Error-Squared Controller

As is occasionally stated in some texts, error-squared does not compensate for the nonlinearity between level indication and liquid volume in horizontal cylindrical drums 

It is not usual to square each error term in the controller individually. The most common approach is to multiply the controller gain by the absolute value of the error. Omitting the derivative term (since we usually do not require this for averaging level control) the control equation becomes  

The effect of the additional l l term is to increase the effective controller gain as the error increases. This means the controller will respond more quickly to large disturbances and 

Tuning is calculated using the same approach as for the linear algorithm. We first determine for a proportional-only controller based on restoring the flow balance when 

Following the same approach as the linear algorithm, the full tuning becomes 

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Figure 4.15 Impact of size of flow disturbance on error-squared controller...

Ti and are determined as in Equation (4.18). Figure 4.16 shows the performance of this control with G set at 5 % either side of SP. In this form it exhibits to behavioiu similar to that of the error-squared controller in that it will never settle at SP. Within the deadband no control action is taken and so any flow imbalance will be maintained until the level reaches the edge of the band. At which point corrective action is taken to reverse the direction. While again this has little impact on the downstream process, it is undesirable for the same reasons as described for the error-squared controller, i.e. operator acceptance and steady state detection. 

Table 4.1 Effect of error squared control on noise...

The most well-known nonlinear algorithm is error-squared. Since the controller works with a dimensionless error scaled between 0 and 1 (or 0 % and 1(X) %) the square of the error will have the same range. Strictly the error is not squared but multiplied by its absolute value, because we need to retain the sign if the error is negative. The effect is illustrated in Figure 4.13. 

It should be noted that the tuning calculation presented as Equations (4.23) and (4.25) are for the control algorithms exactly as described. DCS contain many variations of the error-squared algorithm. Even relatively minor changes to the algorithm can have significant... 

Example 1 Sample Quantity for Composition Quality Control Testing An example is sampling for quality control of a 1,000 metric ton (VFg) trainload of-Ks in (9.4 mm) nominal top-size bentonite. The specification requires silica to be determined with an accuracy of plus or minus three percent for two standard errors (s.e.). With one s.e. of 1.5 percent, V is 0.000225 (one s.e. weight fraction of 0.015 squared). The problem to be solved is thus calculating weight of sample to determine sihca with the specified error variance. 

If the UCKRON expression is simplified to the form recommended for reactions controlled by adsorption of reactant, and if the original true coefficients are used, it results in about a 40% error. If the coefficients are selected by a least squares approach the approximation improves significantly, and the numerical values lose their theoretical significance. In conclusion, formalities of classical kinetics are useful to retain the basic character of kinetics, but the best fitting coefficients have no theoretical significance. 

The P-matrix is chosen to fit best, in a least-squares sense, the concentrations in the calibration data. This is called inverse regression, since usually we fit a random variable prone to error (y) by something we know and control exactly x). The least-squares estimate P is given by... 

Fig. 15.23 Normalized area occupation of BG and anti BG bacterial spores vs. solution concen tration of three different assays (filled triangle) BG/BSA functionalized control surface, (filled diamond) BG/Anti BG immobilized surface and (filled square) E. colitAnti BG immobilized surface. The surfaces were exposed for 60 min to the respective solutions. Error bars represent 1 standard deviation, n 4. Reprinted from Ref. 21 with permission. 2008 American Chemical Society...

In MPC a dynamic model is used to predict the future output over the prediction horizon based on a set of control changes. The desired output is generated as a set-point that may vary as a function of time the prediction error is the difference between the setpoint trajectory and the model prediction. A model predictive controller is based on minimizing a quadratic objective function over a specific time horizon based on the sum of the square of the prediction errors plus a penalty... 

Referring to Exercise 1, use Cohen-Coon settings from Table 2.2 to obtain the best controller settings for P and PI control from the process reaction curve parameters. Try these out in a simulation. Observe the suitability of the settings from the error e and the integral of its square, EINT2. 

Parameter estimation is also an important activity in process design, evaluation, and control. Because data taken from chemical processes do not satisfy process constraints, error-in-variable methods provide both parameter estimates and reconciled data estimates that are consistent with respect to the model. These problems represent a special class of optimization problem because the structure of least squares can be exploited in the development of optimization methods. A review of this subject can be found in the work of Biegler et al. (1986). 

Several statistics from the models can be used to monitor the performance of the controller. Square prediction error (SPE) gives an indication of the quality of the PLS model. If the correlation of all variables remains the same, the SPE value should be low, and indicate that the model is operating within the limits for which it was developed. Hotelling s 7 provides an indication of where the process is operating relative to the conditions used to develop the PLS model, while the Q statistic is a measure of the variability of a sample s response relative to the model. Thus the use of a multivariate model (PCA or PLS) within a control system can provide information on the status of the control system. 

A statistical algorithm, also known as linear regression analysis, for systems where Y (the random variable) is linearly dependent on another quantity X (the ordinary or controlled variable). The procedure allows one to fit a straight line through points xi, y0, X2,yi), x, ys),..., ( n,yn) where the values jCi are defined before the experiment and y values are obtained experimentally and are subject to random error. The best fit line through such a series of points is called a least squares fit , and the protocol provides measures of the reliability of the data and quality of the fit. 

Statistical Analysis. Analysis of variance (ANOVA) of toxicity data was conducted using SAS/STAT software (version 8.2 SAS Institute, Cary, NC). All toxicity data were transformed (square root, log, or rank) before ANOVA. Comparisons among multiple treatment means were made by Fisher s LSD procedure, and differences between individual treatments and controls were determined by one-tailed Dunnett s or Wilcoxon tests. Statements of statistical significance refer to a probability of type 1 error of 5% or less (p s 0.05). Median lethal concentrations (LCjq) were determined by the Trimmed Spearman-Karber method using TOXSTAT software (version 3.5 Lincoln Software Associates, Bisbee, AZ). 

Fig. 9. Graph shows the logarithm of relative tumor volume as a function of days after treatment. Growth of tumors treated with adenovirus vector (dark square) or with radiation therapy (dark triangle) is diminished relative to that in the control group (light circle). Tumors receiving combined treatment (dark diamond) decreased in size over time. Error bars = SD.

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