Minimum-time control problem

It can be shown [173] that the average time for the system to approach S5 is much smaller then the average escape time and thus the optimal escape paths found from the statistical analysis of the escape trajectories is independent of the initial conditions on the attractor and provides an approximation to the global minimum of the corresponding deterministic control problem. 

The minimum time problem is also known as the time optimal control problem. Coward (1967), Hansen and Jorgensen (1986), Robinson (1970), Mayur et al. (1970), Mayur and Jackson (1971), Mujtaba (1989) and Mujtaba and Macchietto (1992, 1993, 1996, 1998) all minimised the batch time to yield a given amount and composition of distillate using conventional batch distillation columns. The time optimal operation is often desirable when the amount of product and its purity are specified a priori and a reduction in batch time can produce either savings in the operating costs of the column itself or permit improved scheduling of other batch operations elsewhere in a process. Mathematically the problem can be written as ... 

For comparison, we now study the classical control problem and determine the field from Eq. (31), where the expectation values are replaced by the values of the classical coordinates and momenta, respectively. This yields the classical trajectory shown in Fig. 16, which is superimposed on the potential energy contours. Here, a perfect transfer is found where the particle stops in the minimum of the target potential well. A comparison with the trajectory derived from the quantum calculation (Fig. 14) shows that the classical orbit follows the quantum orbit closely until the reaction barrier is passed. At later times, due to the missing dispersion in the classical treatment, deviations are found. 

It is desired to change x from xq to a stable value of Xf at time ti with minimum displacement of the control rod. Thus, the optimal control problem is to find the control function u t) that minimizes the objective functional... 

We introduce this method to determine the minimum in the optimal control problem of Section 6.1.1 (p. 153) in which both final time and final state are... 

Several cost functions or criteria of excellence can be proposed for this problem, depending on whether some use can be made of the intermediate flux (which could consist of a limited period at full power between periods of zero power) or the need to allow activity to die away before maintenance work is undertaken, etc. For the purpose of the example here, we shall consider two cost functions minimum time and minimum lost flux-time. In the first, the control period itself is to be kept as short as possible in the second, the criterion is based on the wasted potential for delivering nuclear energy. These two examples are representative problems. Both, of course, have a considerable literature, having been approached by more than just the Pontryagin method. Our treatment follows that of Roberts and Smith (25,26) for the time optimum problem. 

Chapter 8 considers optimal control problems for systems that are either linear or nonlinear in the state variables but are linear in the controls. The solution of this class of problems leads to bang-bang control strategies. The existence of singular or intermediate control must also be investigated. Both time-optimal control and minimum integral square error problems are discussed. 

Minimum-time optimal control problem for continuous time systems can be formulated as follows ... 

Steady state operability is a necessary but not sufficient requirement for a well-designed plant, as the dynamic characteristics should also be considered. The dynamic operability is examined by the use of a dynamic model of the process and considers the issue of whether a given disturbance will be rejected quickly or whether a set-point change can be implemented within a given time interval, or both. This is addressed by solving an optimal control problem to find the minimum time, within which the process can respond to a disturbance or move to a new operating point with the available ranges of inputs. Such performance represents the best possible performance of any feedback controller, and similar to the steady state case, identifies the inherent operability characteristics of the process. 

There are two fundamental problems to be addressed in combining a number of separate software modules into a coherent package which should appear to its users as a single system. Firstly, the data to be handled must be transferable from one software module to another, and secondly it must be possible for run-time control to be switched within the whole system from one module to the next (see Figure 2). Ideally all of this should happen with a minimum of user intervention other than relevant application level selections. 

Literature on the optimization of the batch column is focused mostly on the solution of optimal control problems, which includes optimizing the indices of performance such as maximum distillate, minimum time, and maximum profit. However, literature on optimal design of batch distillation for performing specified operations by using the constant reflux or variable reflux policies is very limited[46]. 

Determination of the optimal temperature (or supersaturation) trajectory for a seeded batch crystallizer is a well studied problem. This is a dynamic optimization or optimal control problem. The process performance is determined by the crystal size distribution and product yield at the final time. For uniformity of shape and size in the crystals in a seeded batch crystallization process, it is essential to ensure that the nucleation phenomena occurs to the minimum and mostly the seeded crystals grow to the desired size at a certain rate. If nucleation occurs in the initial phase, then there is a possibility that the nucleated crystal will compete with the seeded ones, thus if the phenomena is of late growth, then nucleation in the earlier phase is preferred. Thus, depending upon the process operation, many types of objective functions have been proposed [4]. 

The best anti-surge control is the simplest and most basic that will do the job. The most obvious parameter is minimum-flow measurement, or if there is a relatively steep pressure-flow characteristic, the differentia pressure may be used. The latter parameter allows for a much faster response system, as flow measurement response is generally slow however, the speed of response need only be fast enough to accept expected transients. One major problem with the conventional methods of measurement and control is the need to move the set point for initiation of the control signal away from the exact surge point to allow some safety factor for control response time and other parameters not directly included... 

In the second phase, analysts in participating laboratories prepare and analyze a minimum of two conttol samples and two samples fortified at the proposed tolerance concenttation. This phase allows analysts to become familiar with the method before the analysis of samples that will be part of the method validation. Results from the second phase should demonsuate that the control samples are without interference and that the analysts in the participating laboratories can achieve acceptable recovery of analyte from the samples. It is not uncommon for an analyst to have to repeat the second phase several times before adequate results are obtained. Failure at this phase of the trial can cause a method to fail the Uial. Often the problems are related to a poorly written SOP that does not adequately describe the procedure. 

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