The use of function generation in the real-time simulation of stiff systems.

Abstract

One of the most difficult real-time simulation problems is the simulation of stiff dynamic systems. These systems are governed by equations, when linearized, have eigenvalues that are widely separated. Implicit methods which are commonly used in nonreal-time simulation of such systems require iterations within each integration step and are not generally suitable for real-time implementation. On the other hand, explicit integration methods normally used in real-time simulation require very small step sizes and excessive computing power in simulating stiff systems. A new approach, the function generation method, is suggested in this thesis. The equations of the fast subsystem are integrated off-line over a time interval which will be used as the step size for the on-line integration of the slow subsystem. This off-line integration uses a sufficiently small step size to insure both accurate and stable solutions and is repeated for a matrix of initial conditions and inputs. These results are then stored in multivariable function tables. In the on-line real-time simulation, table lookup and linear interpolation are then used to determine each new fast subsystem state based on the old state and inputs. A time-shift scheme is also introduced in the thesis. In this scheme, the frame times for the fast subsystem are shifted half an integration step with respect to the integer frame times for the slow subsystem. This, in turn, means that the output data sequences of the fast and slow subsystems are staggered by one-half of an integration step with respect to each other. This scheme provides substantial improvement in accuracy of the function generation method as well as the multiple frame-state method. Three practical examples are used to demonstrate how the function generation method is implemented. The first example is a flight control system which includes a high speed, nonlinear actuator. The actuator represents a much faster dynamic subsystem than either the airframe by itself or the remainder of the flight control system. The second example is an airframe/landing-gear system with the landing gear representing the fast subsystem and the airframe representing the slow subsystem. The third example is a tracked land vehicle, where the road arm/wheels that support the vehicle have much faster dynamics than the rigid hull. From these examples it is shown that the function generation method has speed and accuracy advantages over both conventional integration of all state equations with a common step size, as well as the multiple frame-rate method. It is also shown that the time-shift scheme can improve the accuracy of both the function generation and multiple frame-rate approaches.