Exact analytical inversion of TSK fuzzy systems with singleton and linear consequents

In literature, exact inversion methods for TSK fuzzy systems exist only for the systems with singleton consequents. These methods have binding limitations such as strong triangular partitioning, monotonic rule bases and/or invertibility check. These extra limitations lessen the modeling capabilities of the TSK fuzzy systems. In this study, an exact analytical inversion method for TSK fuzzy systems with singleton and linear consequents is presented. The only limitation of the proposed method is that the inversion variable should be represented by piecewise linear membership functions (PWL-MFs). In this case, the universe of discourse of the inversion variable is divided into specific regions in which only one linear piece exists for each PWL-MF at most. In the proposed method, the analytical formulation of TSK fuzzy system is expressed in terms of the inversion variable by using linear equations of PWL-MFs. Thus, the fuzzy system output in any region can be obtained by using the appropriate parameters of the linear equations of PWL-MFs defined within the related region. This expression provides a way to obtain linear and quadratic equations in terms of the inversion variable for TSK fuzzy systems with singleton and linear consequents, respectively. So, it becomes very easy to find exact inverse solutions for each region by using explicit analytical solutions for linear or quadratic equations. The proposed inversion method has been illustrated through simulation examples.     

Exact inversion of decomposable interval type-2 fuzzy logic systems

It has been demonstrated that type-2 fuzzy logic systems are much more powerful tools than ordinary (type-1) fuzzy logic systems to represent highly nonlinear and/or uncertain systems. As a consequence, type-2 fuzzy logic systems have been applied in various areas especially in control system design and modelling. In this study, an exact inversion methodology is developed for decomposable interval type-2 fuzzy logic system. In this context, the decomposition property is extended and generalized to interval type-2 fuzzy logic sets. Based on this property, the interval type-2 fuzzy logic system is decomposed into several interval type-2 fuzzy logic subsystems under a certain condition on the input space of the fuzzy logic system. Then, the analytical formulation of the inverse interval type-2 fuzzy logic subsystem output is explicitly driven for certain switching points of the Karnik–Mendel type reduction method. The proposed exact inversion methodology driven for the interval type-2 fuzzy logic subsystem is generalized to the overall interval type-2 fuzzy logic system via the decomposition property. In order to demonstrate the feasibility of the proposed methodology, a simulation study is given where the beneficial sides of the proposed exact inversion methodology are shown clearly.     

Numerical solution of hybrid fuzzy differential equations using improved predictor–corrector method

The hybrid fuzzy differential equations have a wide range of applications in science and engineering. This paper considers numerical solution for hybrid fuzzy differential equations. The improved predictor–corrector method is adapted and modified for solving the hybrid fuzzy differential equations. The proposed algorithm is illustrated by numerical examples and the results obtained using the scheme presented here agree well with the analytical solutions. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated calculations of algorithm.     

Volterra series approximation for rational nonlinear system

Rational nonlinear systems are widely used to model the phenomena in mechanics, biology, physics and engineering. However, there are no exact analytical solutions for rational nonlinear system. Hence, the approximate analytical solutions are good choices as they can give the estimation of the states for system analysis, controller design and reduction. In this paper, an approximate analytical solution for rational nonlinear system is derived in terms of the solution of a polynomial system by Volterra series theory. The rational nonlinear system is transformed to a singular polynomial system with finite terms by adding some algebraic constraints related to the rational terms. The analytical solution of singular polynomial system is approximated by the summation of the solutions of Volterra singular subsystems. Their analytical solutions are derived by a novel regularization algorithm. The first fourth Volterra subsystems are enough to approximate the analytical solution to guarantee the accuracy. Results of numerical experiments are reported to show the effectiveness of the proposed method.     

Inversion of electroencephalography data for a 2‐D current distribution

It is known from the fundamental work of Albanese and Monk that, the recovery of the support of a three dimensional current, within a conducting medium, from measurements of the generated exterior electric potential, is not possible. However, it is possible to recover the support of any other current, which is supported on a set of dimension lower than three. Nevertheless, no algorithm for such an inversion is known. Here, we propose such an algorithm for a two dimensional current distribution, and in particular, we apply this algorithm to the inverse problem of electroencephalography in the case where the neuronal current is restricted to a small disk of arbitrary location and orientation within the brain. The solution of this inverse problem is reduced to the solution of a nonlinear algebraic system, and numerical tests show that the there exists a unique real solution to this system.Copyright © 2014 John Wiley & Sons, Ltd.     

线性Fuzzy方程组(A~)(x~)=(b~)的解

重点研究线性Fuzzy方程组Ax=b的解,其中矩阵A和向量5均以有限Fuzzy数为其元素。文中首先指出,使用扩展原理和Fuzzy数运算规则有时会导致Ax=b没有解。本文以实广义逆矩阵为工具,给出了Ax=b的六个解,证明了它们都是R^n上的Fuzzy向量。     

Analytical inversion of general periodic tridiagonal matrices

In this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices, and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed Toeplitz for both periodic and non-periodic tridiagonal matrices. An efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given, it is suited for implementation using Computer Algebra systems such as MAPLE, MATLAB, MACSYMA, and MATHEMATICA. An example is also given to illustrate the algorithm.     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

复频域卷积的离散算法

导出了适用于计算机进行计算的复频域卷积的离散算法,应用Durbin拉氏变换数值反演法对复频域卷积结果进行数值反演,可获得时域数值解.将该数值解与解析解进行比较表明,数值解具有较高的精度.     

A perturbation treatment for optimal slightly nonlinear systems with linear control and quadratic criteria

Recently, it has been discovered that symbolic algebraic manipulations can be performed on the computer with input/output data in symbolic form. Accordingly, and for slightly nonlinear two-point boundary-value problems, it is feasible to obtain approximate analytical solutions in the form of power series in a small parameter. In these solutions, the boundary values are presented in a literal form. In this paper, the Lie canonical transformation is applied to derive approximate optimal solutions for slightly nonlinear systems with quadratic criteria. The transformation generator is determined by simple partial differential equations of the first order. To determine the arbitrary constants of the transformation in terms of the two-point boundary values, inversion of a vectorial power series in a small parameter is required, and a recursive algorithm for this inversion is given. To express the final solution in terms of these boundary values, a substitution of the inverted vectorial power series into another vectorial power series is also necessary, and a recursive algorithm for this substitution is presented.The authors are indebted to Professor J. V. Breakwell for his suggestions.     

Depth sounding: an illustration of some of the pitfalls of inverse scattering problems

The principal objective of this work is to show how various “connections” between the estimator and the predictor affect the solution of an inverse scattering problem as it is formulated in the frequency domain. We show that when there is little or no connection, it is impossible to obtain a solution. The other extreme, i.e., identity of the estimator and predictor (inverse crime [1]), enables solutions to be obtained, whatever the particular choices of the estimator or predictor, but these solutions are not trivial, as is written in [1], in that they are not unique. Moreover, we show that by a suitable change of external variables (e.g., frequency), one can lift the degeneracy and thereby spot the correct solution, which is unique. In this respect, the inverse crime turns out to be useful in that it enables one to devise methods for resolving the nonuniqueness issue of inverse problems. More generally, we show that successful inversion, in both the frequency and time domains, can be accomplished only when the discrepancy between the estimator and the predictor is small.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

An analytical spectral stiffness method is proposed for the efficient and accurate buckling analysis of rectangular plates on Winkler foundation subject to general boundary conditions (BCs). The method combines the advantages of superposition method, stiffness-based method and the Wittrick–Williams algorithm. First, exact general solutions of the governing differential equation (GDE) of plate buckling considering both elastic foundation and biaxial loading is derived by using a modified Fourier series. The superposition of such general solutions satisfy the GDE exactly and BCs approximately, which guarantees the rapid convergence and high accuracy. Then, based on the exact general solution, the spectral stiffness matrix which relates the coefficients of plate generalized displacement BCs and force BCs is symbolically developed. As a result, arbitrary BCs can be prescribed straightforwardly in the stiffness-based model. As an efficient and reliable solution technique, the Wittrick–Williams algorithm with the J₀ problem resolved is applied to obtain the critical buckling solutions. The accuracy and efficiency of the method are verified by comparing with other methods. Benchmark buckling solutions are provided for plates with all possible boundary conditions. Also, dependence of various factors such as foundation stiffness, load combinations and aspect ratio on the buckling behaviors are investigated.     

Construction of Invertible Input-Output Mappings and Parameter Identification

The problem of continuation of an input-output mapping to a right invertible mapping is solved. The proposed solution is based on transforming the system to a normal form and solving the problem for such systems. The well-known Singh inversion algorithm is modified to calculate the normal forms. It is proved that each step of the modified algorithm can be realized and the result of the algorithm application is a normal form. A new approach to the parameter identification problem based on the inversion of the input-output mapping is proposed to illustrate the application of the results.     

Analytical solutions for heat transfer on fractal and pre-fractal domains

Fractals can be used to represent intricate self-similar geometries, but their application to the representation of physical systems is beset with difficulties which stem from an inability to define traditionally derived-physical quantities such as stress, pressure, strain, heat etc. This paper describes a method for the determination of analytical heat-transfer solutions on pre-fractal and fractal domains. The approach requires the construction of maps from pre-fractal domains to the continuum, which facilitate the application of traditional continuum solution methods. Solutions on fractal domains are achievable with this approach, and are defined to be the limit solution of analytical solutions obtained on the pre-fractals approximating the fractal of interest. This approach avoids many of the complications and technical difficulties arising from the use of measure theory and fractional derivatives, but also infers that the governing heat transfer equations are valid on all pre-fractals. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions presented are limited to one and two-dimensional domains and, in 1-D, are applied to an idealised composite material consisting of relatively small particles of infinitely low thermal conductivity embedded in a relatively large matrix of infinitely high thermal conductivity. The fractal composite system is thus not truly representative of a realistic physical system, but the methods presented do serve to demonstrate how analytical solutions can be attained on dust-like fractal domains. It is demonstrated that a measurable temperature is possible on a fractal structure along with finite measures of heat flux and energy. Transient and steady state thermal solutions are presented. The solutions on a selection of the pre-fractals are compared against finite element predictions to reinforce the validity of the approach.     

LP narrowing: A new strategy for finding all solutions of nonlinear equations

An efficient algorithm is proposed for finding all solutions of systems of n nonlinear equations. This algorithm is based on interval analysis and a new strategy called LP narrowing. In the LP narrowing strategy, boxes (n-dimensional rectangles in the solution domain) containing no solution are excluded, and boxes containing solutions are narrowed so that no solution is lost by using linear programming techniques. Since the LP narrowing is very powerful, all solutions can be found very efficiently. By numerical examples, it is shown that the proposed algorithm could find all solutions of systems of 5000-50,000 nonlinear equations in practical computation time.     

Differential Equations With Fuzzy Parameters

This paper is concerned with systems of ordinary differential equations with fuzzy parameters. Applying the Zadeh extension principle to the equations, we introduce the notions of fuzzy solutions and of componentwise fuzzy solutions. The fuzzy extension of the solution operator is shown to provide the unique fuzzy solution as well as the maximal componentwise fuzzy solution. A numerical algorithm based on monotonicity properties of membership functions is presented, together with a proof of convergence. In an interplay of interval analysis and possibility theory, these methods allow to process subjective information on the possible fluctuations of parameters in models involving ordinary differential equations. This is demonstrated in two engineering applications: a queueing model for earthwork and a model of oscillations of bell-towers.     

Using the Hamming distance to extend TOPSIS in a fuzzy environment

Considering the fact that, in some cases, determining precisely the exact value of attributes is difficult and that their values can be considered as fuzzy data, this paper extends the TOPSIS method for dealing with fuzzy data, and an algorithm for determining the best choice among all possible choices when the data are fuzzy is also presented. In this approach, to identify the fuzzy ideal solution and fuzzy negative ideal solution, one of the Yager indices which is used for ordering fuzzy quantities in [0, 1] is applied. Using Yager’s index leads to a procedure for choosing fuzzy ideal and negative ideal solutions directly from the data for observed alternatives. Then, the Hamming distance is proposed for calculating the distance between two triangular fuzzy numbers. Finally, an application is given, to clarify the main results developed in the paper.     

A Pseudo-rigid model for the inverse dynamics of an Euler beam

Inversion technique has been very successful in the tracking control of nonlinear dynamical systems. However, when applied to manipulators constructed with elastic links, inverse dynamics through direct integration in temporal space causes unbounded controller command. It has been suggested that seeking an inverse dynamics solution for a given tip trajectory with given initial conditions is an ill-posed problem. It has also been suggested that increasing model accuracy by including more terms in a truncated beam model worsens the controller’s ability to stabilize the system dynamics. In this paper, we seek to understand the nature of the inverse dynamics instability and to find an alternative solution. We appeal to the notion of a pseudo-rigid model which describes the beam deflection by a homogeneous displacement field. Particularly, we derive the mode shape in order to yield a bounded inverse dynamics solution. Different from most of the existing solutions where solution stability was achieved through modifying the output function, we modified the inverse dynamics model. A bounded inverse solution and model simplicity provide much needed ease in the design and implementation of an inversion controller. Numerical simulations and experiments have both been conducted to prove the validity of the proposed method.