Set Descriptions of White Noise and Worst Case Induced Norms.

摘要

This paper provides a framework for analyzing white noise disturbances in linear systems. Rather than the usual stochastic approach, noise signals are described as elements in sets and the disturbance rejection properties of the system are described in a worst case setting. This type of modeling of noise and disturbances very much fits the philosophy of both the behavioral and robust control settings. The description is based on properties of finite records of signals, which may be verified directly on experimental data. Bounds of system gain for input signals in these sets are given, and their asymptotic behavior for long data records is analyzed. The presence of low-correlated disturbances (noise) in physical systems has usually been modeled by thinking of the disturbance as the realization of a stochastic process, which is white in the sense of having zero autocorrelations in the expected value sense. The basic result for analysis of linear systems in the presence of stochastic noise is that if unit variance white noise is input to a stable linear system, the output variance (expected value of power) is given by the 2-norm of the system function. Moreover, the spectral characteristics of the output signal are given by the filter. However, if in a real-world situation we want to use results of this type, we will have to convince ourselves that our disturbances can be accurately modeled as a stochastic white noise trajectory. Trying to decide this from experimental data leads to a statistical hypothesis test on a finite record of the signal. In other words, we will accept our signal as white noise if it belongs to a certain set, designed to give us reasonable confidence in the whiteness of the source. This set is typically described in terms of time autocorrelations.