Laguerre coefficients

This section introduces the Laguerre model for representing the process transfer function. The basic idea is to approximate the continuous-time impulse response of the process in terms of the orthonormal Laguerre functions. The Laguerre coefficients themselves will then be defined in terms of both the process impulse response and its frequency response. 

Equation (2.32) corresponds to the original definition of the Laguerre coefficients in Equations (2.6). It can be shown that the solutions of the coefiicients given by Equation (2.32) minimize the integral squared error in... 

The expressions for the Laguerre coefficients in Equations (2.14) in terms of the process frequency response can be derived by minimizing Equation... 

Theorem 2.1 Given that the Laguerre coefficients cj can be obtained from Equations (2.14), and assuming that the true system G s) is L2 stable, then the derivative of the loss function V with respect to the time scaling factor p is given by... 

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. 

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

This is the key equation for estimating the Laguerre coefficients from step response data, and it will be used subsequently for analysis of variance and bias of the estimates as well as for the development of a data pretreatment strategy later in this chapter. However, Equation (2.72) is still not in the final form to be used for computational purposes. Using the estimate of the process settling time Tg, Equation (2.72) can be rearranged into... 

Estim tion of Laguerre Coefficients from Step Response Data... 

Let g to),g ti)f... denote the discretized step response data with sampling interval At = t +x — U for all i, and to = 0. One approach for computing the Laguerre coefficients could be based on the rectangular rule. For instance, the integral equations found in Equations (2.77) would be approximated by... 

Proof The bias of the estimated Laguerre coefficient can be computed using Equations (2.72) and (2.75)... 

By imposing some assumptions on the power spectrum of the disturbance, we can obtain explicit expressions for the variances of the estimated Laguerre coefficients. 

To look at the influence of disturbances on the estimated Laguerre coefficients in the time domain, we rewrite the estimate of the ith coeifficient in Equation (2.72) as... 

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