Modeling and sensitivity analysis methodology for hybrid dynamical system

Abstract

This paper provides a complete mathematical framework to compute the sensitivities with respect to system parameters for any second order hybrid Ordinary Differential Equation (ODE) and ranked 1 and 3 Differential Algebraic Equation (DAE) system. The hybrid system is characterized by discontinuities in the velocity state variables due to an impulsive forces at the time of event. Such system may also exhibit at the time of event a change in the equation of motions, or in the kinematic constraints.The methodology and the tools developed in this study compute the sensitivities of the states of the model and of the general cost functionals with respect to model parameters for both, unconstrained and constrained, hybrid mechanical systems. The analytical methodology that solves this problem is structured based on jumping conditions for both, the velocity state variables and the sensitivities matrix. The proposed analytical approach is then benchmarked against a known numerical method.Finally, this study emphasizes the penalty formulation for modeling constrained mechanical systems since this formalism has the advantage that it incorporates the kinematic constraints inside the equation of motion, thus easing the numerical integration, works well with redundant constraints, and avoids kinematic bifurcations.