Optimal time scaling factor for first order plus delay systems

3 Optimal time scaling factor for first order plus delay systems 

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 transfer function of a first order plus delay system is given by 

Therefore the anal3rtical solution of the Laguerre coefficients for the first order plus delay process is 

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