Abstract
This chapter investigates the gap between positive forms and SOS forms. Conservatism of the LMI relaxations described in Chapter 1 is related to the existence of positive forms which are not SOS, called PNS forms. a priori conditions for non-conservatism of these relaxations are presented for some classes of forms. The class of SMR-tight forms is introduced in order to derive a posteriori tightness conditions. A further contribution of this chapter consists of providing a parametrization of the set of PNS forms. It is shown that the set of PNS forms is dense in the space of forms, that each PNS form is the vertex of a cone of PNS forms, and how PNS forms can be constructed via the SMR.
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© 2009 Springer London
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Chesi, G., Garulli, A., Tesi, A., Vicino, A. (2009). Positivity Gap. In: Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems. Lecture Notes in Control and Information Sciences, vol 390. Springer, London. https://doi.org/10.1007/978-1-84882-781-3_2
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DOI: https://doi.org/10.1007/978-1-84882-781-3_2
Publisher Name: Springer, London
Print ISBN: 978-1-84882-780-6
Online ISBN: 978-1-84882-781-3
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