Dynamic error analysis of numerical methods for simulation of state-variable derivatives with discontinuities.

Abstract

One of the most difficult computer simulation problems is the simulation of dynamic systems which include discontinuous nonlinear elements. When real-time simulation is required, the selection of integration algorithms used to simulate the system is limited to fixed-step methods with inputs compatible with real time. In order to assess the comparative accuracy of the different integration methods and to develop improved algorithms, it is important to have a general method to evaluate the dynamic errors in simulating discontinuous nonlinearities. Procedures to predict dynamic errors in open-loop and closed-loop systems which include discontinuous nonlinearities are introduced in this thesis. The procedures use the concept of a uniform distribution of "step-begin" times over the integration step containing the discontinuity as well as the concept of ergodicity in order to predict mean-square errors in the frequency domain. These procedures are applied to specific systems, but are shown to be more general, i.e., applicable to a wide range of nonlinear systems which, except for the discontinuous derivative function, can otherwise be linearized for analysis purposes. With the procedures developed in the thesis, the accuracy of different integration algorithms is compared. Simulation experiments are performed on both the bang-bang and effort-limited control systems to confirm the accuracy of the analysis procedures. Finally, a time-domain simulation experiment is performed to show that frequency domain analysis can be used to predict errors in the time-domain.