(110a) Sensitivity Analysis of Discontinuous Dynamical Systems

Generalized derivatives provide useful sensitivity information for solving problems involving nondifferentiable functions, including numerical methods for nonsmooth equation-solving (e.g., semismooth Newton methods) and optimization problems (e.g., bundle methods for local optimization). These methods often utilize elements of a generalized derivatives (e.g., Clarke's generalized Jacobian [1]) as a means to determine local sensitivity information at points where the function may be nondifferentiable. Among recent advances are computationally tractable methods for evaluating lexicographic directional derivatives (LD-derivatives), from which lexicographic derivatives can be evaluated. These lexicographic derivatives are no less useful than elements of Clarke's generalized Jacobian in many applications, including nonsmooth equation-solving and optimization methods mentioned above. The LD-derivative approach has been applied to many classes of problems, including inverse and implicit functions, nonsmooth dynamical systems and optimization problems [2].

Discontinuous dynamical systems occur in many applications, including mechanics of moving objects (e.g. friction forces and nonsmooth harmonic oscillators) and control systems (e.g. room temperature controller) [3, 4]. Although discontinuous dynamical systems appear in many applications, the ability to compute generalized derivative sensitivities of solutions of these systems is limited. In our work, we have extended the existing sensitivity theory for nonsmooth (but continuous) dynamical systems to certain classes of discontinuous dynamical systems. In particular, we relate the solutions of discontinuous ordinary differential equation systems (ODEs) to nonsmooth differential algebraic equations (DAEs), by generalizing the idea of underlying ODEs of DAEs. Then, using existing sensitivity theory for nonsmooth DAEs of (generalized) differentiation index-1, for which sensitivity theory exists, we can obtain generalized sensitivities for the related discontinuous ODEs, which can then be used in the nonsmooth numerical methods mentioned above.

References

[1] F.H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990.
[2] Paul I. Barton, Kamil A. Khan, Peter Stechlinski, and Harry A. J. Watson. Computationally relevant generalized derivatives: theory, evaluation and applications. Optimization Methods and Software, 33(4-6):1030-1072, 2018.
[3] Jorge Cortes. Discontinuous dynamical systems. IEEE Control Systems Magazine, 28(3):36-73, 2008.
[4] A. F. Filippov. Differential Equations with Discontinuous Righthand Sides. Mathematics and Its Applications. Springer Netherlands, Dordrecht, 1988.