Optimal control schemes

A proposed control scheme by Drew and Burns (1992) uses an LQG design, whereby the three process parameters are controlled in an optimal manner, their values (particularly the moisture control) being estimated. 

As discussed by M. Shapiro and R Brumer in the book Quantum Control of Molecular Processes, there are two general control strategies that can be applied to harness and direct molecular dynamics optimal control and coherent control. The optimal control schemes aim to find a sef of external field parameters that conspire - through quantum interferences or by incoherent addition - to yield the best possible outcome for a specific, desired evolution of a quantum system. Coherent control relies on interferences, constructive or destructive, that prohibit or enhance certain reaction pathways. Both of these control strategies meet with challenges when applied to molecular collisions. 

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

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. 

However, I do not agree with Prof. Rice s suggestion that the Brumer-Shapiro schemes, as the optimal control schemes, become more difficult to apply as the molecules increase in size. At the moment, barring the question of the role of overlapping levels, where... 

S. R. Jain When Prof. Rice talks about optimal control schemes, his Lagrange function follows a time-reversed Schrodinger equation. Is it assumed in the variational deduction that the Hamiltonian is time reversal invariant that is, is it always diagonalizable by orthogonal transformations ... 

Two types of nonlinear optimizers—the sectionalized linear program and the gradient search—have been successfully implemented in advanced computer control schemes. 

The remainder of this paper is organized as follows In Sect. 5.2, we present the basic theory of the present control scheme. The validity of the theoretical method and the choice of optimal pulse parameters are discussed in Sect. 5.3. In Sect. 5.4 we provide several numerical examples i) complete electronic excitation of the wavepacket from a nonequilibrium displaced position, taking LiH and NaK as examples ii) pump-dump and creation of localized target wavepackets on the ground electronic state potential, using NaK as an example, and iii) bond-selective photodissociation in the two-dimensional model of H2O. A localized wavepacket is made to jump to the excited-state potential in a desirable force-selective region so that it can be dissociated into the desirable channel. Future perspectives from the author s point of view are summarized in Sect. 5.5. 

The trade-offs among process design, optimization and control must be considered. The hierarchical or distributed nature of the plant or process may need to be exploited in an advanced control scheme. The operation of energy-integrated plants requires design of control systems which are decentralized (such as with microprocessors) but which respond to overall plant objectives via a communication link to a larger computer. 

In heat recovery applications there can be a large number of feasible plant configurations. After the configuration is optimized based on steady state considerations (which may not be an easy problem), the evaluation of the effectiveness of various control schemes can be performed. The dynamic plant operation must be evaluated in terms of economics, regulation, reliability, and safety over a broad range of operating regimes. 

At the other extreme, it may be argued that the traditional low-dimensional models of reactors (such as the CSTR, PFR, etc.) should be abandoned in favor of the detailed models of these systems and numerical solution of the full convection-diffusion reaction (CDR) equations using computational fluid dynamics (CFD). While this approach is certainly feasible (at least for singlephase systems) due to the recent availability of computational power and tools, it may be computationally prohibitive, especially for multi-phase systems with complex chemistry. It is also not practical when design, control and optimization of the reactor or the process is of main interest. The two main drawbacks/criticisms of this approach are (i) It leads to discrete models of very high dimension that are difficult to incorporate into design and control schemes. 

These control schemes are very effective for a certain class of processes but are not versatile and ineffective for, for example, multilevel-multilevel transitions we shall consider in this chapter. There exist several mathematical studies that investigate controllability of general quantum mechanical systems [11,12]. The theorem of controllability says that quantum mechanical systems with a discrete spectrum under certain conditions have complete controllability in the sense that an initial state can be guided to a chosen target state after some time. Although the theorem guarantees the existence of optimal fields, it does not tell us how to construct such a field for a given problem. 

Polyatomic molecules are characterized by complex internal motions leading to several possible rearrangements. With the advent of ultrafast lasers, much hope arose for achieving mode selectivity in chemical reactions based on the interaction between the laser pulses and the dynamics of the molecule. In the quest to steer complex systems, an especially attractive control scheme is the adaptive optimal... 

A control mechanism has been proposed on the basis of the joint analysis of the experimental and theoretical information. The control scheme leading selectively to the formation of CpMn(CO)3+ is represented in Figure 6(b). The experiment realized with an optimal laser field is simulated by one pump pulse (at 3.49 eV) followed by a probe pulse (at 4.716eV) designed with the adequate properties of phase, frequency, and duration. Within these specific conditions, the quasi-bound state c A is populated selectively and the CO dissociation... 

In a well-engineered system, measurement noise is unlikely to affect the performance of a PI controller significantly (Section V.A.2), but it may limit the use of derivative control action and will certainly limit the performance of advanced control schemes that attempt to approach ideal control (Section V.A.5). Our approach to PID control has been to optimize PI controllers and leave the possible benefit of derivative action to the commissioning engineers. 

The overall optimization and control scheme is illustrated in Fig. 1, and the procedure can be summarized as follows ... 

To conclude, the technique developed in this study proved to be reliable for the optimization of a complex control law, as well as the determination of its robustness. Further work will focus on the comparison of different usual control schemes. Their multi-objective optimization will help us to have a clear insight into their optimal performances. Other objectives formulations will also be studied that may lead to better understanding of these results by the decision maker. Typically, at the industrial scale, a WWTP manager does not want to discharge pollutant loads as low as possible but his objective is to have the insurance to meet the quality standards (usually in term of... 

However, the problem of analytical transferability remains. The optimal, but usually very unrealistic, situation assumes that the analytical methods, including their calibration and quality assurance, are identical in the laboratories. A more pragmatic approach involves standardization of analytical protocols, common calibration, design of a sufficiently efficient external quality control scheme, and the use of mathematical transfer functions if the results still are not directly comparable. 

In this chapter we have advocated the use of online model-based optimization for the automatic control of SMB plants. This approach has the advantage that the process is automatically operated at its economic optimum while meeting all relevant constraints on purities and flow rates. Application to a pilot-plant-scale reactive SMB process for glucose isomerization showed that implementation at a real plant is feasible - the requirements for additional hardware are moderate (a high-level PC and online concentration measurements in the recycle line). The experiments confirmed the excellent properties of the proposed control scheme. The scheme is extremely versatile, and the cost function and constraints can easily be adapted to any specific separation task. 

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