Applications of statistics in the spectral analysis of time-varying systems

Abstract

Recent advances in the theory of evolutionary spectral analysis of time-varying systems has led to a resurgence in the popularity of frequency domain analysis techniques. Policies for adaptive control of time-varying systems based on state-space and Liaponov techniques require an accurate measurement of the system phase variables. Under inherently noisy conditions, access to the complete system state is seldom possible, and frequency domain analysis requirinq only input/output measurements has an obvious appeal. The sampling properties of short-term spectral estimates are of central importance both in system tracking and in choosing suitable control policies. Goodman (1957) developed some of the sampling properties associated with spectral estimates of complex bivariate Gaussian processes. Akaike (1962-66) extended Goodman's results to multi input/output linear systems with 'Gaussian input forcing functions. Both these authors considered the case where the data sequences were stationary. This thesis reviews and extends the research of these two authors with respect to single input/output linear systems. It is shown that the sampling distributions associated with spectral estimates of stationary open-loop systems are approximately valid for a restricted class of non-stationary systems. Two examples of non-stationary systems are investigated and an adaptive control technique using input compensation in the frequency domain is developed on a hydraulic fatigue loading rig. It is shown that statistical tests developed earlier can successfully identify system variations when estimates are measured in a noisy environment. The sampling distributions associated with spectral estimates of closed-loop systems are developed and the results are applied to the modelling and tracking of the human operator response in a trackinq task situation, for various input signals. With regard to future research, it remains to extend the results for closed-loop systems to the time-varying multi input/output case. In its full complexity this problem remains intractable but by considering uncorrelated Gaussian inputs it reduces to determining the distributions associated with multi-variate complex Gaussian sequences