Optimal control problems simplest

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

Consider the simplest optimal control problem in which it is desired to find a continuous control function u t) that optimizes the objective functional... 

For the simplest optimal control problem with m control functions and n state variables in general, the objective functional to be optimized is... 

To yield the above inequality, a sufficient condition must include at least one inequality, which in fact is another optimal control problem. Thus, for the global minimum in the simplest optimal control problem, a set of sufficient conditions by Mangasarian (1966) requires that... 

Before delving into the proof, let us take the simplest optimal control problem and examine the application of Pontryagin s minimum principle. We will realize that we already have been applying the minimum principle to our optimal control problems. 

The method presented in this paper for selecting controlled variables (task 1) follows the ideas of Morari et al. (1980) and Skogestad and Postlethwaite (1996) and is very simple. The basis is to define mathematically the quality of operation in terms of a scalar cost function J to be minimized. To achieve truly optimal operation we would need a perfect model, we would need to measure all disturbances, and we would need to solve the resulting dynamic optinriza-tion problem on-line. This is unrealistic, and the question is if it is possible to find a simpler implementation which still operates satisfactorily (with an acceptable loss). The simplest operation would result if we could select controlled variables such that we obtained acceptable operation with constant setpoints, thus effectively turning the complex optimization problem into a simple feedback problem and achieve what we call self-optimizing control . 

A quadratic programming problem minimizes a quadratic function of n variables subject to m linear inequality or equality constraints. A convex QP is the simplest form of a nonlinear programming problem with inequality constraints. A number of practical optimization problems are naturally posed as a QP problem, such as constrained least squares and some model predictive control problems. 

For simplest cases the use of the dynamic programming method leads to analytical solutions. Meanwhile, for several phase variables and control parameters the search of optimal solutions is an extraordinarily complicated problem. Therefore, the application of the method of dynamic programming proves to be accurate in numerical computations where powerful computer techniques are used. 

Continuous and batch processes require different strategies for implementation of process analyzers. For continuous processes, startup must be controlled to quickly reach the optimal process conditions. Constant monitoring is needed for both continuous and batch processes to detect when the process is going out of control so that corrective action can be taken. For batch processes, end point detection is an important control parameter. For all analyses, the simplest method that performs the desired analysis is best. As the complexity of a method increases, changes in the instrument, materials, process, and measurement conditions can cause problems (1). 

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