Integrating Networked Process Systems to Solve Dynamic Optimization Problems
B. Erik Ydstie and Michael R. Wartmann. Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213
We introduce a new framework for distributed control and optimization of complex networks based on variational calculus. Conservation laws for extensive quantities and the second law of thermodynamics lead to conditions for stability and optimality of the network. We derive a general way of describing interconnections in networks through matrix representations that capture a network's topology using basic principles from electrical engineering methodologies. This shows how the dynamics of independent entities in a network define the objective function of the optimization problem that is simultaneously solved. A generalized version of Tellegen's theorem from electrical circuit theory plays a central role in developing the objective function of the regarded dynamic networks. These results indicate that we can solve optimization problems using dynamical systems, and how the objective function depends on the choice of feedback control and strategies. We highlight the concept of duality between the extensive variables and intensive variables and their role for the process network. Several examples are presented to illustrate these principles for different types of network connections, for continuous and discrete and linear and nonlinear flow connections. The boundary or terminal connections of the network are investigated in connection with the resulting primal and dual optimization problem and the effect on the dynamic behavior of the system. We illustrate how distributed control and optimization can be implemented in network structures using the proposed framework.