# A REAL ^ LOWER BOUND

3.1 BACKGROUND

The aim of this section is to compute a reliable upper bound of the robustness margin (i. e. a lower bound of the maximal s. s.v. over the frequency range) in the context of a purely real model perturbation, which contains a large number of parametric uncertainties (Ferreres and Bian – nic, 1998b). Note indeed that the primary aim is to compute an interval of the robustness margin, rather than an interval of the s. s.v. n(M(juj)) as a function of ui.

This is not an easy problem, since the real ) lower bound of (Dailey, 1990) is exponential time. On the other hand, the mixed д lower bound of (Young and Doyle, 1990) is polynomial time, but the convergence properties of the power algorithm are very poor in the context of a purely real model perturbation. The result provided by this power algorithm is consequently not reliable in this specific context.

A state-space approach was proposed in (Magni and Doll, 1997): consider the interconnection structureM{s) – Д, and let (А, В, C, 0) a state – space representation ofM(s), which is assumed to be strictly proper just for the ease of notation. The idea is to interpret the model perturbation Д as a fictitious feedback gain which moves the poles of the closed loop, whose state matrix is A + ВАС, from the left half plane through the imaginary axis. The norm of Д is to be minimized during the process of migration of the poles towards the imaginary axis.

The algorithm provides a model perturbation A*, which brings one closed loop pole on the imaginary axis at jui*. An upper bound of the robustness margin is thus obtained as the size of A*, and its inverse is a lower bound of the s. s.v. )(M (M*) ) (and thus a lower bound of the maximal s. s.v. over the frequency range).

Good results are generally obtained with this technique. There are however some technical difficulties. A key issue is especially to choose

which poles of the nominal closed loop (i. e. which poles of the state matrix A) are to be moved towards the imaginary axis.

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