A synthesis theory for multiple-loop oscillating adaptive systems

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Abstract
The multiple-loop self-oscillating adaptive (SOAL) system is presented as a natural, logical means of overcoming a serious limitation of the single-loop self-oscillating system (SOAS). Both structures have the property of zero sensitivity to plant high-frequency gain uncertainty p = Kmax/Kmin, the factor which is generally responsible for large ‘ cost of feedback ’. It is however necessary to design these systems such that the response is essentially quasi-linear to the desired class of command and disturbance signals. In the SOAS, p reappears as a significant factor in the quasi-linear requirements which may, depending on the numbers involved, completely vitiate its banishment as an uncertainty factor. The development of a quantitative design theory for the SOAS pinpoints the two-loop SOAL extension needed to overcome this SOAS limitation, and the development of a similar SOAL quantitative design theory. In the latter, p disappears from both the adaptive and quasi-linear conditions, but reappears as a factor in the rate of adaptation of the system. It may be banished from here too, by means of a three-loop self-oscillating system (SOANL), for which the SOAL design theory is applicable with minor extensions.

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