At CMU we are in the process of developing a new approach for modeling and control of complex, networked systems. The new approach combines elements of the passivity theory of nonlinear control and process thermodynamics (first and second law with invariance in a Gallilean frame of reference). The current presentation focuses on consequences of an important toplogical result for networks called the Tellegen theorem. The Tellegen theorem is similar to the Gibbs-Duhem equation. It expresses orthogonality amongst primal and dual variables - in our context the forces and the fluxes. Forces are defined as gradients of the intensive variables whereas the fluxes are flows. The state space is the linear metric (Hausdorff) space of extensive variables and the dual space is a Banach space of intensive variables. The Tellegen result is independent of constitutive equations. These can be nonlinear, discrete, passive or active. They don't even have to be defined.
The Tellegen theorem allows us to develop a theory for self-organization of distributed networks of chemical processes, sensors actuators and communication systems. A process system is viewed as symmetric system with a convex extension as developed in the Friedrich-Lax theory of hyperbolic systems of PDEs. In our case the extension can be viewed as the entropy of semi-classical statistical mechanics. Its exact definition is not important, however. We establish conditions under which the complex network is isomorphic to a gradient system with a Riemannian structure. Paths of such systems are optimal in a very precise and can be defined using variational principles. For example, if the constitutive equations are linear then we solve quadratic programs. If they are discrete then the network solves mixed integer programs. In the general we solve mixed integeter nonlinear programs to local optimality.
The most interesting feature of the theory is that we change the objective function when we connect the process network with a communication system via an interface layers consisting of actuators sensors and A/D-D/A converters. The interface layer can be viewed as a Legendre transform which imposes a duality pairing between the signal space and space of physical variables. Convexity of this transform can be interpreted as detectability and reachability of the networks and controls. Non-convexity leads to ambiguity and local solutions. The theory applies to discrete event and hybrid systems with linear and nonlinear constitutive relations. Control changes the constitutive equations and leads to new optimality conditions and dynamic behaviors: Complex dynamics can be stabilized, non-convexities can be eliminated or introduced and completely new behaviors can be generated.
It is important to note that the algebraic structures of the process systems and communication systems are different and that signal flow is processed differently in process systems and information systems. For example, information can be copied whereas stream splits obey conservation principles. There is no natural convex extension, like the entropy which imposes constraints on the signal space and the dissipation is generally not observed. The combination of the physics and communication systems can therefore lead to unexpected and very useful dynamic behaviors.
I will compare and contrast the CMU program with the European Union Geoplex program. The goal of the EU sponsored project was to study modeling and control problems for complex systems in different physical domains using a single unified framework. In this framework, called the port-Hamiltonian framework, complex physical systems are interpreted as network interconnections of simpler physical systems that exchange energy. Properties of the complex system are then analyzed and controlled by looking at the properties of the simpler subsystems and the network structure. The focus on energy rather entropy leads to Symplectic rather that Riemannian structure. Significant structural properties inherent in chemical engineering processes due to dissipation are lost when the focus is placed on energy.