Semi-analytical framework for the study of finite-time stability of forced dynamical systems with slowly varying parameters

Framework to analytically approximate the solution of forced dynamical systems with time varying parameters and to analyze their finite-time stability. The work was inspired by an example in robotic machining, where the mechanical parameters of the system can vary over a wide range during the process, and where there are large forces due to an assumed cutting operation. The simplest possible non-autonomous linear system undergoing dynamic stability loss is studied which serves as a solid foundation to explore the mathematical intricacy behind such systems. After defining the differential equation corresponding to this simple system, the complementary function is studied using a frozen-time approach. The particular integral can be evaluated for this system by the asymptotic expansion of error functions. We present a new approach for the approximation of particular integrals, the iterative integration by parts (IIBP) method, which is then extensively studied and compared to the equations describing the exact analytic solution. The convergence and sensitivity of the IIBP method are discussed. The method is extended to multiple degrees of freedom mechanical systems with time varying parameters. It is shown that standard numerical schemes are not suitable for predicting finite-time stability properties even in the simplest case, because small errors accumulate causing large differences from the analytical solution.