Showing papers on "Fourier series published in 2017"

The Fourier decomposition method for nonlinear and non-stationary time series analysis.

Abstract: for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary time series. The proposed FDM decomposes any data into a small number of ‘Fourier intrinsic band functions’ (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a finite number of band-limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.

Recursive Least Squares and Multi-innovation Stochastic Gradient Parameter Estimation Methods for Signal Modeling

Abstract: The sine signals are widely used in signal processing, communication technology, system performance analysis and system identification. Many periodic signals can be transformed into the sum of different harmonic sine signals by using the Fourier expansion. This paper studies the parameter estimation problem for the sine combination signals and periodic signals. In order to perform the online parameter estimation, the stochastic gradient algorithm is derived according to the gradient optimization principle. On this basis, the multi-innovation stochastic gradient parameter estimation method is presented by expanding the scalar innovation into the innovation vector for the aim of improving the estimation accuracy. Moreover, in order to enhance the stabilization of the parameter estimation method, the recursive least squares algorithm is derived by means of the trigonometric function expansion. Finally, some simulation examples are provided to show and compare the performance of the proposed approaches.

Stability and accuracy of three Fourier expansion‐based strong form finite elements for the free vibration analysis of laminated composite plates

Abstract: Summary In the present paper, strong form finite elements are employed for the free vibration study of laminated arbitrarily shaped plates. In particular, the stability and accuracy of three different Fourier expansion-based differential quadrature techniques are shown. These techniques are used to solve the partial differential system of equations inside each computational element. The three approaches are called harmonic differential quadrature, Fourier differential quadrature and improved Fourier expansion-based differential quadrature methods. The improved Fourier expansion-based differential quadrature method implements auxiliary functions in order to approximate functional derivatives up to the fourth order, with respect to the Fourier differential quadrature method that has a basis made of sines and cosines. All the present applications are related to literature comparisons and the presentation of new results for further investigation within the same topic. A study of such kind has never been proposed in the literature, and it could be useful as a reference for future investigation in this matter. Copyright © 2016 John Wiley & Sons, Ltd.

Fourier method for recovering acoustic sources from multi-frequency far-field data

Abstract: We consider an inverse source problem of determining a source term in the Helmholtz equation from multi-frequency far-field measurements. Based on the Fourier series expansion, we develop a novel non-iterative reconstruction method for solving the problem. A promising feature of this method is that it utilizes the data from only a few observation directions for each frequency. Theoretical uniqueness and stability analysis are provided. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed method in both two and three dimensions.

An analytical method for vibration analysis of cylindrical shells coupled with annular plate under general elastic boundary and coupling conditions

Abstract: A unified solution for coupled cylindrical shell and annular plate systems with general boundary and coupling conditions is presented in the study by using a modified Fourier-Ritz method. Under the framework, regardless of the boundary and continuity conditions, each displacement for the cylindrical shell and the annular plate is invariantly expressed as the modified Fourier series composed of the standard Fourier series and auxiliary functions. The introduction of the auxiliary functions can not only remove the potential discontinuities at the junction and the extremes of the combination but also accelerate the convergence of the series expansion. All the expansion coefficients are determined by the Rayleigh-Ritz method as the generalized coordinates. The arbitrary axial position of the annular plate coupling with the cylindrical shell considered in the theoretical formulation makes the present method more general. The theoretical model established by present method can be conveniently applied to cylindr...

FAT-based robust adaptive control of electrically driven robots without velocity measurements

Abstract: Recently, regressor-free control approach has been presented in which uncertainties are estimated using function approximation techniques (FAT) such as the Fourier series expansion or Legendre polynomials. However, FAT-based observer design remains as an open problem. With this in mind, this paper presents a robust adaptive controller for electrically driven robots, without any need for velocity measurements. The mixed observer/control design procedure is based on universal approximation theory and using Stone–Weierstrass theorem. To highlight the contribution of the paper, it should be emphasized that in comparison with previous related FAT-based controllers, the proposed controller is simpler and less computational. In addition, the number of required Fourier series expansions, control laws, and also adaptation rules has been reduced. Moreover, the observer design is free of model. Simulation results of the controller on a 6-DOF industrial robot manipulator have been presented which proves robustness of the proposed controller against various uncertainties. The results are also compared to those obtained from Chebyshev neural network.

Uncertainty estimation in robust tracking control of robot manipulators using the Fourier series expansion

Abstract: This paper presents a novel control algorithm for electrically driven robot manipulators. The proposed control law is simple and model-free based on the voltage control strategy with the decentralized structure and only joint position feedback. It works for both repetitive and non-repetitive tasks. Recently, some control approaches based on the uncertainty estimation using the Fourier series have been presented. However, the proper value for the fundamental period duration has been left as an open problem. This paper addresses this issue and intuitively shows that in order to perform repetitive tasks; the least common multiple (LCM) of fundamental period durations of the desired trajectories of the joints is a proper value for the fundamental period duration of the Fourier series expansion. Selecting the LCM results in the least tracking error. Moreover, the truncation error is compensated by the proposed control law to make the tracking error as small as possible. Adaptation laws for determining the Fourier series coefficients are derived according to the stability analysis. The case study is an SCARA robot manipulator driven by permanent magnet DC motors. Simulation results and comparisons with a voltage-based controller using adaptive neuro-fuzzy systems show the effectiveness of the proposed control approach in tracking various periodic trajectories. Moreover, the experimental results on a real SCARA robot manipulator verify the successful practical implementation of the proposed controller.

Modeling of memory-dependent derivative in a fibre-reinforced plate

Abstract: Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of magneto-thermoelasticity to investigate the transient phenomena for a fibre-reinforced thick plate having a heat source in the context of three-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The upper surface of the plate is free of traction having a prescribed surface temperature while the lower surface rests in a rigid foundation and is thermally insulated. Employing Laplace and Fourier transforms as tools, the problem has been solved analytically in the transformed domain. The inversion of the Fourier transform is carried out using suitable numerical techniques while the numerical inversion of Laplace transform is done incorporating a method on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory dependent derivative, magnetic field and reinforcement also.

On General Properties of Eigenvalues and Eigenfunctions of a Sturm–Liouville Operator: Comments on “ISS With Respect to Boundary Disturbances for 1-D Parabolic PDEs”

Abstract: In the paper “ISS with respect to boundary disturbances for 1-D parabolic PDEs” ( IEEE Transactions on Automatic Control , vol. 61, pp. 3712–3724, 2016), input-to-state stability properties are established for 1-D spatially varying parabolic partial differential equations (PDEs) under certain assumptions, imposed on eigenvalues and eigenfunctions of an associated Sturm–Liouville operator. A key assumption on the absolute convergence of an associated Fourier series, composed of the normalized eigenfunctions and inverse eigenvalues of the Sturm–Liouville operator, is analyzed in the present note. General properties of the Sturm–Liouville operator are carried out to demonstrate that such a key assumption becomes redundant for the underlying PDEs with sign-definite sufficiently smooth coefficients.

Stable super-resolution limit and smallest singular value of restricted Fourier matrices

Abstract: We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given $M$ of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance $\Delta$ between the two closest point sources is less than $1/M$. We introduce a clumps model where the point sources are closely spaced within several clumps. Under this assumption, we derive a non-asymptotic lower bound for the minimum singular value of a Vandermonde matrix whose nodes are determined by the point sources. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. The main novelty is that our lower bound obtains an exact dependence on the {\it Super-Resolution Factor} $SRF=(M\Delta)^{-1}$. As noise level increases, the {\it sensitivity of the noise-space correlation function in the MUSIC algorithm} degrades according to a power law in $SRF$ where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the sensitivity of MUSIC. We also provide lower and upper bounds for a min-max error of super-resolution for the grid model, which in turn is closely related to the minimum singular value of Vandermonde matrices.

Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Multiple Fourier Series, Converging in the Mean

Abstract: The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple Fourier series for expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity $k$ ($k\in\mathbb{N}$) with respect to components of the multidimensional Wiener process is proposed and developed. The obtained expansions contain only one operation of the limit transition in contrast to its existing analogues. In the article it is also obtained the generalization of the proposed method for discontinuous complete orthonormal systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$ as well as for complete orthonormal with weight $r(t_1)\ldots r(t_k)$ systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$. The comparison of the considered method with the well-known expansions of iterated Ito stochastic integrals based on the Ito formula and Hermite polynomials is given. The convergence in the mean of degree $2n$ $(n \in \mathbb{N})$ and with probability 1 of the proposed method is proved.

Benchmark solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions

Abstract: In the present article, a new three-dimensional exact solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. The admissible functions of the thick shells are described as a combination of a three-dimensional (3-D) Fourier cosine series and auxiliary functions. Compared with the traditional Fourier series, the improved Fourier series can eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature and obtained by using ABAQUS which is based on the finite element method. Numerous new results for thick open cylindrical shells on Pasternak foundation with elastic boundary conditions are presented. In addition, comprehensive studies on the effects of the elastic restraint parameters, geometric parameters and elastic foundation coefficients are also reported.

A Posteriori Verification of Invariant Objects of Evolution Equations: Periodic Orbits in the Kuramoto--Sivashinsky PDE

Abstract: In this paper, a method for computing periodic orbits of the Kuramoto--Sivashinsky PDE via rigorous numerics is presented. This is an application and an implementation of the theoretical method introduced in [J.-L. Figueras, M. Gameiro, J.-P. Lessard, and R. de la Llave, “A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations,” SIAM J. Appl. Dyn. Syst., to appear]. Using a Newton--Kantorovich-type argument (the radii polynomial approach), existence of solutions is obtained in a weighted $\ell^\infty$ Banach space of Fourier coefficients. Once a proof of a periodic orbit is done, an associated eigenvalue problem is solved and Floquet exponents are rigorously computed, yielding proofs that some periodic orbits are unstable. Finally, a predictor-corrector continuation method is introduced to rigorously compute global smooth branches of periodic orbits. An alternative approach and independent implementation of [J.-L. Figueras, M. Gameiro, J.-P. Less...

Chaos synchronization using the Fourier series expansion with application to secure communications

Abstract: In this paper, a new method for secure communication based on chaos synchronization is proposed. It is consisted of a state feedback controller and a robust control term using the Fourier series expansion for compensation of uncertainties. In comparison with other uncertainty estimators such as neural networks and fuzzy systems, Fourier series are more efficient, since they have fewer tuning. Thus, their tuning process is simpler. Similar to the parameters of fuzzy systems, Fourier series coefficients are estimated online using the adaptation rule obtained from stability analysis. The case study is the Duffing–Holmes oscillator. Also, observer-based secure communication using the Fourier series expansion has been proposed. Simulation results and comparisons, reveal the superiority of the proposed approach.

Nonlocal Mixed-Value Problem for a Boussinesq-Type Integrodifferential Equation with Degenerate Kernel

Abstract: We consider the problem of unique solvability of a mixed problem for a nonlinear Boussinesq-type fourthorder integrodifferential equation with degenerate kernel and integral conditions. A method of degenerate kernel is developed for the case of nonlinear Boussinesq-type fourth-order partial integrodifferential equation. The Fourier method of separation of variables is used. After redenoting, the integrodifferential equation is reduced to a system of countable systems of algebraic equations with nonlinear and complex right-hand sides. As a result of the solution of this system of countable systems of algebraic equations and substitution of the obtained solution in the previous formula, we get a countable system of nonlinear integral equations. To prove the theorem on unique solvability of the countable system of nonlinear integral equations, we use the method of successive approximations. Further, we establish the convergence of the Fourier series to the required function of the mixed problem. Our results can be regarded as a subsequent development of the theory of partial integrodifferential equations with degenerate kernels.