Showing papers on "Volterra series published in 2023"

Simulation of Nonlinear Dynamic Processes with Volterra Polynomials and Continued Fractions

Abstract: In this paper, we propose a technique for increasing the accuracy of modeling nonlinear dynamics of input-output systems using finite segments (polynomials) of Volterra integral power series. We highlight a case when the Volterra kernel identification problem is solved. The linear part of the Volterra polynomials of power N is formed as an equivalent continued fraction. Assuming that the corresponding Volterra kernel is known with some error, such an approach allows us to obtain a model with a new linear part that depends on the error. At the same time, prognostic properties of the integral model can be improved by the using the connection coefficient. The problem of indentifying such a parameter was solved using a special extremal problem. We analyzed the performance of the approach for some dynamic systems. Tests examples were used to compare our approach with the known modeling method that employs a series of Volterra polynomials whose transient responses are tuned to different test signal heights.

Coherence-Based Input Design for Nonlinear Systems

Abstract: Many off-the-shelf generic non-linear model structures have inherent sparse parametrizations. Volterra series and non-linear Auto-Regressive with eXogeneous inputs (NARX) models are examples of this. It is well known that sparse estimation requires low mutual coherence, which translates into input sequences with certain low correlation properties. This paper highlights that standard optimal input design methods do not account for this requirement which may lead to designs unsuitable for this type of model structure. To tackle this problem, the paper proposes incorporating a coherence constraint to standard input design problems. The coherence constraint is defined as the ratio between the diagonal and non-diagonal entries of the Fisher information matrix (FIM) and can be easily added to any input design problem for nonlinear systems, while the resulting problem remains convex. The paper provides a theoretical analysis of how the range of the optimal objective function of the original problem is affected by the coherence constraint. Additionally, the paper presents numerical evaluations of the proposed approach’s performance on a Volterra series model in comparison to state-of-the-art algorithms.

Optimal Sparse Volterra Modeling for Transient Behavior of Turbofan Engines based on Internet of Things

Abstract: In recent years, Internet of Things (IoT) technologies have been increasingly utilized to collect enormous volumes of performance data for intelligent analysis and modeling of aero-engines. This has aided in the development of numerous data-driven solutions so that a strong knowledge of the intricate operations within the equipment is no longer needed. To characterize the dynamic nonlinear transient behavior of turbofan engines, fast response changes have the possibility of being captured accurately through high-order, long-memory-length Volterra series. However, exponentially increasing coefficients are still challenging to be handled properly. For fast and reliable modeling of turbofan engines, an Optimal Sparse Volterra (OSV) model is developed in this paper by reconstructing sparse nonzero coefficients after a global selection through particle swarm optimization. The OSV model focuses on the optimal sparsity of the Volterra kernels while being insensitive to the signal length. Besides, noise reduction and the correlation analysis method are specifically designed for sensor measurements of low-bypass ratio turbofan engines. The OSV model, as well as retaining the powerful descriptive capability of the Volterra series for nonlinear characteristics, finds the most relevant sets of variables and the set of model parameters automatically under the minimum computing workload. According to the experimental results, when real test data are used for turbofan transient maneuvers, the OSV model ensures that the mean absolute error is less than [Formula: see text] for high-pressure rotor speed, thrust and exhaust temperature. Moreover, the nonzero identification coefficients produced by the OSV model in the experiments are less than 6% of the total coefficients. At the same time, the average running time required by the OSV model is less than 35% of that of traditional identification algorithms.

On the closed-loop Volterra method for analyzing time series

Abstract: The main focus of this paper is to approximate time series data based on the closed-loop Volterra series representation. Volterra series expansions are a valuable tool for representing, analyzing, and synthesizing nonlinear dynamical systems. However, a major limitation of this approach is that as the order of the expansion increases, the number of terms that need to be estimated grows exponentially, posing a considerable challenge. This paper considers a practical solution for estimating the closed-loop Volterra series in stationary nonlinear time series using the concepts of Reproducing Kernel Hilbert Spaces (RKHS) and polynomial kernels. We illustrate the applicability of the suggested Volterra representation by means of simulations and real data analysis. Furthermore, we apply the Kolmogorov-Smirnov Predictive Accuracy (KSPA) test, to determine whether there exists a statistically significant difference between the distribution of estimated errors for concurring time series models, and secondly to determine whether the estimated time series with the lower error based on some loss function also has exhibits a stochastically smaller error than estimated time series from a competing method. The obtained results indicate that the closed-loop Volterra method can outperform the ARFIMA, ETS, and Ridge regression methods in terms of both smaller error and increased interpretability.

Regularised Volterra series models for modelling of nonlinear self-excited forces on bridge decks

Abstract: Abstract Volterra series models are considered an attractive approach for modelling nonlinear aerodynamic forces for bridge decks since they extend the convolution integral to higher dimensions. Optimal identification of nonlinear systems is a challenging task since there are typically many unknown variables that need to be determined, and it is vital to avoid overfitting. Several methods exist for identifying Volterra kernels from experimental data, but a large class of them put restrictions on the system inputs, making them infeasible for section model tests of bridge decks. A least-squares identification method does not restrict the inputs, but the identified model often struggles with noisy (non-smooth) kernels, which is deemed to be unphysical and a sign of overfitting. In this work, regularised least-squares identification is introduced to improve the performance of model identification using least-squares. Standard Tikhonov regularisation and other penalty techniques that impose decaying kernels are also explored. The performance of the methodology is studied using experimental data from wind tunnel tests of a twin deck section. The regularised Volterra models show equal or better results in terms of modelling the self-excited forces, and the regularisation makes the models less prone to overfitting.

Deep Neural Network for the Behavioral Modeling of Memory Effects and Supply Dependency on 10-W Nonlinear Power Amplifiers

Abstract: In this paper, a deep neural network (DNN) model is proposed for the behavioral modeling of nonlinear power amplifiers with supply dependency. Although the conventional nonlinear model, such as the Volterra series, has high accuracy, it is not commonly implemented because of its complexity. However, with manageable complexity, the multidimensional input parameters of the proposed model ensure the modeling of the nonlinear behavior of power amplifiers with supply voltage dependency. The proposed model is trained by multi-tone signals on a 10-W power amplifier and validated by comparing the output spectrum and the third-order intermodulation (IMD3) of the model versus the measured data. The output spectrum shows less than 0.38 dB of error over a bandwidth of 10 MHz and input power from 11 dBm to 17 dBm, and the IMD3 error is less than 0.1 dB over the output power range.

Generalized Pole-Residue Method for Dynamic Analysis of Nonlinear Systems based on Volterra Series

Abstract: Dynamic systems characterized by second-order nonlinear ordinary differential equations appear in many fields of physics and engineering. To solve these kinds of problems, time-consuming step-by-step numerical integration methods and convolution methods based on Volterra series in the time domain have been widely used. In contrast, this work develops an efficient generalized pole-residue method based on the Volterra series performed in the Laplace domain. The proposed method involves two steps: (1) the Volterra kernels are decoupled in terms of Laguerre polynomials, and (2) the partial response related to a single Laguerre polynomial is obtained analytically in terms of the pole-residue method. Compared to the traditional pole-residue method for a linear system, one of the novelties of the pole-residue method in this paper is how to deal with the higher-order poles and their corresponding coefficients. Because the proposed method derives an explicit, continuous response function of time, it is much more efficient than traditional numerical methods. Unlike the traditional Laplace domain method, the proposed method is applicable to arbitrary irregular excitations. Because the natural response, forced response and cross response are naturally obtained in the solution procedure, meaningful mathematical and physical insights are gained. In numerical studies, systems with a known equation of motion and an unknown equation of motion are investigated. For each system, regular excitations and complex irregular excitations with different parameters are studied. Numerical studies validate the good accuracy and high efficiency of the proposed method by comparing it with the fourth-order Runge--Kutta method.

Parameterization of kernels of the Volterra series for systems given by nonlinear differential equations

Abstract: The presented article is devoted on an issue regarding to the transformation of nonlinear models of a certain class to the Volterra functional series. The new identification method based on analytical input and output of a system was developed. The key task of representing kernels was achieved by Frechet functional derivative. Explicit calculation of the functional derivative for the output was solved using mean-value theorem, while the sifting property of the Dirac function was used in order to solve derivative of input. Attention is given to the calculation of the second-order functional derivative. The procedure for adding linear kernels to the composition of a quadratic kernel is described. All techniques of the method to other components of the model are described in detail. The method's outcome is differential equation, which allows representation of kernels. An illustrative example of the transformation of the nonlinear differential Riccati equation is considered. The form of a subsystem consisting of linear and quadratic kernels for adding to a complex system is shown. Linear and quadratic kernels were parameterized using operational calculus within the example. The agreement of the obtained analytical results, with the frequency characteristics, which was obtained by the test signals method, is shown.

Reducing odd-order nonlinear distortions in a low noise amplifier

Abstract: The increase of transmitting devices around leads to the fact that increasing the dynamic range in receivers is becoming more and more important part of the development and design of receiving devices. The aim of the research is to study the possible methods of increasing the dynamic range of receiving devices, to carry out experiments and to evaluate the applicability of these methods. Methods to reduce nonlinear distortions in the receiving path that caused by the operation of a low-noise amplifier (LNA) in a nonlinear mode were investigated. Moreover, the method of modeling nonlinear systems with Volterra series was studied, as well as the application of Volterra series to model nonlinear systems with a polynomial model with memory for low-level signals was considered. Simulations of the receiving channel behavior was described by Volterra series that has been obtained in this work. During experiments with real signals, it was found that the nonlinear equalization effectively suppresses the nonlinear distortions of the LNA.