Relative control optimization example

The amplitude of temperature fluctuations was controlled in a feedback loop by adjusting the relative phase between the primary and secondary forced air flows. A demonstration of the closed-loop performance is illustrated in Fig. 24.12. The controller converged on the optimum phase with a 1/e rise time of approximately 30 control steps (Fig. 24.12a). Figure 24.126 illustrates the difference between the power spectra with control off (i.e., neither primary nor secondary drivers) and control optimized. The response time necessary to reach the optimum phase was slowed by the large variations in the measured coherence (examples shown in Fig. 24.12a) which are attributed to the complex interactions between the inlet mode, the combustor modes, and the preferred mode of the jet. 

Two broad approaches to dynamic operability analysis are the use of so-called open-loop indicators, and the solution of a suitably formulated optimization problem. Characteristics of the former are that they are based on steady-state or linear dynamic models, are relatively easy to compute and seek to provide indications of potential plant-inherent control problems independent of the choice of control system. Examples are the minimum singular value and the plant condition number which reflect sensitivity to input constraints and model uncertainty respectively. More detail may be found in [1] with a good overview of these methods given in [2]. 

In another example of application of the simplex method, McMullen et cd. [38] demonstrated the rapid optimization and scaling of a Heck reaction using an automated microreactor system with HPLC monitoring and feedback control. Optimal reaction conditions in the microreactor were determined after 19 automated experiments and required a relatively small amount of starting material. The reaction was then successfully scaled up 50-fold from a microreactor to a Coming meso-scale glass reactor using the optimal conditions determined by the microreactor system. 

The globally optimal laser field for this example is presented in Fig. 2. The field is relatively simple with structure at early times, followed by a large peak with a nearly Gaussian profile. Note that the control formalism enforces no specific structure on the field a priori. That is, the form of the field is totally unconstrained during the allotted time interval, so simple solutions are not guaranteed. Also shown in Fig. 2 is the locally optimal... 

The first example, also being the example introducing optimal control to solid-state NMR [40] and further elaborated on later [161], is optimal control versions of the DCP experiment. This experiment was a natural choice for numerical improvements as it is widely used and it is well known that this experiment is sensitive to offsets, rf mismatch relative to the MAS-modified Hartmann-Hahn condition, and rf inhomogeneity. In particular the two latter effects may reduce significantly the performance of 15N to 13C transfers, severely complicate setup of such experiments, and render these critically sensitive to altered tuning/rf conditions in the course of potentially long experiments for biological samples. 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

Substrate availability for certain reactions can be optimized by anaplerotic ( topping-up ) reactions. For example, citrate synthase is a key control point of the TCA cycle. The co-substrates of citrate synthase are acetyl-CoA and oxaloacetate (OAA) and clearly, restriction in the availability of either substrate will decrease the rate of the citrate synthase reaction. Suppose, for example, a situation arises when acetyl-CoA concentration is significantly higher than that of OAA, the concentration of the latter can be topped-up and the concentration of acetyl-CoA simultaneously reduced by diverting some of the pyruvate away from acetyl-CoA synthesis (via pyruvate dehydrogenase) to OAA synthesis (via pyruvate carboxylase) as shown in Figure 3.1. The net effect is to balance the relative concentrations of the two co-substrates and thus to promote citrate synthase activity. 

Having obtained a rough estimate of the irradiation time, the time dependence of labeling of the receptor should be measured directly. The optimal photolysis time (e.g. Fig. 4.2), determined by the incorporation of label or by photoinactivation (see below), will be used in many labeling and control experiments, and it is important that it be reproducible. For the results to be useful the sample must be irradiated at a fixed point relative to the lamp. For example, if a long arc is used, the intensity varies both with the distance from the center of the lamp and with the position along the... 

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