Optimal control theory examples

In this section, we provide some examples of optimal control theory calculations using the ENBO approximation. The reader is referred to Ref. [42], from where aU the examples are taken, for further details. 

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. 

We can readily extend bimolecular control to superpositions composed of more than two states. Indeed, we can introduce a straightforward method to optimize the reactive cross section as a function of am for any number of states [252], Doing so is an example of optimal control theory, a general approach to altering control parameters to optimize the probability of achieving a desired goal, introduced in Chapter 4. 

An example of the application of optimal control theory for experiment design is described in Fig. 8, which in panel (a) shows the transfer efficiency achievable by optimal control sequences by the solid line, while corresponding efficiencies for the DCP experiment under ideal and inhomogeneous rf conditions as well as the performance of ramped versions of DCP are illustrated by the various dashed/dotted lines. We should... 

Optimal control theory has a broad spectrum of applications that will not be addressed in their completeness here. This includes research directions such as laser cooling of internal degrees of freedom of molecules and quantum computing (cf. Refs. 307-311 for examples of metallic dimers) since they usually require inclusion of additional methodological aspects. 

We begin in Section 6.4.2.1 by discussing the basic idea of controllability, in relation to optimal control theory, and linear systems. These discussions are facilitated with reference to a simple example—the controlled motion of a rocket to the moon. 

Pontryagin has enriched the classical calculus of variations by the development of optimal control theory, especially in his treatment of problems where some restraint is placed on the state of the system. It is of considerable practical importance that such restraints, for example, the greatest power at which a nuclear reactor may safely be operated, should be fitted into any theory of control. Such a development had been sought for some time (2). 

A robust mathematical tool is needed to perform an optimisation. We have found that optimal control theory provides such a tool. The system is constrained by the requirement to balance energy, momentum and mass. These constraints must be specified for each particular case. In optimal control terminology there are two classes of variables. The first class are the state variables, for instance the temperature, T z,t), the pressure, p z,t), and the concentrations, Cj z,t), in a tubular reactor. The second class are the control variables, which are determined from the outside. An example is the temperature, T (z,t), on the outside along the tubular reactor. Optimal control theory in this case provides a general method to obtain T (z,t) such that the total entropy production is minimal, given certain constraints. 

The first two points are valid for open-loop process nonlinearity measures as well. The third point is new in control-relevant nonlinearity quantification. In a more general context, one has not only to consider the performance criterion but additionally mention the controller design method. Following the idea of Ref 24, optimal control theory with an integral performance criterion will be used here as it represents a benchmark for any achievable performance. Considering nonlinear internal model control with different filter time constants is also possible, see for example Ref 23. 

It was pointed out very early [3] that the natural way to find such optima is through the application of optimal control theory. In fact the first such application was carried out by Rubin [6,7], specifically to find the pathways and optimal performance so obtained for a cyclic engine of the sort described above, Rubin found the conditions for optimum power and for optimum efficiency, which of course are normally different. It was in these works that he introduced the term endoreversible to describe a process that could have irreversible interactions with its environment but would be describable internally in terms of the thermodynamic variables of a system at equilibrium. An endoreversible system comes to equilibrium internally very rapidly compared, whatever heat or work exchange it incurs with the outside. It was here that one first saw the comparison of the efficiency for maximum power of the Curzon-Ahlborn engine compared graphically with the maximum efficiency, in terms of a curve of power vs. heat flow. Figure 14.1 is an example of this. 

By an appropriate extension of the Hamiltonian (9.4), the two-pulse EOM-PMA can be applied to a general electronic AMevel system beyond the weak-pump limit. The method can thus be used, for example, for the calculation of so-called control kernels (see, e.g., ref. 28) within optimal control theory. 

S. A. Rice I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control held, the perturbation theory result can be used as a first guess, for which purpose it is very good. 

The situation is quite different when inequality constraints are included in the MPC on-line optimization problem. In the sequel, we will refer to inequality constrained MPC simply as constrained MPC. For constrained MPC, no closed-form (explicit) solution can be written. Because different inequahty constraints may be active at each time, a constrained MPC controller is not linear, making the entire closed loop nonlinear. To analyze and design constrained MPC systems requires an approach that is not based on linear control theory. We will present the basic ideas in Section III. We will then present some examples that show the interesting behavior that MPC may demonstrate, and we will subsequently explain how MPC theory can conceptually simplify and practically improve MPC. 

If we add a solution of species A to a solution of species B, eddies of solution A in solution B are created. As a first approximation, these eddies can be considered as spherical drops with constant mean radius R. The lifetime of such an eddy can be estimated to be 0.01-1 s. The radius R depends on the intensity of the turbulence created by mixing and may be controlled, for example, by mechanical stirring. From the theory of turbulence, one can estimate the minimum mean size of such elements of liquid. For the common solvents water, methanol, and ethanol, the mean minimum radius R of the eddies in optimal turbulence is approximately 10 to 10 cm. 

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