Optimizing control optimum

Optimal control theory A method for determining the optimum laser field used to maximize a desired product of a chemical reaction. The optimum field is derived by maximizing the objective function, which is the sum of the expectation value of the target operator at a given time and the cost penalty function for the laser field, under the constraint that quantum states of the reactants satisfy the Schrodinger equation. 

In order to interpret the results of our experiments, optimal-control calculations were performed where a GA controlled 40 independent degrees of freedom in the laser pulses that were used in a molecular dynamics simulation of the laser-cluster interactions for Xejv clusters with sizes ranging from 108 to 5056 atoms/cluster. These calculations, which are reported in detail elsewhere [67], showed optimization of the laser-cluster interactions by a sequence of as many as three laser pulses. Detailed inspection of the simulations revealed that the first pulse in this sequence initiates the cluster ionization and starts the expansion of the cluster, while the second and third pulse optimize two mechanisms that are directly related to the behaviour of the electrons in the cluster. We consistently observe that the second pulse in the three-pulse sequence arrives a time delay where the conditions for enhanced ionization are met. In other words, the second pulse arrives at a time where the ionization of atoms is assisted by the proximity of surrounding ions. The third peak is consistently observed at a delay where the collective oscillation of the quasi-free electrons in the cluster is 7t/2 out of phase with respect to the driving laser field. For a driven and damped oscillator this phase-delay represents an optimum for the energy transfer from the driving force to the oscillator. 

OPTIMUM REACTOR DESIGN ECONOMIC EVALUATION FLEXIBILITY STUDY PROCESS OPTIMIZATION/CONTROL... 

Other recent work in the field of optimization of catalytic reactors experiencing catalyst decay includes the work of Romero e/ n/. (1981 a) who carried out an analysis of the temperature-time sequence for deactivating isothermal catalyst bed. Sandana (1982) investigated the optimum temperature policy for a deactivating catalytic packed bed reactor which is operated isothermally. Promanik and Kunzru (1984) obtained the optimal policy for a consecutive reaction in a CSTR with concentration dependent catalyst deactivation. Ferraris ei al. (1984) suggested an approximate method to obtain the optimal control policy for tubular catalytic reactors with catalyst decay. 

The introduction of digital computers for process control has allowed the implementation of optimizing control strategies in chemical plants. The usual mode of implementation is that of supervisory control (see Sections 3.2 and 26.3). Figure 25.11b shows the supervisory control implementation of an optimizing control strategy for the simple process of Figure 25.11a. Notice that the computer calculates the new optimum set point values and communicates them to the two control loops. 

It is desired to determine the optimal control u that optimizes (i.e., either minimizes or maximizes) J. To that end, we need to obtain the variation of J to help determine the optimum. The variation of J is a straightforward generalization of Equation (2.16) and is given by... 

Most important, if the equations are not satisfied, then the optimum cannot exist. The above facts conform to the logic of a conditional statement explained in Section 9.25 (p. 282). For now, let us apply these conditions to optimal control problems. 

The improvement in control functions causes the objective functional value to get closer to the optimum. Therefore, iterative application of the above steps leads to the optimum, i. e., the optimal functional value, and the corresponding optimal control functions. While the state and costate equations are satisfied in each iteration, the variational derivatives are reduced successively. They vanish when the optimum is attained. 

For example, in the problem of Example 3.4 solved above, the optimum obtained is local. Three different optima are obtained when the same problem is solved by initializing the algorithm with different guesses for the control function. Figure 3.2 shows the corresponding optimal control functions Ti, T2, and T3. They are quite different from not only each other but also the optimal T in Figure 3.1. While Ti and T2 yield the same maximum as obtained with T, T3 provides a better maximum of / = 6.70 kmol/m . 

Hence, to increase the confidence on the final optimal solution, the optimal control problem needs to be solved with different initial guesses to the numerical algorithm. If several optima are obtained, then the most optimal among them is adopted. There is no golden rule to ensure that the optimum is global. 

Note that the entities H, F, g, and their partial derivatives are functions of y and u. Such an entity, say, H, is denoted by H when evaluated at the optimum, i.e., for the vector of optimal controls u and the corresponding vector of optimal states y. 

W.H. Ray and M.A. Soliman. The optimal control of processes containing pure time delays — I. Necessary conditions for an optimum. Chem. Eng. Sci., 25 1911-1925, 1970. 

In Section 3.2.1 (p. 59), we had asserted the Lagrange Multiplier Rule that the optimum of the augmented J is equivalent to the constrained optimum of I. This rule is based on the Lagrange Multiplier Theorem, which provides the necessary conditions for the constrained optimum. We will first prove this theorem and then apply it to optimal control problems subject to different types of constraints. 

We will apply the Lagrange Multiplier Rule to obtain the set of necessary conditions for the optimum in an optimal control problem constrained by a differential equation. In Section 3.2, we asserted the rule and obtained the following necessary conditions (see p. 60) ... 

An upper controller is designed to switch between the 625 feedback control gains during earthquake simulations. At each time step, the optimal control force is calculated based on the feedback gain for the system with damping constants that are calculated in the previous step. The force that is required for the i th device is divided by the i th dampers velocity to obtain the optimum damping constant (see Eq. 18.1). Then the closest damping constant within [5,000-25,000 Ns/m at increments of 5,000] is selected for the next time step. 

The method of self-optimizing control is applied to a heat integrated prefractionator arrangement. This system has a total of eleven degrees of freedom with six DOF available for optimization when variables with no-steady state effects have been excluded and the duties of the two columns are matched. From the optimization it is found that there is one degree of freedom left for which there is not an obvious choice of control variable. The method of self- optimizing control will be used to find a suitable control variable that will keep the system close to optimum when there are disturbances. 

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. 

Optimal control theory was a useful tool for finding the pathway, in terms of the time dependence of temperature T and volume V, that yields the optimum power or the optimum efficiency for a given fixed cycle time. We shall only outline the method and show some of the results here the full derivation is available [6]. 

The off-line optimization results when applied in practice often become suboptimal due to ever-existing process disturbances and changes in process dynamics (e.g., when capacity is increased). Online optimal control can circumvent this problem and ensure optimum process operation aU the time. 

Normally the operating conditions of any plant are surrounded by constraints and limitations. It is not surprising to learn that the optimum conditions for many plants lie outside equipment limitations. Many applications would not result in enough remuneration to pay for the computer or the engineering involved. These two facts severely restrict the number of processes that could benefit by optimizing control. 

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