(545e) From Statistical Mechanics to Distributed Process Control

The current process control literature comprises a veritable morass of case studies. Each case shows that out of a large and increasing number of different approaches, methods, and algorithms, the chosen one controls a specific chemical process model. To move our field beyond this point it is necessary to investigate if any basic principles can apply to provide a unified approach to support such case studies and possibly make distinctions between them.

Our research program at CMU has advanced in the direction of developing a theory for process control by basing it on the axioms of statistical mechanics. The aim is to develop a system-theoretic basis for process control as it is applied in industry. Our theory includes explanations for why PI(D) controllers can be used for decentralized control and stabilization and MPC for multivariable control and optimization. Specifically we show that process control can discussed within the framework of dissipative Port-Hamiltonian systems. However, the different algebraic structures for how the manage physical flows (flow sheet algebra) and signal propagation (block diagram algebra) make it challenging to apply these results directly. The main problems is that measurements cannot be obtained incautiously and flows cannot be controlled accurately.

The presentation centers on explain how to solve two problems:

Problem 1: Develop a system theoretic basis for demonstrating stability of the state of an underlying (quantum) system (X,Π). X is a Hausdorff space and Π a set of processes that transform states into states. This problem is referred to as the problem of proving stability of the zero-dynamics in mathematical system theory. We don’t aim to control these microscopic states directly, indeed this is not possible since the state is large. To lift the theory of thermodynamics to a practical level we need to relate the microscopic state x to macroscopic variables such as energy and mol-numbers. To advance this theory we define a measure V that can be identified with the volume of the system. We then define energy, mol-numbers of n_c different chemical species, and a natural number Ω that counts the number of distinct microstates in an open neighborhood of given macroscopic state. Using Boltzmann’s definition of the entropy S=k_B ln(Ω) we demonstrate the existence of an equilibrium manifold. The equilibrium manifold corresponds the subsets of X_EQ where the distribution of states in uniform. We use the availability (exergy) as a Lyapunov function. The LaSalle invariance principle now applies and we conclude that the state of system (X,Π) converges to an ω-limit set. The implication is that the microscopic state fluctuates close to the equilibrium manifold. Using standard results from measure theory we now show that the support of the measure V defines the phase distribution and that the number of phases is bounded by n_c+2 as suggested by Gibbs in his monumental monograph.

Problem 2: The thermodynamic theory reviewed briefly above was developed for an isolated system. To connect such ideas with process control we need to re-interpret some assumptions so that they, at least in an approximate manner, can be applied to an open system. The main idea here is to use the assumption of local equilibrium. It states that at sufficiently small-scales equilibrium is established while fluctuations are averaged out. Continuum theories then apply. In due course we see that in the context of process control extensive (micro-canonical) variables provide a basis for writing (approximate) balance equations and defining fluxes. Whereas intensive variables (the tangents to the concave envelope of the equilibrium manifold) define force variables. The duality between intensive and extensive variables and their different algebraic structures allows us to define the two-port model and take advantage of formalisms developed in circuit theory. The availability function still applies and shows that system (X,Π) at the macroscopic level can be modelled as a dissipative Port-Hamiltonian system. This property allows for inter-connection of systems to form large networks. It also shows that we can define input output pairs that are related in a passive manner. These collocated actuator-measurement pairs consist of temperatures, pressures, and concentrations. The number of extensive variables to be controlled should be no less than the number of phases present in the mixture at that point in the network.

The main challenges in applying the theory are: (1) Variables such as levels, temperature and pressures are accessed through measurements and observers that induce delays and extra dynamics. (2) In a similar way we cannot control fluxes (flows) instantaneously and with great accuracy. At the level of practical process control some advantages of the dissipative Port-Hamiltonian structures are therefore lost. Additional modeling is required to approximate these dynamics and they must be included in the process control design. However, the theory developed above shows very accurate modeling is not needed to achieve stability and a certain level of robustness. The reason for this is that the system itself, due to passivity, has input output pairs with dynamics approximated by first order systems with a (sector bounded) nonlinear gains and no delay. Almost any feedback control strategy can now be applied for stabilization and control provided the measurement and actuator systems are included in the modeling process.