In this paper, the authors present a linear matrix inequality (LMI) approach to the strictly positive real (SPR) synthesis problem: find an output feedback K K K such that the closed-loop system T(s) T(s) T(s) is SPR. The authors establish that if no such constant output feedback K K K exists, then no dynamic output feedback with a proper transfer matrix exists to make the closed-loop system SPR. The existence of K K K to guarantee the SPR property of the closed-loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters. Index Terms-Adaptive control, H 1 control, linear matrix inequality, output feedback, positive real functions.
We present results connecting the shape of the numerical range to intrinsic properties of a matrix A. When A is a nonnegative matrix, these results are to a large extent analogous to the Perron-Frobenius theory, especially as it pertains to irreducibility and cyclicity in the combinatorial sense. Special attention is given to polygonal, circular and elliptic numerical ranges. The main vehicles for obtaining these results are the Hermitian and skew-Hermitian parts of A, as well as Levinger's transformation aA + (1 − a)A * . (J. Maroulas), ppsarr@math.ntua.gr (P.J. Psarrakos), tsat@math.wsu.edu (M.J. Tsatsomeros). 0024-3795/02/$ -see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 -3 7 9 5( 0 1) 0 0 5 7 4 -2
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