Pade approximations

Elimination of Ci and C3 from these equations will result in the desired relation between inlet Cj and outlet Co concentrations, although not in an exphcit form except for zero or first-order reactions. Alternatively, the Laplace transform could be found, inverted and used to evaluate segregated or max mixed conversions that are defined later. Inversion of a transform hke that of Fig. 23-8 is facilitated after replacing the exponential by some ratio of polynomials, a Pade approximation, as explained in books on hnear control theory. Numerical inversion is always possible. 

The dead time transfer function has to be handled differently in classical control, and well use the Pade approximation for this purpose. 

To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called Pade approximation, which puts the function as a ratio of two polynomials. The simplest is the first order (1/1) Pade approximation ... 

There are higher order approximations. For example, the second order (2/2) Pade approximation... 

Again, this form introduces poles in the left-hand plane and at least one zero is in the right-hand plane. At this point, the important task is to observe the properties of the Pade approximation in numerical simulations. 

Example 3.2 Using the first order Pade approximation, plot the unit step response of the first order with dead time function ... 

We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in Eq. (3-30) and match the terms to a Taylor s expansion of the exponential function. If we do, we ll find that the (1/1) Pade approximation is equivalent to a third order Taylor... 

The result is the hairline curve in Fig. E3.2. Note how it dips below zero near t = 0. This behavior has to do with the first order Pade approximation, and we can improve the result with a second order Pade approximation. We will try that in the Review Problems. 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

To arrive at the last term, we have used a simple Taylor expansion (e s 1 — 9s) of the exponential term. This is purposely done to simplify the algebra as shown in the next example. (We could have used the Pade approximation in Eq. (6-26), but the result will not be the simple PI controller.)... 

We use the first order Pade approximation for the dead time and isolate the positive zero term as in Eq. (6-33) ... 

When the system has dead time, we must make an approximation, such as the Pade approximation, on the exponential dead time function before we can apply the Routh-Hurwitz criterion. The result is hence only an estimate. Direct substitution allows us to solve for the ultimate gain and ultimate frequency exactly. The next example illustrates this point. 

Let us first use the first order Pade approximation for the time delay function and apply the Routh-Hurwitz criterion. The approximate equation becomes... 

Now, we switch gears and look into the dead time transfer function approximation. To do a Pade approximation, we can use the MATLAB function 3... 

We have expected q(s) = -0.1s + 1, and p(s) = 0.1s + 1. Obviously, MATLAB normalizes the polynomials with the leading coefficients. On second thought, the Pade approximation is so simple that there is no reason why we cannot do it ourselves as in a textbook. For the first order approximation, we have... 

When we use pade () without the left-hand argument [q, p], the function automatically plots the step and phase responses and compares them with the exact responses of the time delay. Pade approximation has unit gain at all frequencies. These points will not make sense until we get to frequency response analysis in Chapter 8. So for now, keep the [q, p] on the left hand side of the command. 

We want to approximate AM(s) by a Pade approximant Am s). The functional form of Am(s) should be such that the corresponding time-dependent function Am(t) is a series of N exponentials describing the relation of AM(t) to AM(oo) = 0. This implies that Am(s) is an [N — 1. /V]-Pade approximant that can be written in the form... 

G. R. Hadley, Wide angle beam propagation using Pade approximant operators. Optics Letters 17, 1426-1428 (1992). 

Recently, Hentschke [45] and also DuPre and Yang [46] proposed empirical interpolation formulas on the basis of Khokhlov and Semenov s a(N). The former used the Pade approximation, while the latter modified Odijk s a(N) so as to agree with Khokhlov and Semenov s in the asymptotic limits of N 1 and N 1 and derived... 

Yokota and Tanimoto [140] have developed this procedure and from their compact Pade approximant formulae, the rate coefficient is given by... 

E. Brandas, O. Goscinski, Variation-Perturbation Expansions and Pade Approximants to the Energy, Phys. Rev. A1 (1970) 552. 

E. Brandas, R.J. Bartlett, Reduced Partitioning Technique for Configuration Interaction Calculations Using Pade Approximants and Inner Projections, Chem. Phys. Lett. 8 (1971) 153. 

The resonance parameters are then extracted by numerically fitting the formula to the results of the scattering calculation. Numerical instabilities that sometimes arise in fitting the pole structure, either using Eq. (7) for one resonance or Eq. (13) for two resonances, may be assuaged using more sophisticated approaches such as the Pade approximation [64]. 

D. Sokolovski, J.F. Castillo, C. Tully, Semiclassical angular scattering in the F+H2 HF+H reaction Regge pole analysis using the Pade approximation, Chem. Phys. Lett. 313 (1999) 225. 

The most common and usually the fastest way to calculate the exponential function of matrices is done by renormalisation followed by Pade-approximation to fifth order and then successive multiplication.108 109... 

J2 sufficiently flexible Pade approximants have been used,... 

The difficulty encountered in evaluating these integrals originates primarily from the round-off errors caused by a finite number of bits available (64 bits, 128 bits, etc.) in the computer work. I call it the digit-deficiency (or d-d) problem. Although this was the suspected cause from the beginning, it was clearly identified by comparing our numerical work [57] with the result obtained by the Pade approximant method [56],... 

As for the dynamic polarizability, which depends on the imaginary frequency iat, it can be well approximated by the one-term Pade approximant,55... 

For practical purposes, the expansions (33), (34) may be truncated after the first few terms, e.g., from C10 = C10(l, 3) + C10(2,2) + C10(3,1) onwards in the case of the dispersion energy. Values of such dispersion coefficients, which have been calculated using dynamic polarizabilities represented by Pade approximants,56-60 have been reported in the literature for many systems of practical interest (see also Section V.A). 

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