Analysis of time-varying scaled systems via general orthogonal polynomials

A very effective method of using the generalized orthogonal polynomials (GOP) for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the differentiation operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomials. The main advantage of using the differentiation operational matrix is that parameter estimation can be made starting at any time of interest, i.e. without the restriction of starting at zero time. In addition, the present computation algorithm is simpler than that of the integration operational matrix. Using the concept of GOP expansion for state and control functions, the differential input-output equation is converted into a set of linear algebraic equations. The unknown parameters are evaluated by a weighted least-squares estimation method. Very satisfactory results for illustrative example are...

An operational matrix for the integration of Chebyshev vector whose elements are Chebyshev polynomials is first derived in this paper. Using the operational matrix of integration, a time‐invariant linear system can be represented by a set of simultaneous linear algebraic equations with the approximation of time functions by Chebyshev series. Analysis and parameter identification of linear system can thus be made by solving the algebraic equations.

The Taylor series is introduced to analyse and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the derived scaled operational matrix, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the Taylor series expansion can be determined recursively by the derived algorithm. The parameter identification problems of scaled systems are also investigated. Owing to the simplicity of the operational matrix of integration and the scaled operational matrix, the Taylor series presents considerable computational advantages when compared with the other possible polynomial series, provided that both the input and output signals are analytic functions of t.

Analysis of linear distributed parameter systems via double Fourier series

Chapter 2 - Simultaneous linear algebraic equations

Piecewise linear functions arc applied to solve the state equations of linear system science with the introduction of the methods of weighted residuals (MWR). There are four different weighted procedures of MWR to be derived: (a) collocation, (b) subdomain, (c) Galerkin's and (d) least-squares methods. Applying the MWR, a linear system can be represented by a set of simultaneous linear algebraic equations, and a recursive algorithm and/or an operational matrix are available. Analysis and parameter identification of stale space equations of linear systems can thus be converted into a set of linear algebraic equations and solved using a recursive algorithm.

The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.

The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.

Based on a hybrid orthogonal function, a new linear, continuous and bounded operator, piecewise general orthogonal polynomials operator (PGOPO), is proposed. Its main properties and operational rules have been strictly constructed based on the theory of convergence in the mean square, and are useful to discuss the orthogonal polynomial approach in the proper mathematics frame. Then applying the PGOPO method to solve the identification problem of nonlinear systems, the convergence analysis of PGOPO approximation method is given. Finally, using the PGOPO method to solve the parameter identification of two kinds of time-varying systems, the simulation studies show that the algorithm is simple and more effective than that of general orthogonal polynomials.

Many attempts have been made to evaluate the buckling loads of plates with various support conditions to predict the strength of plate girders. These attemps have been based on the assumption that the deflections of webplates are relatively small compared with the thickness of the webplates. Due to this assumption, many inappropriate results have been obtained, and the current design practice of plate girders is still based on these results. For this reason, many experimental investigations, and theoretical investigations based on the large deflection theory of plates have been performed.In this paper, a theoretical method of analysis of webplates with large deflections is proposed. A plate girder panel is simulated by an ideal mechanical model consisting of a thin plate and four beams representing the webplate, two vertical stiffeners, respectively. The loading is a general combination of bending moments and shearing stresses. A method of expanding the unknown displacement components into polynomial series is employed with the magnitude of load taken as the expansion parameter. The sets of linearized partial differential equations are then converted into sets of simultaneous linear algebraic equations by means of finite differences and numerically solved by a digital computer.

Axisymmetric nuclei of strain and their applications for multilayered solids

This paper presents a fully complex-valued relaxation network (FCRN) with its projection-based learning algorithm. The FCRN is a single hidden layer network with a Gaussian-like sech activation function in the hidden layer and an exponential activation function in the output layer. For a given number of hidden neurons, the input weights are assigned randomly and the output weights are estimated by minimizing a nonlinear logarithmic function (called as an energy function) which explicitly contains both the magnitude and phase errors. A projection-based learning algorithm determines the optimal output weights corresponding to the minima of the energy function by converting the nonlinear programming problem into that of solving a set of simultaneous linear algebraic equations. The resultant FCRN approximates the desired output more accurately with a lower computational effort. The classification ability of FCRN is evaluated using a set of real-valued benchmark classification problems from the University of California, Irvine machine learning repository. Here, a circular transformation is used to transform the real-valued input features to the complex domain. Next, the FCRN is used to solve three practical problems: a quadrature amplitude modulation channel equalization, an adaptive beamforming, and a mammogram classification. Performance results from this paper clearly indicate the superior classification/approximation performance of the FCRN.

Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

The operational matrix consisting of the product of two time functions, and the operational matrices for forward or backward integration consisting of general orthogonal polynomials are derived, respectively, for the analysis and optimal control of linear time-varying systems with a quadratic performance measure. The present results include results obtained via Chebyshev, Legendre, Laguerre, Jacobi, Hermite and ultraspherical polynomials as special cases.

We derive and factorize the fourth-order q-difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations and express the modified families in terms of the starting ones.