Recursive Algorithm Research Articles

Recursive state estimation for stochastic nonlinear non-Gaussian systems using energy-harvesting sensors: A quadratic estimation approach

In this paper, the recursive quadratic state estimation problem is investigated for a class of stochastic nonlinear systems (SNSs) subject to non-Gaussian noises through energy-harvesting sensors. Stochastic nonlinearities are taken into account in both state and measurement equations. The energy-harvesting technique is employed to mitigate the phenomenon of energy depletion and supply energy for data transmissions between the sensors and the remote estimator. By augmenting the state/measurements and their respective second-order Kronecker powers, the original SNS is converted into a new nonlinear system that exploits more information about the non-Gaussian noises, and the resultant quadratic estimator can then be designed by developing an efficient recursive variance-minimization algorithm. Subsequently, an upper bound is proven to exist on the estimation error covariance matrix via solving certain matrix difference equations, and such an upper bound is minimized by resorting to the design of appropriate gain parameters of the quadratic estimator. Moreover, we analyze the monotonicity of the trace of the minimal upper bound with respect to the probability of measurement transmission. Finally, a simulation example is given to demonstrate the effectiveness of the proposed quadratic estimation algorithm.

A digital twin-based framework for multi-element seismic hybrid simulation of structures

This paper proposes a digital twin-based multi-element hybrid simulation (DMHS) framework to predict the nonlinear cyclic response of structural components (digital twin), e.g., seismic fuses, that are not physically tested due to laboratory limitations by leveraging the experimental test data collected from the physical test specimen (physical twin) during hybrid simulation. This data-based simulation approach can address biased results of hybrid simulation of structures that contain multiple critical components while improving the efficiency of the seismic hybrid simulation. The digital twin is trained in two phases: (1) passive (initial) training phase using past experimental test data before the hybrid simulation starts, and (2) recursive model updating phase using the active (real-time) data produced by the physical specimen during hybrid simulation. The passive training is achieved using the Prandtl–Ishlinskii (PI) hysteresis model combined with the sparse identification technique, while the recursive least-squares algorithm is used in the second phase as the model updating scheme. In particular, the sparse identification technique facilitates the selection of the optimal number of hysteretic model parameters in the passive training phase, which are then tuned in the model updating phase. The architecture of the proposed DMHS framework is first presented, followed by digital twin training steps. The application of the proposed DMHS is then demonstrated, and its simulation accuracy is assessed through virtual hybrid simulation of a two-storey steel buckling-restrained braced frame, which consists of a digital twin (second-storey brace) and a virtual experimental specimen (first-storey brace) integrated into the numerical model of the structure that is subjected to a set of earthquake ground motion accelerations. The results obtained from the verification study serve to validate the proposed architecture of the DMHS framework and evaluate the accuracy and efficiency of this technique in simulating the nonlinear seismic response of structural systems.

A recursive algorithm for an efficient and accurate computation of incomplete Bessel functions

In a previous work, we developed an algorithm for the computation of incomplete Bessel functions, which pose as a numerical challenge, based on the \(G_{n}^{(1)}\) transformation and Slevinsky-Safouhi formula for differentiation. In the present contribution, we improve this existing algorithm for incomplete Bessel functions by developing a recurrence relation for the numerator sequence and the denominator sequence whose ratio forms the sequence of approximations. By finding this recurrence relation, we reduce the complexity from \({\mathcal O}(n^{4})\) to \({\mathcal O}(n)\). We plot relative error showing that the algorithm is capable of extremely high accuracy for incomplete Bessel functions.

Hysteresis Segmentation Modeling and Experiment of Piezoelectric Ceramic Actuator

Piezoelectric ceramics have been widely used in ultraprecision displacement drive systems because of their excellent performance such as fast response and high positioning accuracy. However, its tracking control accuracy is limited by its inherent hysteresis nonlinearity. To date, the parameter identification of the hysteresis model that has been developed is complex. This situation has resulted in the difficulty for the model to be used in the real-time control system of piezoelectric ceramic actuators. This article proposed a new mathematical model that can accurately describe hysteresis nonlinearity. The hysteresis main loop in the model is approximated using high-order rational fractions. The hysteresis segmentation model is completed through the recursive algorithm. Simulation and experimental results show that the hysteresis characteristics between the input voltages and output displacements have been clearly modeled. This segmentation modeling process is simple. In addition, the computation is low and accuracy is high. The model proposed in this article can complete the modeling of complex hysteresis phenomena of the multidegree-of-freedom micromotion stage with force–voltage coupling. Through a series of experiments to verify the effectiveness of the model in this article, thereby laying the model foundation for the subsequent control of the piezoelectric ceramic actuator. This result also provides a strong scientific basis for future research on the application of piezoelectric ceramic actuators in ultraprecision motion systems and further improves the positioning accuracy of the system.

Form-Finding Method of Site Road from the Perspective of Meta-Design

Site road alignment is one of the main elements of architectural environmental design [B. Holdsworth, Refocus 6(1) (2005) 58–60]. It is difficult for traditional methods to achieve both qualitative exploration and quantitative optimization objectives, and optimization algorithms can only optimize quantitative objectives. On the other hand, the shape of the site road is subject to various other human interventions besides the designer, which is a multi-objective problem. Based on the idea of meta-design, this paper proposes a new method for “form-finding” of site roads. This method develops the architecture and components of an expert system, which can handle both qualitative and quantitative objectives in conceptual design, and can realize various human interventions. Component development combines site specification, recursive algorithm, and genetic algorithm, which treats human intervention as a factor of variation, allows for qualitative exploration of different preferences, and finally demonstrates the capabilities of this new approach through case studies.

Probabilistic-Constrained Distributed Set-Membership Estimation Over Sensor Networks: A Dynamic Periodic Event-Triggered Approach

This article is concerned with the event-triggered probabilistic-constrained distributed set-membership estimation problem for a discrete-time nonlinear system over sensor networks. For saving communication resource, a novel discrete-time dynamic periodic event-triggered mechanism (ETM) is first developed for the sensor network. Under the proposed method, the sensor node calculates the ETM in a periodic manner and the threshold is adjusted dynamically. Thereafter, a distributed set-membership estimator is constructed, and a probability-based estimated ellipsoidal constraint is put forward to acquire a more flexible set-membership estimation algorithm. Simultaneously, an auxiliary-function-dependent approach is proposed to derive the criterion for the co-design of the probabilistic-constrained set-membership estimator and the event-triggered parameter such that the system states reside in the estimated ellipsoid with a pre-specified probability. The auxiliary function is constructed in a piecewise style aiming to deal with the sawtooth constraint of sampling signals. Furthermore, a recursive convex optimization algorithm with regard to the estimated ellipsoid is presented. Finally, a simulation example is employed to verify the validity of the developed method.

Thickness-Independent Measurement of Grain Moisture Content by Attenuation and Corrected Phase Shift of Microwave Signals at Multiple Optimized Frequencies

Inadequate drying and overdrying of grains due to the absence of timely and reliable information on moisture content (MC) aggravates grain postharvest loss. In this study, we investigate the real-time acquisition of MC using microwave free-space transmission measurements. A multifrequency swept signal is used to acquire the attenuation and phase-shift spectrum (2.00–10.00 GHz) of the paddy samples (9.553%–29.633% w.b.) for six thicknesses (1–6 cm). Based on the four rules that are followed in the phase-shift measurement, a phase-shift correction algorithm without any restrictions on the sample thickness is proposed to solve the phase-shift ambiguity that occurs when the sample thickness exceeds the wavelength. To choose the most effective frequencies, 17 candidate frequency subsets are generated by the recursive feature elimination algorithm, and the optimal frequency set, containing eight individual frequencies, is selected by using the Friedman and Nemenyi post hoc tests. The sample thickness and microwave characteristics are used as the input variables to establish the prediction models to achieve a thickness-independent measurement of the MC, and the support vector machine model yielded the best performance ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R^2$</tex-math></inline-formula> = 0.992, RMSE = 0.555%, and MAE = 0.398%). The results of this study should encourage future research on the real-time acquisition of reliable MC information in food processing and agriculture-related industries.

A Composite State of Charge Estimation for Electric Vehicle Lithium-Ion Batteries Using Back-Propagation Neural Network and Extended Kalman Particle Filter

Accurate estimation of battery state of charge (SOC) plays a crucial role for facilitating intelligent battery management system development. Due to the high nonlinear relationship between the battery open-circuit voltage (OCV) and SOC, and the shortcomings of traditional polynomial fitting approach, it is an even more challenging task for predicting battery SOC. To address these challenges, this paper presents a composite SOC estimation approach for lithium-ion batteries using back-propagation neural network (BPNN) and extended Kalman particle filter (EKPF). First, a second order resistance capacitance model is established to make parameters identification of a lithium-ion battery cell using recursive least squares algorithm with forgetting factors (FFRLS) approach. Then, BPNN is used to fit the desired OCV-SOC relationship with relatively high precision. Next, by incorporating the extended Kalman filter (EKF) into the particle filter (PF), an expected EKPF approach is presented to realize the SOC estimation. Last, the performances of SOC estimation using different methods, namely the PF, EKF and the EKPF are compared and analyzed under constant current discharge and urban dynamometer driving schedule working conditions. The experimental results show that the proposed method has higher accuracy and robustness compared to the other two SOC estimation methods.

Adaptive Loss Model Control for Robustness and Efficiency Improvement of Induction Motor Drives

Indirect field-oriented control (IFOC) can provide an excellent dynamic response in three-phase induction motor drives and is widely used in the industry. Still, as a considerable part of the electrical energy is used to drive motors, intense research has been carried out to reduce the energy consumption of the induction motor drives and improve their efficiency. In this article, a high-efficiency smooth IFOC (SIFOC) scheme is proposed and implemented experimentally with improved current controllers and the loss model control (LMC) technique, which achieves up to 41% reduction in energy consumption compared to the traditional IFOC approach. The main contribution is the new adaptive LMC (ALMC) control scheme, which includes two recursive adaptation algorithms to minimize the LMC technique sensitivity to the variation of the motor parameters. The algorithms are derived considering the model reference adaptive system concept and both active and reactive power to estimate the stator and rotor bandwidths, which are used to update the ALMC controller. The SIFOC-ALMC approach is verified experimentally, with up to 7.2% energy saving compared to SIFOC-LMC. Moreover, since the traditional IFOC controller is sensitive to rotor bandwidth variation, estimating this parameter improves the robustness of the SIFOC drive system.

Frequency Reflection Modulation for Reconfigurable Intelligent Surface Aided OFDM Systems

Reconfigurable intelligent surface (RIS) based reflection modulation (RM) has been considered as a promising information delivery mechanism, and has the potential to realize passive information transfer of a RIS without consuming any additional radio frequency chain and time/frequency/energy resource. The existing on-off RM (ORM) schemes are based on manipulating the “on/off” states of RIS reflection elements, which may lead to the degradation of RIS reflection efficiency. This paper proposes a frequency RM (FRM) method for RIS-aided OFDM systems. The FRM-OFDM scheme modulates the frequency of the incident electromagnetic waves, and the RIS information is embedded in the frequency-hopping states of RIS elements. Unlike the ORM-OFDM scheme, the FRM-OFDM scheme can achieve higher reflection efficiency, since the latter does not turn off any reflection element in RM. We show that, for the RIS phase shift optimization, the multiplicative multiple access channel in the FRM-OFDM system can be converted to an equivalent RIS-aided multiple-input multiple-output channel. Then, we propose an alternating optimization AO) algorithm for sum rate maximization of the FRM-OFDM system. A low-complexity recursive AO algorithm is further developed to avoid direct channel matrix inversion in the AO algorithm with negligible performance degradation. In addition, we design a bilinear message passing (BMP) algorithm for the bilinear recovery of both the user symbols and the RIS data. Numerical simulations verify the efficiency of the designed optimization algorithms for system optimization and the BMP algorithm for signal detection, as well as the superiority of the proposed FRM-OFDM scheme over the existing ORM-OFDM scheme and the RIS-aided OFDM system.

A Generalized Recursive Vogler Algorithm for Multiple Bridged Knife-Edge Diffraction

In this article, we consider the problem of estimating diffraction attenuation from the approximation of terrain using a multiple bridged knife-edge model. This model can be considered as a generalization of the well-known multiple knife-edge (MKE) one where spaces between knife-edges are bridged by reflecting surfaces. A series-based-standard solution is presented in the literature but suffers from its very high computational complexity. We, thus, propose to generalize the recursive Vogler algorithm developed for the MKE model to tackle this problem. To reduce complexity, the proposed algorithm exploits the recursive form to avoid repeated calculations of the existing solution. Moreover, we provide a complexity analysis of both the standard algorithm and the proposed algorithm based on the number of computed integrals. Both theoretical and numerical results show that our proposed algorithm is faster than the original one while having identical accuracy, hence proving the effectiveness of our solution.

Managing equipment rentals: Unreliable fleet, impatient customers, and finite commitment capacity

In this paper, we discuss fleet size decisions of an equipment rental firm. The availability of the rental equipment depends on the fleet size of the firm and has a direct impact on its profitability. In our setting, we allow for partial backordering, reneging, and finite commitment capacity. Moreover, we explicitly consider the breakdown possibility of the available equipment fleet. We develop an efficient recursive algorithm to solve the underlying two‐dimensional stochastic single‐player model. Our algorithm determines the global optimal fleet size of the firm for the same reneging and equipment return rates. Our extensive managerial insights quantify the behavior of various performance measures in the single‐player model with regard to repair performance of the firm, customer impatience level, traffic intensity, and equipment rental revenue. We demonstrate that by applying our model to a real case, there is a potential of more than a 4% (400,000 USD) increase in total daily profits. Extending our model to a two‐player game, we propose an approximation heuristic to derive closed‐form solutions to estimate equilibrium fleet sizes under complete information. Using our heuristic as the initial solution, we develop a simulation model to determine the exact equilibrium fleet sizes and draw a detailed comparison between the two‐player and single‐player models.

Identifying Reliability-Critical Primary Inputs of Combinational Circuits Based on the Model of Gate-Sensitive Attributes

The identification of reliability-critical primary input leads (RCPIs) plays an important role in the testing and prediction of reliability boundaries of logic circuits. This article presents a gate-sensitive-attributes-based approach to estimate the criticality of the primary input leads in combinational circuits to their reliability. Oriented to the input vector, a subcircuit-based traversal method marks the critical input leads of each gate in a circuit. Gate-sensitive attributes and a reverse recursive algorithm quantify the effect of each RCPI on circuit reliability under the input vector. A parallel calculation method based on subcircuits with only one primary output reduces the computational complexity to accelerate the calculation process. Similarity-based clustering avoids unnecessary calculations, and a self-adaptive strategy is used to check convergence. Experimental results on benchmark circuits show that the average accuracy of this approach is 0.9634 with Monte Carlo (MC) as the reference and it is 3445 times faster than the MC on average while its average memory cost is 1.67 greater than the MC model. Although the fitness of the worst input vector obtained by other reference methods is 1.09 times better than that of this approach on average, this approach is approximately 21 times faster than that reference method on average.

Cost Optimal Hybrid Communication Model for Smart Distribution Grid

There have been significant changes and dynamic growth in power grids since the inception of the smart grid. To ensure a self-sustainable grid, the power grid should facilitate the real-time sharing of the dynamic attributes of the power grid between the spatially distributed power grid elements. This demands a flawless design of the overlay communication network for a smart grid. However, instrumenting the entire grid with intelligent communication devices would be prohibitively expensive. A cost-optimal model is necessary to select the actual cardinality of intelligent communication devices required for the data transmission by ensuring its Quality of Service (QoS) metrics. This research work presents a smart distribution grid architecture with a network of microgrids and details the need for hybrid communication for intra-microgrid communication. A cost-optimal model is proposed to offer an optimal combination of technologies for intelligent devices in the hybrid communication overlay network. The formulation considers QoS metrics such as data packet latency, bandwidth requirement, link reliability, packet drops, and communication range of the technology to derive a cost-optimal solution. Based on this model, a Recursive Algorithm for Cost Optimal Combination of Communication Technology (RACOCCT) for a power grid topology is presented in this work. The simulation study performed on selected power grid topology based on the Standard IEEE 33 bus network reveals the cost-optimal combination of hybrid wireless communication technologies for the varying probability of link unreliability.

Dependency maximization forward feature selection algorithms based on normalized cross-covariance operator and its approximated form for high-dimensional data

Supervised feature selection (FS) for classification aims at finding a more discriminative subset from original features, to facilitate classifier training, improve classification performance and enhance model interpretability. Primary normalized cross-covariance operator (NOCCO) is a nonlinear kernel-based dependency measure between features and labels, including two inverse matrices, and its approximated version (ANOCCO) is simplified via exploiting linear kernel for features, delta kernel for labels and Moore–Penrose inverse. In this paper, we apply NOCCO and ANOCCO to FS task. According to sequential forward selection, a forward NOCCO-based FS algorithm (i.e., FoNOCCO) is directly implemented via various accelerating strategies, but its computational complexity is extremely high. To this end, we propose its fast version via maximizing ANOCCO (i.e., FoANOCCO), where each candidate feature is evaluated efficiently according to column recursive algorithm for Moore–Penrose inverse. Theoretical analysis shows that evaluating a candidate feature needs a time complexity of O(N2) in FoANOCCO and O(N3) in FoNOCCO, where N is the number of instances. On eight small-size high-dimensional benchmark data sets, our detailed experiments illustrate that our two forward FS algorithms are superior to six state-of-the-art FS methods with four baseline classifiers, and FoANOCCO runs 70 times faster than FoNOCCO with a 1.79 % accuracy drop.

Trends and Analysis of Graduate Programs

Due to the influx of students in higher education institutions, this study was designed to generate, process, and analyze students’ information using online data retrieval. The system was crafted using data mining techniques. The PHP scripting language for web development and the V-Model Design were employed during the course of the study. To analyze the data, simple recursive algorithm was used. The procedure solved the base cases that directly recurs with a simpler sub problem and does some extra work to provide the solution to the simpler sub problems. In this study, it combined the records from one table to another to generate results and determine all the registered enrollees in the system. Respondents included new students, old students, and the graduates of the academic institution under survey. The objective is to replace the actual rigorous paper works during every transaction engaged by the students. The system used to generate information on enrolment, vertical alignments, and program trends. The study found out that the system was deemed effective in terms of system, information, and service quality.

Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts

Almost every supplier faces uncertain and time-varying demand. E-commerce and online shopping have given suppliers unprecedented access to data on customers’ behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized supply channel members to devise optimal long-term (multi-period) supply, pricing, and timing strategies while catering to stochastic demand in a diverse set of market scenarios. Despite its ubiquity in potential applications, the time-dependent channel optimization problem in its general form has received limited attention in the literature due to its complexity and the highly nested structure of its ensuing equilibrium problems. However, there are many scenarios where a single-period channel optimization solution may turn out to be myopic as it does not consider the after-effects of current pricing on future demand. To remedy this typical shortcoming, using general memory functions, we include the strategic customers’ cognitive bias toward pricing history in the supply channel equilibrium problem. In the form of two constructive theorems, we provide explicit solution algorithms for the ensuing Nash–Stackelberg equilibrium problems. In particular, we prove that our recursive solution algorithm can find equilibria in the multi-periodic variation of many standard supply channel contracts such as wholesale, buyback, and revenue-sharing contracts.

Identification of the Dynamic Properties of the Coal Flotation Process as a Control Object with the Use of the Kalman Filter

For various sorts of hard coal, enrichment by flotation is used for feed consisting of grains smaller than 0.5 mm. Regarding process automation, coal flotation is a multidimensional, dynamic nonlinear object of control, for which the main control signal is the flow rate of the flotation agent. Typically, in Polish coal-processing facilities the monitoring and control systems of the flotation process can only measure the parameter of the waste quality (content of ash in flotation tailings). This naturally becomes an output signal, enabling an indirect assessment of the ongoing process. Therefore, knowledge of the dynamic properties of the flotation process, analysed as an object with one control input (the flow rate of the flotation agent) and with one output for measuring (content of ash in flotation tailings) may be material in designing automatic control systems for this operation. It is important to use an appropriate identification method when developing a model of the dynamics of the flotation process, especially if the model parameters are to be determined on an ongoing basis. This article discusses the research method and presents the results of applying the method of identifying the dynamic properties of the coal flotation process with the use of the Kalman filter. We carried out a comparative analysis of the results obtained by this method based on the Kalman algorithm and the method of least squares, taken as the reference method. The presented parameters of the dynamic models were calculated based on actual data obtained from industrial tests conducted at the coal-processing plant at one of the Polish mines. It was demonstrated that, for control purposes, the Kalman algorithm can be successfully applied in identification of the coal flotation process. This is due to the fact that it gives satisfactory results in relation to the adopted reference method despite the fact that it is a recursive algorithm.

A new inexact stochastic recursive gradient descent algorithm with Barzilai–Borwein step size in machine learning

A new inexact stochastic recursive gradient descent algorithm with Barzilai–Borwein step size in machine learning

Numerical strategies for recursive least squares solutions to the matrix equation AX = B

The recursive solution to the Procrustes problem -with or without constraints- is thoroughly investigated. Given known matrices and , the proposed solution minimizes the square of the Frobenius norm of the difference when rows or columns are added to and . The proposed method is based on efficient strategies which reduce the computational cost by utilizing previous computations when new data are acquired. This is particularly useful in the iterative solution of an unbalanced orthogonal Procrustes problem. The results show that the computational efficiency of the proposed recursive algorithms is more significant when the dimensions of the matrices are large. This demonstrates the usefulness of the proposed algorithms in the presence of high-dimensional data sets. The practicality of the new method is demonstrated through an application in machine learning, namely feature extraction for image processing.