(341e) Sensitivity Analysis for Nonsmooth Hybrid Systems

Dynamic chemical process systems exhibiting qualitative changes in behavior may be modeled as hybrid discrete/continuous systems, which are systems of differential equations that are punctuated by discrete events.  At these events, state variables are permitted to jump, and the functions governing the state variables’ continuous evolution may be switched.  These events can be used to describe transitions in multi-stage processes, for example, or changes in flow regime or thermodynamic phase.  This presentation describes new numerical methods for sensitivity analysis of hybrid systems, thereby broadening the class of hybrid systems for which sensitivity analysis is possible.

Firstly, a numerical method is presented for computing Nesterov’s lexicographic derivatives [2] for implicit functions that are described in terms of nonsmooth residual functions.  Lexicographic derivatives are analogs of classical derivatives of smooth systems, and provide local sensitivity information to methods for nonsmooth equation-solving and optimization.

Secondly, a numerical method is presented for computing lexicographic derivatives for nonsmooth hybrid systems.  In these systems, the functions describing continuous dynamics, event timing, and post-event jumping are not required to be differentiable everywhere.  This method combines the sensitivity analysis of nonsmooth implicit functions with our recent treatment [1] of lexicographic derivatives for nonsmooth dynamic systems. The desired sensitivities are described and computed as the state variables of an auxiliary hybrid system. This method permits sensitivity analysis of certain hybrid systems for which classical sensitivity analysis methods fail, including certain systems in which the order of discrete events encountered depends on system parameters.   Examples are presented for illustration.

[1] K. Khan and P. Barton, Generalized derivatives for solutions of parametric ordinary differential equations with non-differentiable right-hand sides, J. Optim. Theory App., 163 (2014), pp. 355-386. 

[2] Y. Nesterov, Lexicographic differentiation of nonsmooth functions, Math. Program. B, 104 (2005), pp. 669-700.