REDUCED ORDER FSF MODEL

Based on these properties, we will refer to n as the reduced order of the FSF model, which represents the number of significant parameters in the FSF model. We must qualify the use of the term reduced order to distinguish the number of significant parameters in the FSF model (n) from the order of the individual FSF filters (JV). This reduced nth order FSF model can be written in the following form 

Intuitively, it would seem that the reduced model order n is related to the properties of the underlying continuous-time system. For instance, it would appear that its value depends on how fast the magnitude of the process frequency response rolls off to zero. We will illustrate this point in the following example. 

The reduced FSF model order n corresponding to maximum truncation levels of 10%, 5% and 1% are listed in Table 4.1 for the above examples, where the neglected frequency response coefficients all have magnitudes less them the indicated percentage of the steady state gain. Note that the reduced orders given in Table 4.1 are independent of the time constant T, 

Effect of FSF Model Reduction on Process Frequency Response 

We can attempt to construct the process frequency response using the reduced order FSF model given in Equation (4.15) by letting = and evaluating 

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This chapter consists of six sections. Section 4.2 introduces the FSF model structure. Section 4.3 examines the properties of the FSF model with a fast data sampling rate. Section 4.4 introduces the concept of a reduced order FSF model. Section 4.5 discusses the use of least squares for estimating the FSF model parameters from input-output data. Section 4.6 excunines the nature of the correlation matrix that arises when using a least squares estimator with an FSF model and the relationship between the elements of this matrix and the energy content of the input signal. 

Figure 4.5 Construction of frequency responses using reduced order FSF models for Example 4-3 (solid true response dash-dotted n = 3 dotted n = 5)...

This section deals with the problem of estimating the parameters of a reduced order FSF model from process input-output data using the least squares algorithm. 

Equation (5.4) gives an explicit relationship between the parameters of the FSF model and the step response coefficients. The step response coefficients, evaluated using a reduced order FSF model, can be obtained from... 

Figure 5.6 Comparison of true step response with that generated from a reduced order FSF model for Example 5.2 (solid true response dotted n = 99 ...

This section is devoted to a simulation example that illustrates the problems associated with obtaining an estimate of the process step response using an FIR model and motivates the use of a reduced order FSF model instead. 

This suggests that the errors in the estimated high frequency parameters are the reason for the lads of smoothness in the step response estimates. To confirm this conjecture, we estimate step response models using various reduced order FSF models. The model orders selected aure n = 99, 49, 25 and 11. Figure 5.11 shows the estimated step response models for these four choices of n where it can be seen that, as more high frequency parameters are deleted from the estimated FSF model, the step response model becomes smoother. Also, as the number of estimated high frequency parameters is decreased, the numerical conditioning of the FSF correlation matrix improves. When n = 49, the condition number is 1684.7, with n = 25 the... 

Figure 5.11 Step response estimates obtained using various reduced order FSF models and an input signal with slow switches for Example 5.3 (a n = 99 b n = 49 c n = 25 d n = 11 ...

We assume that the process being identified is stable, linear and time invariant and can be accurately represented by a reduced nth order FSF model. For an arbitrary process input u k) and the measured process output y k), the frequency sampling filter model can be written as... 

Table 4.1 Reduced FSF model orders corresponding to different truncation levels for the second order system...

For each process output, the p inputs are denoted as ui k), U2 k),..., Up k), the times to steady state for the individual subsystems are given by Ni, N2,- Np, and the reduced orders for each subsystem represented by its own FSF model are chosen to be ni, ri2,..., np. In the matrix representation for this MISO system, the first input ui k) is passed through a set of n frequency sampling filters based on N to form the first n columns in the data matrix, followed by passing the second input U2 k) through a set of U2 frequency sampling filters based on N2 to form the next U2 columns in the data matrix, etc. The parameter vector 9 contains the n firequency response parameters associated with the first subsystem, followed by the n2... 

Estimates of the times to steady state are required for application of the FSF identification method. In order to compare with the DMI results, we have chosen the same times to steady state presented in Table 5.1. The reduced FSF process model orders that were selected through application... 

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