulously compared with the RLS algorithm and Convolutional Neural Networks (CNN). It exhibited significantly superior performance, with mean squared errors that were 4.68 and 12.02 times lower, respectively, compared to the aforementioned techniques.

Convex Regularized Recursive Minimum Error Entropy Algorithm

It is well known that the recursive least squares (RLS) algorithm is renowned for its rapid convergence and excellent tracking capability. However, its performance is significantly compromised when the system is sparse or when the input signals are contaminated by impulse noise. Therefore, in this paper, the minimum error entropy (MEE) criterion is introduced into the cost function of the RLS algorithm in this paper, with the aim of counteracting the interference from impulse noise. To address the sparse characteristics of the system, we employ a universally applicable convex function to regularize the cost function. The resulting new algorithm is named the convex regularization recursive minimum error entropy (CR-RMEE) algorithm. Simulation results indicate that the performance of the CR-RMEE algorithm surpasses that of other similar algorithms, and the new algorithm excels not only in scenarios with sparse systems but also demonstrates strong robustness against pulse noise.

System Identification of a Structure Equipped with a Cable-Bracing Inerter System Using Adaptive Extended Kalman Filter

An innovative cable-bracing inerter system (CBIS) has been proposed and shown to be effective in mitigating the structural response under dynamic excitation. The CBIS comprises an inerter element, an eddy current damping element, and a pair of tension-only cables that can transfer the story drift to rotating flywheels. To further investigate the characteristics of the CBIS, a system identification approach based on an adaptive extended Kalman filter (AEKF) and a recursive least-squares (RLS) algorithm is proposed. Depending on the CBIS model’s availability, the proposed approach uses two strategies: the AEKF identifies the parameters of the structure and the CBIS when the model is specific; alternatively, when the model is unspecific, the KF combined with an RLS algorithm identifies the restoring force generated by the CBIS as an unknown fictitious input. In addition, the AEKF incorporates a time-variant fading factor to track the target adaptively. The proposed approach is validated through free vibration and shaking table tests, demonstrating the accuracy in identifying structural parameters and restoring force provided by the CBIS. The identification process involves two stages: initially, the AEKF identifies the parameters of the bare structure without the CBIS, followed by a dual strategy using either AEKF or KF-RLS for identifying the parameters of the CBIS or its restoring force, respectively. The findings also verify the feasibility and validity of the mechanical model and operating principle of the CBIS, thereby contributing to the advancement and application of the CBIS in future studies.

Simulations of real-time system identification for superconducting cavities with a recursive least-squares algorithm

Simulations of real-time system identification for superconducting cavities with a recursive least-squares algorithm

Identification of Wiener Systems with Recursive Gauss-Seidel Algorithm

The Recursive Gauss-Seidel (RGS) algorithm is presented that is implemented in a one-step Gauss-Seidel iteration for the identification of Wiener output error systems. The RGS algorithm has lower processing intensity than the popular Recursive Least Squares (RLS) algorithm due to its implementation using one-step Gauss-Seidel iteration in a sampling interval. The noise-free output samples in the data vector used for implementation of the RGS algorithm are estimated using an auxiliary model. Also, a stochastic convergence analysis is presented, and it is shown that the presented auxiliary model-based RGS algorithm gives unbiased parameter estimates even if the measurement noise is coloured. Finally, the effectiveness of the RGS algorithm is verified and compared with the equivalent RLS algorithm by computer simulations.

Efficient enhanced Recursive Least Square algorithm adaptive filtering scheme for artifacts removal in ECG signals

This paper proposes an improved Recursive Least Square (RLS) algorithm to address the limitations of the conventional RLS algorithm, which suffers from poor performance when dealing with non-stationary noise. The proposed algorithm, along with various algorithms, have been designed using Xilinx System Generator (XSG). These algorithms have been tested to denoise ECG signals that have been corrupted with four types of real noise obtained from the MIT-BIH dataset. The improved algorithm utilizes a systolic architecture that enables faster processing and better noise reduction capabilities. The experimental results demonstrate that the proposed algorithm with and without systolic architecture outperforms the conventional RLS algorithm and other state-of-the-art algorithms in terms of signal-to-noise ratio (SNR), mean squared error (MSE), and convergence speed.

Variable step-size convex regularized PRLS algorithms

The proportionate updating (PU) and zero-attracting (ZA) mechanisms have been applied independently in the development of sparsity-aware recursive least squares (RLS) algorithms. Recently, we propose an enhanced l1-proportionate RLS (l1-PRLS) algorithm by combining the PU and ZA mechanisms. The l1-PRLS employs a fixed step size which trades off the transient (initial convergence) and steady-state performance. In this letter, the l1-PRLS is improved in two aspects: first, we replace the l1 norm penalty by a general convex regularization (CR) function to have the CR-PRLS algorithm; second, we further introduce the variable step-size (VSS) technique to the CR-PRLS, leading to the VSS-CR-PRLS algorithm. Theoretical and numerical results were provided to corroborate the superiority of the improved algorithm.

Research on the Prediction Model of Engine Output Torque and Real-Time Estimation of the Road Rolling Resistance Coefficient in Tracked Vehicles.

Road parameter identification is of great significance for the active safety control of tracked vehicles and the improvement of vehicle driving safety. In this study, a method for establishing a prediction model of the engine output torques in tracked vehicles based on vehicle driving data was proposed, and the road rolling resistance coefficient f was further estimated using the model. First, the driving data from the tracked vehicle were collected and then screened by setting the driving conditions of the tracked vehicle. Then, the mapping relationship between the engine torque Te, the engine speed ne, and the accelerator pedal position β was obtained by a genetic algorithm-backpropagation (GA-BP) neural network algorithm, and an engine output torque prediction model was established. Finally, based on the vehicle longitudinal dynamics model, the recursive least squares (RLS) algorithm was used to estimate the f. The experimental results showed that when the driving state of the tracked vehicle satisfied the set driving conditions, the engine output torque prediction model could predict the engine output torque T^e in real time based on the changes in the ne and β, and then the RLS algorithm was used to estimate the road rolling resistance coefficient f^. The average coefficient of determination R of the T^e was 0.91, and the estimation accuracy of the f^ was 98.421%. This method could adequately meet the requirements for engine output torque prediction and real-time estimation of the road rolling resistance coefficient during tracked vehicle driving.

Turbo-shaft engine adaptive neural network control based on nonlinear state space equation

Intelligent Adaptive Control (AC) has remarkable advantages in the control system design of aero-engine which has strong nonlinearity and uncertainty. Inspired by the Nonlinear Autoregressive Moving Average (NARMA)-L2 adaptive control, a novel Nonlinear State Space Equation (NSSE) based Adaptive neural network Control (NSSE-AC) method is proposed for the turbo-shaft engine control system design. The proposed NSSE model is derived from a special neural network with an extra layer, and the rotor speed of the gas turbine is taken as the main state variable which makes the NSSE model be able to capture the system dynamic better than the NARMA-L2 model. A hybrid Recursive Least-Square and Levenberg-Marquardt (RLS-LM) algorithm is advanced to perform the online learning of the neural network, which further enhances both the accuracy of the NSSE model and the performance of the adaptive controller. The feedback correction is also utilized in the NSSE-AC system to eliminate the steady-state tracking error. Simulation results show that, compared with the NARMA-L2 model, the NSSE model of the turbo-shaft engine is more accurate. The maximum modeling error is decreased from 5.92% to 0.97% when the LM algorithm is introduced to optimize the neural network parameters. The NSSE-AC method can not only achieve a better main control loop performance than the traditional controller but also limit all the constraint parameters efficiently with quick and accurate switching responses even if component degradation exists. Thus, the effectiveness of the NSSE-AC method is validated.

Recursive least-squares algorithm based on a third-order tensor decomposition for low-rank system identification

A recently developed third-order tensor (TOT) decomposition-based method has proved to be working very well in linear system identification problems that target the estimation of long length impulse responses. This technique exploits the decomposition of the impulse response based on the nearest Kronecker product and low-rank approximations, but it also owns a specific advantage. Its main feature is related to the way of handling the rank of a third-order tensor, which is controlled in terms of the rank of a matrix and, as a result, is limited to small values. In this paper, we further develop a recursive least-squares (RLS) algorithm based on the TOT decomposition technique. The resulting solution combines the estimates provided by three shorter adaptive filters, thus gaining in terms of both performance and complexity. The three component filters are updated in parallel, which represents an important practical advantage, in terms of the modularity of implementation. Also, as compared to the conventional RLS algorithm, a significant reduction of the computational complexity can be achieved for the common setup of the decomposition parameters. Besides, since the length of the adaptive filter highly influences the main performance criteria (e.g., convergence, tracking, and accuracy), the proposed RLS version based on the TOT decomposition outperforms both the conventional algorithm and a previously developed RLS-based solution that involves a second-order decomposition. These performance gains mainly reflect in a faster tracking capability and a better accuracy of the estimate (i.e., lower misalignment). Simulations performed in the framework of echo cancellation support the performance features of the proposed tensorial RLS-based algorithm.

Battery-SOC Estimation for Hybrid-Power UAVs Using Fast-OCV Curve with Unscented Kalman Filters.

Unmanned aerial vehicles (UAVs) have drawin increasing attention in recent years, and they are widely applied. Nevertheless, they are generally limited by poor flight endurance because of the limited energy density of their batteries. A robust power supply is indispensable for advanced UAVs; thus hybrid power might be a promising solution. State of charge (SOC) estimation is essential for the power systems of UAVs. The limitations of accurate SOC estimation can be partly ascribed to the inaccuracy of open circuit voltage (OCV), which is obtained through specific forms of identification. Considering the actual operation of a battery under hybrid conditions, this paper proposes a novel method, "fast OCV", for obtaining the OCVs of batteries. It is proven that fast OCV offers great advantages, related to its simplicity, duration and cost, over traditional ways of obtaining OCV. Moreover, fast-OCV also shows better accuracy in SOC estimation than traditional OCV. Furthermore, this paper also proposes a new method, "batch mode", for talking-data sampling for battery-parameter identification with the limited-memory recursive least-square algorithm. Compared with traditional the "single mode", it presents good de-noising effect by making use of all the sampled battery's terminal current and voltage data.

Equalizer of Amplitude-Frequency Response of Acoustic Channels and its Adaptive Filter Based on Recursive Least Squares Algorithm with Diagonalized Correlation Matrix

The paper presents an equalizer of the amplitude-frequency response of the acoustic wave propagation channels in the rooms. The considered equalizer uses the so-called modified architecture based on the adaptive filter with the input signal filtering, which weights are calculated using the Recursive Least Squares (RLS) algorithm. The equalizer of the acoustic channels amplitude-frequency response usually requires the usage of the adaptive filters with the large number of weighs that means the large computational complexity of the adaptive filter. The adaptive filters based on the fast RLS algorithms are efficient in the terms of their complexity but often unstable if the number of adaptive filter weights is large. Therefore, the algorithms are poorly suited for usage in the amplitude-frequency response equalizers of the acoustic channels. Adaptive filters based on the RLS algorithm, which use the Matrix Inversion Lemma (MIL), are stable, but computationally complex. The equalizer presented in this paper uses a simplified adaptive filter based on the MIL RLS algorithm with a diagonalized correlation matrix of the filter input signal. In addition, in the filter, each of the square submatrices on the simplified correlation matrix diagonal is inverted using a computationally efficient version of the MIL that takes into account the symmetrical structure of the mentioned submatrices relatively their main diagonals. Computer simulation confirms the performance of the proposed equalizer and demonstrates its efficiency comparing to the unsimplified equalizer. The demo includes the graphs of the equalizer steady-state amplitude-frequency response, the acoustic channel amplitude-frequency responses before and after the equalization, and the power spectral density plots of the equalizer input signal and the signals passed through the equalizer and the acoustic medium. The equalizer can be used in the equipment for the high-quality voice and music reproduction in the conditions of the limited computing resources for the implementation of the equalizer.

A scalable implementation of the recursive least-squares algorithm for training spiking neural networks.

Training spiking recurrent neural networks on neuronal recordings or behavioral tasks has become a popular way to study computations performed by the nervous system. As the size and complexity of neural recordings increase, there is a need for efficient algorithms that can train models in a short period of time using minimal resources. We present optimized CPU and GPU implementations of the recursive least-squares algorithm in spiking neural networks. The GPU implementation can train networks of one million neurons, with 100 million plastic synapses and a billion static synapses, about 1,000 times faster than an unoptimized reference CPU implementation. We demonstrate the code's utility by training a network, in less than an hour, to reproduce the activity of > 66, 000 recorded neurons of a mouse performing a decision-making task. The fast implementation enables a more interactive in-silico study of the dynamics and connectivity underlying multi-area computations. It also admits the possibility to train models as in-vivo experiments are being conducted, thus closing the loop between modeling and experiments.

Fault-Tolerant Self-Tuning Control for Three-Phase Three-Level T-Type Rectifier

This paper proposes a sensorless self-tuning control algorithm for a three-phase T-type rectifier. The proposed sensorless algorithm consists of three layers including an observer, which estimates grid voltages or currents based on the Kalman filter algorithm (KFA), an online estimator, which estimates the state-space matrix parameters, and a linear quadratic tracker (LQT), which is the main controller in this study. On the other hand, the performance of the LQT depends on the gains that need to be adjusted according to the system parameters, which is a challenge. To overcome this problem, a self-tuning method based on recursive least square algorithm (RLSA) is also proposed to tune the gains of LQT online. An experimental prototype is established, and the results, including the steady-state and dynamic performances, are carried out and presented to validate the effectiveness of the proposed controller.

Robustness Enhanced Predictive Control for Dual Three Phase PMSM with Duty Ratio Modulation and Parameter-Free Solution

The classical model predictive control not only relies on the motor model but also is sensitive to the system parameters. To improve the robustness of the model predictive control system, this article proposes a parameter-free predictive control for a dual three-phase PMSM with duty ratio modulation. First, the free-model theory is employed to reconstruct the prediction model of the motor, of which no parameters are needed. So, the problem of parameter mismatch between the motor and the inverter can be handled, and enhance the robustness of the system as well. Moreover, the ultra-local model is optimized with the forgetting factor recursive least-squares algorithm to improve the identification accuracy of the estimated value. Finally, the deadbeat duty ratio modulation solution is introduced to further improve operational performance while ensuring the robustness of the system. Simulation and experiment verify the feasibility and effectiveness of the proposed algorithm.

Chatter detection for micro milling considering environment noises without the requirement of dominant frequency

Existing chatter detection methods were mainly developed for conventional milling processes, and did not consider the influences of environment noises. Thus, they are not suitable for micro milling, in which the widely used small cutting parameters easily lead to the occurrence of non-negligible noises. This article proposes a new variable forgetting factor recursive least-squares (VFF-RLS) algorithm to filter out the chatter-independent component, which is composed of environment noises and periodic components, for the signals sampled from the micro milling process. It is theoretically found that with the aid of VFF-RLS algorithm, there is almost nothing in the filtered signal for stable cuts, while chatter components will be dominant in the filtered signal for unstable cuts. This important finding confirms that some features can be easily extracted from the filtered signal to directly reflect the stable or unstable machining states without the requirement to measure the system’s dominant frequency, which is necessary for some existing methods to obtain the frequency band where chatter vibrations appear. Subsequently, a streaming feature selection considering feature interaction (SFS-FI) method is established to select optimal features and remove irrelevant and redundant features. Then, a chatter detection model is developed based on the selected features. Finally, a series of micro milling tests with various axial depths of cut and spindle speeds prove that the proposed model can accurately identify the cutting state, even for slight chatter and stable cuts with small little depths of cut, which are more easily disturbed by environment noises. Results confirm that the detection accuracy is larger than 99 %, and this fact means that the proposed method is effective and practical for detecting chatter occurring in micro milling processes.

Robust Adaptive Control of Hypersonic Flight Vehicle With Aero-Servo-Elastic Effect

This paper investigates the notch filter based robust adaptive control for the longitudinal dynamics of flexible hyper- sonic flight vehicle using finite-time design and online estimation. Considering the aero-servo-elastic effect caused by the elastic vibration at sensor position, the recursive least-square algorithm is employed to track the natural frequencies of the flexible modes, which then be used as the center frequencies for the cascade-type notch filter. Based on the filtered signal, the robust adaptive control is studied where the neural network is used to handle aerodynamics uncertainty and the disturbance observer is employed to compensate flexible coupling. The prediction error is constructed for the update laws while the nonlinear function is further developed to achieve the finite-time tracking. The uniformly ultimately bounded stability is guaranteed under the proposed approach. The effectiveness of the design is illustrated through the simulation study.

An Improvement of DV-Hop Localization Algorithm Based on Cyclotomic Method in Wireless Sensor Networks

Location information is one of the crucial and essential elements for monitoring data in wireless sensor networks. The distance vector-hop (DV-Hop) localization algorithm is of practical importance in improving its localization performance. To achieve global optimization, a DV-Hop algorithm based on the cyclotomic method and weighted normalization, also known as CMWN-DV-Hop, is nominated in this paper. Therefore, the segmentation and weighting factors are introduced and normalized. The weighted recursive least-squares (WRLS) algorithm is chosen to compute the coordinates of the unknown nodes. The effects of various factors on this algorithm are tested, including the number of nodes, the anchor node ratio, and the communication radius. The simulation results show that the proposed algorithm has a super performance in reducing the localization error.

Cooperative Filtering and Parameter Identification for Advection–Diffusion Processes Using a Mobile Sensor Network

This article presents an online parameter identification scheme for advection–diffusion processes using data collected by a mobile sensor network. The advection–diffusion equation is incorporated into the information dynamics associated with the trajectories of the mobile sensors. A constrained cooperative Kalman filter is developed to provide estimates of the field values and gradients along the trajectories of the mobile sensors so that the temporal variations in the field values can be estimated. This leads to a co-design scheme for state estimation and parameter identification for advection–diffusion processes that is different from comparable schemes using sensors installed at fixed spatial locations. Using state estimates from the constrained cooperative Kalman filter, a recursive least-square (RLS) algorithm is designed to estimate unknown model parameters of the advection–diffusion processes. Theoretical justifications are provided for the convergence of the proposed cooperative Kalman filter by deriving a set of sufficient conditions regarding the formation shape and the motion of the mobile sensor network. Simulation and experimental results show satisfactory performance and demonstrate the robustness of the algorithm under realistic uncertainties and disturbances.

Path Tracking Control of Intelligent Vehicles Considering Multi-Nonlinear Characteristics for Dual-Motor Autonomous Steering System

In the path tracking control of intelligent vehicles, the traditional linear control method is prone to high tracking errors for uncertain parameters of the steering transmission system and road conditions. Therefore, considering the mechanical friction in the dual-motor autonomous steering system and the nonlinearity of tires, this paper proposes a path tracking control strategy of intelligent vehicles for the dual-motor autonomous steering system that considers nonlinear characteristics. First, a dual-motor autonomous steering system considering mechanical friction and the variation of tire cornering stiffness under different tire–road friction coefficients was established based on the structure of an autonomous steering system. Second, a tire–road friction coefficient estimator was designed based on a PSO-LSTM neural network. The tire cornering stiffness under different tire–road friction coefficients was estimated through the recursive least-square algorithm. Then, the control strategy of the dual-motor autonomous steering system was designed by combining the LQR path tracking controller with the adaptive sliding mode control strategy based on field-oriented control. Here, mechanical friction and the variation of tire cornering stiffness were considered. Finally, simulation and HiL tests validated the method proposed in this paper. The results show that the proposed control strategy significantly improves the tracking accuracy and performance of the dual-motor autonomous steering system for intelligent vehicles.