Z-transform techniques

As in any simation where probability distributions are involved, useful averages are defined that act as hallmarks for related properties. Some of the important distributions conunonly found in polymer populations are outlined in the following section, as derived from the binomial distribution (BD). Complementary approaches to these and other distributions can be found in Chapters 1 and 16 of this book. In particular. Chapter 16 uses the method of moments and z-transform techniques to treat MWD. 

Application of z-Transform Techniques to Polymerization Modeling Consider the polymerization system described by Equations 16.1 and 16.2 and modeled by Equations 16.3-16.8, 16.20, and 16.21. For simphdty, we will assume isothermal polymerization. In addition, one can make the following variable transformation which has the effect of removing monomer concentration from the live chain equations ... 

In free radical polymerization, the product is made up of dead, rather than live chains, since the lifetime of a live chain in only 1-10 s. Therefore, the statistics of the MWD of the dead chains is critical. An approach based on population balances for M, the concentration of dead chains containing n monomer units can be used. This approach is given in detail in Reference [3]. It requires the use of z-transform techniques, and contains a number of assumptions. Here, we will take a simplified approach. If termination occurs solely by disproportionation, the MWD of the dead chains will be identical to that of the live chains. However, since live chains exist for only very short times, while dead chains... 

This system can be solved via z-transform techniques, as was done by KiUcson [11]. Only the results are given here however, by inspection, we can see that the concentration of P goes down, while the concentration of M remains constant. Then the initial concentration of monofunctional monomer (Mjj) controls the ultimate NACL. 

Equation 17.9 may be solved directly Equations 17.10 and 17.11 may be solved by z-transform techniques to give ... 

Equations 17.20 and 17.21 can be solved by z-transform techniques to give the same results for the moment of the live chain NCLD as in the batch case, with a defined by Equation 17.24 (including the residence lime) ... 

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. 

In Chap. 18 we will define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary) and develop transfer functions in the z domain. These fundamentals are then applied to basic controller design in Chap. 19 and to advanced controllers in Chap. 20. We will find that practically all the stability-analysis and controller-design techniques that we used in the Laplace and frequency domains can be directly applied in the z domain for sampled-data systems. 

C. LONG DIVISION. The most interesting and most useful z-transform inversion technique is simple long division of the numerator by the denominator of The ease with which z transforms can be inverted by this technique is one of the reasons why z transforms are often used. 

Typically, one specifies the desired response, C(z)/R(z), which yields from equation (12) the required design of the controller, D(z). In practice, however, this design technique results in a controller which requires excessive valve movement, an undesirable situation. Consequently, Kalman (12) developed a Z-transform algorithm which specifies the desired output, C(z), and the desired valve travel, M(z) for a setpoint change. The desired response and valve travel for a unit step change in setpoint is shown in Figure 22. The system response,... 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

Z-Transform. Z- (zeta) transforms, defined as follows, are the most recent and widely used technique for calculating molecular weight distributions. 

P — Pseudo-steady-state assumption C = Continuous variable for chain length L = Laplace transforms Z = Z-transforms E = Eigenzeit transform M = Moments of distribution N = Numerical techniques... 

Batch Polymerization. The case of stepwise polymerization without termination was originally treated by Dostal and Mark (16) by using the Eigenzeit transformation to linearize the equations, as discussed earlier. Gee and Melville (21) extended this by the same technique to a case where the propagation rate constant varied with molecular size, contrary to the usual assumption. In the case of stepwise polymerization without termination, batch reactions can give a very narrow (Poisson) distribution. Abraham (2) and Kilkson (35) both showed that the use of the Z-transform simplified the handling of this type of mechanism. 

In general, phases (and hence distance Rj) can be determined at present with greater accuracy than amplitudes i.e. co-ordination numbers Aj) the uncertainties in distances, for higher-Z elements are of the order of 0.001-0.002 nm, and in number of atoms in the first co-ordination shell 20%. It has been shown recently that a modification of the usual Fourier transform technique can give accuracies of better than 0.001 nm in distances. Also, the phase problem may be circumvented by considering the beats between two scattering shells. 

Economics in process control, 3, 10-11, 15, 26, 532-34 Environmental regulations, 3 Equal-percentage valve, 254, 255 Equations of state, 57 Equilibria, 56, 78 chemical, 56 phase, 56-57, 71, 75, 78 Error criteria (see Time integral criteria) Euler s identities, 131-32, 149 Experimental modeling, 45, 656 frequency response techniques, 668 process identification, 657-62 time constant determination, 228, 232 Exponential function, 130 approximations, 215-16 Laplace transform, 130 z-transform, 592... 

These were solved thus using generating function techniques, though Laplace transform technique could equally well have been used. The most important quantity of interest for comparison with other analysis is the average number of reactants present at any time, t, (AT) = Z NP,. This... 

This equation can be solved by Laplace transform techniques and Mi expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the Z = 0 (i.e. 

Physically, the rotational transformation is to untwist the twisted helical structure, so that in the rotating frame the medium now appears to be a simple birefringent material with a dielectric tensor Tj. = R T(2) R independent of z. This technique is analogous to the rotational-transformation technique used in magnetic resonance. After the transformation, Eq. (2) becomes... 

The solution by Laplace transform techniques is described in the Appendix (A 1.1.2 Example 4) and with the approximation that Z)r = Dq = Z> is given by... 

The advantage of using a z-transform approach Ues in the fact that very large tables of z-transforms exist, allowing one to take the z-transform of rather complex sununations. The simple example of Uving anionic polymerization with very rapid initiation in a batch reactor will be used to illustrate the technique. 

The discrete counterpart of the Laplace transform is the z-transform. Both techniques, together with some examples will be discussed in this chapter. 

An alternative to discrete PI or PID algorithms is one that is determined by sampled-data techniques using the z-transformation. This algorithm does not have parameters K Tr, and td, but is expressed as a ratio of polynomials in powers of Z whose coefficients are specified to achieve a certain response. Some criteria for selecting coefficients are like the methods described in the previous section while others select the coefficients to obtain a specified type of closed-loop response. For example, Dahlin s method specifies the response to a step change in set point to be a first-order lag with dead time. The minimal prototype sampled-data algorithm is a type of optimal control in which the output is specified to reach set point, that is, zero error, in the fastest time without exlubiting oscillations. 

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