Stochastic Functional Approach Research Articles

Membership-Function-Dependent Design of Quantized Fuzzy Sampled-Data Controller for Semi-Markovian Jump Systems With Actuator Faults

This article concerns a fuzzy sampled-data controller for nonlinear semi-Markovian jump systems (SMJS) via the fuzzy dependent stochastic looped-Lyapunov functional (FSLF) approach. To do this, with the help of the Takagi–Sugeno (T–S) method, the stochastic jump parameters, actuator faults, and logarithmic quantizer are all considered in a unified framework. The stochastic parameters are generated by the semi-Markov process, whose transition rates depend upon the sojourn time. The nonlinear SMJS depicts real-time models subject to unpredictable structural changes, such as single-link robotic arm (S-LRA) systems. A key issue under consideration is how to design a fuzzy sampled-data controller to ensure stochastic stability for the nonlinear SMJS in the presence of actuator faults and state quantization. For this purpose, by using the weak infinitesimal operator of constructed FSLF, sufficient conditions are determined to realize the stochastic stability of the proposed nonlinear systems. Then, the stochastic stability conditions for the proposed sampled-data controller scheme can be derived under a linear matrix inequality framework. The validity of the theoretical findings is verified by Rossler’s chaotic systems. The S-LRA model is demonstrated to illustrate the applicability and efficacy of the proposed controller scheme.

Analytical approach for the Mott transition in the Kane-Mele-Hubbard model

The description of interactions in strongly correlated topological phases of matter remains a challenge. Here, we develop a stochastic functional approach for interacting topological insulators including both charge and spin channels. We find that the Mott transition of the Kane-Mele-Hubbard model may be described by the variational principle with one equation. We present different views of this equation from the electron Green's function, the free-energy, and the Hellmann-Feynman theorem. In particular, we show the stability of the transition line towards fluctuations, in good agreement with numerical results. The band gap remains finite at the transition and the Mott phase is characterized by antiferromagnetism in the $x\text{\ensuremath{-}}y$ plane. The interacting topological phase is described through a ${\mathbb{Z}}_{2}$ number related to helical edge modes. Our results then show that improving stochastic approaches can give further insight on the understanding of interacting phases of matter.

White noise functional integral for exponentially decaying memory: nucleotide distribution in bacterial genomes

We utilize a stochastic functional integral approach that forms a natural framework for analyzing ubiquitous complex sequences of fluctuations with underlying non-Markovian stochastic process beyond fractional Brownian motion. We demonstrate how Hida white noise calculus, guided by mean square deviation (MSD) analysis of empirical data, allows derivation of single nucleotide occurrence probability distributions for whole genomes of four significant species of bacteria: (a) freshwater cyanobacteria Synechococcus elongatus PCC7942, 2.7 Mbp, (b) marine cyanobacteria Prochlorococcus marinus subsp. marinus str. CCMP1375, 1.8 Mbp, (c) pathogenic bacteria Staphylococcus aureus subsp. aureus NCTC 8325, 2.8 Mbp, and (d) Staphylococcus aureus ILRI Eymole1/1, 2.9 Mbp. Here, the stochastic variable is chosen to represent separation distances between succeeding identical single nucleotides where distance is defined as the number of steps through intervening bases. The stochastic parameter set takes values of nucleotide occurrence count along the genome length. The probability density function (PDF) is derived in closed form for the associated stochastic process with exponentially damped memory kernel, and is shown to satisfy a modified diffusion equation with a parameter-dependent diffusion coefficient. The PDF yields an analytical result for MSDs that match empirical plots, showing a rising nonlinear curve that flattens to a plateau starting close to 1 kb, similar to restricted diffusion. The plots exhibit compliance with Chargaff’s second parity rule for nucleotides. The same PDF describes occurrences of single nucleotides adenine, guanine, cytosine, and thymine for all four bacterial genomes considered.

TE Plane Wave Reflection and Transmission from a Two-Dimensional Random Slab - Slanted Fluctuation -

This paper deals with reflection and transmission of a TE plane wave from a two-dimensional random slab with slanted fluctuation by means of the stochastic functional approach. Such slanted fluctuation of the random slab is written by a homogeneous random field having a power spectrum with a rotation angle. By starting with the previous paper [IEICE Trans. Electron., Vol. E92-C, no.1, pp.77-84, January 2009], any statistical quantities are immediately obtained even for slanted fluctuation cases. The first-order incoherent scattering cross section is numerically calculated and illustrated in figures. It is then newly found that shift and separation phenomena of the leading or enhanced peaks at four characteristic scattering angles take place in the transmission and reflection sides, respectively.

A stochastic functional approach to difference scattering from a three-dimensional perfect electric conductor target on randomly rough surface

A stochastic functional approach (SFA) is developed to calculate electromagnetic composite scattering from a three-dimensional perfectly electric conducting (PEC) target on a randomly rough PEC surface. The stochastic Green's function for randomly rough surface is derived, and a novel four-path model is presented to describe the scattering mechanism of target-surface interactions. The SFA is used to obtain the difference scattering cross section due to the rough surface of target. As a simple example of a spherical target with rough surface, the SFA is numerically compared with other numerical approaches, showing its accuracy and effectiveness. Finally, the difference bistatic scattering from a complex electric large target with rough surface and its functional dependence on the parameters are discussed.

A Further Improved Technique on the Stochastic Functional Approach for Randomly Rough Surface Scattering-Analytical-Numerical Wiener Analysis-

This paper proposes a further improved technique on the stochastic functional approach for randomly rough surface scattering. The original improved technique has been established in the previous paper [Waves in Random and Complex Media, vol.19, no.2, pp.181-215, 2009] as a novel numerical-analytical method for a Wiener analysis. By deriving modified hierarchy equations based on the diagonal approximation solution of random wavefields for a TM plane wave incidence or even for a TE plane wave incidence under large roughness, large slope or low grazing incidence, such a further improved technique can provide a large reduction of required computational resources, in comparison with the original improved technique. This paper shows that numerical solutions satisfy the optical theorem with very good accuracy, by using small computational resources.

Stochastic functional analysis of propagation and localisation of cylindrical waves in a two-dimensional random medium

Propagation and localisation of cylindrical waves in a two-dimensional (2D) isotropic and homogeneous random medium is studied using the stochastic functional approach. By expanding the random permittivity fluctuation in the form of a Wiener integral equation, and representing the wave fields by a linear combination of outgoing and incoming waves, the scalar Helmholtz equation is solved in the cylindrical coordinates system. An analytical expression of the cylindrical wave is derived and demonstrates the localisation phenomenon, as well as the wavenumber fluctuation in the random medium. Comparing with the waves in non-random medium, the wave transfer equation between plane wave and cylindrical wave in random medium shows an additional exponential factor to indicate the modulation effect owing to the medium randomness in both the amplitude and phase. Numerical simulations are presented to illustrate the functional dependence of the localisation phenomena.

Stochastic functional analysis of propagation and localization of cylindrical wave in a two-dimensional random medium

Propagation and localization of cylindrical wave in a two-dimensional isotropic and homogeneous random medium is studied. By expanding the random permittivity fluctuation in the form of a Wiener integral equation in the frequency domain, and representing the wave fields by a linear combination of outgoing and incoming waves, the scalar Helmholtz equation is solved by means of stochastic functional approach to obtain the analytical expression of cylindrical wave. The spatial wave energy distribution is derived to demonstrate the localization phenomenon, and the localization length is also calculated. Compared with the waves in non-random medium, the wave transfer equation between plane wave and cylindrical wave in random medium shows an additional exponential factor to indicate the modulation effects due to the medium randomness in both the amplitude and the phase. Numerical simulations are presented to illustrate the functional dependence of the localization phenomenon.

An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis†

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.

TE Plane Wave Reflection and Transmission from a Two-Dimensional Random Slab

This paper reexamines reflection and transmission of a TE plane wave from a two-dimensional random slab discussed in the previous paper [IEICE Trans. Electron., Vol.E79-C, no.10, pp.1327-1333, October 1996] by means of the stochastic functional approach with the multiply renormalizing approximation. A random wavefield representation is explicitly shown in terms of a Wiener-Hermite expansion. The first-order incoherent scattering cross section and the optical theorem are numerically calculated. Enhanced scattering as gentle peaks or dips on the angular distribution of the incoherent scattering is reconfirmed in the directions of reflection and backscattering, and is newly found in the directions of forward scattering and `symmetrical forward scattering.' The mechanism of enhanced scattering is deeply discussed.

Reflection and Transmission of a TE Plane Wave from a Two-Dimensional Random Slab-Anisotropic Fluctuation-

This paper studies reflection and transmission of a TE plane wave from a two-dimensional random slab with statistically anisotropic fluctuation by means of the stochastic functional approach. By starting with a representation of the random wavefield presented in the previous paper [IEICE Trans. Electron., vol.E92-C, no.1, pp.77-84, Jan. 2009], a solution algorithm of the multiple renormalized mass operator is newly shown even for anisotropic fluctuation. The multiple renormalized mass operator, the first-order incoherent scattering cross section and the optical theorem are numerically calculated and illustrated in figures. The relation between statistical properties and anisotropic fluctuation is discussed.

Diffraction and scattering of TM plane waves from a binary periodic random surface

This paper deals with the diffraction and scattering of a TM plane wave from a binary periodic random surface generated by a stationary binary sequence using the stochastic functional approach. The scattered wave is represented by a product of an exponential phase factor and a periodic stationary process. Such a periodic stationary process is regarded as a stochastic functional of the binary sequence and is expressed by an orthogonal binary functional expansion with band-limited binary kernels. Then, hierarchical equations for the binary kernels are derived from the boundary condition without approximation. We point out that binary kernels obtained by a single scattering approximation diverge unphysically when the periodic random surface is zero on average, thus the effects of multiple scattering should be taken into account. The expressions of such binary kernels are obtained using the multiply renormalizing approximation. Then, statistical properties such as differential scattering cross-section and the optical theorem are numerically calculated with the first two order binary kernels and illustrated in the figures. It is found that the incoherent Wood's anomaly appears in the angular distribution of scattering even when the surface has zero average.

TM Plane Wave Reflection and Transmission from a One-Dimensional Random Slab

This paper deals with a TM plane wave reflection and transmission from a one-dimensional random slab with stratified fluctuation by means of the stochastic functional approach. Based on a previous manner [IEICE Trans. Electron. E88-C, 4, pp. 713-720, 2005], an explicit form of the random wavefield is obtained in terms of a Wiener-Hermite expansion with approximate expansion coefficients (Wiener kernels) under small fluctuation. The optical theorem and coherent reflection coefficient are illustrated in figures for several physical parameters. It is then found that the optical theorem by use of the first two or three order Wiener kernels holds with good accuracy and a shift of Brewster's angle appears in the coherent reflection.

Propagation and localization of random plane waves in two‐dimensional homogeneous and isotropic random medium

Random plane wave in two‐dimensional (2D) homogeneous and isotropic random medium is studied by means of stochastic functional approach. The random medium is assumed to be an omnidirectional Gaussian random field with isotropic narrowband spectrum centered at 2k twice as much as the wave number. The approximate stochastic solutions are obtained based on some theoretic considerations of the translation operator associated with the homogeneity of the random medium. The random plane wave is exponentially decaying or expanding with distance in the propagating direction indicating that the wave is in the cutoff state as was expected in the study of random cylindrical waves. Comparisons among these solutions and those obtained earlier by Ogura are given to demonstrate that the statistical behaviors of propagating waves have nothing to do with the shape of the wavefront, but are solely determined by the spectral form of the random medium. The presence of wave localization is verified by computer simulations for such a random plane wave field. Also the average and variance for the phase shift and log‐amplitude of the wave are measured from the simulated data to compare with the theoretical values. The agreements between the theory and the simulation are shown to be satisfactory in spite of the approximate theoretical formulas.

Scalar plane wave scattering from a thin film with two-dimensional fluctuation**

This paper deals with a scalar plane wave scattering from a thin film with two-dimensional fluctuation by means of the stochastic functional approach. The refractive index of the thin film is written as a Gaussian random field in the transverse directions with infinite extent, and is invariant in the longitudinal direction with finite thickness. An explicit form of the random wavefield involving effects of multiple scattering is obtained in terms of a Wiener–Hermite expansion under small fluctuation. The first- and second-order incoherent scattering cross-sections are calculated numerically and illustrated in figures. In the incoherent scattering, scattering ring, quasi-anomalous scattering, enhanced scattering and gentle enhanced scattering may occur. ** Part of this paper was presented at the 2006 Progress in Electromagnetics Research Symposium [1].

Propagation and localization of plane waves in two-dimensional homogeneous and isotropic random medium

Propagation and localization of plane waves in two-dimensional homogeneous and isotropic random medium is studied. The approximate stochastic solution of the plane wave and the propagation behaviors are obtained by means of stochastic functional approach. The results verify the phenomenon of localization of waves in random medium. The agreements between the theory and the computer simulations are shown to be satisfactory in spite of the use of approximate theoretical formulas.

TM plane wave scattering and diffraction from a randomly rough half-plane: (part I) Derivation of the random wavefield representation*

This paper deals with a scattering and diffraction problem by a one-dimensionally random rough half-plane with perfect conductivity when a TM plane wave is incident. Since a divergence difficulty appears in a perturbation analysis for such a problem, a new formulation by a combination of the stochastic functional approach and the Wiener-Hopf technique is presented. By the D a -Fourier transformation, an inhomogeneous scattered wavefield is transformed into an s-homogeneous random field, where s is a complex wavenumber parameter. Such an s-homogeneous random field is represented in terms of a Wiener-Hermite expansion with unknown coefficients called Wiener kernels. Hierarchical equations for the Wiener kernels are derived by the boundary condition. First, a Wiener-Hopf equation is derived from the hierarchical equations by approximate evaluations of scattering processes due to the surface roughness. Next, the Wiener-Hopf equation is solved by the Wiener-Hopf technique, and then the Wiener kernels are determined. By the inverse D a -Fourier transformation, the inhomogeneous random wavefield is made up of three types of Fourier integrals representing effects of multiple scattering due to surface roughness, single edge diffraction by the edge and interactions between them. As a result, any statistical properties of the random wavefield may be calculated reasonably. *A part of this paper was presented at the 2000 Progress in Electromagnetics Research Symposium [1].

A double Kirchhoff approximation for very rough surface scattering using the stochastic functional approach

On the basis of the stochastic functional approach, an analytical derivation for the incoherent scattering angular distribution contributed from the double Kirchhoff solution is presented for very rough surface, that is, with both the surface roughness and the correlation length comparable to the wavelength. By the second‐order Kirchhoff approximation and using the propagation shadowing function, the incoherent scattering distribution can be numerically calculated by only the two‐dimensional integrals. The numerical calculations are made for the normalized roughness σ/λ = 1.0, 1.5, and 3.5 and the normalized correlation length ℓ/λ = 1.8 and 5.0. The results show that the backscattering enhancement comes from the cross terms in the second‐order Kirchhoff approximation. However, in order to obtain the calculations agreement well with the Monte Carlo simulation, the angular shadowing effect should also be included, in addition to the propagation shadowing effect.

Enhanced scattering from a thin film with one-dimensional disorder

This paper deals with a TE plane wave scattering from a thin film with one-dimensional disorder by means of the stochastic functional approach. The thin film is one-dimensionally inhomogeneous in the horizontal direction with infinite extent, and is homogeneous in the vertical direction with finite thickness. Based on an approximate wavefield representation in terms of a Wiener–Hermite expansion in a preceding paper (Tamura et al., 2004, Waves in Random Media, 14, 435–465), the first- and second-order incoherent scattering cross-sections are presented in explicit forms and scattering properties are discussed. The scattering properties vary entirely with the film thickness. In a case where the thickness is smaller than a few wavelengths in the thin film, enhanced scattering and associated enhanced scattering may appear as sharp peaks or dips on the second-order incoherent scattering distribution if the thin film has guided wave modes. When the thickness becomes sufficiently larger than the wavelength inside the film, a new enhanced scattering phenomenon appears as gentle peaks on the second-order incoherent scattering distribution in four special directions. Such four directions are the directions of forward scattering, specular reflection, backscattering, and the symmetrical direction of forward scattering with respect to the normal of the film surface.The first-order incoherent scattering occurs distinctly in four such directions. Such enhanced scattering is independent of the existence of the guided wave modes inside the thin film, and deeply relates to the structure of the thin film with one-dimensional disorder that has infinite correlation in the vertical direction. For SiC and glass thin films having one-dimensional disorder with a Gaussian correlation and three types of exponential correlation, the first- and second-order incoherent scattering cross-sections are illustrated in figures. The narrow enhanced scattering peaks appear for the glass film in a thin case. The gentle enhanced peaks turn up for both the SiC and glass films in a thick case. Furthermore, the optical theorem is calculated for several cases. It is then found that the error of the optical theorem decreases and the performance of the wavefield is improved by taking into account the second-order incoherent scattering.

Enhanced backscattering from a metallic cylinder with a random rough surface

The scattering of a TE-polarized electromagnetic wave from a metallic cylinder with a slightly random rough surface is studied by means of the stochastic functional approach. The incoherent scattering distribution can be calculated from the Wiener coefficients up to the second order and demonstrates the enhanced backscattering peak, which is attributed to the participation of the surface plasmon waves supported by the cylindrical metal surface as the intermediate scattering processes in multiple scattering.