Robustness parameter 748 Subject

The pH value of the bioreactor is subjected to external disturbances (also called load changes), and the task of suppressing or rejecting the effects of disturbances is called regulatory control. Implementation of a controller may lead to instability, and the issue of system stability is a major concern. The control system also has to be robust such that it is not overly sensitive to changes in process parameters. 

Relative robustness will have values between 0 (no robustness) and 1 (ideal robustness). Burns et al. [2005] defines the relative robustness of an analytical procedure as the ratio of the ideal signal for an uninfluenced method compared to the signal for a method subject to known operational parameters determined in an intra-laboratory experiment (Burns et al. [2005]). 

Figure 47. The robustness of metabolic states. Shown is the probability that a randomly chosen state is unstable. Starting with initially 100% stable models, the parameters are subject to increasing perturbations of strength p, corresponding to a random walk in parameter space. (A) The initial states are chosen randomly from the parameter space. (B) The initial states are confined to a small region with 0.01 < < 0. Note that the state Catp exhibits a rapid decay in stability. The data...

One final comment should be made about model-based control before we leave the subject. These model-based controllers depend quite strongly on the validity of the model. If we have a poor model or if the plant parameters change, the performance of a model-based controller is usually seriously affected. Model-based controllers are less robust than the more conventional PI controllers. This lack of robustness can be a problem in the single-input-single-output (SISO) loops that we have been examining. It is an even more serious problem in multi-variable systems, as we will find out in Chaps. 16 and 17. 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

Parameter selection is a crucial process in multiparametric screening and a fiill discussion on the subject is beyond the scope of this chapter. Briefly, parameters are sought that allow best separation between negative control and positive control. The parameters should describe various aspects of the phenotype and should therefore not be too much correlated to each other. Furthermore, the parameters should be robust. 

AMANDA has yet to observe an extraterrestrial neutrino source, but she has demonstrated the cost-effectiveness and robustness of the technique. The detector is very versatile it addresses many different neutrino physics subjects and sets the most stringent upper limits on Galactic and extragalactic neutrino fluxes. The improved search for diffuse fluxes, which has ruled out several predictions, along with the extended four-year search for point sources has started to constrain the enormous parameter space that exist in many models of neutrino production. The reported experimental limits on the diffuse neutrino flux are less than an order of magnitude above the Waxman-Bahcall bound ( Waxman and Bahcall, 1999). As more of the data on tape is analyzed, AMANDA sensitivities will continue to improve. This is a very exciting time in neutrino astronomy and we look forward to neutrino astrophysics with next generation of neutrino telescopes. 

The SA may also provide rationale for simplifying a model. This may occur if the outcome is shown to be robustly tolerant to wide ranges of parameter uncertainty, which may allow for the removal of some parameters and thus lead to a more parsimonious simulation model. Conversely, the SA may reveal that insufficient information currently exists to define a precise or reliable range of trial outcomes. In this latter case, either more time may be required to obtain additional informative experimental data and thus reduce the uncertainty to an acceptable range, or separate sets of plausible assumptions may need to be considered and subsequently tested for their own sensitivity. Such decisions need buy-in from the subject matter experts and should be considered in the full context of the development program. 

As a last comment, caution should be exercised when fitting small sets of data to both structural and residual variance models. It is commonplace in the literature to fit individual data and then apply a residual variance model to the data. Residual variance models based on small samples are not very robust, which can easily be seen if the data are jackknifed or bootstrapped. One way to overcome this is to assume a common residual variance model for all observations, instead of a residual variance model for each subject. This assumption is not such a leap of faith. For GLS, first fit each subject and then pool the residuals. Use the pooled residuals to estimate the residual variance model parameters and then iterate in this manner until convergence. For ELS, things are a bit trickier but are still doable. 

The linear mixed effect model assumes that the random effects are normally distributed and that the residuals are normally distributed. Butler and Louis (1992) showed that estimation of the fixed effects and covariance parameters, as well as residual variance terms, were very robust to deviations from normality. However, the standard errors of the estimates can be affected by deviations from normality, as much as five times too large or three times too small (Verbeke and Lesaffre, 1997). In contrast to the estimation of the mean model, the estimation of the random effects (and hence, variance components) are very sensitive to the normality assumption. Verbeke and Lesaffre (1996) studied the effects of deviation from normality on the empirical Bayes estimates of the random effects. Using computer simulation they simulated 1000 subjects with five measurements per subject, where each subject had a random intercept coming from a 50 50 mixture of normal distributions, which may arise if two subpopulations were examined each having equal variability and size. By assuming a unimodal normal distribution of the random effects, a histogram of the empirical Bayes estimates revealed a unimodal distribution, not a bimodal distribution as would be expected. They showed that the correct distributional shape of the random effects may not be observed if the error variability is large compared to the between-subject variability. 

Wade et al. (1993) simulated concentration data for 100 subjects under a one-compartment steady-state model using either first-or zero-order absorption. Simulated data were then fit using FO-approximation with a first-order absorption model having ka fixed to 0.25-, 0.5-, 1-, 2-, 3-, and 4 times the true ka value. Whatever value ka was fixed equal to, clearance was consistently biased, but was relatively robust with underpredictions of the true value by less than 5% on average. In contrast, volume of distribution was very sensitive to absorption misspecification, but only when there were samples collected in the absorption phase. When there were no concentration data in the absorption phase, significant parameter bias was not observed for any parameter. The variance components were far more sensitive to model misspecification than the parameter estimates with some... 

An area related to model validation is influence analysis, which deals with how stable the model parameters are to influential observations (either individual concentration values or individual subjects), and model robustness, which deals with how stable the model parameters are to perturbations in the input data. Influence analysis has been dealt with in previous chapters. The basic idea is to generate a series of new data sets, where each new data set consists of the original data set with one unique subject removed or has a different block of data removed, just like how jackknife data sets are generated. The model is refit to each of the new data sets and how the parameter estimates change with each new data set is determined. Ideally, no subject should show... 

FO-approximation as the amount of data per subject decreases. This data set was fairly sparse with most subjects having about two samples per subject. It would be expected that the methods would produce fairly different estimates but this was not the case. All methods generated roughly the same parameter estimates indicating that the model was fairly robust to choice of estimation algorithm. 

The aforementioned methods can be applied to evaluate the reliability of engineering systems subjected to stochastic input with a given mathematical model. On the other hand, if a parametric model of the underlying system is available and the probability density function of these parameters is obtained by Bayesian methods, the uncertain parameter vector can be augmented to include the model parameters and the uncertain input components. Then, robust reliability analysis can proceed for stochastic excitation with an uncertain mathematical model. This allows for more realistic reliability evaluation in practice so that the modeling error and other types of uncertainty of the mathematical model can be taken into account. 

For example, in reliability analysis, the quantity Q is considered as the probability that the structure with parameter vector 0 would fail, i.e., Q(0) = P(F 0, C). Then, the updated integral becomes the updated robust probability of failure for the structure, when it is subjected to some stochastic excitation [198] ... 

Inevitably such calculations will be based on assumptions which may differ from real parameters in particular circumstances. Models should be robust and conservative. However, those that use all worst case parameters may result in recommendations leading to unnecessary practical difficulties or financial penalties. That the application of the resulting segregation distances leads to acceptably low doses is more important than the basis on which the distances were calculated. However, transport patterns are subject to change and doses should be kept under review. 

In some cases it is observed that, under the experimental conditions used (mobile phase composition, ionization and API interface parameters), more than one ionized form of the intact analyte molecule is observed, i.e. adduct ions of various kinds (see Section 5.3.3 and Table 5.2). An example is shown in Figure 9.6, in which a well known anticancer drug (paclitaxel, Figure 9.6(a)) was analyzed by positive ion ESI-MS (infusion of a clean solution). The first spectrum (Figure 9.6(b)) shows four different adducts (with H+, NH, Na+ and K+). Adjustment of the cone (skimmer) potential (Section 5.3.3a), to lower values in this case, enabled production of the ammonium ion adduct to dominate the MS spectrum (Figure 9.6(c)) in a robust fashion, and this ion yielded a useful product ion spectrum (that appeared to proceed via a first loss of ammonia to give the protonated molecule) which was exploited to develop an MRM method that was successfully validated and used. It is advisable to avoid use of analyte adducts with alkali metal ions (commonly Na+ and to some extent K+) since, when subjected to colli-sional activation, these adducts frequently yield the metal ion as the dominant product ion with only a few low abundance product ions derived from the analyte molecule. However, when feasible, both the ammonium adduct and protonated molecule should be investigated as potential precursor ions at least until it becomes clear that one will provide superior performance (sensitivity/selectivity compromise) than the other. 

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