Optimal-control

So far, we have only discussed control loops that keep process conditions constant or reject distuibances, thereby avoiding undesirable process conditions. 

One could go one step further and try to keep the process in an optimal operating point. A checklist for items to be addressed is shown in Table 33.6. 

Since the optimal operating point depends on the mode of operation , high demands are imposed on the optimal control structure. The control scheme must be able to switch to different constraints if they become active. 

For the desulphurization process (Fig. 33.5), this investigation yields the following list of variables  

The separator level does not appear in this list, since the level does not have any impact on the operation of the process (there is no optimal value of the level). The oxygen controller takes care of a local optimization by maintaining a small excess air to fuel ratio. 

How do we find the best (t) that minimizes (5.144) We describe first a direct approach for an open-loop problem in which we compute the entire optimal trajectory for a specific initial state. Then, we outline an alternative dynamic programming approach that turns the integral equation (5.144) into a corresponding time-dependent partial differential equation, and generates a closed-loop optimal feedback control law. 

As a first approach, let us parameterize u(t) as a piecewise-constant function by splitting [to, fn] into Ns subintervals separated by the time points 

We tiien minimize (5.150) using the techniques described above. As U can be of quite high a dimension, this optimization problem can be costly. 

FUN is a structure containing the names of the user-supphed routines that define the optimal control problem. FUN.f is the routine that returns the time derivative vector for 

PARAM is an optional user-specified structure of fixed system parameters. FUN.sigma returns the integrand a t, x, u) of the cost functional, 

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Bryson, A. E., Ho, Y. B., "Applied Optimal Control Optimization, Estimation and Control," Blaisdell Publishing, Waltham, Mass. (1969). ... 

Such a model can be developed to a new design to get a feedback (FB) and build up a quality control system for materials. This scheme also includes smart block (SB) for optimal control and generation of a feedback function (Figure 1). 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

Peirce A P, Dahleh M A and Rabitz H 1988 Optimal control of quantum mechanical systems - Existence, numerical approximations and applications Phys. Rev. A 37 4950... 

Yan Y J, Gillilan R E, Whitnell R M, Wilson K R and Mukamel S 1993 Optimal control of molecular dynamics -Liouville space theory J. Chem. Phys. 97 2320... 

Closely related to tliese experimental approaches are optimal control procedures, in which one simulates... 

Thus, the function / solves the optimal control problem (2.19). Theorem 2.2 is proved. 

Different optimal control problems can be found in the monographs and papers (Khludnev, Sokolowski, 1997 Banichuk, 1980 Barbu, 1984 Cea, 1971 Lions, 1968a, 1968b Litvinov, 1987 Mignot, 1976 Fuel, 1987 Bock, Lovisek, 1987, Haslinger et ah, 1986). 

As we know the vertical displacements of the plate defined from (2.7), (2.8) can be found as a limit of solutions to the problem (2.9)-(2.11). Two questions arise in this case. The first one is the following. Is it possible to solve an optimal control problem like (2.19) when w = w/ is defined from (2.9)-(2.11) The second question concerns relationships between solutions of (2.19) and those of the regularized optimal control problem. Our goal in this subsection is to answer these questions. 

First of all let us formulate the regularized optimal control problem. If the set F is introduced in similar way and w/ = w is found from the equation... 

Theorem 2.7. Under the above conditions, there exists a solution of the optimal control problem (2.49). 

In the sequel we shall study an optimal control problem. Let C (fl) be a convex, bounded and closed set. Assume that ( < 0 on T for each G. In particular, this condition provides nonemptiness for Kf. Denote the solution of (2.131) by % = introduce the cost functional... 

In the next two subsections the parameter c is supposed to be fixed. The convergence of solutions of the optimal control problem (2.134) as —> 0 will be analysed in Section 2.5.4. For this reason the -dependence of the cost functional is indicated. 

Let the set be the same as in Section 2.5.2. Consider the optimal control problem... 

Suppose that 5 is fixed for the time being. We shall prove that a solution of the optimal control problem (2.189), (2.188) exists. We choose a minimizing sequence Um U. It is bounded in and so we can assume... 

An existence theorem to the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. The solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C °° near crack points provided the crack opening is equal to zero. The results of this section are published in (Khludnev, 1996c). 

Further, in Section 3.1.4, an optimal control problem is analysed. The external forces u serve as a control. The solution existence of the optimal control problem with a cost functional describing the crack opening is proved. Finally, in Section 3.1.5, we prove C°°-regularity of the solution near crack points having a zero opening. 

The goal of this subsection is to prove an existence theorem for the optimal control problem. 

The result given below provides the solvability of the optimal control problem formulated. 

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