Establishing tightness in robust H∞ analysis via homogeneous parameter-dependent lyapunov functions

Several methods have been proposed to provide upper bounds to the robust H(infinity) performance of a linear system with polytopic state space uncertainty. A common problem of these methods is the lack of a test for establishing if the found upper bound is tight or not, which is essential to determine if the search of the sought performance is finished or should continue. This paper provides a necessary and sufficient condition for establishing the tightness of the upper bound provided by homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs) which recently have been shown to be a powerful tool for computing these upper bounds. This condition is based on the existence of particular vectors in suitable linear spaces. Techniques for finding these vectors are also provided. (C) 2007 Elsevier Ltd. All rights reserved.