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Increasing the energy efficiency of an Internet of Things (IoT) system is a major challenge for its successful implementation. To reduce the computation and storage burden and enhance the efficiency of traditional IoT, an energy-efficient diffusion-based algorithm for state estimation in multi-agent networks is proposed in this paper. In the proposed algorithm (referred to as reduced-link diffusion Kalman filter (RL-diffKF)) the nodes (agents) can communicate only with a fraction of their neighbors and each node runs a local Kalman filter to estimate the state of a linear dynamic system. This algorithm results in a significant reduction in communication cost during both adaptation and aggregation processes albeit at the expense of possible degradation in the network performance. To justify the stability and convergence of the RL-diffKF algorithm, an in-depth analysis of the performance is reported. We also consider the problem of optimal selection of combination weights and use the idea of minimum variance estimation to analytically derive the adaptive combiners. The theoretical findings are verified through numerical simulations.

Hypercholesterolemia is one of the major risk factors associated with the emergence and development of cardiovascular diseases (CVD) and atherosclerosis. The hypocholesterolemic effects of probiotics have been indicated by numerous studies. The 3-hydroxy-3-methylglutaryl-coenzyme A reductase (HMGCR) and cytochrome P450 7A1 or cholesterol 7 alpha-hydroxylase (CYP7A1) are two important genes in choletsterol metabolism. In this study, the effects of Bacillus subtilis JQ61816 on lipid panel, hepatic enzymes and expression levels of HMGCR and CYP7A1 were investigated. Twenty-one male Wistar rats were randomly allotted to 3 experimental groups; a) negative control group (ND) fed with normal diet, b) high-fat diet group (HFD) fed with high cholesterol diet, and c) probiotic group (BS) fed with high cholesterol diet supplemented with probiotic B. subtilis. Serum analysis of treatment groups was performed to measure fasting blood sugar (FBS), lipid profile parameters, hepatic enzymes, urea, and uric acid. Our results showed that B. subtilis could reduce the level of total cholesterol, triglycerides, and LDL and it also could increase high-density lipoprotein (HDL) level. Moreover, alanine transaminase (ALT), aspartate transaminase (AST), and uric acid were significantly lower in BS group compare to HFD group. Furthermore, up-regulation of HMGCR and down-regulation of CYP7A1 were observed in BS group. The results of our study suggest that consumption of probiotic B. subtilis JQ61816 may prevent or decline the development of hypercholesterolemia and other cardiovascular diseases.

Joint energy consumption and trading management is still a major challenge in smart (micro) grids. The main goal of solving such problems is to flatten the aggregate power consumption-generation curve and increase the local direct power trading among the participants as much as possible. Here, an inclusive formulation for energy management and trading of a micro/nano-grid (M/NG) is proposed in this article. Subsequently, a holistic solution to jointly optimizing the internal energy consumption management and external local energy trading for a smart grid including several M/NGs is provided. As the problem is computationally intractable, the proposed approach involves three hierarchical stages. First, a game-theoretic online stochastic energy management model is provided with a reinforcement learning solution by which the M/NGs can schedule their power consumptions. Second, an effective incentive-compatible double-auction is formulated by which the M/NGs can directly trade with each other. Third, the central controller develops an optimal power allocation program to reduce the power transmission loss and the destructive effects of local energy trading. The simulation results validate the efficiency of the proposed framework.

Demand-side management (DSM) involves a group of programs, initiatives, and technologies designed to encourage consumers to modify their level and pattern of electricity usage. This is performed following the methods such as financial incentives and behavioral change through education. While the objective of the DSM is to achieve a balance between energy production and demand, effective and efficient implementation of the program rests within effective use of emerging Internet-of-Things (IoT) concept for online interactions. Here, a novel DSM framework based on the diffusion and alternating direction method of multipliers (ADMM) strategies, repeated under a model predictive control (MPC) protocol, is proposed. On the demand side, the customers autonomously and by cooperation with their immediate neighbors estimate the baseline price in real time. Based on the estimated price signal, the customers schedule their energy consumption using the ADMM cost-sharing strategy to minimize their incommodity level. On the supply side, the utility company determines the price parameters based on the customer’s real-time behavior to make a profit and prevent infrastructure overload. The proposed mechanism is capable of tracking drifts in the optimal solution resulting from the changes in supply/demand sides. Moreover, it considers all classes of appliances by formulating the DSM problem as a mixed-integer programming (MIP) problem. Numerical examples are provided to show the effectiveness of the proposed framework.

In this letter a robust incremental adaptation algorithm is presented to solve distributed estimation for a Hamiltonian network, where the measurements at each node may be corrupted by heavy-tailed impulsive noise. In the proposed algorithm, each node employs an error-nonlinearity into the update equation to mitigate the detrimental effects of impulsive noise. Moreover, the algorithm estimates both the optimal error non-linearity and the unknown parameter together, which in turn, obviates the requirement of prior knowledge about the statistical characteristics of measurement noise. In addition to algorithm development, its steady-state performance as well as convergence analysis have been provided. Simulation results validate the correctness of the analysis and reveal the superiority of the proposed algorithm over some existing algorithms.

A novel pricing and scheduling mechanism is proposed in this paper for plug-in electric vehicles (PEVs) charging/discharging to track and synchronize with a renewable power generation pattern. Moreover, the proposed mechanism can be used in the demand-side management and ancillary service applications for the peak shaving and frequency regulation responding, respectively. We design a fully distributed stochastic optimization mechanism using a Bayesian pure strategic repeated game by which the PEVs optimally schedule their demands. We also use a mixed Bayesian–diffusion–Kalman filtering strategy for the customers to collaboratively estimate and track the stochastic price and regulation signals for the upcoming scheduling window. In this paper, all the characteristics of the PEVs as well as the uncertainty about their deriving patterns are considered. As our framework converges to an equilibrium even with incomplete information, it is agent-based, and the agents share the information only with their optional neighbors, it is scale free, robust, and secure.

Adaptive estimation of optimal combination weights for partial-diffusion Kalman filtering together with its mean convergence and stability analysis is proposed here. The simulations confirm its superior performance compared with the existing combiners. Sensor networks with limited accessible power highly benefit from this design.

Adaptive filters are useful solutions for system identification problem where an optimization problem is used to formulated the estimation of the unknown model coefficients. The nonnegativity constraint is one of the most frequently used constraint which can be imposed to avoid physically unreasonable solutions and to comply with physical characteristics. In this letter, we propose a new variant of non-negative least mean square (NNLMS) that has a less mean square error (MSE) value and faster convergence rate. We provide both mean weight behavior and transient excess mean-square error analysis for proposed algorithm. Simulation results validate the theoretical analysis and show the effectiveness of our proposed algorithm.

This work presents a novel fully distributed and cooperative demand-side management framework based on the adaptive diffusion strategy, wherein, each customer autonomously and without any need to the global information, minimizes his incommodity function. Our framework has ability to track drifts resulting from the changes in the customer preferences and conditions or rapidly updated price parameter coming from the wholesale market. In the considered scenario, the customers aim at maximizing their individual utility functions, while the utility company aims at minimizing the smart grid total payment (i.e., maximization of the social welfare). We show that there is no need for the utility company to participate in the scheduling program for maximizing the social welfare, and this measurement is maximized automatically while the customers minimizing their incommodity. Numerical results show that the proposed framework works well, is scale free, and can achieve lower peak-to-average ratio of the total energy demand compared with game theoretical methods.

In partial diffusion-based least mean square (PDLMS) scheme, each node shares a part of its intermediate estimate vector with its neighbors at each iteration. In this paper, besides studying the general PDLMS scheme, we figure out how the noisy links deteriorate the network performance during the exchange of weight estimates. We investigate the steady state mean square deviation (MSD) and derive a theoretical expression for it. We also derive the mean and mean-square convergence conditions for the PDLMS algorithm in the presence of noisy links. Our analysis reveals that unlike the PDLMS with ideal links, the steady-state network MSD performance of the PDLMS algorithm is not improved as the number of entries communicated at each iteration increases. Strictly speaking, the noisy links condition imposes more complexity to the MSD derivation that has a noticeable effect on the overall performance. This term violates the trade-off between the communication cost and the estimation performance of the networks in comparison with the ideal links. Our simulation results substantiate the effect of noisy links on PDLMS algorithm and verify the theoretical findings. They match well with theory.

In this paper, our aim is to propose a fully distributed adaptive algorithm for learning the parameters of a widely linear autoregressive moving average model by measurements collected by a network. To this end, we consider a connected network where every node uses the augmented complex adaptive infinite impulse response (ACA-IIR) filter as the learning rule. We firstly formulate the learning problem as an optimization problem and resort to stochastic gradient optimization argument to solve it and derive the proposed algorithm, which will be referred to as diffusion ACAIIR (DACA-IIR) algorithm. We also introduce a reduced-complexity version of the DACA-IIR algorithm. We use both synthetic and real-world signals in our simulations where the results show that the proposed cooperative algorithm outperforms the noncooperative solution.

The incremental least-mean-square (ILMS) algorithm is a useful method to perform distributed adaptation and learning in Hamiltonian networks. To implement the ILMS algorithm, each node needs to receive the local estimate of the previous node on the cycle path to update its own local estimate. However, in some practical situations, perfect data exchange may not be possible among the nodes. In this paper, we develop a new version of ILMS algorithm, wherein in its adaptation step, only a random subset of the coordinates of update vector is available. We draw a comparison between the proposed coordinate-descent incremental LMS (CD-ILMS) algorithm and the ILMS algorithm in terms of convergence rate and computational complexity. Employing the energy conservation relation approach, we derive closed-form expressions to describe the learning curves in terms of excess mean-square-error (EMSE) and mean-square deviation (MSD). We show that, the CD-ILMS algorithm has the same steady-state error performance compared with the ILMS algorithm. However, the CD-ILMS algorithm has a faster convergence rate. Numerical examples are given to verify the efficiency of the CD-ILMS algorithm and the accuracy of theoretical analysis.

In this paper, we consider the demand response problem in smart grid consisting of a retailer and multiple residential consumers, where the retailer determines consumers’ payments based on their power consumption profile. Our aim is to propose a fully distributed algorithm that is able to optimize the aggregate cost, utility, and retailer's profit simultaneously. To this end, we first formulate the consumer-side trend as a constrained convex optimization problem and propose a fully distributed adaptive diffusion algorithm to solve it. In addition, we design a one-leader $N$ -follower Stackelberg game to model interactions among the retailer and consumers. The proposed framework is able to continuously track the drifts resulting from the changes in the real-time pricing or the consumer preferences. Moreover, it is scalable and does not require network-wide information or rely on central controller. We provide comprehensive simulation results to show the effectiveness of the proposed framework.

We study the effect of fading in the communication channels between sensor nodes on the performance of the incremental least mean square (ILMS) algorithm, and derive steady state performance metrics, including the mean-square deviation (MSD), excess mean-square error (EMSE) and mean-square error (MSE). We obtain conditions for mean convergence of the ILMS algorithm, and show that in the presence of fading channels, the ILMS algorithm is asymptotically biased. Furthermore, the dynamic range for mean stability depends only on the mean channel gain, and under simplifying technical assumptions, we show that the MSD, EMSE and MSE are non-decreasing functions of the channel gain variances, with mean-square convergence to the steady states possible only if the channel gain variances are limited. We derive sufficient conditions to ensure mean-square convergence, and verify our results through simulations.

Partial diffusion-based recursive least squares (PDRLS) is an effective way of lowering computational load and power consumption in adaptive network implementation. In this method, every single node distributes a fraction of its intermediate vector estimate with its immediate neighbours at each iteration. In this study, the authors examine the steady-state performance of PDRLS algorithm in the presence of noisy links by means of an energy conservation argument. They consider the mean-square-deviation (MSD) as the performance metric in the steady-state and derive a theoretical expression for PDRLS algorithm with noisy links. The authors' analysis reveals that unlike the established statements on PDRLS scheme under ideal links, the trade-off between MSD performance and the number of selected entries of the intermediate estimate vectors, as a sign of communication cost, is mitigated. They further examine the convergence behaviour of the PDRLS algorithm. The obtained results show that under certain statistical assumptions for the measurement data and noise signals, under noisy links the PDRLS algorithm is stable in both mean and mean-square senses. Finally, they present some simulation results to verify the theoretical findings.

In this paper, we develop a new algorithm for recovery of block sparse signals in compressed sensing framework based on orthogonal matching pursuit. Furthermore, we point out that a major issue in conventional sparse signal recovery is the lack of prior knowledge about the order of sparsity. Consequently, block sparse signal recovery algorithms suffer even more from the same problem since two parameters are needed for exact recovery of the signal, order of active blocks and within-block sparsity order. Therefore, we propose a new approach to determining both of the active blocks and within-block sparsity order which is embedded in the proposed block sparse signal recovery algorithm. The simulation results illustrate the improved performance of the proposed method for recovery of block sparse signals compared to the conventional methods which are not aware of the prior information. We also apply our proposed algorithm to ECG signal compression, where the obtained results reveal its efficiency.

We propose an event-triggered game-theoretic strategy for managing the power grids demand side, capable of responding to changes in consumer preferences or the price parameters coming from the wholesale market. The relationship between the retailer and the residential consumers is modeled as one-leader, N-follower Stackelberg game. We provide a detailed characterization of the household appliances to reflect the reality and improve the efficiency of the demand response (DR). Moreover, to consider all the appliances’ essentials, the consumer's objective function is formulated as a mixed integer non-linear program (MINLP), which, unlike conventional procedures, is solved via an integrated method. The proposed method consists of a day-ahead stage, in which the DR problem is solved for the next scheduling horizon, and a real-time stage which runs repeatedly to tackle the change in the parameters and adapt to the new condition. For any change in the grid, the consumers use the estimated optimal parameters (given by the original objective function) and develop another Stackelberg game based solution to maximize the satisfaction level. Given the appliances of multi-class nature, the proposed method is shown to be very tractable for ancillary services and reducing the mismatch between the renewable power generation and the load demand.

The performance of a partial diffusion Kalman filtering (PDKF) algorithm for the networks with noisy links is studied here. A closed-form expression for the steady-state mean square deviation is then derived and theoretically shown that when the links are noisy, the communication–performance tradeoff, reported for the PDKF algorithm, does not hold. Additionally, optimal selection of combination weights is investigated, and a combination rule along with an adaptive implementation is motivated. The results confirm the theoretical outcome.

Many problems in multiagent networks can be solved through distributed learning (state estimation) of linear dynamical systems. In this paper, we develop a partial-diffusion Kalman filtering (PDKF) algorithm, as a fully distributed solution for state estimation in the multiagent networks with limited communication resources. In the PDKF algorithm, every agent (node) is allowed to share only a subset of its intermediate estimate vectors with its neighbors at each iteration, reducing the amount of internode communications. We analyze the stability of the PDKF algorithm and show that the algorithm is stable and convergent in both mean and mean-square senses. We also derive a closed-form expression for the steady-state mean-square deviation criterion. Furthermore, we show theoretically and by numerical examples that the PDKF algorithm provides a trade-off between the estimation performance and the communication cost that is extremely profitable.

In this paper, we propose a distributed adaptive learning algorithm to train the coefficients of a widely linear autoregressive moving average model by measurements collected by the nodes of a network. We assume that each node uses the augmented complex adaptive infinite impulse response (ACA-IIR) filter as the learning rule, and nodes interact with each other under an incremental mode of cooperation. To derive the proposed algorithm, called the incremental ACAIIR (IACA-IIR), we firstly formulate the distributed adaptive learning problem as an unconstrained minimization problem. Then, we apply stochastic gradient optimization argument to solve it and derive the proposed algorithm. We further find the step size range where the stability of the proposed algorithm is guaranteed. We also introduce a reduced-complexity version of the IACA-IIR algorithm. Since the proposed algorithm relies on the augmented complex statistics, it can be used to model both types of complex-valued signals (proper and improper signals). To evaluate the performance of the proposed algorithm, we use both synthetic and real-world complex signals in our simulations. The results exhibit superior performance of the proposed algorithm over the non-cooperative ACA-IIR algorithm.

This study addresses the tracking performance of the incremental augmented complex least mean square (IAC-LMS) algorithm, operating in the presence of model non-stationarities. The authors consider the mean-square deviation and excess mean square error as performance metrics and use energy conservation argument to derive closed-form expressions for the mentioned metrics. The expression describes how the IAC-LMS algorithm performs under such non-stationary conditions. The authors further find the step size range where the mean-square stability of the IAC-LMS algorithm is guaranteed. The authors provide some simulation results to support the theoretical derivations.

This letter studies the tracking performance of a stochastic gradient-based adaptive algorithm, namely the maximum correntropy criterion algorithm, where a random walk is used to model the non-stationarity. In our analysis, we use the energy conservation argument to derive expressions for the steady-state excess mean square error (EMSE). We consider two different cases for measurement of noise distribution including the Gaussian noise and general non-Gaussian noise. For the Gaussian case, we derive a fixed-point equation that can be solved numerically to find steady-state EMSE value. For the general non-Gaussian case, we derive an approximate closed-form expression for EMSE. For both cases, unlike the stationary environment, the EMSE curves are not increasing functions of step size parameter. We use this observation to find the optimum step size learning parameter for general non-Gaussian case. The validity of the theoretical results are justified via simulation results.

Purpose This paper aims to propose a new approach for determining a decoupling point in leagile chain, based on Lean and agile criteria regarding market and customer demands and internal capabilities of the chain with the ultimate goal of fulfilling customer needs and increasing chain profit. Design/methodology/approach In the new approach, Lean and agile criteria have been defined for assessing the effectiveness and efficiency of supply chain. The efficiency and effectiveness ratios have been calculated for Lean and agile processes using input- and output-oriented Banker, Charnes and Cooper (BCC) methods, respectively. Based on the results, inefficient and ineffective units have been addressed and the decoupling point has been determined. Findings Findings indicate that the decoupling point can be regarded as a borderline between two strategies of Lean and agile production, and fuzzy decoupling point and lean–agile distance can provide the basis for distinguishing the two strategies. Practical implications Determining the decoupling point has an important role in dynamic performance of a supply chain. By the proposed approach, managers can estimate the most probable area for the decoupling point. Moreover, by appropriate determination of decoupling point, an organization can increase its public responsibility by appropriate usage of its resources and responding faster to customers’ requirements. Originality/value In this study, in addition to determining a decoupling point in a supply chain with the aim of increasing productivity, the subject of leagile strategy of supply chains has been developed.

Adaptive networks with mobile nodes possess the distributed adaptation abilities in addition to the collective patterns of motion. Thus, mobile adaptive networks have been used in new applications such as modeling the biological networks and source localization. The original mobile adaptive networks needs full cooperation among the neighbor nodes meaning that each node gathers the information from all of its neighbour nodes. This strategy requires large amount of communications per iteration per node. To address this problem, in this paper we propose a mobile diffusion adaptive network with selective cooperation. In the proposed algorithm each node selects a subset of its neighbors so that its steady-state performance be as close as possible to the traditional mobile diffusion network. Since the selective cooperation reduces the learning rate we also use affine projection algorithm (APA) as the learning rules at the nodes. Our simulation results reveal that the proposed algorithm is able to achieve the same mean-square deviation (MSD) as the original mobile adaptive network but with a lower communication per iteration per node.

Augmented adaptive filters provide superior performance over their conventional counterparts when dealing with noncircular complex signals. However, the performance of such filters may change considerably when they are implemented in finite-precision arithmetic due to round-off errors. In this paper, we study the performance of recently introduced augmented complex least mean-square (ACLMS) algorithm when it is implemented in finite-precision arithmetic. To this aim, we first derive a model for the finite-precision ACLMS updating equations. Then, using the established energy conservation argument, we derive a closed-form expression, in terms of the excess mean-square error (EMSE) metric which explains how the quantized ACLMS (QACLMS) algorithm performs in the steady-state. We further derive the required conditions for mean stability of the QACLMS algorithm. The derived expression, supported by simulations, reveals that unlike the infinite-precision case, the EMSE curve for QACLMS is not monotonically increasing function of the step-size parameter. We use this observation to optimize the step-size learning parameter. Simulation results illustrate the theoretical findings. HighlightsWe firstly derive the quantized version of the QACLMS algorithm.Starting with an energy conservation relation we derive closed-form expression for mean-square performance curve in terms of EMSE.We show that the steady-state curve is not monotonically increasing functions of the step size.We find the optimum value of the step-size learning parameter. We also derive the mean stability condition in terms of the step size parameter.Simulation results illustrate the theoretical findings.

In extracellular neural recording experiments, detecting neural spikes is an important step for reliable information decoding. A successful implementation in integrated circuits can achieve substantial data volume reduction, potentially enabling a wireless operation and closed-loop system. In this paper, we report a 16-channel neural spike detection chip based on a customized spike detection method named as exponential component-polynomial component (EC-PC) algorithm. This algorithm features a reliable prediction of spikes by applying a probability threshold. The chip takes raw data as input and outputs three data streams simultaneously: field potentials, band-pass filtered neural data, and spiking probability maps. The algorithm parameters are on-chip configured automatically based on input data, which avoids manual parameter tuning. The chip has been tested with both in vivo experiments for functional verification and bench-top experiments for quantitative performance assessment. The system has a total power consumption of 1.36 mW and occupies an area of 6.71 mm $^2$ for 16 channels. When tested on synthesized datasets with spikes and noise segments extracted from in vivo preparations and scaled according to required precisions, the chip outperforms other detectors. A credit card sized prototype board is developed to provide power and data management through a USB port.

In this study, the effects of noisy links are investigated on the steady-state performance of mobile adaptive networks with diffusion least mean-squares strategies. The authors derive theoretical relations which explain how the steady-state performance metrics, including the steady-state network mean-square deviation and steady-state velocity mean-square-error is affected by noisy links. The provided analysis relies on the spatial–temporal energy conservation argument. The proposed simulation results reveal that although the noisy links degrade the performance of mobile adaptive networks; however, for suitably chosen combination coefficients the mobile adaptive network with noisy links provides a bounded estimation error. Finally, the proposed simulations verify that the derived theoretical analysis closely matches the actual steady-state performance observed in a network.

Naturally complex-valued information or those presented in complex domain are effectively processed by an augmented complex least-mean-square (ACLMS) algorithm. In some applications, the ACLMS algorithm may be too computationally- and memory-intensive to implement. In this paper, a new algorithm, termed partial-update ACLMS (PU-ACLMS) algorithm is proposed, where only a fraction of the coefficient set is selected to update at each iteration. Doing so, two types of partial-update schemes are presented referred to as the sequential and stochastic partial-updates, to reduce computational load and power consumption in the corresponding adaptive filter. The computational cost for full-update PU-ACLMS and its partial-update implementations are discussed. Next, the steady-state mean and mean-square performance of PU-ACLMS for non-circular complex signals are analyzed and closed-form expressions of the steady-state excess mean-square error (EMSE) and mean-square deviation (MSD) are given. Then, employing the weighted energy-conservation relation, the EMSE and MSD learning curves are derived. The simulation results are verified and compared with those of theoretical predictions through numerical examples., 6

In this paper the steady-state tracking performance of minimum kernel risk-sensitive loss (MKRSL) in a non-stationary environment is analyzed. In order to model a non-stationary environment, a first-order random-walk model is used to describe the variations of optimum weight vector over time. Moreover, the measurement noise is considered to have non-Gaussian distribution. The energy conservation relation is utilized to extract an approximate closed-form expression for the steady-state excess mean square error (EMSE). Our analysis shows that unlike for the stationary case, the EMSE curve is not an increasing function of step-size parameter. Hence, the optimum step-size which minimizes the EMSE is derived. We also discuss that our approach can be used to extract steady-state EMSE for a general class of adaptive filters. The simulation results with different noise distributions support the theoretical derivations.

This paper presents a game-theoretic decentralized electric vehicle charging schedule for minimizing the customers’ payments, maximizing the grid efficiency, and providing the maximum potential capacity for ancillary services. Most of the available methods for electric vehicle charging assume that the customers are rational, there is a low-latency perfect two-way communication infrastructure without communication/computation limitation between the distribution company and all the customers, and they have perfect knowledge about the system parameters. To avoid these strong assumptions and preserve the customers’ privacy, we take advantage of the regret matching and the Nash Folk theorems. In the considered game, the players (customers) interact and communicate locally with only their neighbors. We propose a mechanism for this game, which results in a full Nash Folk theorem. We demonstrate and prove that the on-off charging strategy provides the maximum regulation capacity. However, our mechanism is quite general, takes into account the battery characteristics and degradation costs of the vehicles, provides a real-time dynamic pricing model, and supports the vehicle-to-grid and modulated charging protocols. Moreover, the developed mechanism is robust to the data disruptions and takes into account the long/short-term uncertainties.

In this study, the authors propose a robust adaptive algorithm for frequency estimation in three-phase power systems when the voltage readings are corrupted by random noise sources. The proposed algorithm employs the Clarke's transformed three-phase voltage (a complex signal) and augmented complex statistics to deal with both of balanced and unbalanced system conditions. To derive the algorithm, a widely linear predictive model is assumed for the Clarke's transformed signal where the frequency of system is related to the parameters of this model. To estimate the model parameters with noisy voltage reading, they utilise the notions of maximum correntropy criterion and gradient-ascent optimisation. The proposed algorithm has the computational complexity of the popular complex least-mean-squares (CLMS) algorithm, along with the robustness that is obtained by using higher-order moments beyond just second-order moments. They compare the performance of the proposed algorithm with a recently introduced augmented CLMS (ACLMS) algorithm in different conditions, including the voltage sags and presence of impulsive noises and and higher-order harmonics. Their simulation results demonstrate that the proposed algorithm provides improved frequency estimation performance compared with ACLMS especially when the measured voltages are corrupted by impulsive noise.

Successful implementation of incremental adaptive networks requires that the sampling speed of the network be faster than the time between two consecutive measurements taken by nodes. This issue affects the performance of incremental based algorithm in networks with large number of nodes as well as non-stationary environments. In this paper, we introduce a modified incremental least mean-square (ILMS) algorithm that resolves this problem. In the proposed algorithm, every node updates its local estimate when the measurement data are available (temporal processing). When the estimate from the previous node is also available, an adaptive strategy is used to combine the available estimates to further improve the network learning performance (spatial cooperation). We examine the performance of the proposed algorithm in two scenarios including the network with noisy links and non-stationary environment. Simulation results show the superior performance of the proposed algorithm in comparison with the original ILMS algorithm.

The augmented complex least mean-square ACLMS algorithm is a suitable algorithm for the processing of both second-order circular proper and noncircular improper signals. In this paper, we provide tracking analysis of the ACLMS algorithm in the non-stationary environments. Using the established energy conservation argument, we derive a variance relation that contains moments that represent the effects of non-stationary environment. We evaluate these moments and derive closed-form expressions for the excess mean-square error EMSE and mean-square error MSE. The derived expressions, supported by simulations, reveal that unlike the stationary case, the steady-state EMSE, and MSE curves are not monotonically increasing functions of the step-size parameter. We also use this observation to optimize the step-size learning parameter. Simulation results illustrate the theoretical findings and match well with theory. Copyright © 2015 John Wiley & Sons, Ltd.