Frequency domain parameter identification and the statistical properties of frequency response estimates

Frequency domain techniques in systems theory have their origins in Heavyside's operational calculus (Heavyside, 1889). Such work was later developed by Foster and Campbell (1931), Brune (1931), Nyquist (1932), Black (1934), Darlington (1939) and subsequently Bode (1948). This interest in the frequency domain was due to its appeal to the intuition of the engineer. The dominance of frequency domain techniques was subsequently eroded from the late 1950s through the 1960s by the influence of the space programmes. The space systems being analysed were based on strong theoretical foundations with well-defined sets of differential equations. The analysis led to the development of the state-space methods which were able to cope with the multivariable problems and were amenable to numerical solution. As a result of these developments, control engineering was largely dominated by the state-space approach and the associated areas of LQG optimal control, Kaiman-Bucy filters, observability and controllability. Two factors led to a resurgence of interest amongst academics in the development of frequency domain techniques in the 1970s and 1980s. The first was the development of the Fast Fourier Transform (FFT) (Cooley & Tookey, 1965). This provided an efficient method of analysing the Fourier transforms of signals and allowed the development of spectral methods of obtaining frequency response estimates. The collection of data was greatly speeded up and this enabled frequency domain methods to be increasingly applied to on-line control problems. The second factor was that the developments in the time domain were never fully embraced by practicing engineers in traditional control environments