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.     

On a multidimensionalS-transform and the realization problem for homogeneous nonlinear systems

A multidimensionalS-transform, related to the multidimensional Laplace transform, is defined in an algebraic fashion. Some properties of this transform with regard to nonlinear system representation are discussed. In particular theS-transform representation leads to a solution of a realization problem for stationary, homogeneous, degreennonlinear systems. For ease of exposition the results are presented for the two-dimensional case only.     

On transfer function representations for homogeneous nonlinear systems

For stationary, homogeneous, nonlinear systems described via the Volterra functional representation, the Laplace transform of the system kernel is commonly used as a "transfer function." Corresponding to the symmetric and triangular forms of the kernel, special forms are obtained for the transfer function. We propose a new form for both the time-domain and transform-domain representation of homogeneous systerns. Properties of these so-called regular forms in relation to the other two forms and in regard to input/output computation, stability, and bilinear realization theory are discussed.     

A scale-invariant internal representation of time

We propose a principled way to construct an internal representation of the temporal stimulus history leading up to the present moment. A set of leaky integrators performs a Laplace transform on the stimulus function, and a linear operator approximates the inversion of the Laplace transform. The result is a representation of stimulus history that retains information about the temporal sequence of stimuli. This procedure naturally represents more recent stimuli more accurately than less recent stimuli; the decrement in accuracy is precisely scale invariant. This procedure also yields time cells that fire at specific latencies following the stimulus with a scale-invariant temporal spread. Combined with a simple associative memory, this representation gives rise to a moment-to-moment prediction that is also scale invariant in time. We propose that this scale-invariant representation of temporal stimulus history could serve as an underlying representation accessible to higher-level behavioral and cognitive mechanisms. In order to illustrate the potential utility of this scale-invariant representation in a variety of fields, we sketch applications using minimal performance functions to problems in classical conditioning, interval timing, scale-invariant learning in autoshaping, and the persistence of the recency effect in episodic memory across timescales.     

分数阶线性系统的内部和外部稳定性研究

介绍了分数阶线性定常系统的状态方程描述和传递函数描述．运用拉普拉斯变换和留数定理，给出并证明了分数阶线性定常系统的内部和外部稳定性条件，并讨论了其相互关系．以一个粘弹性系统的实例验证了上述方法的正确性．     

Distribution of busy period for the bulk-service queueing system Ek/Ma,b/1

Using the Erlangian technique the busy-period equations for the single-server bulkservice system E_k/M^a,b/1. are solved to obtain the Laplace transform of the probability density function (pdf) of the busy period. The Laplace transform is expressed in terms of an easily computable real root of the characteristic equation. Expressions for the mean and variance of the busy-period distribution are given. Explicit results for the pdf of the busy period are obtained for the special systems M/M^a,b/1 and E_k/M^a,a/1.     

Explicit form for impedance of cauer networks

In this paper a new procedure is established to obtain in closed form the poles and zeros of the impedance function of a first Cauer Network. The technique involves the continued fraction representation of the corresponding Laplace transform as a rational function whose numerators and denominators can be represented by tridiagonal determinants.     

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。     

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,     

Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

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.     

STRUCTURED REPRESENTATION OF FMS INTEGRATING SI-NETS AND HIGH LEVEL PETRI NETS

In this paper a structured knowledge-based approach to the representation and scheduling of flexible manufactoring systems (FMSs) is described. Our approach is based on a structured conceptual representation (a KL-ONE-like Si-net representation formalism), extended with an instant-based temporal reasoning formalism. Furthermore, the approach integrates a particular extension to high-level Petri nets (PNs), structured timed colored Petri nets (STCPNs), for the modeling and simulation of the FMS. Such a representation scheme allows us to use SI-nets' good properties related to inference (classification and inheritance), which are lacking in PNs, and at the same time provides an extension toward an explicit representation for time. The integration of Si-nets with PNs is necessary because simulation and low-level coordination of FMSs require a procedural approach that is not within the aims of Si-nets. Therefore, procedural and symbolic levels, corresponding to the different hierarchical levels of the representation and control system of the FMS, coexist in the system. Using a qualitative terminology, we may also call them analog and symbolic knowledge. We assume that such a hybrid representation system may be useful, since a procedural representation, integrated within a logic formalism, can increase the expressive power without complicating the notation or the representation itself. The paper describes both the representational aspects and the modeling of the control system of the FMS, focusing on the interaction mechanisms among the different levels of representation. In particular, we show how an STCPN-based model can be automatically derived starting from the symbolic component of the system. A particular FMS case study, regarding a class of problems of resource-constrained multiproject scheduling (where projects are sets of tasks temporally related), is discussed.     

Investigation of some probabilistic characteristics of one class of semi-Markov wandering with delaying screens

The process of semi-Markov wandering with delaying screens “b” and “a” (a > b > 0) is constructed by the given sequence of independent and identically distributed random vectors (ξ _i , η _i ), i ≥ 1. The integral equation for the Laplace transform by time and the Laplace-Stieltjes transform by the phase of its conditional distribution is derived. If the wandering occurs by a complicated Laplace distribution, the ergodic distribution of the process and its moments are found. Then, the integral equation for the generating function of the conditional distribution of the number of process steps at which it firstly reaches the level a is derived. When the wandering occurs by the simple Laplace distribution, its generating functions and moments are found.     

Teletraffic analysis of cellular communication systems with general mobility based on hyper-Erlang characterization

This paper proposes a new analytical model for the performance evaluation of cellular communication systems. Our model allows more general cases of the distributions of the call time and the cell residence time. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of the channel occupancy time distribution is derived as a function of the Laplace transform of the cell residence time distribution. In particular, the essential quality of service measure such as the forced termination probability is exactly obtained by using a delayed renewal process. From numerical results, it is shown that the non-exponential model has an obvious difference in performance as compared to exponential models.     

On the equivalence between stable inversion for nonminimum phase systems and reciprocal transfer functions defined by the two-sided Laplace transform

In this paper, the two-sided Laplace transform, a classical but not very common mathematical tool, is revived to express the stable inversion for linear nonminimum phase systems that was recently proposed from the viewpoint of state-space representations. It is demonstrated that those two different expressions for the stable inversion are mathematically equivalent. Simple examples are presented to illustrate the two-sided Laplace transform as a direct and intuitive approach to stable inversion. The two-sided Laplace transform approach is also applied to the development of an iterative learning control for nonminimum phase systems that needs neither a precise inversion model nor Fourier-Transform computations, but instead requires only measuring the system response with time reversals.     

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.     

Thermally induced vibrations of beam structures

A general numerical method is developed for determining the dynamic response of beam structures to rapidly applied thermal loads. The method consists of formulating and solving the dynamic problem in the Laplace transform domain with the aid of dynamic stiffness influence coefficients defined for a beam element in that domain and of obtaining the response by a numerical inversion of the transformed solution. Thus, the solution of the associated heat conduction problem, usually obtained by Laplace transform and needed for the computation of the thermal load, can be used in its transformed form. The effects of damping and of axial compressive forces on the structural response are also studied. Three examples are presented in detail to illustrate the proposed method and demonstrate its advantages.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.