Simple cost function

Outlet Damper mth Constant Fan Speed. The system resistance is varied with this damper. The volume of gas delivered from the fan is changed as a function of the movement of the damper. It is low in first cost and simple to operate but does require more horsepower than other methods of control. 

We have, on the other hand done simple simulations for the case of one-step ahead and two-step ahead scheduling. In the latter case, the revisit times and waveforms are calculated while the target states are propagated forward over two measurements, with the cost function being the absolute value of the determinant of the track error covariance after the second measurement. Only the first of these measurements is done before the revisit calculation is done again for that target, so that the second may never be implemented. 

The difficulty with this procedure is that simple refinement routines, such as simplex or least squares, lead only to the nearest minimum in the cost function which is unlikely to be the global minimum. The refinement procedure therefore has to be one that randomly samples different parts of configuration space so as to be able to reach different minima, ultimately selecting the global minimum. Two refinement methods have been proposed, simulated annealing and the genetic algorithm. 

The final series of five procedures presents optimized preparations of a variety of useful organic compounds. The first procedure in this group describes the preparation of 3-BROMO-2(H)-PYRAN-2-ONE, a heterodiene useful for (4+2] cycloaddition reactions. An optimized large scale preparation of 1,3,5-CYCLOOCTATRIENE, another diene useful for [4+2] cycloaddition, is detailed from the readily available 1,5-cyclooctadiene. Previously, the availability of this material has depended on the commercial availability of cyclooctatetraene at reasonable cost. A simple large scale procedure for the preparation of 3-PYRROLINE is then presented via initial alkylation of hexamethylenetetramine with (Z)-1,4-dichloro-2-butene. This material serves as an intermediate for the preparation of 2,5-disubstituted pyrroles and pyrrolidines via heteroatom-directed metalation and alkylation of suitable derivatives. The preparation of extremely acid- and base-sensitive materials by use of the retro Diels-Alder reaction is illustrated in the preparation of 2-CYCLOHEXENE-1.4-DIONE, a useful reactive dienophile and substrate for photochemical [2+2] cycloadditions. Functionalized ferrocene derivatives... 

At the price of about 5.57, all firms will have a zero profit. With the simple cost functions used in this example, no one firm can make a profit while another loses money. In reality this may not always be the case, as can easily be shown if we assume some economy of scale for fixed cost ... 

The evaluation of industrial reaction paths depends on a detailed understanding of the costs associated with each reaction step. A simple cost function is... 

In the example, the gradient of the least squares cost function with respect to the target fault parameters has been computed numerically. For the simple network, ARRs in closed symbolic form can be derived from the diagnostic bond graph so that partial derivatives of the cost function with respect to parameters can be provided in symbolic form. 

As to the previously considered simple hybrid network (Fig. 4.1), ARRs in closed symbolic form could be derived from its diagnostic bond graph (Fig. 4.2). As a result, analytical expressions for the ARR residuals to be used in the formulation of the least squares cost function are available and symbolic partial derivatives of ARR residuals with respect to the targeted fault parameters can be exploited. 

---