FIR model

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. 

It is clear that the foregoing problem formulation necessitates the prediction of future outputs y[k + i k]. This, in turn, makes necessary the use of a model for the process and external disturbances. To start the discussion on process models, assume that the following finite-impulse-response (FIR) model describes the dynamics of the controlled process ... 

With the dimension of multivariable MFC systems ever increasing, the probability of dealing with a MIMO process that contains an integrator or an unstable unit also increases. For such units FIR models, as used by certain traditional commercial algorithms such as dynamic matrix control (DMC), is not feasible. Integrators or unstable units raise no problems if state-space or DARMAX model MFC formulations are used. As we will discuss later, theory developed for MFC with state-space or DARMAX models encompasses all linear, time-invariant, lumped-parameter systems and consequently has broader applicability. 

A sampling period of 0.3 and an FIR model with n = 50 coefficients are used. The following MFC system is used to control the process ... 

Both of these methods likely suffer from errors due to assumptions made. Virial equilibrium may very well be a poor assumption for molecular clouds. On the other hand, the FIR calibration depends on the assumption that the dust-to-gas ratio is the same in neutral gas and in molecular clouds however, grains may be preferentially destroyed by shocks in the lower density neutral component. The FIR model is also very sensitive to the dust temperature. The relation between I(CO) and N(H2) likely depends on a variety of factors besides metallicity (Maloney Black 1988). More work needs to be done to determine the best method to obtain H2 masses. 

Since this work deals with the aggregated simulation and planning of chemical production processes, the focus is laid upon methods to determine estimations of the process models. For process control this task is the crucial one as the estimations accuracy determines the accuracy of the whole control process. The task to find an accurate process model is often called process identification. To describe the input-output behaviour of (continuously operated) chemical production plants finite impulse response (FIR) models are widely used. These models can be seen as regression models where the historical records of input/control measures determine the output measure. The term "finite" indicates that a finite number of historical records is used to predict the process outputs. Often, chemical processes show a significant time-dynamic behaviour which is typically reflected in auto-correlated and cross-correlated process measures. However, classic regression models do not incorporate auto-correlation explicitly which in turn leads to a loss in estimation efficiency or, even worse, biased estimates. Therefore, time series methods can be applied to incorporate auto-correlation effects. According to the classification shown in Table 2.1 four basic types of FIR models can be distinguished. 

This finite number of regressors can be chosen quite large which is indicated by the term non-parsimonious FIR models, see Dayal and MacGregor (1996). 

Alternatively, the noise type can be ignored applying so-called non-parsimoniovs finite impulse-response (FIR) models by increasing N, the number of lagged observations of the control variable(s). This method has some serious drawbacks e.g. biased and instable estimates/forecasts in case of finite samples. See e.g. Dayal and MacGregor (1996) and references therein. 

Recognizing that the reason for lack of smoothness of the FIR models lies with the type of input signals used for identification experiments in the process industries, we have chosen to focus on an alternative model structure for process identification in Chapters 4 and 5. Our approach is fundamentally different from the RR, PLS and SVD approaches in the sense that we approach this problem by first performing a frequency decomposition... 

This section illustrates application of the PRESS statistic for process model structure selection. Ljung (1987) used data collected from a laboratory-scale Process Trainer to illustrate various identification techniques and examined the sum of squared conventional residuals and Akaike s information theoretic criterion (AIC) for model structure selection. Two sets of input-output data collected from this process are available within MATLAB. We use the entire first set of data M = 1000), called DRYER2 in MATLAB, for this study. Two different model structures are examined here, namely the ARX and FIR model structures, with the objective to find the model within a pcurtic-ular structure that produces the smallest PRESS. 

Figure 3.1 Estimated step responses for Process Trainer (solid ARX model dashed FIR model)...

The best FIR model, in terms of predictive capability, for the Process Trainer is a 27th order model (N = 27) corresponding to a PRESS of 9.13. Figure 3.1 shows the step response of the best FIR model and the step response of the best ARX model with d = 2 where it can be seen that the estimated step responses from the two models are almost identical. 

Example 4.4. We will illustrate the properties of the diagonal and off-diagonal elements in the correlation matrix by compauring the FSF result with the correlation matrix that would result with an FIR model structure using an identical input signal. Consider the process described by the transfer function... 

This process has an approximate settling time of 199 sec and is sampled with an interval of 1 sec. Thus the parameter N is chosen to be 199 for both the FSF and FIR models. For the identification experiment, we have used a binary input signal with amplitude equal to unity. The input signal has been taken as a generalized random binary signal (GRBS) with probability... 

Figure 4.9 Elements of the correlation matrix using an FIR model structure (dimension 199 X 199 for Example 4-4- Upper diagram the diagonal elements of the correlation matrix lower diagram row sums of the absolute values of the off-diagonal elements...

In comparison with the FIR model, the FSF model is able to represent... 

These two examples demonstrate that the number of parameters required by the FSF model to accurately construct a step response model is determined by the underlying continuous-time process frequency response. If the process frequency response is relatively smooth and decays quickly to zero at higher frequencies, such as the process in Example 5.1, then the FSF model structure can capture the process dynamics with a very small number of parameters. If the frequency response is complicated and decays slowly to zero at higher frequencies, such as the process in Exaunple 5.2, the number of parameters required in the FSF structure increases. Nevertheless, the FSF structure provides an effective means to describe various types of process dynamics without the need for process structural information and is, in many cases, more efficient than the FIR model structure in terms of the number of parameters required to represent a process with a given accuracy. 

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. 

In the first experiment, a binary input signal with an amplitude of 1 and switching probability of 0.5 is used. This type of input signal, which has the same spectral characteristics as white noise, was shown by Levin (1960) to be the optimal input signal for the estimation of an FIR model. The correlation matrix associated with the least squares estimates of the FIR model parameters is well-conditioned with a condition number of 10.8. Without any noise added to the process output, the corresponding estimated... 

Figure 5.7 Comparison of step responses estimated using an FIR model and a white input signal for Example 5.3 (solid true response dash-dotted estimate with output noise (a = 0.1) dashed estimate toith output noise (a = 0.3 ...

The remaining subsections will examine the results obtained using the FSF approach, FIR models obtained using the least squares method, and models obtained using DMI, a commercially available process identification software package. With this process, there are a total of 15 input-output relationships to be estimated. For brevity, we will only examine a subset of the results to highlight the features of the FSF approach. One key difference between the various approaches that needs to be mentioned at the outset is that the initial value of each step response model (go) has been estimated with the FSF approach but has been set equal to zero with the FIR and DMI approaches. 

DMI is a commercial product of DMC Corporation (Cutler and Yocum, 1991). The selection of DMI models is an interactive process that involves analyzing results for a number of different times to steady state. Smooth and non-smooth step response models, which in the latter case correspond to step response models generated from least squares estimated FIR models, are presented graphically to the user for each input-output pair. A proprietary smoothing method is used to reduce the effect of noise on the non-smooth models while minimizing the residual errors in the fit of the data. A particular model is selected for a controller application if there is a reasonable match between the smooth and non-smooth curves, and if the response appears to have reached a steady state. The selected times to steady state for this system are summarized in Table 5.1. 

For the FIR model estimate, we decided to work with exactly the same set of data collected from the relay experiment. As a result, 662 parameters needed to be estimated based on the estimated value for Tg and the sampling rate for this process (i.e. N = 662). A batch least squares algorithm has been used to estimate these FIR model parameters. The estimated step response models are compared with the true step response in Figme 8.9. After... 

Figure 8.9 Step response for Process A (solid true response dashed estimated response using FIR model after (a) Ts (no model available), (b) 2Ta, (c) and (d)4Ta)...

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