A new criteria for the update speed needed for the correct real-time simulation
of the continuous dynamics of automotive systems.Summary A new criteria is derived for the update frequency needed to simulate the dynamics of systems represented by coupled ordinary nonlinear differential equations. The frequency response of the Euler Numerical Integration Rule is computed and compared to that of an ideal integrator. The result is a table relating the ratio of simulation update frequency to the highest frequency being simulated versus deviation. For example, for less than a 0.1 db deviation from an ideal integrator's frequency response the simulation should run at 12.8 times the highest frequency. This conclusion dictates the computer power needed. INTRODUCTION The real-time simulation of the dynamics of automotive systems is finding wide application throughout the industry. A common application is hardware-in-the-loop (HIL) simulation where mixtures of real and simulated components interact in real time in order to test the components as well as the overall behavior of the automobile. Examples include the simulation of vehicle dynamics including tire-road interaction to test antilock braking systems (ABS) or the simulation of engine and drive train dynamics to test fuel-injection electronic control units. The benefits include faster time to market and the reduction of costly testing on the test track with real vehicles. XANALOG's XANALOG/RT Series of real-time hardware-in-the-loop and controller prototyping systems are currently in use for testing ABS, fuel injection control, cruise control, transmission control and other automotive systems in Europe, Japan and the United States. In working with these users as well as users in the aerospace and industrial automation industries the question often arises as to how fast a simulation must be updated given the highest frequency components present in the dynamics of the mechanism being simulated. For example, if the simulation must handle frequencies up to 100 Hz, with what frequency must the simulation be computed? When this question was posed to experienced simulationists their responses ranged from 10 to 20 times the highest frequency that the simulation needs to represent. Neither the simulationists or the literature had a derivation from first principles of why 10 to 20 times was good value with which to work. This paper starting from first principles shows that the experienced simulationists were correct. The derivation which follows computes the frequency response of Euler integration and compares it to the frequency response of an ideal integrator. The result is a table which relates the ratio of update frequency to highest frequency in the simulation to deviation from ideal integration for the case of the Euler Integration Rule. DISCUSSION DERIVATION OF THE FREQUENCY RESPONSE OF EULER INTEGRATION The frequency response of the Z-Domain circuit of Figure 1 which is equivalent to the Euler Integration Rule can be computed as follows: CONCLUSION This paper derives from first principles a table which relates a ratio (the highest frequency in a simulation to the update rate of the simulation) with the deviation from ideal integration for the case of Euler integration. This ratio has major economic implications in specifying a simulation system. REFERENCES 1. Hamming, R., "Numerical Methods for Scientists and Engineers", McGraw-Hill, New York, NY, 1962 2. Gelb, A., "Applied Optimal Estimation", The MIT Press, Cambridge, MA, 1974, pp. 294-303 3. Franklin, G., "Digital Control of Dynamic Systems", Addision-Wesley, Reading, MA, 1980, pp.54-59 |
