Differential-algebraic nonlinear optimization

As the plant to be optimized considers a process operating at steady state, then the variation of the phase concentrations with time is zero. For this reason, the mathematical model that describes the plant is a set of ordinary differential equations, as the phase concentrations depend only on the module axial position. In the tanks, the concentrations are constant. The differential-algebraic nonlinear optimization (DNLP) problem PI to be solved includes the ordinary differential equations that represent the mass balances for the phases in the membrane module. The objective function to be maximized is the amount of metal processed FeC , where Fe is the effluent flow rate whose Cr(VI) concentration after dilution from wastewaters is C . The problem has the following form ... 

Off-line analysis, controller design, and optimization are now performed in the area of dynamics. The largest dynamic simulation has been about 100,000 differential algebraic equations (DAEs) for analysis of control systems. Simulations formulated with process models having over 10,000 DAEs are considered frequently. Also, detailed training simulators have models with over 10,000 DAEs. On-line model predictive control (MPC) and nonlinear MPC using first-principle models are seeing a number of industrial applications, particularly in polymeric reactions and processes. At this point, systems with over 100 DAEs have been implemented for on-line dynamic optimization and control. 

The application of simultaneous optimization to reactor-based flowsheets leads us to consider the more general problem of differentiable/algebraic optimization problems. Again, the optimization problem needs to be reconsidered and reformulated to allow the application of efficient nonlinear programming algorithms. As with flowsheet optimization, older conventional approaches require the repeated execution of the differential/algebraic equation (DAE) model. Instead, we briefly describe these conventional methods and then consider the application and advantages of a simultaneous approach. Here, similar benefits are realized with these problems as with flowsheet optimization. 

Note that many problems of different kinds (i.e., the solution of differential-algebraic equation systems or constrained optimization problems) lead to the numerical solution of an underdimensioned nonlinear system. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

This text first presents a fundamental discussion of linear algebra, to provide the necessary foundation to read the applied mathematical literature and progress further on one s own. Next, a broad array of simulation techniques is presented to solve problems involving systems of nonlinear algebraic equations, initial value problems of ordinary differential and differential-algebraic (DAE) systems, optimizations, and boundary value problems of ordinary and partial differential equations. A treatment of matrix eigenvalue analysis is included, as it is fundamental to analyzing these simulation techniques. 

Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation, using nonlinear programming techniques, is discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities and thus requires efficient and novel solution techniques. 

The estimation of model parameters is an important activity in the design, evaluation, optimization, and control of a process. As discussed in previous chapters, process data do not satisfy process constraints exactly and they need to be rectified. The reconciled data are then used for estimating parameters in process models involving, in general, nonlinear differential and algebraic equations (Tjoa and Biegler, 1992). 

The major change is that the model constraint consists of nonlinear differential equations rather than linear or nonlinear algebraic equations as in the previous sections. The differential equations make it much more expensive to evaluate the constraints while solving the optimization problem. We can Increase the efficiency of the optimizer if... 

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