Abstract
This paper is concerned with the approximation of unstable input-output plants and their feedback interconnections.
Systems are represented by closed operators with restricted domains in a Hilbert space. A metric on systems is introduced in terms of the gap (or aperture) between their graphs.
It is shown that under the gap metric the mapping from the open to the closed loop system is structurally stable; i.e., the mapping (and its inverse) is continuous. Furthermore, it is shown that all metrics which preserve the structural stability of a feedback interconnection generate the same topology as the gap metric.
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References
A. El-Sakkary, The Gap Metric for Unstable Systems, Ph.D. Dissertation, Department of Electrical Engineering, Mc Gill University, 1981.
J. Davis, Mean-square Gain Criteria for the Stability and Instability of Time-Varying Systems, IEEE Trans. Automat. Contr. Vol. AC-17, No. 2, April, 1972.
C. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, 1975.
T. Kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin, 1976.
G. Zames and A. El-Sakkary, Unstable Systems and Feedback: The Gap Metric, Proc. 16th Allerton Conf. October 1980.
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© 1982 Springer-Verlag
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El-Sakkary, A. (1982). Structural stability of a feedback system: An operator theoretic approach. In: Bensoussan, A., Lions, J.L. (eds) Analysis and Optimization of Systems. Lecture Notes in Control and Information Sciences, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0044383
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DOI: https://doi.org/10.1007/BFb0044383
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