(612c) Computing Sensitivities for Nonsmooth Dynamic Systems

Nonsmoothness can enter chemical engineering models through intrinsic nonsmooth phenomena such as flow reversals, incorporation of safety mechanisms such as check and relief valves, thermodynamic phase transitions, and flow regime changes.  Conventional derivative-based methods for equation solving and optimization can fail when applied to these systems, since required derivative information may not be available, finite-difference approximations may no longer provide useful information, and convergence results for smooth systems may no longer hold.

Specialized methods for nonsmooth equation solving and optimization make use of the (Clarke) generalized Jacobian [1], which is a set-valued mapping containing sensitivity information.  In particular, semismooth Newton methods for equation solving and bundle methods for local optimization require computation of generalized Jacobian elements at each iteration.  Though techniques have been developed recently to evaluate generalized Jacobian elements for closed-form nonsmooth models [2,3], there are currently no methods for computing generalized Jacobian elements for Caratheodory ODEs whose right-hand sides are not differentiable with respect to the independent variable.  These elements would be useful for local dynamic optimization, and for solving boundary-value problems involving nonsmooth dynamic systems.  The latter arise, for example, when determining cyclic steady states and limit cycles. 

In this presentation, a method is described for evaluating elements of the plenary hull of the generalized Jacobian for a nonsmooth dynamic system, which are argued to be as useful as generalized Jacobian elements in providing sensitivity information.  This method is essentially a generalization of classical sensitivity analysis for smooth systems: it describes sensitivity information as the states of an auxiliary dynamic system, which is then simulated in conjunction with the original dynamic system.  For illustration, this method is applied to solve a local optimization problem and a boundary-value problem involving underlying nonsmooth dynamic systems.

[1]          F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983.

[2]          A. Griewank, On stable piecewise linearization and generalized algorithmic differentiation, Optimization Methods and Software, In press (2013).

[3]          K. A. Khan and P. I. Barton, Evaluating an element of the Clarke generalized Jacobian of a composite piecewise differentiable function, ACM T. Math. Software, In press (2013).