Identification of Nonlinear and Multivariable Systems by Means of Periodic Step Sequence.

The thesis deals with theoretical aspects of the measurement, by correlation, of the kernels of time-invariant multivariable and nonlinear dynamical systems, using periodic-step-sequence inputs of the pseudorandom type. The crosscorrelation of equal-length pairs of binary maximum-length sequences is examined in detail. The frequency distributions of the correlation coefficients are listed for all such pairs of period < 255, and formulae are derived for the first four moments of the distribution. A limited amount of information about the correlation sequences is obtained from a study of the generating polynomials. The sampling property of maximum-length sequences is used, in an alternative approach, to classify the frequency distributions. Finally, recent work by Gold (1968) is related to the author's work on ternary correlation sequences, and suggestions are made for extending this. The problem of calculating the linear kernels of a multi-input system, from input/output crosscorrelations evaluated at sample time intervals, is found to reduce to the solution of a finite set of linear algebraic equations in the system 'weights'. Two cases of practical importance are found to yield exactly N independent equations, where N is the period of the output signal. From this result, and a review of other work, a suggestion is made as to the best type of input for any multiple linear identification. The same reduction to N equations is found to hold for a singleinput nonlinear system, defined by a Volterra series of any order, when the test signal is a binary or inverted-binary maximum-length sequence, or a ternary maximura-length sequence. A non-rigorous argument is adduced to show why this reduction to N equations may hold for any form of identification by sample-interval correlation using synchronized periodic step sequences. It is proved that all binary maximum-length sequences, but not all ternary sequences, can be started at such a point that the first moment vanishes