Basis Functions Research Articles

Neural Network-Based Fixed-Time Tracking and Containment Control of Second-Order Heterogeneous Nonlinear Multiagent Systems.

This study concentrates on the fixed-time tracking consensus and containment control of second-order heterogeneous nonlinear multiagent systems (MASs) with and without measurable velocity under directed topology. By defining a time-varying scaling function and approximating the unknown nonlinear dynamics with radial basis function neural networks (RBFNNs), a novel distributed protocol for solving the fixed-time tracking consensus and containment control problems of second-order heterogeneous nonlinear MASs with full states available is proposed based on a nonsingular sliding-mode control method constructed by designing a prescribed-time convergent sliding surface. For the scenario of immeasurable velocity, a fixed-time convergent states' observer is designed to reveal the velocity information when the unknown linearity is bounded. Subsequently, a distributed fixed-time consensus protocol based on observed velocity information is proposed for the extended results. Ultimately, the acquired results are verified by three simulation examples.

Neural Network-Based Adaptive Tracking Control for Denitrification and Aeration Processes With Time Delays.

Wastewater treatment process (WWTP), consisting of a class of physical, chemical, and biological phenomena, is an important means to reduce environmental pollution and improve recycling efficiency of water resources. Considering characteristics of the complexities, uncertainties, nonlinearities, and multitime delays in WWTPs, an adaptive neural controller is presented to achieve the satisfying control performance for WWTPs. With the advantages of radial basis function neural networks (RBF NNs), the unknown dynamics in WWTPs are identified. Based on the mechanistic analysis, the time-varying delayed models of the denitrification and aeration processes are established. Based on the established delayed models, the Lyapunov-Krasovskii functional (LKF) is used to compensate for the time-varying delays caused by the push-flow and recycle flow phenomenon. The barrier Lyapunov function (BLF) is used to ensure that the dissolved oxygen (DO) and nitrate concentrations are always kept within the specified ranges though the time-varying delays and disturbances exist. Using Lyapunov theorem, the stability of the closed-loop system is proven. Finally, the proposed control method is carried out on the benchmark simulation model 1 (BSM1) to verify the effectiveness and practicability.

Adaptive Decentralized Control for Constrained Strong Interconnected Nonlinear Systems and Its Application to Inverted Pendulum.

This work is dedicated to adaptive decentralized tracking control for a class of strong interconnected nonlinear systems with asymmetric constraints. Currently, there are few related studies on unknown strongly interconnected nonlinear systems with asymmetric time-varying constraints. To deal with the assumptions of the interconnection terms in the design process, which include upper functions and structural restrictions, the properties of Gaussian function in radial basis function (RBF) neural networks are applied to overcome this difficulty. By constructing the nonlinear state-dependent function (NSDF) and using a new coordinate transformation, the conservative step that the original state constraint converts into a new boundary of the tracking error is removed. Meanwhile, the virtual controller's feasibility condition is removed. It is proven that all the signals are bounded, especially the original tracking error and the new tracking error, which are both bounded. In the end, simulation studies are carried out to verify the effectiveness and benefits of the proposed control scheme.

Accuracy of using the axial length of the fellow eye for IOL calculation in retinal detachment eyes undergoing silicone oil removal

PurposeEvaluate whether the axial length of the fellow eye can be used to calculate the intraocular lens (IOL) in eyes with retinal detachment.DesignRetrospective, consecutive case series.MethodsOur study was conducted at...

Geodesic Basis Function Neural Network.

In the learning of existing radial basis function neural networks-based methods, it is difficult to propagate errors back. This leads to an inconsistency between the learning and recognition task. This article proposes a geodesic basis function neural network with subclass extension learning (GBFNN-ScE). The geodesic basis function (GBF), which is defined here for the first time, uses the geodetic distance in the manifold as a measure to obtain the response of the sample with respect to the local center. To learn network parameters by back-propagating errors for the purpose of correct classification, a specific GBF based on a pruned gamma encoding cosine function is constructed. This function has a concise and explicit expression on the hyperspherical manifold, which is conducive to the realization of error back propagation. In the preprocessing layer, a sample unitization method with no loss of information, nonnegative unit hyperspherical crown (NUHC) mapping, is proposed. The sample can be mapped to the support set of the GBF. To alleviate the problem that one-hot encoding is not effective enough in the differential expression of data labels within a class, a subclass extension (ScE) learning strategy is proposed. The ScE learning strategy can help the learned network be more robust. For the working of GBFNN-ScE, the original sample is projected onto the support set of GBF through the NUHC mapping. Then the mapped samples are sent to the nonlinear computation units composed of GBFs in the hidden layer. Finally, the response obtained in the hidden layer is weighted by the learned weight to obtain the network output. This article theoretically proves that the separability of the data with ScE learning is stronger. The experimental results show that the proposed GBFNN-ScE has a better performance in recognition tasks than existing methods. The ablation experiments show that the ideas of the GBFNN-ScE contribute to the algorithm performance.

Event-Triggered Adaptive Containment Control for Heterogeneous Stochastic Nonlinear Multiagent Systems.

This article investigates the event-triggered adaptive containment control problem for a class of stochastic nonlinear multiagent systems with unmeasurable states. A stochastic system with unknown heterogeneous dynamics is established to describe the agents in a random vibration environment. Besides, the uncertain nonlinear dynamics are approximated by radial basis function neural networks (NNs), and the unmeasured states are estimated by constructing the NN-based observer. In addition, the switching-threshold-based event-triggered control method is adopted with the hope of reducing communication consumption and balancing system performance and network constraints. Moreover, we develop the novel distributed containment controller by utilizing the adaptive backstepping control strategy and the dynamic surface control (DSC) approach such that the output of each follower converges to the convex hull spanned by multiple leaders, and all signals of the closed-loop system are cooperatively semi-globally uniformly ultimately bounded in mean square. Finally, we verify the efficiency of the proposed controller by the simulation examples.

Command Filtered Neuroadaptive Fault-Tolerant Control for Nonlinear Systems With Input Saturation and Unknown Control Direction.

This article studies the tracking control of a class of nonlinear systems with input saturation, subject to nonaffine faults and unknown control direction. A fault-tolerant command filtered control (CFC) method based on adaptive neural networks (NNs) is proposed for this kind of nonlinear system. First, the combination of CFC and error compensation overcomes the "explosion of complexity" issue and alleviates the impact of filter errors. Then, a set of radial basis function NNs is constructed to approximate the unknown nonlinear items containing the nonaffine fault function. Additionally, the issue of unknown control direction in the system is effectively resolved by using Nussbaum gain technology. It is proven that the designed controller can ensure that all signals in the closed-loop system are bounded and convergent, and the upper bound of the absolute value of system tracking error is given. Finally, three comparative simulation results are illustrated to show the effectiveness of the proposed method.

The error term of the sum of digital sum functions in arbitrary bases

Let $k$ be a non-negative integer and $q > 1$ be a positive integer. Let $s_q(k)$ be the sum of digits of $k$ written in base $q.$ In 1940, Bush proved that $A_q(x)=\sum_{k \leq x} s_q (k)$ is asymptotic to $\frac{q-1}{2}x \log_q x.$ In 1968, Trollope proved an explicit formula for the error term of $A_2(n-1),$ labeled by $-E_2(n),$ where $n$ is a positive integer. In 1975, Delange extended Trollope's result to an arbitrary base $q$ by another method and labeled the error term $nF_q(\log_q n).$ When $q=2,$ the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope's method to extend $-E_2(n)$ to $-E_q(n)$ with a general base $q,$ and we also proved directly that $nF_q(\log_q n)$ and $-E_q(n)$ are equal for any $q.$

Analyzing physical activity impact based on ubiquitous nonlinear dynamics and electroencephalography data.

Understanding complex systems is made easier with the tools provided by the theory of nonlinear dynamic systems. It provides novel ideas, algorithms, and techniques for signal processing, analysis, and classification. Presently, these ideas are being applied to the investigation of how physiological signals evolve. The study applies nonlinear dynamics theory to electroencephalogram (EEG) signals to better comprehend the range of alcoholic mental states. One of the main contributions of this paper is an algorithm for automatically distinguishing between sober and drunken EEG signals based on their salient features. The study utilized various entropy-based features, including ApEn, SampEn, Shannon and Renyi entropies, PE, TS, FE, WE, and KSE, to extract information from EEG signals. To identify the most relevant features, the study employed ranking methods like T-test, Wilcoxon, and Bhattacharyya, and trained SVM classifiers with the selected features. The Bhattacharyya ranking method was found to be the most effective in achieving high classification accuracy, sensitivity, and specificity. Classification accuracy of 95.89%, the sensitivity of 94.43%, and specificity of 96.67% are achieved by the SVM classifier with radial basis function (RBF) for polynomial Kernel using the Bhattacharyya ranking method. From the result, it is clear that the model serves as a cost-effective and accurate decision-support tool for doctors in diagnosing alcoholism and for rehabilitation centres to monitor the effectiveness of interventions aimed at mitigating or reversing brain damage caused by alcoholism.

Knowledge-based Data Processing for Multilingual Natural Language Analysis

Natural Language Processing (NLP) aids the empowerment of intelligent machines by enhancing human language understanding for linguistic-based human-computer communication. Recent developments in processing power, as well as the availability of large volumes of linguistic data, have enhanced the demand for data-driven methods for automatic semantic analysis. This paper proposes multilingual data processing using feature extraction with classification using deep learning architectures. Here, the input text data has been collected based on various languages and processed to remove missing values and null values. The processed data has been extracted using Histogram Equalization based Global Local Entropy (HEGLE) and classified using Kernel-based Radial basis Function (Ker_Rad_BF). These architectures could be utilized to process natural language. We present solutions to the multilingual sentiment analysis issue in this research article by implementing algorithms, and we compare precision factors to discover the optimum option for multilingual sentiment analysis. For the HASOC dataset, the proposed HEGLE_ Ker_Rad_BF achieved an accuracy of 98%, a precision of 97%, a recall of 90.5%, an f-1 score of 85%, RMSE of 55.6%, and a loss curve analysis attained 44%. For the TRAC dataset, the accuracy of 98%, the precision attained is 97%, the Recall is 91%, the F-1 score is 87%, and the RMSE of the proposed neural network is 55%.

Time Efficient Ultrasound Localization Microscopy Based on A Novel Radial Basis Function 2D Interpolation.

Ultrasound localization microscopy (ULM) allows for the generation of super-resolved (SR) images of the vasculature by precisely localizing intravenously injected microbubbles. Although SR images may be useful for diagnosing and treating patients, their use in the clinical context is limited by the need for prolonged acquisition times and high frame rates. The primary goal of our study is to relax the requirement of high frame rates to obtain SR images. To this end, we propose a new time-efficient ULM (TEULM) pipeline built on a cutting-edge interpolation method. More specifically, we suggest employing Radial Basis Functions (RBFs) as interpolators to estimate the missing values in the 2-dimensional (2D) spatio-temporal structures. To evaluate this strategy, we first mimic the data acquisition at a reduced frame rate by applying a down-sampling (DS = 2, 4, 8, and 10) factor to high frame rate ULM data. Then, we up-sample the data to the original frame rate using the suggested interpolation to reconstruct the missing frames. Finally, using both the original high frame rate data and the interpolated one, we reconstruct SR images using the ULM framework steps. We evaluate the proposed TEULM using four in vivo datasets, a Rat brain (dataset A), a Rat kidney (dataset B), a Rat tumor (dataset C) and a Rat brain bolus (dataset D), interpolating at the in-phase and quadrature (IQ) level. Results demonstrate the effectiveness of TEULM in recovering vascular structures, even at a DS rate of 10 (corresponding to a frame rate of sub-100Hz). In conclusion, the proposed technique is successful in reconstructing accurate SR images while requiring frame rates of one order of magnitude lower than standard ULM.

Codesign of FDI Attacks Detection, Isolation, and Mitigation for Complex Microgrid Systems: An HBF-NN-Based Approach.

The primary purpose of this article is to design an intelligent false data injection (FDI) attacks detection, isolation, and mitigation scheme for a class of complex microgrid systems with electric vehicles (EVs). First, a networked microgrid with an EV model is well established, which takes load disturbance, wind generation fluctuation, and FDI attacks into account so as to truly reflect the operation process of the complex system. Then, an intelligent hyper basis function neural network (HBF-NN) observer is designed to accurately estimate the state of the microgrids, learn, and reconstruct the possible attack signal online. Subsequently, a novel HBF-NN-based H∞ controller is skillfully designed to mitigate the negative impact of FDI attacks online, so as to ensure the normal operation of the complex systems in an unreliable network environment. Finally, a two-stage integrated intelligent detection and maintenance algorithm is summarized and one simulation is presented to provide tangible evidence of the feasibility and superiority of the proposed FDI attacks detection, isolation, and mitigation methodology.

A Hybrid Controller for Musculoskeletal Robots Targeting Lifting Tasks in Industrial Metaverse.

In manufacturing, musculoskeletal robots have gained more attention with the potential advantages of flexibility, robustness, and adaptability over conventional serial-link rigid robots. Focusing on the fundamental lifting tasks, a hybrid controller is proposed to overcome control challenges of such robots for widely applications in industry. The metaverse technology offers an available simulated-reality-based platform to verify the proposed method. The hybrid controller contains two main parts. A muscle-synergy-based radial basis function (RBF) network is proposed as the feedforward controller, which is able to characterize the phasic and the tonic muscle synergies simultaneously. The adaptive dynamic programming (ADP) is applied as the feedback controller to address the optimal control problem. The actor-critic structure is applied in the ADP-based controller, where the critic network is trained to approximate the optimal performance index and the actor network is trained to compute the optimal muscle excitations. Furthermore, the convergence and stability of the ADP algorithm are also analyzed. Finally, experiments have been designed to verify the effectiveness of this hybrid controller on an upper limb musculoskeletal system, and the comparisons with other controllers are also illustrated. The results show that the proposed controller can obtain a satisfactory performance for lifting tasks.

Prediction-Based Power Consumption Monitoring of Industrial Equipment Using Interpretable Data-Driven Models

Efficient energy optimization and scheduling in industrial factories depend on accurate, reasonable, and real-time monitoring of equipment power consumption. However, the power prediction of industrial equipment requires a large number of process data, which could be inevitably contaminated by some imperfect data, due to the harsh environments. Monitoring or predicting equipment power consumption is usually not feasible using data-driven black-box models with imperfect data. This paper proposes a prediction-based approach for power consumption monitoring using an interpretable data-driven model. First, a data preprocessing method is used to remove outliers and fill in missing values. Then, a Volterra polynomial basis function (VPBF) model is built to predict equipment power consumption. This model decomposes power values into a series of basis functions consisting of input parameters. Moreover, to compensate for data dropouts during the power consumption monitoring process, a networked predictive monitoring system is also proposed. Finally, this paper presents two case studies based on actual production equipment in an industrial manufacturing factory. The results demonstrate that the proposed approach can achieve satisfactory monitoring accuracy and adaptation ability. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the problem of power consumption monitoring of industrial equipment in harsh production environments. A conventional solution is to predict the power consumption using data-driven black-box models, with limited feasibility and interpretability. This paper proposes a new power consumption monitoring approach, utilizing an interpretable data-driven model and a networked predictive method. This approach accurately reveals the transparent relationships between the output and input parameters. The power prediction result is consequently interpretable and more reasonable. Meanwhile, this approach actively compensates for data dropouts in the network, which can help operators real-time grasp the power consumption of equipment. Furthermore, this approach can be integrated into energy optimization systems as a basis of optimization and decision-making. Practical applications in an aluminum manufacturing company located in Guangdong Province demonstrate that this approach is feasible and applicable. Future research is to expand this approach further for energy optimization and scheduling.

Adaptive Neural Network-Based Event-Triggered SOC Observer With Application to a Stochastic Battery Model.

Accurate state of charge (SOC) is crucial to achieving safe, reliable, and efficient use of batteries. This article proposes an adaptive neural network (NN)-based event-triggered observer to estimate SOC. First, a stochastic battery equivalent circuit model (ECM) is established, where an adaptive NN is employed to approximate the unknown nonlinear part. The learning process of network weight is conducted online to observe the variations of model parameters and avoid time-consuming processes for parameter extraction. Besides, for the purpose of saving computational cost, an event-triggered mechanism (ETM) is employed in the weight updating law, which means the weights only update when it is necessary. Then, an adaptive radial basis function (RBF) NN-based SOC observer is designed, and its stability is proven by the Lyapunov theory. Moreover, the strictly positive lower bound of interevent time is derived, and undesirable Zeno behavior can be excluded. Finally, the accuracy and robustness of the proposed observer are evaluated by experiments and simulations. Results show that the proposed method can estimate SOC accurately in the presence of initial deviation and sensor noises.

Distributed Adaptive Forwarding Finite-Time Output Consensus of High-Order Multiagent Systems via Immersion and Invariance-Based Approximator.

A finite-time output consensus control problem is investigated in this article for an uncertain nonlinear high-order multiagent systems (MASs). For this class of MASs, the order of individual follower is reduced gradually by implementing the immersion and invariance (I&I) control theory repeatedly, and a requirement of solving partial differential equations (PDEs) in I&I control theory is obviated. Furthermore, an I&I-based radial basis function neural network (RBFNN) approximator is developed, where an extra cross term is added in the approximation mechanism, and the form of an update law for weights is transformed into a proportional and integral one. This I&I-based RBFNN approximator does not rely on a cancellation of the perturbation term, and these uncertainties are reconstructed by the I&I manifold adaptively, which is for improvement of approximation behaviors of traditional RBFNNs. On this basis, a distributed adaptive forwarding finite-time output consensus control strategy is proposed by combining a sign function, and the convergence time of the MAS can be adjusted with appropriate finite-time parameters. Finally, two illustrative examples verify the effectiveness of the theoretical claims.

A Computationally Efficient Technique Using Characteristic Basis Functions and Large Block Sizes

A Computationally Efficient Technique Using Characteristic Basis Functions and Large Block Sizes

Machine learning-based electricity theft detection using support vector machines

Electricity theft is a serious issue that many nations face, especially in developing areas where non-technical losses can make up a significant percentage of the overall losses sustained by utilities. Electricity theft detection (ETD) is a very challenging task because it frequently introduces irregularities in customer electricity consumption patterns. In recent times, machine learning (ML) techniques have been investigated as a potential solution for ETD. In this research, author propose electricity theft detection based on four kernel functions of support vector machines (SVM). The proposed method analyzes the electricity consumption patterns and then predicts the category of the user. The kernel functions utilized includes polynomial, sigmoid, radial basis function (RBF) and linear kernel function. For experimentation and model training, a dataset of Pakistani utility company is used, which contains the electricity consumption information. The results highlight SVM method works well for accurate ETD. The detection accuracy of the various kernel functions of SVM is 83%, 79%, 80%, and 76% for RBF, polynomial, sigmoid, and linear kernel functions, respectively, demonstrating the effectiveness of the proposed SVM-based method for theft detection. By leveraging these ML-based methods, utility companies can strengthen their ability to detect and prevent electricity theft, leading to improved revenue management and dependability of services.

Observer-Based Neural Control of N-Link Flexible-Joint Robots.

This article concentrates on the adaptive neural control approach of n -link flexible-joint electrically driven robots. The presented control method only needs to know the position and armature current information of the flexible-joint manipulator. An adaptive observer is designed to estimate the velocities of links and motors, and radial basis function neural networks are applied to approximate the unknown nonlinearities. Based on the backstepping technique and the Lyapunov stability theory, the observer-based neural control issue is addressed by relying on uplink-event-triggered states only. It is demonstrated that all signals are semi-globally ultimately uniformly bounded and the tracking errors can converge to a small neighborhood of zero. Finally, simulation results are shown to validate the designed event-triggered control strategy.

Adaptive Intelligent Minimum Parameter Singularity Free Sliding Mode Controller Design for Quadrotor

This paper presents a singularity free fast terminal sliding mode control (SFFTSMC) for position and attitude tracking of a quadrotor with unknown dynamics. The contribution of this work is threefold, namely: devising an SFFTSMC strategy using an auxiliary function to eliminate the singularity in the control law, a cost-saving minimum parameter update scheme using a fully connected recurrent neural network (FCRNN) to handle the unknown dynamics of quadrotor and development of an adaptive control law to compensate for the loss of effectiveness of the actuator and saturation effects, considered explicitly, without the requirement of any fault detection and diagnosis (FDD) unit. The proposed approach offers a direct singularity free control action in the presence of unknown dynamics, actuator faults, and saturation. Overall closed-loop stability is guaranteed in finite-time using Lyapunov stability theory and update laws for minimum parameters of FCRNN and fault compensator are derived using the same. Extensive simulations are presented for the trajectory tracking tasks using the proposed method and compared with the existing piece-wise fast terminal sliding mode controller (PFTSMC) and Radial basis function network (RBFN) based fault compensation approach. The proposed method is also validated in the Gazebo simulator via Pixhawk autopilot to demonstrate the feasibility in real-time. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This research is motivated by a cost-effective control scheme for quadrotors in cases of unknown/uncertain dynamics, external disturbances, actuator faults, and saturation. Precise quadrotor dynamics are difficult to obtain in real time. Also, there may be parametric uncertainties in quadrotor parameters like mass, inertia, and aerodynamic coefficients. External disturbances like wind gusts are always present in outdoor environments, and it may be the case that quadrotor motors lose their effectiveness due to defective motors or propeller damage while flying. To address these issues, first, it requires a precise control scheme to control the quadrotor while guaranteeing stability to ensure the safety of the quadrotor itself and people in the surrounding environment. Second, the approach should not be computationally heavy, which may sluggish the overall response of the quadrotor. Thus, we propose a control scheme to address these two issues simultaneously using a new fast terminal sliding mode control scheme augmented with FCRNN by updating minimum parameters instead of all the weight parameters of FCRNN. A compensator is introduced to mitigate the effect of unexpected actuator fault and saturation. The proposed approach can be helpful in various quadrotor applications, like pesticide spraying in agriculture, payload transportation, search and rescue, construction, etc., where unknown quadrotor dynamics and unknown payload situations occur.