Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work, uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or off-set, the resulting Pareto optimal trade-off curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.