Optimal control problems final state

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. 

Finally, finite elements are added as decision variables in (27) not just to ensure accurate approximation (of the state and control profiles), but also to provide optimal points of discontinuity for the control profile. This dual purpose led Cuthrell and Biegler (1987) to distinguish some elements as finite-and super-elements. These roles can be combined, however, if one considers the NLP formulation of the optimal control problem given below ... 

Subject to the satisfaction of Equations (1.13)-(1.15), the optimal control problem is to And the control function T t) that brings in time the final unsteady state fluid temperature closest to the steady state wall temperature. Hence it is desired to minimize the objective functional... 

This is the optimal control problem of Section 5.2 (p. 126) in which both the final time and the final state are unspecified or free. Using vectors, the objective is to minimize the functional... 

Consider the optimal control problem of Section 6.1.1 (p. 153). If the final state is fixed, say, at y(ff) = yf, then the variation 5yt must be zero. Consequently, Equation (6.9) simplifies to... 

In this case, we have the optimal control problem of the last section, but with the final state fixed. Thus, Equation (6.14) becomes... 

Integral constraints could be equality or inequality constraints. We first consider integral equality constraints in an optimal control problem with free state and free final time. 

Table 7.1 lists the parameters used to solve the optimal control problem with the above algorithm. In this problem, the final time and final states are free. It is desired to find two controls, u t) and U2 t), and the final time tf that maximize the final product concentration 2/4(tf). 

We explain the shooting Newton Raphson method with the help of an optimal control problem having one state, one control, and fixed final time. The objective of the problem is to find the control function u t) that minimizes the functional... 

The control landscape, 7 i f[s(0], only has extrema corresponding to either no control or perfect control Furthermore, as the control problem li) If) only specifies traversal from the initial state li) to the final state If), generally an infinite number of optimal paths... 

A periodicity condition implies that the initial and final values of a state (or costate) variable are equal to a single value. Thus, in a optimal periodic control problem, the set of state as well as costate equations poses a two point boundary value problem. Either successive substitution or the shooting Newton-Raphson method may be used to integrate the periodic state and costate equations. 

Since optimal control theory was applied in the history matching problem, an adjoint system of equations similar to the state system of equations (equations 1 to 3) was derived. In the adjoint system of equations, Q was substituted for the state primary dependent variable, P. The adjoint system of equations was solved backward in time. The final adjoint equation, the final condition and the associated boundary conditions are shown in equations (4), (5) and (6), respectively. 

Finally, Figures 14.9 through 14.11 show a comparison between optimal control and state variable profiles, as function of the objective function weights. In the first case, we have run the problem with co, = 1 x 10 and 0)2=2x 1(F, while in the second... 

In this paper an industrial semibatch polymerisation process is considered. In order to guarantee the product quality particularly controlled reaction conditions are necessary. The general aim of this work is to ascertain optimal state and control profiles and to develop a model-based control scheme. As a first step, this paper introduces the dynamic model, which is validated with experimental data, and describes the optimisation approach. An aim of the work is to assess the possibilities of the commercial flowsheet simulator CHEMCAD in the optimisation of the performance of semibatch polymerisation processes. Finally the formulation of the mathematical optimisation problem, solution strategies and their implementation in CHEMCAD are discussed. 

One final point about closed-loop process control Economic considerations dictate that to derive optimum benefits, processes must invariably be operated in the vicinity of constraints. A good control system must drive the process toward these constraints without actually violating than. In a polymerization reactor, the initiator feed rate may be manipulated to control monomer conversion or Mj, however, at times when the heat of polymerization exceeds the heat transfer capacity of the kettle, the initiator feed rate must be constrained in the interest of thermal stability. In some instances, there may be constraints on the controlled variables as well. Identification of constraints for optimized operation is an important consideration in control system design. Operation in the vicinity of constraints poses problems because the process behavior in this region becomes increasingly nonhnear. In many cases, the capabihty to control polymerizations is severely limited by the state of the art in measurement instrumentation. In other cases, the dynamic response of the instruments dictates the design strategy for the process. 

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