Sensitivity analysis for nonsmooth dynamic systems

Abstract

Nonsmoothness in dynamic process models can hinder conventional methods for simulation, sensitivity analysis, and optimization, and can be introduced, for example, by transitions in flow regime or thermodynamic phase, or through discrete changes in the operating mode of a process. While dedicated numerical methods exist for nonsmooth problems, these methods require generalized derivative information that can be difficult to furnish. This thesis presents some of the first automatable methods for computing these generalized derivatives. Firstly, Nesterov's lexicographic derivatives are shown to be elements of the plenary hull of Clarke's generalized Jacobian whenever they exist. Lexicographic derivatives thus provide useful local sensitivity information for use in numerical methods for nonsmooth problems. A vector forward mode of automatic differentiation is developed and implemented to evaluate lexicographic derivatives for finite compositions of simple lexicographically smooth functions, including the standard arithmetic operations, trigonometric functions, exp / log, piecewise differentiable functions such as the absolute-value function, and other nonsmooth functions such as the Euclidean norm. This method is accurate, automatable, and computationally inexpensive. Next, given a parametric ordinary differential equation (ODE) with a lexicographically smooth right-hand side function, parametric lexicographic derivatives of a solution trajectory are described in terms of the unique solution of a certain auxiliary ODE. A numerical method is developed and implemented to solve this auxiliary ODE, when the right-hand side function for the original ODE is a composition of absolute-value functions and analytic functions. Computationally tractable sufficient conditions are also presented for differentiability of the original ODE solution with respect to system parameters. Sufficient conditions are developed under which local inverse and implicit functions are lexicographically smooth. These conditions are combined with the results above to describe parametric lexicographic derivatives for certain hybrid discrete/ continuous systems, including some systems whose discrete mode trajectories change when parameters are perturbed. Lastly, to eliminate a particular source of nonsmoothness, a variant of McCormick's convex relaxation scheme is developed and implemented for use in global optimization methods. This variant produces twice-continuously differentiable convex underestimators for composite functions, while retaining the advantageous computational properties of McCormick's original scheme. Gradients are readily computed for these underestimators using automatic differentiation.