Advised by B. Erik Ydstie
Batch carbothermic process for aluminum production has been proposed as an alternative to the conventional electrolytic process and is currently under joint development by Alcoa and CMU. Two problems associated with this process are of particular academic interests. The fist problem is on the stability of the system dynamics under thermodynamic equilibrium conditions, as many metallurgical reactors are operated at temperatures high enough to assume thermodynamic equilibrium. The second issue is the control and optimization of batch processes with model inaccuracies.
The first problem is a classic one and has been investigated by Doherty and Perkins (1982), Lucia (1986) and Rouchon(1990) among others. After the discussion of the limitations in their works, a new analysis method based on the geometric insight is presented. The computationally challenging problem of finding the global minimum of the Gibbs energy is shown to be equivalent to the construction of the convex hull for the Gibbs energy functions of candidate phases. In convex analysis, a convex hull is defined with Caratheodary theorem. The infimum operator in the theorem can be dropped under mild assumptions by invoking the stronger result in Greiwank and Rabier (1990). We then demonstrate that each n-phase region is composed of mutually-exclusive (n-1)-dim simplexes and show the uniqueness and stability of the system steady state based on this observation.
The second problem has received relatively less attention. The control and optimization of batch processes is challenging due to the inherent system nonlinearities and model inaccuracies. In our work, barrier reformulation is applied so that the original discontinuous control profile can be approximated with a smooth one with required error bounds in the L2 norm sense. The control profile is then projected onto basis functions and updated online based on the sensitivity of the objective function to the decomposition coefficients. We further investigate the interplay between the parameter identification and the optimum seeking. The parameters in batch processes normally vary from batch to batch and their identification has to be carried on in real time. A mechanism to coordinate the conflict between the parameter identification and optimal control will be introduced. Standard examples from the batch control literature are used to illustrate the performance of our controller.