Optimal control advantage

It was felt that a nonisothermal policy might have considerable advantages in minimizing the reaction time compared to die optimal isothermal policy. Modem optimal control theory (Sage and White (1977)), was employed to minimize the reaction time. The mathematical development is presented below. 

As mentioned in Section IV. A, a straightforward way to deal with optimal control problems is to parameterize them as piecewise polynomial functions on a predefined set of time zones. This suboptimal representation has a number of advantages. First, the approaches developed in the previous subsection can be applied directly. Secondly, for many process control applications, control moves are actually implemented as piecewise constants on fixed time intervals, so the parameterization is adequate for this application. 

One advantage in the sequential approach is that only the parameters that are used to discretize the control variable profile are considered as the decision variables. The optimization formulated by this approach is a small scale NLP that makes it attractive to apply for solving the optimal control with large dimensional systems that are modeled by a large number of differential equations. In addition, this approach can take the advantage of available IVP solvers. However, the limitation of the sequential method is a difficulty to handle a constraint on state variables (path constraint). This is because the state variables are not directly included in NLP. 

It is clear from Table 5.9 that the results obtained are in very good agreement with the objectives set for each individual optimisation problem. Table 5.9 also deary shows the advantages of optimal reflux policies over the conventional constant reflux operation. Table 5.9 shows that the time optimal control policy (variable reflux) saves about 63% of the operation time compared to that required in the simulation (Table 4.6). Even the time optimal constant reflux policy saves about 33% of the operation time compared to the original simulation... 

Summary. An efficient semiclassical optimal control theory for controlling wave-packet dynamics on a single adiabatic potential energy surface applicable to systems with many degrees of freedom is discussed in detail. The approach combines the advantages of various formulations of the optimal control theory quantum and classical on the one hand and global and local on the other. The efficiency and reliability of the method are demonstrated, using systems with two and four dimensions as examples. 

As this approach deals with a set of classical trajectories, its numerical cost remains reasonable for multidimensional systems. Contrary to the classical approach, which controls only the averaged classical quantities, the present semiclassical method can control the quantum motion itself. This makes it possible to reproduce almost all quantum effects at a computational cost that does not grow too rapidly as the dimensionality of the system increases. The new approach therefore combines the advantages of the quantum and classical formulations of the optimal control theory. 

We have used sensitivity equation methods (Leis and Kramer, 1985) for gradient evaluation as these are simple and efficient for problems with few parameters and constraints. In general, the balance in efficiency between sensitivity and adjoint methods depends on the type of problem being addressed. Adjoint methods are particularly advantageous for optimal control problems in which the inputs are represented as a large number of piecewise constant input values and few interior point constraints exist. Sensitivity methods are preferable for problems with few parameters and many constraints. 

The salient feature of optimal control is that it uses functions as optimization parameters. These functions are called control functions or simply controls. The routine, static optimization is a special case of optimal control using uniform or single-valued controls such as Td(f) or Td in Figure 1.2. Had we prescribed the invariance of temperature with respect to time in the previous example, the problem would have been that of the routine optimization with the goal to find the optimal time invariant temperature from all possible choices restricted to be time invariant like Td- Because this restriction (or invariance with respect to independent variable) does not exist in optimal control, it has a significant advantage over routine optimization. Let us get more details. 

A distributed control system may involve the use of microcomputers at the local level and the use of more powerful rruchines to coordinate overall plant control objectives. In this context, it has been suggested that process control might be described better as process management." " Local control of the operation of individual separators is still important but, with the use of distributed control, reliability is maintained (microcomputers are dedicated to particular process units) while overall technical and economic objectives are pursued (mainframe computers can perform complex on-line/off-line optimization). The advantage to distributed control is that it makes effe ve use of current technology and provides a framework within which control and optimization developments can be implemented. These developments probably will include better simulation and optimization routines that will help to assess the current state of the process plant and to suggest improvements. 

For particular cases, it maybe required to add more complex phenomena with additional effects or more evolved descriptions of the same mechanisms. In general, however, reduced models are appropriate and desirable. Historically, this stemmed from the shorter computational effort and time required for the numerical solution of such models. Today this is also an advantage for optimization, control, and real-time simulation applications, and reliable simplified models are still used for almost all purposes due to the lower number of dimensionless parameters requiring estimation and to the success found in the description of experimental results. On the other hand, complex detailed models fulfill the most generic purpose of reactor simulation, which is related to the prediction of the actual behavior from fundamental, independently measured parameters. Therefore, it is important to understand the equivalence and agreement between both detailed and reduced models, so as to take advantage of their predictive power without unnecessary effort. 

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