(110d) Advances in Bounding and Comparison Methods for Dynamic Process Models

Parametric systems of ordinary differential equations (ODEs) are widely used to model dynamic process systems such as batch reactors and closed-loop control systems. While there have been several recent advances in methods for optimization and reachable set generation for dynamic systems, further advances somewhat limited by the implicit nature of an ODE system. Since the effect of perturbations of initial conditions or parameters on a dynamic model’s state variables is only accessible indirectly via simulation, and since existing theory concerning ODEs is limited, methods for dynamic optimization and sensitivity analysis can benefit significantly from a stronger theoretical understanding of how ODE solutions behave.

This presentation describes new sufficient conditions under which one ODE system’s state variables will never exceed another’s; these conditions are significantly less stringent than previously proposed conditions. As an application of these conditions, it is shown that a state-of-the-art method by Scott and Barton [1] for constructing useful convex enclosures of the reachable set of a parametric ODE system has the following property: if tighter enclosures of the ODE right-hand side function are available, then these enclosures will necessarily translate into tighter enclosures of the reachable set. While plausible, this property was previously unknown and surprisingly difficult to prove. This result shows that it is worthwhile from a dynamic optimization standpoint to seek tighter enclosure methods for closed-form functions and models in general, since doing so translates into superior descriptions of reachable sets for dynamic systems, which are in turn useful in robust control and global dynamic optimization. Moreover, this result enables direct comparison of competing methods for dynamic reachable set generation; specific examples of this are presented. Further implications and examples are discussed.

Reference

[1] J.K. Scott and P.I. Barton, Improved relaxations for the parametric solutions of ODEs using differential inequalities, J. Glob. Optim. 57:143-176, 2013.