Discrete Laplace transform

It transpires that the general solution of the recombination equations may be found using a discrete Laplace transform. If a is a vector with components ap j = 1,2,..., then the discrete Laplace transform of a is defined as... 

Taking the discrete Laplace transform of this equation and noting the relation mentioned in the preceding text regarding the convolution operator, gives d(p /dx = (p)2, which has solution... 

A more advanced method is the inverse (discrete) Laplace transformation (or probability generating function [pgf]) method (Asteasuain et al., 2002a,b, 2004). In this method, the CLD is reconstracted from the integrated moment equations, as illustrated in Fig. 10.6. A number and mass probability are first introduced for the living polymer molecules ... 

Numerical Laplace Transform Discrete Laplace Transform... 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

To analyze systems with discontinuous control elements we will need to learn another new language. The mathematical tool of z transformation is used to desigii control systems for discrete systems, z transforms are to sampled-data systems what Laplace transforms are to continuous systems. The mathematics in the z domain and in the Laplace domain are very similar. We have to learn how to translate our small list of words from English and Russian into the language of z transforms, which we will call German. 

This Exercise demonstrates that some care is needed in utilizing (4.3) for finding n and x because pRtm(X) may have a singularity X = 0. Let us suppose in particular that the M-equation has a discrete eigenvalue spectrum, as in V.7. The lowest eigenvalue is zero, with eigenfunction psn = pn m(oo). Then the Laplace transform has a pole,... 

The transfer function approach will be used where appropriate throughout the remainder of this chapter. Transfer functions of continuous systems will be expressed as functions of s, e.g. as G(j) or H(s). In the case of discrete time systems, the transfer function will be written in terms of the z-transform, e.g. as G(z) or H(z) (Section 7.17). An elementary knowledge of the Laplace transformation on the part of the reader is assumed and a table of the more useful Laplace transforms and their z-transform equivalents appears in Appendix 7.1. 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

To solve Equation (11.8), we introduce the generating function, which is essentially the Laplace transform for discrete functions. We define... 

Sudicky, E. A., and R.G. McLaren. 1992. The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations. Water Resour. Res. 28 499-514. 

In Chapter 28 we will introduce z-transforms, which constitute the main tool for the analysis of discrete-time systems and play the same role as Laplace transforms for continuous systems. 

Input/Output interface, 557-61 Input-output models, 81 discrete-time, 609-26 examples, 81-85, 162, 163, 166 using Laplace transforms, 159-66 Input variables, 12-14 Integral of absolute error, 302 Integral control action, 273, 277-78 advantages and drawbacks, 274-75, 307... 

The Laplace transforms allowed us to develop simple input-output relationships for a process and provided the framework for easy analysis and design of loops with continuous analog controllers. For discrete-time systems we need to introduce new analytical tools. These will be provided by the z-transforms. 

In other words, z-transforms play the same role for discrete-time systems as that played by Laplace transforms for the dynamic analysis and design of continuous open- or closed-loop systems. 

Like the Laplace transforms, z-transforms possess certain properties that we will find very useful in dealing with discrete-time systems. Let us examine these properties. 

Consider the block diagram of a direct digital feedback control loop shown in Figure 29.9. Such loops contain both continuous- and discrete-time signals and dynamic elements. Three samplers are present to indicate the discrete-time nature of the set point j/Sp( ), control command c(z), and sampled process output y(z). The continuous signals are denoted by their Laplace transforms [i.e., y(s), Jn(s), and d(s)]. Furthermore, the continuous dynamic elements (e.g., hold, process, disturbance element) are denoted by their continuous transfer functions, H(s), Gp(s), and GAs), respectively. For the control algorithm, which is the only discrete element, we have used its discrete transfer function, D(z). 

Finally, the Laplace transform of the time-dependent friction is given in discretized form as... 

A discrete view of the transport considers either random trajectories of discrete particles, or random streamtubes of tracer "parcels" carried by a constant flow rate through a network of fractures. For pulse injection of mass Af, we compute the tracer discharge as J(t) = My(t,T), the Laplace transform of y is... 

The multiple-trapping model can also be solved analytically for the transit time by the method of Laplace transforms. The one-dimensional transport equations for the fiee-electron density n(x, t) in a semiconductor with a distribution of discrete trapping levels are... 

Other variants are due to Fano [76], Anderson [77], Lee [78], and Friedrichs [79] and have been successfully applied to study, for example, autoionization, photon emission, or cavities coupled to waveguides. The dynamics can be solved in several ways, using coupled differential equations for the time-dependent amplitudes and Laplace transforms or finding the eigenstates with Feshbach s (P,Q) projector formalism [80], which allows separation of the inner (discrete) and outer (continuum) spaces and provides explicit expressions ready for exact calculation or phenomenological approaches. For modern treatments with emphasis on decay, see Refs. [31, 81]. Writing the eigenvector as [31, 76]... 

In the discrete space the Laplace transform of the memory-function equation (5.38) reads... 

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