Some improved methods for real-time integration of state variable derivatives with discontinuities

Abstract

In this paper we consider an analytic averaging technique for integration of discontinuous nonlinear functions. Functions with both displacement and slope discontinuities are treated. The analytic averaging method is shown to provide much better accuracy than conventional integration algorithms. This is especially true when fixed integration step sizes are used, as in real-time simulation. A simple but practical example of a bangbang control systems is used to verify the superior performance of the analytic averaging method. It is also shown how averaging formulas for unit step and unit ramp nonlinear functions can by superposition be used to construct analytic averaging formulas for any nonlinear function which has displacement and slope discontinuities. A modified form of Euler integration is shown to be especially compatible with the analytic averaging method. In real-time simulation of dynamic systems the time derivatives of state variables sometimes have discontinuities. For example, this is clearly the case when simulating a spacecraft attitude control system which uses on-off reaction control thrusters. It is also true in the simulation of continuous controllers with effort limiting, controllers with dead-zone, etc. In general the discontinuities occur at times which are asynchronous with respect to integration step times. Because of this the use of conventional integration methods can result in substantial dynamic errors. Methods have been proposed using variable integration step size to improve the accuracy when discontinuities are present(1.2.3). However, in real-time simulation the integration step size must be fixed and the errors introduced by discontinuous derivatives can become very serious unless the step size is made inordinately small. A technique compatible with real time simulation which utilizes an intermediate step to the discontinuity has been described and shown to exhibit high accuracy(4). However, this method can require considerable computation time when many discontinuous functions are present in a simulation. A less accurate but faster method for handling discontinuous nonlinear functions in fixed step integrations has also been' described(5). The method, which uses an analytic averaging technique, is introduced in the next section. A general formula for the analytic averaging function for any nonlinearity consisting of straight line segments and displacement discontinuities is developed in Section