Second-order Volterra Research Articles

Comparative Study of Statistical Analysis Methods for Nonlinear Hydrodynamic Responses of Offshore Structures

AbstractThe present study compares the statistical analysis methods for nonlinear hydrodynamic responses in offshore engineering. In particular, the Kac–Siegert and Hermite-moment methods were compared for estimating the probability distribution of the second-order responses represented via the two-term Volterra series. The Kac–Siegert method analytically formulates the probability density function (PDF) of the second-order Volterra series using an eigenvalue problem constructed with the frequency-domain transfer functions and the wave spectrum, whereas the Hermite-moment method utilizes the statistical moments to determine the coefficients of the fitting function. In addition, the probability distribution of the peak values in the second-order Volterra series with high spectral bandwidth was derived explicitly. The fatigue damage rate and the extreme response were estimated analytically. The accuracy and applicability of each method were investigated by comparing the methods with the results of the direct sampling obtained from the time series.

An hp-version of the C0-continuous Petrov-Galerkin method for second-order Volterra integro-differential equations

We present an hp-version of the C0-continuous Petrov-Galerkin method for linear second-order Volterra integro-differential equations. We combine the continuous and discontinuous Galerkin methodologies to obtain a natural discretization of the second-order derivative of arbitrary order. We derive a-priori error bounds in the L2- and H1-norm that are completely explicit in the local time steps, local approximation orders, and local regularity of the exact solution. Numerical experiments are provided to illustrate the theoretical results.

Recursive second-order Volterra filter based on Dawson function for chaotic memristor system identification

In this paper, we propose a modified version of exponential cost function to improve the stability of adaptive algorithm, where the recursive algorithm is based on the Dawson function. The second-order Volterra (SOV) filter is incorporated into the proposed recursive algorithm, resulting the SOV-ExRLS algorithm, to achieve the improved performance in both $$\alpha $$-stable and Gaussian environments. Moreover, the mean and mean-square behavior of the SOV-ExRLS algorithm is analyzed. In particular, the proposed cExRLS-IDLMS method is convexly combined with the functional link artificial neural network filter to flexibly model the chaotic memristor system. Simulation studies verify the analytical findings and reveal enhanced identification performance of the proposed algorithms over the existing nonlinear algorithms.

A Model for Joint Probabilistic Forecast of Solar Photovoltaic Power and Outdoor Temperature

In this paper, a stochastic model is proposed for a joint statistical description of solar photovoltaic (PV) power and outdoor temperature. The underlying correlation emerges from solar irradiance that is responsible in part for both the variability in solar PV power and temperature. The proposed model can be used to capture the uncertainty in solar PV power and its correlation with the electric power consumption of thermostatically controlled loads. First, a model for solar PV power that explicitly incorporates the stochasticity due to clouds via a regime-switching process between the three classes of sunny, overcast and partly cloudy is proposed. Then, the relationship between temperature and solar power is postulated using a second-order Volterra model. This joint modeling is leveraged to develop a joint probabilistic forecasting method for solar PV power and temperature. Real-world datasets that include solar PV power and temperature measurements in California are analyzed and the effectiveness of the joint model in providing probabilistic forecasts is verified. The proposed forecasting methodology outperforms several reference methods thus portraying that the underlying correlation between temperature and solar PV power is well defined and only requires a simple lower-complexity sampling space.

Low-Complexity Second-Order Volterra Equalizer for DML-Based IM/DD Transmission System

Volterra nonlinear equalizers (VNLEs) are known to be effective in compensating for nonlinear distortions. However, the major drawback of these equalizers is the huge implementation and computation complexity. In this article, we propose and demonstrate low-complexity quadratic VNLEs specifically designed for directly modulated laser (DML)-based intensity-modulation/direct-detection (DD) systems. We first derive an analytical expression of the quadratic Volterra model for DML/DD systems and then identify the nonlinear distortion terms mainly contributing to the system performance. We show that the interplay between the DML's adiabatic frequency chirp and fiber chromatic dispersion produces the quadratic beating between the signal and the sum of time-sampled signals, which, in turn, makes the coefficients of some beating terms in Volterra series similar. Based on these findings, we reduce the number of coefficients to be determined in the quadratic VNLEs by grouping those beating terms. The performance of the proposed VNLEs is evaluated experimentally on a 56-Gb/s 4-ary pulse amplitude modulation link implemented by using a 1.55-μm DML. We measure the bit-error ratio performance of various equalizers, including feedforward equalizer, polynomial nonlinear equalizer, diagonally-pruned (DP) VNLE, and proposed quadratic VNLE, and compare them in terms of receiver sensitivity and implementation complexity. The results show that the proposed VNLE outperforms the DP-VNLE, although the implementation complexity of the two equalizers is similar. Also presented in this article is an algorithm for optimizing the parameter required in the proposed VNLE.

Control of VSC for enhancement of power quality in off‐grid distributed power generation

In this study, a control algorithm based on non-linear adaptive second-order Volterra filter (NLVF) is utilised to operate the voltage-source converter (VSC). The main role of VSC is to supply compensating current to improve the power quality in wind-based off-grid distributed power generation system. This off-grid system consists of three-phase induction generator, VSC and non-linear loads. The proposed control algorithm is adaptive as well as non-linear in nature. The characteristic of Volterra series is exploited to generate the reference source current for VSC operation. The NLVF estimates the active and reactive weights of fundamental load current for reference current generation. It mitigates the power quality problems created by disturbances in non-linear load or wind oscillations on off-grid system. The solution of power quality problems such as the presence of harmonics in supply current, load unbalance and reactive power compensation is obtained. Nevertheless, voltage and frequency control are also achieved. The VSC is used as a shunt compensator in the system. The entire off-grid system is designed and validated through the laboratory prototype development. The objective of the system is achieved and the steady state and dynamic performance is found satisfactory.

Overestimation problem with ANN and VSTF in optical communication systems

The authors investigated the problem of overestimation with the Volterra series transfer function (VSTF) and an artificial neural network (ANN), which are used for non-linear equalisers in optical communication systems. The results revealed that the risk of predicting a pseudo-random binary sequence (PRBS) pattern, which causes overestimation of the equaliser performance, occurs not only with an ANN but also with the VSTF. When using PRBS9, PRBS11 and PRBS15, the number of taps of a feedforward tapped delay line, which is required in the VSTF to predict the PRBS pattern, was the same as that with the ANN. When the second-order Volterra kernels were omitted, a larger number of taps was required in the VSTF to observe the overestimation.

Heat Conduction at a Variable Heat-Transfer Coefficient

Some practically important problems of unsteady heat conduction with a time-variable relative heat-transfer coefficient are considered. Various approaches to finding a solution to the analytical problem are systematized: decomposition of the generalized integral Fourier transform, the serial expansion of a sought temperature function, and the reduction of the problem to an integral, second-order Volterra equation. It is demonstrated that the solution is reduced to an infinite series of successive approximations of different functional form in all cases, and the main objective of each of these approaches is to find the most advantageous first approximation. Particular cases of the time dependence of the relative heat-transfer coefficients are considered: linear, exponential, power, and root cases. Analytical solutions and numerical experiments are described, and some specific features of the temperature curves of a number of mentioned dependences are revealed. It is established that the picture of the change in the temperature curve for the linear time law of the heat-transfer coefficient becomes appreciably different from the classic case of a constant coefficient, whereas the exponential dependence does not introduce any essential changes.

Adaptive Volterra Filter for Parallel MRI Reconstruction

Parallel magnetic resonance imaging (MRI) technique is able to accelerate MRI speed for reducing costs and enhancing patient’s comfortability. Parallel MRI can be categorized into two types: image-based and k-space-based methods. For k-space-based parallel MRI, missing k-space data is reconstructed by interpolating existing acquired k-space data with appropriate coefficients, which is generally considered as a linear process. However, noise cannot be suppressed or removed during the linear reconstruction process and therefore reconstructed image often suffers serious noise, especially when the acceleration factor is high. Non-linear filters are known to remove non-linear noise better. Based on the Volterra series that discovers and removes the second-order non-linear noise, we proposed a non-linear reconstruction strategy called adaptive Volterra generalized autocalibrating partial parallel acquisition (AV-GRAPPA) to reconstruct the unacquired k-space signals. For the proposed AV-GRAPPA, optimal selection of the second-order Volterra series terms is adjusted and determined for optimizing reconstruction quality. Experimental results show that the proposed method is able to better remove the reconstruction noise and suppress aliasing artifacts.

Recursive Geman–McClure Estimator for Implementing Second-Order Volterra Filter

The second-order Volterra (SOV) filter is a powerful tool for modeling the nonlinear system. The Geman-McClure estimator, whose loss function is non-convex and has been proven to be a robust and efficient optimization criterion for learning system. In this paper, we present a SOV filter, named SOV recursive Geman-McClure, which is an adaptive recursive Volterra algorithm based on the Geman-McClure estimator. The mean stability and mean-square stability (steady-state excess mean square error (EMSE)) of the proposed algorithm is analyzed in detail. Simulation results support the analytical findings and show the improved performance of the proposed new SOV filter as compared with existing algorithms in both Gaussian and impulsive noise environments.

RPSOVF Prediction Model for Speech Signal Series Based on UPSO

In this paper, we propose a nonlinear prediction model of speech signal series with an explicit structure. In order to overcome some intrinsic shortcomings, such as traps at the local minimum, improper selection of parameters, and slow convergence rate, which are always caused by improper parameters generated by, typically, the low performance of least mean square (LMS) in updating kernel coefficients of the Volterra model, a uniform searching particle swarm optimization (UPSO) algorithm to optimize the kernel coefficients of the Volterra model is proposed. The second-order Volterra filter (SOVF) speech prediction model based on UPSO is established by using English phonemes, words, and phrases. In order to reduce the complexity of the model, given a user-designed tolerance of errors, we extract the reduced parameter of SOVF (RPSOVF) for acceleration. The experimental results show that in the tasks of single-frame and multiframe speech signals, both UPSO-SOVF and UPSO-RPSOVF are better than LMS-SOVF and PSO-SOVF in terms of root mean square error (RMSE) and mean absolute deviation (MAD). UPSO-SOVF and UPSO-RPSOVF can better reflect trends and regularity of speech signals, which can fully meet the requirements of speech signal prediction. The proposed model presents a nonlinear analysis and valuable model structure for speech signal series, and can be further employed in speech signal reconstruction or compression coding.

Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black–Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton–Jacobi–Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

An efficient algorithm for nonlinear active noise control of impulsive noise

Nonlinear active noise control (NANC) systems employing Volterra filter suffer from the stability issues in the presence of impulsive noise. To solve this problem, we combine the second-order Volterra (SOV) filter and maximum correntropy criterion (MCC) in this paper. The Volterra filter-x maximum correntropy criterion (VF × MCC) algorithm and Volterra filter-x recursive maximum correntropy (VF × RMC) algorithm are applied to reduce the impulsive noise of NANC. We find that VF × MCC algorithm has a low computational complexity and VF × RMC algorithm converges fast. In order to extract their advantages, we further propose a hybrid algorithm based on the VF × MCC and VF × RMC algorithms. In addition, the normalize step-size version of VF × MCC (VF × nMCC) algorithm is developed to improve the robustness and performance. Meanwhile, we adaptively adjust the kernel size of MCC online based on the sample variance of reference signal to improve the performance of the proposed algorithms. Simulation results in the context of nonlinear active impulsive noise control demonstrate that the proposed algorithms achieve much better performance than the existing algorithms in various noise environments.

HEREDITARY EFFECTS OF EXPONENTIALLY DAMPED OSCILLATORS WITH PAST HISTORIES

This paper presents hereditary effects of exponentially damped oscillators with past histories. Unlike the classical viscously damped oscillators, the nonviscously damped ones involve damping forces which depend on time-histories of vibrating motions via convolution integrals. As a result, equations of motion of such systems are a set of coupled second-order Volterra integro-differential equations. In this work, initial value problems for the integro-differential equations are revisited. The initial conditions should contain time-histories of vibrating motions. Then, initialization response of exponentially damped oscillators is obtained. It is used to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved from the theoretical viewpoint and verified by numerical simulations. This reveals that the hereditary effects gradually recede with increasing of time.

Kernel Estimation of Volterra Using an Adaptive Artificial Bee Colony Optimization and Its Application to Speech Signal Multi-Step Prediction

In order to solve parameters selection problem when applying recursive least square (RLS), least mean square (LMS) or normalized LMS (NLMS) algorithms to estimate kernels of second-order Volterra filter (SOVF), a novel adaptive gbest-guide artificial bee colony (AGABC) optimization algorithm is used to derive kernels of Volterra, that is a type of the AGABC-SOVF prediction model with an explicit configuration for speech signal is proposed. The AGABC algorithm modifies the solution search equation of ABC algorithm and combines the best solution with neighborhood information at present iteration, which not only ensures the exploration of the global optimization algorithm but also improves the exploitation. The AGABC-SOVF model is performed to predict speech signal series of the given English phonemes, sentences, and chaotic time series. Simulation results based on benchmark function show that AGABC algorithm performs faster convergence in achieving higher quality solutions than original ABC and other improved ABC algorithms. Prediction results of applying the AGABC-SOVF model to multi-step predictions for Lorenz time series reveal its stability and convergence properties. For the measured multi-frame speech signals, prediction accuracy and length of multi-step prediction using the AGABC-SOVF model are better than that of the ABC-SOVF model. The AGABC-SOVF model can better predict chaotic time series and the real measured speech signal series.

Kernel Estimation of Truncated Volterra Filter Model Based on DFP Technique and Its Application to Chaotic Time Series Prediction

In order to overcome some problems caused by improper parameters selection when applying Least mean square (LMS), Normalized LMS (NLMS) or Recursive least square (RLS) algorithms to estimate coefficients of second-order Volterra filter, a novel DavidonFletcher-Powell-based Second-order Volterra filter (DFPSOVF) is proposed. Analysis of computational complexity and stability are presented. Simulation results of system parameter identification show that the DFP algorithm has fast convergence and excellent robustness than LMS and RLS algorithm. Prediction results of applying DFPSOVF model to single step predictions for Lorenz chaotic time series illustrate stability and convergence and there have not divergence problems. For the measured multiframe speech signals, prediction accuracy using DFPSOVF model is better than that of Linear prediction (LP). The DFP-SOVF model can better predict chaotic time series and the real measured speech signal series.

On the Complexity Reduction of the Second-Order Volterra Nonlinear Equalizer for IM/DD Systems

To cope with the various nonlinear signal distortions in intensity-modulation/direct-detection transmission systems, a theoretical analysis on the Volterra nonlinear equalizer (VNLE) is provided, focusing on computational complexity aspects. The analysis yields a simple reduced-complexity scheme for the second-order VNLE (R2-VNLE) based on a performance-complexity tradeoff. An experimental verification is performed with single-sideband 28-GBaud PAM-4 signals, generated by an electroabsorption modulated distributed-feedback laser, over transmission distances of up to 80 km of standard single-mode fiber in the C-band. A comparison of the results for different equalization schemes, including a signal-signal beat interference mitigation technique, shows superior performance for the R2-VNLE.

Improved adaptive recursive even mirror Fourier nonlinear filter for nonlinear active noise control

In this paper, we analyze the theoretical mechanism for the effectiveness of recursive filters including both linear and nonlinear feedforward and feedback subsections to accurately approximate nonlinear distributions in the active noise control (ANC) system. As a result, we improve the recursive even mirror Fourier nonlinear (REMFN) filter by adding an additional linear section termed REMFNL filter together with its channel reduced form (CRREMFNL) to explore better control performance. As for the linear feedback section introduced, we have studied the bound-input bound-output (BIBO) stability condition for the REMFNL filter and designed a stable control scheme to guarantee the filter satisfying the stability criteria. The performance of the proposed filters equipped with a filtered-x least mean square (FXLMS) algorithm is validated through analyses of computational complexity and simulations of various nonlinearities for nonlinear ANC (NANC) systems. Simulation results demonstrate that the proposed REMFNL and CRREMFNL filters can achieve better performance over the bilinear, recursive second-order Volterra (RSOV), recursive functional link artificial neural network (RFLANN) and standard REMFN filters based on the FXLMS algorithm, which often outperform the conventional Volterra FXLMS (VFXLMS) and filtered-s least mean square (FSLMS) algorithms.

Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations

In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W_{2}^{3} [ a,b ] oplus W_{2}^{.3} [ a,b ]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.

Identification of Volterra kernels for improved predictions of nonlinear aeroelastic vibration responses and flutter

Aeroelastic structural systems are intrinsically nonlinear and accurate predictions of dynamic responses of nonlinear aeroelastic systems have become of paramount importance since these directly affect the accuracy and reliability of subsequent stability analyses. Such nonlinear systems can be generally represented with Volterra series whose kernels have been found to be effective in their dynamic characterizations. This paper examines how first- and second-order Volterra kernels of nonlinear aeroelastic systems can be accurately identified and then incorporated into the theoretical models of aeroelastic analyses such as predictions of dynamic response and onset of aeroelastic flutter. A novel identification method based on correlation analysis to extract frequency components has been developed which can be applied to general nonlinear aeroelastic systems to obtain accurately the required Volterra transfer functions. The method is very accurate and extremely robust against measurement noise contaminations in both input and output signals, due to the correlation scheme which effectively filters uncorrelated signal components. Detailed aeroelastic behavior of a representative pitch-plunge airfoil dynamic model with nonlinear pitch stiffness has been examined. Its Volterra transfer functions are then identified which are found to be close to their exact analytical counterparts, though interaction between kernels becomes apparent as input level increases. Once inverse Fourier transformed, these identified Volterra kernels are then included in the modeling of the dynamics of aeroelastic systems for vibration response and flutter. Extensive numerical simulation results have demonstrated that the proposed method is very accurate and resilient to measurement errors when applied to the identification of second-order Volterra kernels, and the improvement in predictions of vibration response and flutter become significant when the contributions of these second-order Volterra kernels are included in the overall aeroelastic system dynamics. The identification and subsequent inclusion of second-order Volterra kernels into system dynamics model offer improved design capabilities of nonlinear aeroelastic structural systems.