Tools for dynamic model development

[摘要] For this thesis, several tools for dynamic model development were developed and analyzed. Dynamic models can be used to simulate and optimize the behavior of a great number of natural and engineered systems, from the movement of celestial bodies to projectile motion to biological and chemical reaction networks. This thesis focuses on applications in chemical kinetic systems. Ordinary differential equations (ODEs) are sufficient to model many dynamic systems, such as those listed above. Differential-algebraic equations (DAEs) can be used to model any ODE system and can also contain algebraic equations, such as those for chemical equilibrium. Software was developed for global dynamic optimization, convergence order was analyzed for the underlying global dynamic optimization methods, and methods were developed to design, execute, and analyze time-varying experiments for parameter estimation and chemical kinetic model discrimination in microreactors. The global dynamic optimization and convergence order analysis thereof apply to systems modeled by ODEs; the experimental design work applies to systems modeled by DAEs. When optimizing systems with dynamic models embedded, especially in chemical engineering problems, there are often multiple suboptimal local optima, so local optimization methods frequently fail to find the true (global) optimum. Rigorous global dynamic optimization methods have been developed for the past decade or so. At the outset of this thesis, it was possible to optimize systems with up to about five decision variables and five state variables, but larger and more realistic systems were too computationally intensive. The software package developed herein, called dGDOpt, for deterministic Global Dynamic Optimizer, was able to solve problems with up to nine parameters with five state variables in one case and a single parameter with up to 41 state variables in another case. The improved computational efficiency of the software is due to improved methods developed by previous workers for computing interval bounds and convex relaxations of the solutions of parametric ODEs as well as improved branch-and-bound heuristics developed in the present work. The convergence order and prefactor were analyzed for some of the bounding and relaxation methods implemented in dGDOpt. In the dGDOpt software, we observed that the empirical convergence order for two different methods often differed, even though we suspected that both had the same analytical convergence order. In this thesis, it is proven that the bounds on the solutions of nonlinear ODEs converge linearly and the relaxations of the solutions of nonlinear ODEs converge quadratically for both methods. It is also proven that the convergence prefactor for an improved relaxation method can decrease over time, whereas the convergence prefactor for an earlier relaxation method can only increase over time, with worst-case exponential dependence on time. That is, the improved bounding method can actually shed conservatism from the relaxations as time goes on, whereas the initial method can only gain conservatism with time. Finally, it is shown how the time dependence of the bounds and relaxations explains the difference in empirical convergence order between the two relaxation methods. Finally, a dynamic model for a microreactor system was used to design, execute, and analyze experiments in order to discriminate between models and identify the best parameters with less experimental time and material usage. From a pool of five candidate chemical kinetic models, a single best model was found and optimal chemical kinetic parameters were obtained for that model.

[发布机构] Massachusetts Institute of Technology