Optimizing control search direction

Using a direct search technique on the performance index and the steepest ascent method, Seinfeld and Kumar (1968) reported computational results on non-linear distributed systems. Computational results were also reported by Paynter et al. (1969). Both the gradient and the accelerated gradient methods were used and reported (Beveridge and Schechter, 1970 Wilde, 1964). All the reported computational results were carried out through discretization. However, the property of hyperbolic systems makes them solvable without discretization. This property was first used by Chang and Bankoff (1969). The method of characteristics (Lapidus, I962a,b) was used to synthesize the optimal control laws of the hyperbolic systems. 

As for all optimization procedures, the function minimized need not be directly related to the potential energy of the physical system. For example, the residual R of equation (1), a measure of the fit of measured and observed diffraction patterns, can form the object function. Similarly the temperature of the system simply determines the relative probability of the simulation permitting two successive states, and provides a convenient means of controlling the degree of searching permitted during the course of the simulation. As discussed in Chapter 1, it should not be equated in this context with the thermodynamic temperature. 

As an alternative to RSM, simulation responses can be used directly to explore the sample space of control variables. To do so, a lot of combinatorial optimization approaches were adapted for simulation optimization. In general, there are four main classes of methods that have shown a particular applicability in (multi-objective) simulation optimization Meta-heuristics, gradient-based procedures, random search, and sample path optimization. Of particular interest are meta-heuristics as they have shown a good performance for a wide range of combinatorial optimization approaches. Therefore, commercial simulation software primarily uses these techniques to incorporate simulation optimization routines. Among meta-heuristics, tabu search, scatter search, and genetic algorithms are most widely used. Table 4.13 provides an overview on aU aforementioned techniques. 

The second important difference between human and automated controllers is that, as noted by Thomas [199], while automated systems have basically static control algorithms (although they may be updated periodically), humans employ dynamic control algorithms that they change as a result of feedback and changes in goals. Human error is best modeled and understood using feedback loops, not as a chain of directly related events or errors as found in traditional accident causality models. Less successful actions are a natural part of the search by operators for optimal performance [164]. 

The goal will be to determine optimal equipment parameters and an optimal steady-state operating point such that feasible operation is maintained for all realizations of uncertain parameters within a specified uncertainty region, with a set of outputs controlled at their nominal values. The use of controller parametrization provides a performance limit for linear control. Feasibility with respect to imcertain parameter variation is handled by posing the problem directly within a multi-period framework. The plant will be assumed to be open-loop stable at the nominal operating point, permitting use of the control structure of Fig. 5. Note that while the search is restricted to linear controllers, path constraints are enforced for the nonlinear plant... 

The constructive method, which is considered as a major breakthrough in control theory, was developed in the last decade. As it stands, the method is intended for feedback control design, and its application to the batch motion case requires the nominal output to be tracked and a suitable definition of finite-time batch motion stability. In a more applied eontext, the inverse optimality idea has been applied to design the nominal motion of homo [11] and copolymer [12] reactor, obtaining results that are similar to the ones drawn from direct optimization [4]. The motion was obtained from the recursive application of the process dynamical inverse [13], and the inverse yielded a nonlinear SF controller [9, 10] that was in turn used to specify a conventional feedforward-feedback industrial control scheme. However, the issues of motion stability and systematized search were not formally addressed. 

In this section the joint process and control design problem of batch processes is addressed. The problem is formulated within an optimization framework, including the search of the equipment, the motion, and the controller. As stated in the introduction, the emphasis will be placed on the motion and control problem design via the inverse optimality, while the complementary role of the direct optimization framework will be outlined only, in the understanding that the corresponding tools are known and have been employed in batch process studies [3-5, 6]. First, the problem is stated. Then, a passivated dynamical inversion is drawn, and the result is applied to construct the output-feedback controller, and to set the algorithm to design the nominal batch motion. 

The shift parameter can be used to ensure that the optimization proceeds downhill even if the Hessian has negative eigenvalues. In addition, it can be chosen such that the step size is lower or equal to a predefined threshold. Popular methods using a shift parameter are the rational function optimization (RFO) [48] and Trust Radius (TR) methods [49, 50]. A finer control on the step size and direction can be achieved using an approximate line search method, which attempts to fit a polynomial function to the energies and gradients of the best previous points [51]. 

The process optimization includes the initial condition search. In order to optimize the process, the control variables should be selected properly. When the control variables are poorly selected, the process optimization may end up with unexplainable result. For the best result, the characteristics of the process variables should be carefully understood. The process variables can be classified as machine variables and material variables, as shown in Table 1. The state of the resin in the mold is closely related with material variables such as resin temperature, resin pressure and viscosity, etc. However, the material variables are hardly measured and controlled directly. The machine variables can be classified as independent, dependent and restrictive variables. The control variables for the process optimization should be chosen among the indepmdait variables. The depmdmt variables such as switchover point and holding time are to be properly found. These variables can be set in the initial condition searching stq>. For the purpose, the PMS provides the detailed information in the mold. 

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