Set-point trajectory

As presented in the earlier chapters, the operating policy for a batch distillation column can be determined in terms of reflux ratio, product recoveries and vapour boilup rate as a function of time (open-loop control). Under nominal conditions, the optimal operating policy may be specified equivalently in terms of a set-point trajectory for controllers manipulating these inputs. In the presence of uncertainty, these specifications for the optimal operating policy are no longer equivalent and it is important to evaluate and compare their performance. 

In general, the operations optimization task is to find suitable set point trajectories for the controllers. As the controllers are omitted from our simplified TMP system model, no setpoint optimization is included in the study. However, the refiner scheduling optimization can also be considered as operations optimization with refiner activity set point trajectory as binary valued (on/off) function of time. 

In the control law part, the MFC eontroller has to calculate the set of control moves (Am) into the future that allows the system to follow a predefined set-point trajectory (from off-line optimization). This is done by solving for the quadratic cost function... 

Feedback control is the preferred method for controlling the wafer temperature trajectory. The design of feedback controllers for RTF is a challenging problem because of the nonlinear nature of the radiative and other heat transfer phenomena, the fast process dynamics, and the additional constraints that are placed on the wafer temperature response. The wafer temperature is controlled by adjusting the power output from an array of lamp heaters above and below the wafer to track the set-point trajectory shown in Fig. 22.16. The three key requirements for control are as follows ... 

Ramp error. It is important that the wafer temperature follow the set-point trajectory as closely as possible. Therefore, one measure of controller performance is the ramp error, defined as the difference between the set point and wafer temperature at any given time during the ramp. It should be less than 10 °C during the ramp. 

Overshoot. The corner of the set-point trajectory should be turned without overshoot. 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

The density is a maximum in all directions perpendicular to the bond path at the position of a bond CP, and it thus serves as the terminus for an infinite set of trajectories, as illustrated by arrows for the pair of such trajectories that lie in the symmetry plane shown in Fig. 7.2. The set of trajectories that terminate at a bond-critical point define the interatomic surface that separates the basins of the neighboring atoms. Because the surface is defined by trajectories of Vp that terminate at a point, and because trajectories never cross, an interatomic surface is endowed with the property of zero-flux - a surface that is not crossed by any trajectories of Vp, a property made clear in Fig. 7.2. The final set of diagrams in Fig. 7.1 depict contour maps of the electron density overlaid with trajectories that define the interatomic surfaces and the bond paths to obtain a display of the atomic boundaries and the molecular structure. 

In MPC a dynamic model is used to predict the future output over the prediction horizon based on a set of control changes. The desired output is generated as a set-point that may vary as a function of time the prediction error is the difference between the setpoint trajectory and the model prediction. A model predictive controller is based on minimizing a quadratic objective function over a specific time horizon based on the sum of the square of the prediction errors plus a penalty... 

The second point depends on the nature of the dynamics. Say we begin with a limiting case of truly ergodic trajectories. Then I really do not need to compute the actual trajectories. I would get the very same result if I simply postulate that my final set of trajectories is the set of all possible trajectories (where by possible I mean that they conserve energy, etc.). Information theory starts from this limit, which we refer to as the prior distribution. [1,3,23]... 

The method proposed for improving the batch operation can be divided into two phases on-line modification of the reactor temperature trajectory and on-line tracking of the desired temperature trajectory. The first phase involves determining an optimal temperature set point profile by solving the on-line dynamic optimization problem and will be described in this section. The other phase involves designing a nonlinear model-based controller to track the obtained temperature set point and will be presented in the next section. 

Model predictive control (MPC) was developed in the 1970s and 1980s to meet control challenges of refineries. The advantages of MPC are most evident when it is used as a multivariable controller integrated with an optimizer. The greatest MPC benefits are realized in applications with dead-time dominance, interactions, constraints, and the need for optimization. As opposed to a traditional control loop, where the controller responds to a difference (error) between the set point and measurement, the predictive controller uses a vector difference between the future trajectory of the set point and the predicted trajectory of the controlled variable as its input (Figure 2.52). 

Sofar the imaging results of Fig. 3.1 were discussed in very classical terms, using the notion of a set of trajectories that take the electron from the atom to the detector. However, this description does not do justice to the fact that atomic photoionization is a quantum mechanical proces. Similar to the interference between light beams that is observed in Young s double slit experiment, we may expect to see the effects of interference if many different quantum paths exist that connect the atom to a particular point on the detector. Indeed this interference was previously observed in photodetachment experiments by Blondel and co-workers, which revealed the interference between two trajectories by means of which a photo-detached electron can be transported between the atom and the detector [33]. The current case of atomic photoionization is more complicated, since classical theory predicts that there are an infinite number of trajectories along which the electron can move from the atom to a particular point on the detector [32,34], Nevertheless, as Fig. 3.2 shows, the interference between trajectories is observable [35] when the resolution of the experiment is improved [36], The number of interference fringes smoothly increases with the photoelectron energy. 

If the end points are defined, the solution to the classical equations of motion corresponding to an elementary process can be sought the resulting system point trajectories represent the realistic evolution of the polyatomic system from reactants to products on the given potential energy surface Wm(QJ) at the total energy H — T + W(Q ). The solutions to the equations of motion can thus be considered as transformations leading from a set of boundary conditions in the reactant asymptote to a set of boundary conditions in the product asymptote. 

Although each set of boundary conditions defines a unique trajectory, not all of the IF quantities at each of the end points can be controlled in an experiment (4,47,48) these quantities usually have random distributions (impact parameter, vibrational phase, etc.). Consequently, it is useful to choose the values of the uncontrollable boundary conditions randomly. For a sufficiently high number of the randomly chosen values (on a relevant interval), all boundary conditions are included and the resulting set of trajectories (related to an elementary process) represents a dynamical picture of the elementary process within the quasiclassical approach (6,44). 

Twenty configurations from a 35 ps ground state adiabatic trajectory, chosen to be on resonance with the laser bandwidth corresponding to the experiments, were selected as the starting points for non-adiabatic excited state trajectories. A corresponding set of trajectories was run in D2O, with a model identical in all respects to the work described previously except that the mass of the H atom was changed from 1 to 2 amu, and preliminary results of the behavior in D2O are included here. 

In the classical concept of predictive control, the trajectory (or set-point) of the process is assumed to be known. Control is implemented in a discrete-time fashion with a fixed sampling rate, i.e. measurements are assumed to be available at a certain frequency and the control inputs are changed accordingly. The inputs are piecewise constant over the sampling intervals. The prediction horizon Hp represents the number of time intervals over which the future process behavior will be predicted using the model and the assumed future inputs, and over which the performance of the process is optimized (Fig. 9.1). Only those inputs located in the control horizon H, are considered as optimization variables, whereas the remaining variables between Hr+1 and Hp are set equal to the input variables in the time interval Hr. The result of the optimization step is a sequence of input vectors. The first input vector is applied immediately to the plant. The control and the prediction horizon are then shifted one interval forward in time and the optimization run is repeated, taking into account new data on the process state and, eventually, newly estimated process parameters. The full process state is usually not measurable, so state estimation techniques must be used. Most model-predictive controllers employed in industry use input-output models of the process rather than a state-based approach. 

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