Proportional-on-error

Many believe therefore that this algorithm should be applied on processes where the MV should be adjusted slowly. However, if this performance were required, it could be achieved by tuning the more conventional proportional-on-error algorithm. Conversely it is important to recognise that the proportional-on-PV algorithm can be retuned to compensate for lack of the proportional kick and so respond well to SP changes. This is illustrated in Figure 3.13. 

Figure 3.14 shows the behaviour of each part of the proportional-on-error control algorithm in response to the SP change above. The proportional kick is clear with the proportional part of the controller returning to zero as the error returns to zero. The derivative action is the greatest as the PV peaks, and so permits more proportional and integral action to be used. It too returns to zero as the rate of change of PV returns to zero. 

Figjure 3.14 Response to SP change (proportional-on-error algorithm)... 

Switching the algorithm between proportional-on-error and proportional-on-PV has no affect on the way it responds to load changes. The difference we see is due to the difference in tuning. The more tightly tuned algorithm deviates from SP by less than half and the duration of the upset is also halved. 

Figure 3.17 shows the breakdown of the control action for the load tuning case above. Since the SP is constant, the response would the same for both the proportional-on-error and proportional-on-PV algorithm. 

We will cover later different measures of control performance but the most commonly used is integral over time of absolute error (ITAE). The higher the value of ITAE, the poorer the controller is at eliminating the error. Figure 3.19 shows the impact that switching from proportional-on-PV to proportional-on-error has on ITAE. Both algorithms have been tuned... 

Figure 3.19 Impact of switching from proportional-on-PV to proportional-on-error...

Remember that if a proportional-only controller is configured as proportional-on-PV, it will not respond to changes in SP. This might be considered advantageous since it prevents the operator changing the SP to a value where the offset violates an alarm. However it might create problems with operator acceptance, in which case the proportional-on-error algorithm can be used. 

This is the equation for the proportional-on-error, derivative-on-error noninteractive controller. We can however choose coefficients to produce almost any control algorithm. For a first order plus deadtime processes we use... 

Modern DCS include a number of versions of the PID controller. Of particular importance in the proportional-on-PV algorithm. It is probably the most misunderstood option and is frequently dismissed as too slow compared to the more conventional proportional-on-error version. In fact, if properly tuned, it can make a substantial improvement to the way that process disturbances are dealt with - often shortening threefold the time it takes the process to recover. This is fully explained in Chapter 3. 

A proportional determinate error, in which the error s magnitude depends on the amount of sample, is more difficult to detect since the result of an analysis is independent of the amount of sample. Table 4.6 outlines an example showing the effect of a positive proportional error of 1.0% on the analysis of a sample that is 50.0% w/w in analyte. In terms of equations 4.4 and 4.5, the reagent blank, Sreag, is an example of a constant determinate error, and the sensitivity, k, may be affected by proportional errors. 

From (9.27), we see that this approach will work nicely if the variance is always small Taylor s theorem with remainder tells us that the error of the first-derivative - mean-field - contribution is proportional to the second derivative evaluated at an intermediate A. That second derivative can be identified with the variance as in (9.27). If that variance is never large, then this approach should be particularly effective. For further discussion, see Chap. 4 on thermodynamic integration, and Chap. 6 on error analysis in free energy calculations. 

The annnal corrective maintenance costs from approximately 250 compnter apphcations are surprisingly consistent. Not snrprisingly, the maintenance effort decreased for older applications on the basis that an increasing proportion of errors is corrected over time. Annnal corrective maintenance costs would seem as a rule of thumb to decrease by about one sixth every year (see Figure 17.8). The initial corrective maintenance costs were more dependent on the size of the application than on initial error rate. This is because fewer but bigger errors tend to be addressed in the early years of operation. These maintenance figures assume there are no other user-driven enhancements or system platform upgrades, etc. 

In regression analysis with proportional random errors, we wiU also consider the dispersion around the line in the vertical direction. SD2i is dependent on the measured concentration, Supposing a proportional error relationship and an intercept of the regression Une close to zero, SDji can be characterized by a proportionality factor. ... 

Figure 14-29 Distances from data points to the line in weighted Deming regression assuming proportional random errors in xl and x2.The symmetric case is illustrated with equal random errors and a slope of unity yielding orthogonal projections onto the line. (From Linnet K. Necessary sample size for method comparison studies based on regression analysis. Clin Chem 1999 45 882-94.)...

As mentioned earlier, the matrix-related random interferences may not be independent. In this case, simple addition of the components is not correct, because a covariance term should be included. However, we can estimate the combined effect corresponding to the bracket term, which then strictly refers to the CV of the differences (CV b2-rb])- As in the case with constant standard deviations, information on the analytical components is usually available, either from duplicate sets of measurements or from quality control data, and the combined random bias term in the second bracket can then be derived by subtracting the analytical component from CV21. Systematic and random errors can then be determined, and it can be decided whether a new field method can replace an existing one. Figure 14-31 shows an example with proportional random errors around the regression line. 

Additional information about the nature of the systematic error is obtained when there are two different control materials analyzed by each laboratory. For example, the laboratory s observed mean for material A is plotted on the y axis versus its observed mean for material B on the x-axis these graphs are called Youden plots. Ideally the point for a laboratory should fall at the center of the plot. Points falling away from the center but on the 45° line suggest a proportional analytical error. Points falling away from the center but not on the 45° line suggest either an error that is constant for both materials or an error that occurs with just one material. 

A large proportion of errors in research and engineering as well as in the classroom are due to treating units as if they were not part of the number. In all scientific disciplines, it is essential that a unit be associated with every value in every calculation unless the value is a dimensionless quantity. On September... 

Quantitation by the standard addition technique Matrix interferences result from the bulk physical properties of the sample, e.g viscosity, surface tension, and density. As these factors commonly affect nebulisation efficiency, they will lead to a different response of standards and the sample, particularly with flame atomisation. The most common way to overcome such matrix interferences is to employ the method of standard additions. This method in fact creates a calibration curve in the matrix by adding incremental sample amounts of a concentrated standard solution to the sample. As only small volumes of standard solutions are to be added, the additions do not alter the bulk properties of the sample significantly, and the matrix remains essentially the same. Since the technique is based on linear extrapolation, particular care has to be taken to ensure that one operates in the linear range of the calibration curve, otherwise significant errors may result. Also, proper background correction is essential. It should be emphasised that the standard addition method is only able to compensate for proportional systematic errors. Constant systematic errors can neither be uncovered nor corrected with this technique. 

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