Precision of impulse-response identification based on short, normal operating records

The characteristics of a process may be estimated from an observation over a finite interval of time of the input and output variables of the system during a period of normal operation. The determination of the most effective method of analyzing the observed data and of estimating the probable errors in such an analysis is an important problem in the study of complex processes and in the design of adaptive controllers for time-varying systems. For adaptive control systems, it is desirable to base this system "identification" analysis on as short an operating record as possible, consistent with the specified degree of accuracy to be obtained. This paper is concerned with the problem of impulse-response estimation based on such short "normal operating" records. If the measurements of the system variables are corrupted by noise, the impulse-response parameter estimates will be random variables, since, for a given record length, these estimates vary from one sample of the observed data to the next, depending on the variation in the characteristics of the noise and the input signal during each short segment of the record. The expected "integrated-squared-error" between the actual and the computed impulse responses is shown to depend only on the input signal and the noise characteristics. A method of computing the expected-integrated-squared error for a given input signal is developed to provide a test of the reliability of the identification routine for each analysis. With assumptions on the statistical nature of the input signal, this "sufficient signal" criterion is transformed to a "sufficient record length" criterion. Examples are given for two such specific assumptions: 1) An input signal with a Gaussian amplitude distribution. 2) A switching-type input signal which jumps between +1 and -1 with a random distribution of switching times. Results are presented in sampled-data form.