On-line calculation of dynamic errors in real-time simulation

In the real-time simulation of dynamic systems, including hardware-in-the-loop (HITL), the direct determination of simulation errors resulting from finite numerical integration step sizes and input/output frame rates has always been elusive. The method for dynamic error measurement introduced in this paper is based on comparing two real-time simulations, one with the nominal integration step size and the second with an augmented step size. Accurate interpolation formulas are then used to convert the discrete times associated with the data sequence outputs from the second simulation to the discrete times for the first simulation. This permits a direct calculation of the difference between the two solutions at common discrete times, from which the deviation of either simulation from the solution for zero step size can be calculated with a simple formula. Although only approximate, this procedure results in calculated dynamic errors that are surprisingly accurate, as is demonstrated in example simulations in the paper. Furthermore, it permits direct determination of the dynamic errors associated with each of the constituents making up the total simulation error. For example, in a multi-rate simulation run on a single processor, it allows separate examination of the dynamic errors associated with each frame rate, which in turn permits the determination of an optimal frame-rate ratio. In a multi-processor simulation, it allows the dynamic errors associated with each processor simulation to be determined separately. It also permits direct determination of the effect of input/output sample rates on the overall dynamic accuracy. When lack of run-to-run repeatability becomes a problem for HITL simulations due to the presence of noise in the hardware, logged hardware output data from the first simulation can be used to provide computer inputs for subsequent simulations in order to restore the repeatability needed for on-line circulation of simulation errors. With the use of extrapolation methods already developed for asynchronous simulation, all of the above techniques can be implemented on-line in a real-time simulation environment.