# (504a) Control and Optimization of Process Networks Using Structural Invariants

- Conference: AIChE Annual Meeting
- Year: 2009
- Proceeding: 2009 AIChE Annual Meeting
- Group: Computing and Systems Technology Division
- Session:
- Time:
Thursday, November 12, 2009 - 8:30am-8:55am

The theoretical problem we address in this paper can be described as follows: We have given two process systems, A and B. Process system A has a feedback control designed to minimize the objective J_A. System B is designed to minimize the objective J_B. What is the overall objective minimized when the two systems are combined to form a larger system?

The systems we are interested are the class called process systems. The dynamic behavior of the process system can be described by its inventories. Conservation laws for each inventory holds. Moreover, there exists a convex extension, which we related to the negative of the entropy. More generally there exists a storage function which is homogeneous degree one and continuous. These conditions turn out to be sufficient broad to develop a very rich theory for stability and optimality of the process networks.

Interconnections in networks are described by a matrix related to the so-called "cut-set matrix" of electrical circuit theory. The framework of circuit theory allows us to derive a potential function which can be interpreted in terms of the content and co-content. This function, which is equivalent to the non-equilibrium entropy production in a steady state steady system, serves as a starting point to derive a Lyapunov function for analyzing the stability of the process networks. The first and second order conditions determine existence, uniqueness and stability of the equilibrium points. A generalized version of Tellegen's theorem plays the central role in developing the objective function when a large number of distributed processes components are connected together to form a complex process network.

The content/co-content formulation together with Tellegen's theorem leads to a variational formulation which provides Euler Lagrange equations which define the dynamic trajectory of the process network. These trajectories are unique for a given set of initial and (terminal) boundary conditions provided the network connection satisfy a certain conic sector condition.

The theory we develop can be used in many application domains including design of plant-wide decentralized control systems, design distributed nonlinear observers, decentralized optimization, stability analysis of hybrid systems and asynchronous dynamic simulation. The basic requirement is that the flow conditions can be expressed in terms of conic sector (passivity) conditions.

Examples will be presented to illustrate application to different types of networks. Combinations of continuous and discrete connections (hybrid systems) will be described. We illustrate how distributed control and optimization can be implemented in network structures using the proposed framework and the proposed objective function can be shaped through control laws and algorithms. Finally, we demonstrate how Tellegen's theorem for process networks can strengthen the relaxation for global deterministic optimization by introducing generalized redundant constraints.