Passivity based control of transport reaction systems

Inventory control is based on the idea manipulating process flows so that the inventories follow their set points. The operator mapping flows to inventories in a coarse grained (macroscopic) system is passive and any input strictly passive (ISP) feedback controller can, therefore, be used to achieve input-output stability. Examples of ISP controllers include the PID controller, parameter adaptive feedforward control, optimal controllers and many nonlinear and gain scheduling controllers. Input-output stability and convergence of inventories, such as total mass and energy can be used to show that other internal state variables are bounded and belong to an invariant set. However, this does not necessarily imply convergence of all state variables to stationary valves because a stabilizability condition, known as strict state passivity, must be satisfied. The 2(nd) law of thermodynamics is used to develop sufficient conditions for strict state passivity in the space of intensive variables. The theory relies on following two assumptions concerning nonequilibrium Systems: (1) The hypothesis of local equilibrium, and (2) That a local entropy is defined using semiclassical statistical mechanics. These assumptions allow us to define a local entropy function for the coarse grained system which is homogeneous degree one, concave, and has positive temperature. Subject to these conditions stability theory is developed for conjugate variables, such as temperature, pressure and chemical potential in infinite dimensional reaction-diffusion-convection systems. In classical it-reversible thermodynamics this type of analysis assumes linearity and symmetry of transport relations. The use of Gibbs tangent plane condition allows us to define a Lyapunov like storagefiinctionfor passivity design, which gives stability criteria for nonlinear problems. The resulting sufficient condition for stability can be expressed in terms of dimensionless groups, similar to the (second) Damkohler number. Simple simulation examples illustrate the application of the theory. (c) 2005 American Institute of Chemical Engineers.