System Laplace-transform estimation from sampled data

This paper presents a method for estimating the Laplace transform of a dynamic system, given its input and output in sampled-data form and corrupted by noise. The estimate is made by first estimating the coefficients of the pulse transfer function relating the input and output and then by converting these estimates to estimates of the Laplace-transform coefficients. Whenever Laplace-transform coefficients are estimated from sampled data, certain knowledge about the signals between the sampling instants must be known a priori or be assumed. In the proposed method this knowledge is used explicitly to relate the coefficients of the Laplace transform to those of theztransform. When this knowledge is correct the estimate Laplace-transform coefficients are asymptotically unbiased. As an illustration, the proposed method has been used to estimate the transfer function of a second-order dynamic system. In this example the variances of the estimates have been compared with the Cramer-Rao bound for unbiased estimates.     

Parametric identification of linear systems using Laplace transform

Proposed was a method for identification of the transfer function coefficients of the stationary one-input one-output linear system using the Laplace images of the input and output measurements. The measurement processes are considered over the given set of values of the transformation parameter. The a posteriori density of distribution of the vector of identified parameters was determined with regard for the autocovariance functions of the measurement errors. A generalization to the multiple-input multiple-output linear system with perturbations and measurements of the state coordinates was described.     

Functional approximation for inversion of Laplace transforms via polynomial series

A method for finding the inverse of Laplace transforms using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into Taylor polynomials by use of a suitable transformation. In this paper, the cross product of a polynomial series basis vector is derived in terms of Taylor polynomials, and as a result the inverse of the Laplace transform is obtained, using the most commonly used polynomial series such as Legendre, Chebyshev, and Laguerre. Properties of Taylor series are first briefly presented and the required function is given as a Taylor series with unknown coefficients. Each Laplace transform is converted into a set of simultaneous linear algebraic equations that can be solved to evaluate Taylor series coefficients. The inverse Laplace transform using other polynomial series is then obtained by transforming the properties of the Taylor series to other polynomial series. The method is simple and convenient for digital computation. Illustrative examples are also given,     

Extension of analytical design techniques to multivariable feedback control systems

Analytical design techniques are developed for multivariable feedback control systems. The design includes a saturation constraint and provides for a disturbance at the system output and additive noise at the system input. The system inputs (signal, noise, and disturbance) are assumed to be generated by independent, stationary, stochastic processes that are adequately represented by rational power-spectral-density matrices. System elements are represented by rational transfer function matrices using the bilateral Laplace transform. The design is applicable to linear time-invariant systems. Design formulas are derived for the general case where the transfer function matrix representing the fixed elements of the system may not be square. The basic design consists of minimizing a weighted sum of the output mean-square errors and the mean-square values of a selected set of saturation signals. A variational technique is used in the optimization, and the technique of spectral factorization is used to obtain a solution. An example is presented to illustrate the design procedure.     

Symbolic-numerical solution of systems of linear ordinary differential equations with required accuracy

In the paper, a symbolic-numerical algorithm for solving systems of ordinary linear differential equations with constant coefficients and compound right-hand sides. The algorithm is based on the Laplace transform. A part of the algorithm determines the error of calculation that is sufficient for the required accuracy of the solution of the system. The algorithm is efficient in solving systems of differential equations of large size and is capable of choosing methods for solving the algebraic system (the image of the Laplace transform) depending on its type; by doing so the efficiency of the solution of the original system is optimized. The algorithm is a part of the library of algorithms of the Mathpar system [15].     

An FFT-based algorithm for 2D power series expansions

An effective numerical algorithm based on inverting a specialized Laplace transform is derived for computing the two-dimensional power-series expansion coefficients of a two-variable function. Due to the special structure of the constructed 2D Laplace transform, the accuracy of the inverted function values can be assured effectively by the generalized Riemann zeta function evaluation and the multiple sets of 2D FFT computation. Therefore, the algorithm is particularly amenable to modern computers having multiprocessors and/or vector processors.     

On a functional approximation for inversion of Laplace transforms via shifted Chebyshev series

A functional representation for inversion of the Laplace transform of a function is considered. The function is given as a shifted Chebyshev series expansion. Using special operational properties, each Laplace transform is converted into a set of simultaneous linear algebraic equations that are then easily solved to give the coefficients of the Chebyshev series. The method is simple and very suitable for computer programming. Applications to rational and irrational Laplace transforms are presented to demonstrate the satisfactory results that the method provides.     

一种新的测量系统频率响应的方法

以指数衰减的组合正弦信号作为系统输入信号,同时实时采集系统输出,在已知系统输入及输出的基础上,运用离散傅立叶变换求得系统频率响应,有效期采用频域补偿法对所求得的系统频率响应,并彩频域补偿法对所求得的系统频率响应进行处理,解决了系统的频率响应在高频端不准确这一特殊问题,从而得到了一套对单输入单输出系统频率响应的测量方法,具有较大的实用价值。     

Evaluation of a linear system impulse response and its Fourier transform†

Impulse response of a linear time invariant system is partitioned by dividing the time axis into equal intervals of time. Then the impulse response is expressed as a sum of these partitioned portions. Each individual portion is approximated by a finite sum of orthogonalized sinusoids satisfying integral squared error criteria. Four different sets are given for this purpose. If the time reversed functions from these sets are applied to the system then the sampled values of the system response at the partitioning instants directly yield the system coefficients as required for the least integral squared error. Knowing these coefficients the best approximation to the impulse response can be constructed as illustrated by the examples considered. Sampled values of the Fourier transform of system impulse response are obtained as a by-product.     

Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity.     

Evaluating failure time probabilities for a Markovian wear process

We present simplified analytical results for the numerical evaluation of failure time probabilities for a single-unit system whose cumulative wear over time depends on its external environment. The failure time distribution is derived as a one-dimensional Laplace–Stieltjes transform with respect to the temporal variable using a direct solution approach and by inverting an existing two-dimensional result with respect to the spatial failure threshold variable. Two numerical examples demonstrate that accurate cumulative probability values can be obtained in a straightforward manner using standard computing environments.Scope and purposeReliability models that incorporate the effect of a stochastic and dynamic environment on a unit's lifetime have attracted a moderate amount of attention in the past decade. However, evaluating failure time probabilities using such models is nontrivial in all but a few cases. Kharoufeh [1] provided a closed-form lifetime distribution for a continuous Markovian wear process as a two-dimensional Laplace transform. The main purpose of this paper is to reduce the lifetime distribution to a one-dimensional Laplace transform in order to facilitate simpler numerical implementation.     

Approximation of Laplace transform of fractional derivatives via Clenshaw–Curtis integration

In this paper, we approximate the Laplace transform of fractional derivatives via Clenshaw–Curtis integration. The idea of applying Chebyshev polynomial to the numerical computation of integrals is extended to Laplace transform of fractional derivatives. The numerical stability of forward recurrence relations is considered, which depends on the asymptotic behaviour of the coefficients. Error estimation for the Laplace approximation of the fractional derivatives is also considered. Finally, from the numerical examples, the method seems to be promising for approximation of the Laplace transform of fractional derivative.     

Hybrid laplace transform/weighting function scheme for transient multidimensional conservation equations

A hybrid Laplace transform/weighting function scheme is developed for solving time-dependent multidimensional conservation equations. The new method removes the time derivatives from the governing differential equations using the Laplace transform and solves the associated equation with the weighting function scheme. The similarity transform method is used to treat the complex coefficient system of the equations, which allows the simplest form of complex number functions to be obtained, and then to use the partial fractions method or a numerical method to invert the Laplace transform and transform the functions to the physical plane. Three different examples have been analyzed by the present method. The present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods. It is found that the present method is a reliable and efficient numerical tool.     

Laplace transform analysis of the carbon cycle

A Laplace transform representation is used to describe the changes in atmospheric CO₂ in response to emissions. The formalism gives an explicit representation of generic relations that are less clear when model results are presented as numerical integrations with particular parameter values. In particular, the Laplace transform formalism clarifies some issues involved in inversion of ice-core data and analysis of geosequestration. The airborne fraction is expressed as the emission growth rate multiplied by the Laplace transform of the atmospheric response function, evaluated at the growth rate. This representation emphasises that historical data only capture carbon cycle dynamics over a limited range of time-scales. The Laplace transform formalism provides a basis for expressing uncertainties in the response function in terms of the Padé–Laplace transformation used for fitting sums of exponentials.     

Numerical inversion of the Laplace transform

An extension of Bellman's method for the numerical inversion of the Laplace transform is discussed. This extension is theoretically equivalent to the method of Lanczos. Tables of coefficients are given which facilitate the inversion of the Laplace transform with the aid of a desk computer.     

过程控制常用连续模型的直接辨识法及应用

针对工业过程中最常用的一阶加滞后、二阶加滞后、二阶加零点、二阶加零点及滞后、积分惯性加滞后等环节,给出了基于阶跃响应的连续模型参数直接辨识算法。由传递函数的拉普拉斯逆变换式和对象阶跃响应的采样数据构成模型参数回归表达式,用最小二乘法或辅助变量法直接辨识对象的连续时间传递函数模型参数。仿真与实际应用结果表明,该算法提高了模型辨识精度,减小了对过程的扰动,并且对输出测量噪声不敏感,鲁棒性强,容易编程实现,可提高实际PID控制器参数整定质量。     

A Flexible Inverse Laplace Transform Algorithm and its Application

A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotient-difference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotient-difference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm.     

Matrix operators for numerically stable representation of stifflinear dynamic systems

A new transformation having features similar to the Laplace transform (but numerically oriented) is developed from the Chebyshev polynomials theory. Signals are represented as vectors of Chebyshev coefficients, and linear subsystems as precomputed matrices. The original problem is preprocessed only once to yield matrix invariants for fast recurrent computations. Theoretical implications of the exact digitizing of a tenth-order transfer function and the reduced-order modeling of a stiff system are discussed     

Dynamic response of framed underground structures

The dynamic response of framed underground structures under conditions of plane strain is numerically determined in this work. The soil deposit surrounding such structures is assumed to be horizontally layered and resting on a rigid base from which shear waves originate, and to exhibit linear elastic or viscoelastic material behavior. The methodology consists of applying the Laplace transform with respect to time to the governing equations of motion of the soil and the structure and subsequently constructing dynamic stiffness influence coefficients for typical soil and structure elements. A numerical inversion of the solution obtained by the finite element methodology employing these influence coefficients in the transformed domain yields the response as a function of time. Numerical examples to illustrate the method and demonstrate its advantages are presented.     

Recursive identification of errors-in-variables Wiener systems

This paper considers the recursive identification of errors-in-variables (EIV) Wiener systems composed of a linear dynamic system followed by a static nonlinearity. Both the system input and output are observed with additive noises being ARMA processes with unknown coefficients. By a stochastic approximation incorporated with the deconvolution kernel functions, the recursive algorithms are proposed for estimating the coefficients of the linear subsystem and for the values of the nonlinear function. All the estimates are proved to converge to the true values with probability one. A simulation example is given to verify the theoretical analysis.