Multiplier Theory and Operator Square Roots: Application to Robust and Time-Varying Stability
This paper considers the extension of a number of passive multiplier theory based results, previously known only for linear time invariant scalar systems, to time varying and multivariable settings. The extensions obtained here have important applications to the stability of both adaptive systems and linear systems in general. We demonstrate in this paper that at the heart of the extensions carried out here lies the result that if a stable multivariable and/or linear time varying system is stable under all scalar constant, positive feedback gains, then it has a well defined square root. The existence of this square root is demonstrated through a constructive Newton-Raphson based algorithm. The extensions provided here (dealing with robust stability and introduction of time-varying gains) though different in form from their linear time invariant scalar counterparts, do recover these as a special case.
KeywordsLinear Time Invariant System Single Input Single Output Linear Time Invariant Matrix Transfer Function Linear Time Vary System
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- R. W. Brockett and J. L. Willems, “Frequency domain stability criteria-part I,” IEEE Trans. Auto. Contr., vol. 10, pp.255–271–1965.Google Scholar
- C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, 1975.Google Scholar
- S. Dasgupta, “Strictly positive realness of matrix products”, Contribution to “Open Problems”, in Robustness of Dynamic Systems with Parameter Uncertainties„ M. Mansour, W.Truol and S. Balemi, Ed.s, p 307, Birkhauser, 1992.Google Scholar
- I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators Vol. 1, Birkhauser Verlag, 1990.Google Scholar
- F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederic Ungar Publishing Company, New York, 1955.Google Scholar
- B.D.O. Anderson, R.R. Bitmead, C.R. Johnson, Jr., P.V. Kokotovic, R.L. Kosut, I.M.Y. Mareels, L. Praly and B. Riedle, Stability of Adaptive systems, MIT press, 1986.Google Scholar