Dynamic optimization problem

Biegler, L. T., Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Comp, and Chem. Eng. 8(3/4), 243-248 (1984). 

Logsdon, J. S., and Biegler, L. T., Decomposition strategies for large scale dynamic optimization problems, in press, Chem. Eng. Sci. (1991). 

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. 

Computational techniques for the solution of a dynamic optimization problem as formulated above have been an active area of research. There are a number of different techniques that have been proposed in the literature to solve this class of problems. In general, they are mainly classified into three classes. 

Selecting the best time-temperature trajectory is a challenging dynamic optimization problem with constraints. There are rigorous nonlinear programming approaches to this problem, but there are also some more simple and practical methods that can be employed, as discussed in Chapter 4. 

Vassiliadis, V.S., Computational Solution of Dynamic Optimization Problems with General Differential-Algebraic Constraints. PhD thesis, (Imperial College, London, 1993). 

We have chosen to use an SQP method (Chen, 1988) as the basis for local solution of the dynamic optimization problem. This choice is based on the po-... 

Local approximations (linear or quadratic) are often particularly poor in dynamic optimization problems. For instance, this situation is found to occur when taking the full step predicted from the local approximation, 6, causes a path constraint to become active or the system to become unstable. 

The dynamic optimization problem of interest in this contribution can be stated as follows ... 

The objeetive of the dynamic optimization problem should be stated before the model reduction is performed, in order to choose the right variables to be kept in the redueed model. The objeetive of the dynamic optimization problem will be stated as follows Increase the plant production by 20% with minimal energy consumption in the... 

Hong, W., Wang, S., Li, P., Wozny, G., Biegler, L.T., A Quasi-Sequential Approach to Large-Scale Dynamic Optimization Problems, AlChe Journal 52, No. 1 (255), 2006. Dorneanu, B., Bildea, C.S., Grievink, L, On the Application of Model Reduction to Plantwide Control, IT " European Symposium on Computer Aided Process Engineering, Bucharest, Romania, 2007. 

V.S. Vassiliadis, R. W. H. Sargent, C. C. Pantelides, 1994, Solution of a class of multistage dynamic optimizations problems. 1-Problems without path constraints, Ind. Eng. Chem. Res, 33,2111-2122. 

The prediction horizon is discretized in cycles, where a cycle is a switching time tshift multiplied by the total number of columns. Equation 9.1 constitutes a dynamic optimization problem with the transient behavior of the process as a constraint f describes the continuous dynamics of the columns based on the general rate model (GRM) as well as the discrete switching from period to period. To solve the PDE models of columns, a Galerkin method on finite elements is used for the liquid... 

Seeded batch crystallization process Three problems with two or three objectives from (1) maximization of the weight mean size of the crystal size disuibution, (2) minimization of the nucleated product, (3) minimization of total time of operation, and (4) minimization of coefficient of variation. NSGA-n Dynamic optimization problems were solved to find the optimal temperature profile. Sarkar et al. (2006)... 

Semibatch reactive crystallization process. Maxinrization of weight mean size while minimizing coefficient of variation NSGA-n Dynamic optimization problems were solved to find the optimal feed addition profile. Sarkar et al. (2007)... 

Equation 8.19 can be used for analysis of both steady-state and dynamic optimization problems. In time-varying processes, actual implementation of optimum operation policies requires discretization of the dynamic trajectories [ 165]. In these cases, the vectors of state variables, end-use properties and manipulated variables include the set of discretized values along the whole dynamic trajectory. This means, for example, that NX equals the number of state variables multiplied by the number of discretized intervals (or sampling intervals). Finally, it must be clear that some of the weighting values can be equal to zero, which means that some of the available data may not be relevant for operation of the analyzed polymerization problem. 

If we consider once again the example of minimizing muscle stress (cf. Sec. 6.6.3), an analogous dynamic optimization problem may be posed as follows. Find the time histories of all actuator forces that minimize the sum of the squares of actuator stresses ... 

A better approach involves parameterizing the input muscle activations (or controls) and converting the dynamic optimization problem into a parameter optimization problem (Pandy et al., 1992). The procedure is as follows. First, an initial guess is assumed for the control variables a. The system dynamical equations [Eq. (6.7) and (6.1)] are then integrated forward in time to evaluate the cost function in Eq. (6.14). Derivatives of the cost function a constraints are then calculated and... 

FIGURE 6.23 Compuiaiional algorithm used to solve dynamic optimization problems in human movement studies. The algorithm computes the muscle excitations (controls) needed to produce optimal perfoimance (e.g., maximum jump height). The optimal controls are found by parameter optimization. 

FIGURE <>.29 Joint contact forces acting at the hip, knee, and ankle during gait. The results were obtained by solving a dynamic optimization problem for normal walking (see Fig. 6.28 for details). [Modifiedfrom Anderson and Pandy (2001a).]... 

Note 3.3. Mixed-integer dynamic optimization problem formulation (Bansal et al., 2000 Bansal, 2000). 

Description of the Dynamic Optimization Problem in Reactive Distillation... 

Formulating design as a dynamic optimization problem, we found that for the synthesis of MTBE, a tradeoff between control and economic performances exists. We solved this multiobjective optimization problem by incorporating appropriate time-invariant parameters e.g. column diameter, heat transfer areas and controllers parameters) in the frame of a dynamic optimization problem in the presence of deterministic disturbances. The design optimized sequentially with respect to dynamic behavior leads to a RD process with a total annualized cost higher than that obtained using simultaneous optimization of spatial and control structures. 

This dynamic optimization problem can be solved using a standard dynamic programming recursion, and for brevity we will only consider the infinite horizon case T oo. Let V(x,r) denote the principal s infinite horizon optimal... 

Moreover, if we let /i (x,r) denote the production rate that maximizes the right hand side of (4.44), then the second best production policy fi (t) solving the dynamic optimization problem (4.44) is given by ... 

However, for the practical solution of realistic, large-scale dynamic optimization problems from the field of chemical engineering, the solution methods must be tailored to the domain-specific, inherent problem structures. Also, they must be interfaced with the models generated by dynamic flowsheeting packages. In this paper, we discuss some of what is currently available, and we identify major areas for future research. 

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