Response specification algorithms

The computational power and flexibility of the computer is much used now to simulate controllers having characteristics other than the standard P, PI, etc., modes. Controllers are described in the following for which the design algorithm is derived directly from a specification of the discrete time character of the response of the controlled variable to a given change in set point. 

Consider the closed-loop transfer function for the system shown in Fig. 7.92a. The closed-loop transfer function (z)/ 3l(z) is derived as in Section 7.17.3, i.e.  

There is a variety of specifications that can be imposed on the system closed-loop response for a given change in set point. These lead to a number of alternative discrete-time control algorithms—the best known of which are the Deadbeat and Dahlin s algorithms. 

Design of a Discrete Time Controller Based Upon a Deadbeat Response 

It is required, in this case, that the response of the controlled variable to a step change in set point exhibits zero error at all sampling instants after the first. Such a response would be described theoretically by a step change of the same magnitude but delayed by one sampling instant. Now, for a step change in set point of unit magnitude (from Appendix 7.1)  

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Fig. 7.96. Comparison of response of the controlled variable using deadbeat and Dahlin s response specification algorithm...

Standardizing the spectral response is mathematically more complex than standardizing the calibration models but provides better results as it allows slight spectral differences - the most common between very similar instruments - to be corrected via simple calculations. More marked differences can be accommodated with more complex and specific algorithms. This approach compares spectra recorded on different instruments, which are used to derive a mathematical equation, allowing their spectral response to be mutually correlated. The equation is then used to correct the new spectra recorded on the slave, which are thus made more similar to those obtained with the master. The simplest methods used in this context are of the univariate type, which correlate each wavelength in two spectra in a direct, simple manner. These methods, however, are only effective with very simple spectral differences. On the other hand, multivariate methods allow the construction of matrices correlating bodies of spectra recorded on different instruments for the above-described purpose. The most frequent choice in this context is piecewise direct standardization... 

There are a variety of specifications that we can impose on the closed-loop response y(z) for a given step change in ySp(z). It is clear that depending on the response specifications, we can derive a series of alternative digital control algorithms. Let us now examine the most commonly used among them. 

For more complex molecular spectra, particularly those collected under Static SIMS conditions, the situation becomes more complicated. In such cases, comparative analysis of spectra or even the use of the Gentle SIMS approach may be required. The original parent molecule responsible for the respective signals can also be defined when the Gentle SIMS approach is combined with the FPM along with the use of the Simplified Molecular Input Line Entry Specification algorithm. 

DAHLIN<44) suggested that, in order to avoid the large overshoots and oscillatory behaviour which are characteristic of the deadbeat algorithm, the specification of the system closed-loop response to a step change in set point should be the same as that for a first-order system with dead time. The first-order time constant can then be employed as a design parameter which can be adjusted to give the desired closed-loop response. Hence ... 

Calibration is the process by which a mathematical model relating the response of the analytical instrument (a spectrophotometer in this case) to specific quantities of the samples is constructed. This can be done by using algorithms (usually based on least squares regression) capable of establishing an appropriate mathematical relation such as single absorbance vs. concentration (univariate calibration) or spectra vs. concentration (multivariate calibration). 

Concentration (or dose) addition (CA) and response addition (RA) both apply algorithms that combine the results of single substance evaluations to produce an estimate of the mixture risk. The uncertainty in the estimate is a combination of the uncertainties in the individual components. The hazard index (HI), which is a specific case of CA, adds the ratios between the exposure and reference values of the individual substances ... 

The risk assessment for mixtures shows much similarity with that for single substances, but there also are some important differences. In order to make accurate risk predictions, risk assessment should pay specific attention to all aspects of mixture exposures and effects. The establishment of a safe dose or concentration level for mixtures is useful only for common mixtures with more or less constant concentration ratios between the mixture components and for mixtures of which the effect is strongly associated with one of the components. For mixtures of unknown or unique composition, determination of a safe concentration level (or a dose-response relationship) is inefficient, because the effect data cannot be reused to assess the risks of other mixtures. One alternative is to test the toxicity of the mixture of concern in the laboratory or the field to determine the adverse effects and subsequently determine the acceptability of these effects. Another option is to analyze the mixture composition and apply an algorithm that relates the concentrations of the individual mixture components to a mixture risk or effect level, which can subsequently be evaluated in terms of acceptability. 

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