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On spectral mappings, higher order circle criteria, and periodically varying systems

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2 Author(s)
Zames, G. ; Department of Transportation, Cambridge, MA, USA ; Kallman, R.

For feedback equations of the forme = x - He, in whichHis a linear operator, stability boundaries obtained by the circle criterion method can frequently be improved by finding a functionf(H)which conformally maps the spectrum ofHfrom its initial region in the complex plane into a more suitable region. This idea, which is based on the spectral mapping theorem of Banach algebras, is applied here to a class of equations consisting of a time-invariant part and a periodically time-varying gain. A stability bound is derived which has a minimum whenever the frequency response of the time-invariant part peaks at a frequency near one half that of the time-varying gain.

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Automatic Control, IEEE Transactions on  (Volume:15 ,  Issue: 6 )