Weighted sensitivity function

In equation (9.131), sup is short for supremum, which means the final result is the least upper bound. Thus the //qo-optimal controller minimizes the maximum magnitude of the weighted sensitivity function over frequency range uj, or in mathematical terms, minimizes the oo-norm of the sensitivity function weighted by fE(jtj). 

Fig. 9.26 Bode magnitude plot of weighted sensitivity function for Example 9.4.

Miscible blends of high molecular weight polymers often exhibit LOST behavior (3) blends that are miscible only because of relatively low molecular weights may show UCST behavior (11). The cloud-point temperatures associated with Hquid—Hquid phase separation can often be adequately determined by simple visual observations (39) nevertheless, instmmented light transmission or scattering measurements frequendy are used (49). The cloud point observed maybe a sensitive function of the rate of temperature change used, owing to the kinetics of the phase-separation process (39). 

Because some patients are more sensitive to side effects with digoxin, the dosage is selected carefully and adjusted as the clinical condition indicates. Adverse reactions were more common and severe in past years before careful attention to weight, renal function, and the concurrent administration of certain medications was given. The incidence and severity of digoxin toxicity has decreased significantly in recent years. 

In conjunction with molecular weight sensitive detection systems, SEC can be used for determining various molecular weight parameters, such as molecular size, conformation and branching, as a function of MW. Furthermore, by interfacing SEC with spectrometry, polymer compositional heterogeneity can be determined. 

Independent of the ligand system, two different activation methods have been used in performing the propylene polymerization experiments. In both cases, the catalytic activities and molecular weights of the polymers are a sensitive function of the aluminum content provided by the activators. This dependence suggested an additional reversible chain transfer to aluminum when activating with MAO. As lower contents of A1 are provided in the polymerization system in the case of in situ activation with TIBA/borate, the only mechanism occurring is the chain back-skip. Furthermore, the differences in the polymer microstructures prepared with MAO and borate as cocatalysts are reflected. They sustain the proposed reversible chain transfer. 

The interesting question is whether in dilute polymer solutions there exists a low frequency dispersion region described by the RB theory, so that just as in the bulk polymer we have to contend with two dispersion regions. The existence of a RB dispersion region in polymer solutions may be associated with entanglement and hence a very sensitive function of polymer concentration and molecular weight. 

For most proteins vp is about 0.75 ml/g, so its value does not present much of a problem. The frictional coefficient, however, is a sensitive function of the shape, varying over a wide range, and we must usually know its value if we need a serious estimate of the molecular weight. The value of/is usually found by working with the diffusion constant D, which is related to the frictional constant by... 

Resin as the Disperse Phase. Several kinds of resins (10) have been used to reinforce rubbers—e.g., phenolic or coumarone resins for natural rubber, styrene-butadiene resin for styrene-butadiene rubber, etc. One other important system, pressure-sensitive adhesive, also belongs to this class. These adhesives generally contain a low molecular weight resin functioning as a tackifier. In 1957, Wetzel (68) and Hock (19) found that these adhesives were actually two-phase systems (Figure 1). Under... 

The computer models described provide a functional simulation of SEC-viscometry-LS analysis of linear polymers. The results for the Flory-Schulz MWD are in qualitative agreement with previous results for the Wesslau MWD. Both models emphasize the importance of determining the correct volume offset between the concentration detector and molecular weight-sensitive detectors. For the Flory-Schulz model, the peak shape, as well as the peak elution volume, can provide information about molecular weight polydispersity. Future work will extend the model to incorporate peak skew and polymer branching. 

Polymer molecular weights may be measured by vapor phase or solution osmometry, or more usually by size exclusion chromatography (SEC or GPC). If the polymer is hydrophilic but neutral, measurements will usually be made in a polar organic solvent such as 1,4 dioxane or dimethyl formamide. If insoluble in organic solvents, measurements may be performed in water, a less than ideal solvent for this type of measurement. The situation is further complicated if dealing with a polyelectrolyte polymer (especially containing weak pH sensitive functionality). SEC measurements become a function of nominal charge... 

It is apparent that the magnitude of the enhancement in the UCFT above the 0i -temperature was a sensitive function of the molecular weight of the polymer, at least for low molecular weight polymers. The enhancement increased with decreasing molecular weight of the stabilizing moieties. 

Other terminology used for material density sensitivity traverses include efficiency functions (52, 55), switching functions (55), statistical weight (55), sensitivity coefficients (90), and effectiveness functions (57, 69). These are, perhaps, the earliest sensitivity functions in use they were introduced by the Russian school (52, 55) for optimization studies. 

Section 5.3.1 has shown that parameter sensitivities of ARR residuals may be obtained from an incremental bond graph. The latter bond graph can be systematically developed from an initial bond graph with nominal parameters by replacing elements with parameters to be estimated by their incremental component model. Inputs into the incBG are variations of the parameters to be estimated multiplied by a power variable of the initial BG. Outputs may be parameter variations of ARR residuals. They are a weighted sum of the parameter variations and the weighting factors are just the residual sensitivity functions. 

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