Laguerre model

THE LAGUERRE MODEL PROCESS IDENTIFICATION FROM STEP RESPONSE DATA... 

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. 

The process transfer function given by Equations (2.9) and (2.10) is called the Laguerre model. The Laguerre filters in Equation (2.10) have a simple... 

Prom this block diagram, we can derive the Laguerre model in its state space form. Defining the state vector... 

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. 

The integral squared error between the unit impulse response of the process and that of the Nth Laguerre model is defined as... 

Example 2.2. Irrational transfer functions have been approximated in the literature using truncated infinite partial fraction expansions (Partington et al., 1988) and the Lagrange interpolation formula (Olivier, 1992). Here, we will illustrate that this class of linear systems can be efficiently approximated by a Laguerre model based on the minimization of the fi-equency domain loss function in Equation (2.34). We will consider the following system (Partington et al, 1988)... 

The equations for estimating the Laguerre model coefficients using step response data can now be summarized as follows... 

Figure 2.5 Comparison of step responses for Example 2.4 (solid true response dashed Laguerre model)...

This is a high order process with severe nonminimum phase behaviour. Its noise-free unit step response is shown in Figure 2.5. This process step response is sampled with an interval At = 1.5 sec. The key to success with the Laguerre model for such a complicated process is to find the optimal time scaling factor p for a given model order N, particularly when the model order is small. Figure 2.6 shows a 3-dimensional plot of the loss function V — for iV = 1,2,..., 10 and 0 < p < 0.1, where the coefficients... 

Example 2.5. To illustrate the effect of disturbances on the accuracy of the Laguerre model coefficients, we examine the problem of estimating a Laguerre model from the following step response data... 

The first step in estimating a Laguerre model for this process from this step response data is to determine the time scaling factor p and choose a model order N. It is seen from Figure 2.10 that the process has an approximate settling time of 100 sec, which gives us an estimate of the lower bound... 

Table 2.1 lists the estimated coefficients of the bth order Laguerre model obtained from the step response data shown in Figure 2.10, along with the coefficients obtained from the noise-free step response. The step response of the Laguerre model obtained from the noisy step response data is compared with the measured step response data in Figure 2.10 and the model s frequency response is compared in Figure 2.12 with the true process frequency response. 

Since the disturbance power spectrum S w) in this example is limited to a narrow frequency band and it can be shown that the spectrum does not significantly overlap any of the weighting functions Wt( u ) for z = 1,..., 5, it is expected from the variance analysis for the band-limited noise case (Equation (2.95)) that the accuracy of the estimated coefficients should not not be significantly compromised by the disturbance. This is confirmed by the results given in Table 2.1 and Figures 2.10 and 2.12 where it can be seen that the estimated Laguerre model gives a very accurate representation of the true process. 

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