CHOICE OF THE TIME SCALING FACTOR

Example 2.1. Consider the construction of a Laguerre model for the first order process described by 

We can compute the coefficients of the Laguerre model using Equations (2.14) for a positive time scaling factor p 

Therefore, the Nth order Laguerre model for this first order system is 

We can see from Equation (2.28) that this Laguerre model serves only as an approximation to the original system unless p = a. 

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The optimal choice of the time scaling factor p described by Clowes (1965) is generalized here for any L2 stable system. 

Clowes, G. J. (1965), Choice of the time-scaling factor for linear system approximations using orthonormal Laguerre functions , IEEE Transactions on Automatic Control 10, 487-489. 

Hence, two basic factors are to be taken into account in the analysis of electro-optical data. First is the pulse duration. At certain values of the pulse length the response shows steady-like behavior, which is due to the separation of the time scales involved and the restricted sensitivity of the optical response to induced changes in the system. Second is the choice of frequency range for the determination of the Kerr constant. The latter is related to the low-field regime, where the acoustic modes contribute significantly. Adequate separation of orientation and... 

Our results indicate that x-ray fluorescence is highly useful for rapid and reasonably accurate analyses of whole coal for trace elements. Because of the speed and simplicity of the method, it is highly adaptable to large-scale surveys of coal resources. A suite of 24 samples can be analyzed for 21 elements in 3 days by manual instrumentation. While this simple procedure can not be used to determine certain elements, the time-saving factor over other methods (40 or 50 to 1 in the case of bromine by neutron activation) without loss of accuracy may well make x-ray fluorescence the method of choice for many elements. Improved equipment, such as nondispersive systems and automation, could extend the application of x-ray analysis to a dominant position for determining trace elements in whole coal. 

First order plus delay systems are commonly encoimtered in the process industries and therefore it is important to consider the choice of an optimal time scaling factor for this class of systems. Our intention is to derive some empirical rules based on the process time delay and time constant so that a near optimal time scaling factor can be found with little computational effort. 

Example 2.1 illustrated that the optimal value of p for a first order system is equal to the inverse of the process time constant. If the process is higher order but without time delay, satisfactory results can be obtained if p is chosen based on the dominant time constant of the process. However, the presence of delay can greatly affect the optimal choice of p. To examine this problem, we shall first derive an analytical solution for the Laguerre coefficients associated with a first order plus delay system and then find empirical rules for choosing the optimal time scaling factor p. 

The choice of 1 s as the time scale is somewhat arbitrary and is based on the amount of time one typically would be willing to allow for a PSA to bond to judge it tacky. However, in some real applications, it may be necessary for a PSA to form a bond faster, such as in high speed splicing on a paper machine. In such cases, it appears that the PSA will be able to form a bond if the compliance at the actual required time scale meets this same numerical factor. 

In favourable circumstances, over 30 elements can be analysed without recourse to chemical separation, but the time scale of the analysis might be several weeks. In many practical cases, for example the analysis of forensic specimens, time may be the determining factor in the choice of an analytical technique. 

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