Sampling and data analysis properties of Prony's method applied to model identification†

Abstract

The advantages and limitations of Prony's (1795) method as a scheme for model identification are explored. A description of the data requirements is followed by a presentation of a theory for bounds on the sample period for data collection. The validity of the existence of a sample period window is tested by Monte Carlo simulations. Many of the results presented in this paper are compared with those obtained with regression analysis techniques. Finally, using measured data, Prony'a method is applied to the identification of a dynamic model for the movement of toxic material through an aquatic microcosm. IntroductionThis paper is primarily concerned with an exploration of the relationships between the state model of a continuous time system and the data set obtained by periodically sampling the system state. The results obtained are based upon the properties of Prony's (1795) method, a scheme for fitting exponential functions to equally spaced data, as applied to state model indentification. Similar data versus model relationships using general regression analysis are invoked throughout the paper for purposes of comparison.The motivation for this work rests on a desire to determine the effect of the data sample period on the method for identifying the parameters of a state model for a continuous time system. Apart from noisy measurements, the accuracy of an identification procedure is determined by the discrete nature of the measured data, both in time and amplitude. As a practical matter, algorithms developed for model identification depend upon the time scale of action for the system response. For periodic sampling schemes, the sampling interval for collecting equally spaced discrete time data of a prescribed arithmetic precision must be carefully chosen in order to adequately capture the dynamics of continuous time systems. This problem has been solved for a bandlimited system response where relationships between the sample period and amplitude quantization have been discovered; however, these results do not generalize.