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

By P. W. Davall


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

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