Multiple frame rate integration

Dynamic systems can often be separated into fast and slow subsystems. The speed and accuracy of a simulation of such systems can frequently be improved by using a frame rate for numerical integration of the fast system which is an integer multiple of the frame rate used for the slow system. The technique of multiple frame-rate integration can be especially important in real-time simulation. In this paper the multiple frame-rate method is introduced, including techniques for converting slow data sequence outputs from slow subsystems to fast data sequence inputs for fast systems. The suitability of various integration algorithms for multiple framing is discussed. The implementation of multiple frame-rate integration using the simulation language ADSIM for the AD 100 computer is described, including software which allows, without program recompiling, choice of multiple-frame ratios and choice of different interpolation or extrapolation algorithms for slow-tofast system interfacing. The paper concludes with an example of multiple framing applied to the simulation of a combined air frame and flight control system in order to improve both the accuracy and stability of the simulation. Dynamic systems can often be separated into fast and slow subsystems. One example is a combined air frame and flight control system, where the rigid airframe represents a slow subsystem, and both elastic structural modes and the flightcontrol system, including control-surface actuators, represent fast subsystems. Another example is a helicopter when modeled by the blade element method, where the rigid airframe again is the slow subsystem and the rotors are fast subsystems. Multiple frame rate integration refers to the technique of making an integer multiple of integration passes through one or more fast subsystems for each pass through the slow subsystem. This reduces the integration step size for the fast subsystem. Since the dynamic errors in a digital simulation will be dominated by the integration truncation errors associated with the fast subsystem, the use of multiple framing can improve significantly the simulation accuracy for a given real-time processor. The accuracy improvement when using multiframing is much more substantial when the fast subsystem is considerably less complex and therefore requires much less processor time than the slow subsystem. The overall concept of multiple frame rate integration is described in Section 2, along with the requirement to use extrapolation or interpolation to interface slow subsystems to fast subsystems. The section also introduces dynamic error measures, following which the compatibility of specific integration methods with multiple framing is discussed. Section *Professor, Department of Aerospace Engineering Member AIAA 3 presents various interpolation and extrapolation algorithms for slow to fast data sequence conversion, as well as the dynamic errors associated with this conversion process. Often it may not be clear exactly how a dynamic system should be partitioned into fast and slow subsystems in order to make most effective use of multiple framing. It may also be difficult to predetermine the optimal frame rate multiple for dynamic accuracy improvement. Analytic methods based on both time and frequency domain considerations, as introduced in Section 2, help in making these choices. However, in Section 4 an interactive software system is described which permits the user to experiment with different problem partitioning, frame rates, and interface extrapolation and interpolation methods. In Section 5 a combined air frame and flight-control system is used to illustrate the multiple framing analysis and synthesis techniques described in the earlier sections. Section 6 contains the concluding remarks. 2. Descriwtion of Multide Frame Rate Intenation The separation of a dynamic system into slow and fast subsystems is illustrated in Figure 1. The slow system utilizes an integration step size denoted by T , whereas the fast system employs a step size denoted by h, where h = TIN and N is an integer. Hereafter we will refer to N as the frame ratio. In Figure 1 the output data sequence { r , ) with sample period T from the slow subsystem is converted to a fast data sequence {fk) with sample period h by means of an interpolator (or extrapolator). This is necessary to provide the fast subsystem with inputs having a sample period h equal to the fast subsystem integration step size. Examples of the generation of a fast sequence from a slow sequence are shown in Figure 2 for a frame ratio N = 4. In Figure 2a first-order interpolation is illustrated; in Figure 2b first-order extrapolation is used. Clearly the interpolation gives a more accurate result than extrapolation. In Section 3 we will see how we can quantify the dynamic accuracy of these and other interpolation and extrapolation algorithms in terms of equivalent gain and phase shift for sinusoidal data sequences. Slow data Fast data sequence from sequence Fast Subsystem fast subsystem Integration { f k ) step-size =h = T/N ( u ,