Abstract
In Section 6.1 we saw that the internal stability of a feedback system {L, C} can be formulated in geometric terms relating the graph of L and C. In this chapter we introduce and study a precise geometric tool, the gap metric, which will allow us to study internal stability from a geometric point of view. This will prepare the grounds for a study of robust stabilization from a point of view seemingly different from that given in Chapter 8.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References, Notes, and Remarks
Krasnosel’skii, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitskii, Ya. B., Stetsenko, V. Ya.Approximate Solutions of Operator EquationsGroningen, Wolters-Noordhoff, 1972.
Kato, T.Perturbation Theory for Linear OperatorsNew York, Springer-Verlag, 1966.
Foias, C., Georgiou, T. T., Smith, M., Robust stability of feedback systems: A geometric approach using the gap metricSIAM J. Cont. and Optim.31, 6 (1993), 1518–1537.
Nikol’skii, N.Treatise on the Shift OperatorNew York, Springer-Verlag, 1986.
Georgiou, T., On the computation of the gap metricSystem Control Lett. 11 (1988), 253–257.
Georgiou, T., Smith, M., Optimal robustness in the gap metricIEEE Trans. Aut. Cont.35, (1990), 673–686.
El-Sakkary, A., The gap metric: Robustness of stabilization of feedback systemsIEEE Trans. Aut. Cont.30 (1985), 240–247.
Ober, R., Sefton, J., Stability of linear systems and graphs of linear systemsSystem Control Lett. 17 (1991), 265–280.
Zhu, S. Q., Hautus, L. J., Praagman, C., Sufficient conditions for robust BIBO stabilization: Given by the gap metricSystem Control Lett.11 (1988), 53–59.
Zhu, S. Q., Graph topology and gap topology for unstable systemsIEEE Trans. Aut. Cont.34 (1989), 848–855.
Feintuch, A., The time-varying gap and co-prime factor perturbationsMCSS8 (1995), 352–374.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Feintuch, A. (1998). The Gap Metric and Internal Stability. In: Robust Control Theory in Hilbert Space. Applied Mathematical Sciences, vol 130. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0591-3_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0591-3_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6829-1
Online ISBN: 978-1-4612-0591-3
eBook Packages: Springer Book Archive