Disturbance model

Like MLR, however, one must be careful to avoid the temptation of overfitting the PCR model. In this case, overfitting can occur through the use of too many principal components, thus adding unwanted noise to the model and making the model more sensitive to unforeseen disturbances. Model validation techniques (discussed in Section 12.4) can be used to avoid overfitting of PCR models. 

In equation (13) b and d represent the time delays in the system, (p, q) is the order of the noise model, (r, s) is the order of the deterministic model, and z and b are shift operators defined as z iyt = Biyt = Yt-i The component et or at represents uncorrelated white noise which is passed through a transfer function to describe the noise or disturbance model. ut is the input and yt is the output. Gp and Gn are discrete transfer functions. 

For the disturbance model given in equation (ll) the minimum variance forecast is given by... 

For processes which operate over a wide range of conditions or production rates it is to be expected that the process and disturbance model parameters will be changing with time. With a very minor change in the recursive estimation algorithm one can use this scheme to track these slowly moving processes and thereby keep the controller well tuned at all times. 

Stochastic disturbance model There are various possibilities for using stochastic disturbance models other than the zero-order model shown in Eq. (13) (Ljung, 1987). 

MacGregor and Tidwell (1979) illustrate some of the steps involved in running plant experimentation, building these process and disturbance models, and implementing simple optimal controllers on some continuous condensation polymerization processes. A number of similar applications to continuous emulsion polymerization processes have also been made. 

Adjustments for a Jump Disturbance Model. Journal of Quality Technology, 32(4), 379-394. 

The essential step in the LQG benchmark is the calculation of various control laws for different values of A and prediction (P) and control (M) horizons (P = M). This is a case study for a special type of MPC (unconstrained, no feedforward) and a special parameter set (M = P) to find the optimal value of the cost function and an optimal controller parameter set. Using the same information (plant and disturbance model, covariance matrices of noise and disturbances), studies can be conducted for any t3q>e of MPC and the influence of any parameter can be examined. These studies... 

A Raich and A Cinar. Multivariate statistical methods for monitoring continuous processes Assessment of discrimination power of disturbance models and diagnosis of multiple disturbances. Chemometrics Intel Lah. Sys., 30 37-48, 1995. 

A Rigopoulos, Y Arkun, and F Kayihan. Identihcation of full prohle disturbance models for sheet forming processes. AIChE J., 43(3) 727-739, 1997. 

All of the linear C R measures use the approximations, F s and which describe the effects of the control variables and disturbances, respectively, on the process outputs. A commonly used controllability measure is the relative-gain array (RGA Bristol, 1966), which relies only on F s. The disturbance condition number (DCN Skogestad and Morari, 1987) and the disturbance cost (DC Lewin, 1996) are resiliency measures that require a disturbance model, in addition to P s. These C R measures are especially useful in... 

If, in continuous polymerization, the samples are sufficiently infrequent that the process settles between samples, and if the samples are not autocorrelated, the procedures outlined above can be used. This amounts to manual steady-state control with the need for control identified by the control chart. If the process is fast compared with the sampling interval, but the samples are autocorrelated (as is often the case), a controller can be developed which specifies the correction to be made to the process. A process model is needed for a process which is fast compared with the sampling interval, this can be simply a gain between the manipulated input and the process output. A disturbance model is also necessary a simple but effective model for continuous process systems is that of an integrated white noise sequence. With these assumptions, a minimum variance controller may be derived [45]. This takes the form of the following discrete pure integral controller ... 

There are several elaborations of the basic ARX model, where different disturbance models are introduced. These include well known model types, such as ARMAX, Output-Error, and Box-Jenkins (Knudsen, 1994 Stoicaeto/., 1985 Van Overschee and DeMoor, 1996). 

The general structure is defined by giving the time-delays nk and the orders of these polynomials (i.e., the number of poles and zeros of the dynamic model from m to y, as well as of the disturbance model from e to y). 

Note thatzl( ) corresponds to poles that are common between the dynamic model and the disturbance model. Likewise Fi q) determines the poles that are imique for the dynamics from input i, and D q) the poles that are unique for the disturbances. 

Feedback inventory controllers Eq. 7-8 Stochastic disturbance model... 

Sensor (Gj) This gives the model for how the sensor or measurement device works and responds to changes in the process. In most cases, since only the measured values are available, it is useful to lump this model together with the process and disturbance models. This block is useful to remind the reader that unless a variable can be measured, then it caimot be used for control. 

Together the process and disturbance models create the plant model. 

As when analysing a time series, the stationarity of the disturbance signal is an important characteristic to consider. If the output is not stationary, then all the data must be differenced in order to obtain a stationary model. If the data is differenced k times, then the disturbance model will be of the form... 

If the data obtained are routine operating data, then the process model, Gp, can only be identified if the controller transfer function has a higher order than the process and the effect of an incorrect model on the controller transfer function is larger than on the disturbance model or if there is significant nonlinearity in the controller and the error caused by an incorrect controller model is larger than the disturbance error. 

When using the direct approach to closed-loop identificatimi for routine operating process data, it is important to note that due to the weak excitations present, the length of the data series is important for obtaining a good estimate of the parameters. For first-order models, about 2,000 data points are required (Shardt and Huang 2011). Furthermore, small, but consistent, changes in the overall disturbance model can render the identification of the process difficult. 

In closed-loop identification, it is not necessary to accurately specify both the process and disturbance models. 

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