Necessity of the small gain theorem for multidimensional systems

In this paper, we study the necessity of linear matrix inequality conditions derived using scaled small-gain arguments for multidimensional system analysis. Since multidimensional analysis conditions involve the computation of structured singular values, the linear matrix inequalities are in general conservative; we show that the conservatism can be described using multidimensional unitary operators. The proofs extend to standard S-procedure based necessity results for scaled small-gain conditions to the case where some of the perturbations are restricted to be unitary (and not just contractive), and to a descriptor generalization of linear fractional transformations.