On the complexity of purely complex μ computation and related problems in multidimensional systems

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Abstract
In this paper, the following robust control problems are shown to be NP-hard: given a purely complex uncertainty structure /spl Delta/, determine if: 1) /spl mu//sub /spl Delta//(M)<1, for a given rational matrix M; 2) /spl par/M(/spl middot/)/spl par//sub /spl mu//<1, for a given rational transfer matrix M(s); and 3) inf(Q/spl isin/H/sup /spl infin//)/spl par/F(T,Q)/spl par//sub /spl mu//<1, for a given linear fractional transformation F(T,Q) with rational coefficients. In other words, purely complex /spl mu/ computation, analysis, and synthesis problems are NP-hard. It is also shown that checking stability and computing the H/sup /spl infin// norm of a multidimensional system, are NP-hard problems. Therefore, it is rather unlikely to find nonconservative polynomial time algorithms for solving the problems in complete generality.

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