Complementary sensitivity function 7

Robust performance then means that the closed-loop system will meet the performance specification given in equation (9.145) if and only if the nominal system is closed-loop stable (equation (9.141)) and that the sensitivity function Sm(jar) and complementary sensitivity function for the nominal system satisfy the rela-... 

The third class of techniques include a frequency-domain method based on the identification of the sensitivity function S s)) and the complementary sensitivity function T s)) from plant data or CPM of multivariable systems [140]. Robust control system design methods seek to maximize closed-loop performance subject to specifications for bandwidth and peak... 

We specify the desired performance of a closed-loop system by specifying its desired (ideal) output complementary sensitivity function, T, which relates the reference signal, r, and the output signal, y, in the one degree of freedom (DOF) control configuration ( see Fig. 1 ). For the SISO case, a classical approach is to map time-domain specifications such as settling time, rise time, maximal overshoot, and steady-state error into the parameterization of a transfer function of the form... 

Stein and Doyle [35] developed an expression to calculate ft for the Robust Performance Problem in the case where the plant is minimum phase and is controlled by an inverse-based decoupling controller. The modeled uncertainty is described by a complex unstructured input block with weighting function w, and performance requirement Wj measured by the closed-loop sensitivity function S. The decoupling controller K is based on the inverse of G in the form. (s) = G (s) (s), where k(s) is a scalar transfer function which makes K s) proper and gives a stable closed-loop system. Note that G s) is a linear stable system with stable inverse (i. e. G is minimum-phase).This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements, namely... 

Equation J-28 indicates that the sensitivity is equal to S. For this reason, S is referred to as the sensitivity function. In view of the important relationship in (J-16), T is called the complementary sensitivity function,... 

The complementary sensitivity function for the nominal model and the PID controller is... 

This HHT procedure is particularly convenient for laboratory-scale syntheses of a wide variety of glyphosate derivatives and intermediates (2). In many cases, fairly sensitive functionalities can often be accommodated because of the mild thermal conditions and the complete absence of water. This method is therefore quite complementary to the other aqueous Mannich procedures, since groups that would not normally tolerate aqueous strongly acidic conditions would frequently decompose or would be difficult to recover in high yield. 

A frequent error encountered in evaluating the performance of an analytical system is to confuse the concepts of sensitivity and detectability. Although both concepts address facets of a system s response, they are not identical, but rather complementary. Sensitivity relates to the ability of the system to respond to changes in analyte concentration and is most typically reflected as the slope of the method s response function. Detectability, as has been noted previously, is the ability of the method to distinguish between two responses (those responses that arise in the presence and absence of the analyte in the sample matrix of interest). It is possible to have a very sensitive method that has a relatively poor detection limit, especially if the method is very unselective or prone to high blanks. However, an insensitive method is very unlikely to exhibit a low LOD. Thus sensitivity is a necessary— but not sufficient—condition for the achievement of a low LOD. 

The results described above are somewhat conflicting and demonstrate that PHEMA has a range of pore sizes, and that a porous network probably exists on a number of size scales. Of particular relevance to this study is the knowledge that the pore structure of PHEMA is a sensitive function of the method of preparation, for example the amount of water, and crosslinking monomer in the polymerization mixture, as well as the rate of polymerization determined by the initiator fraction 20-23, 25, 26). It is clear that no one single technique will provide a complete picture of such a complex morphology, and therefore in this study, we use the three complementary techniques, Xe NMR, PALS and H NMR, to examine the free volume and pore structure in PHEMA prepared under a range of different conditions. These techniques allow the study of porous... 

Infrared and Raman spectroscopy each probe vibrational motion, but respond to a different manifestation of it. Infrared spectroscopy is sensitive to a change in the dipole moment as a function of the vibrational motion, whereas Raman spectroscopy probes the change in polarizability as the molecule undergoes vibrations. Resonance Raman spectroscopy also couples to excited electronic states, and can yield fiirtlier infomiation regarding the identity of the vibration. Raman and IR spectroscopy are often complementary, both in the type of systems tliat can be studied, as well as the infomiation obtained. 

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