On the /spl Nscr//spl Pscr/-hardness of the purely complex /spl mu/ computation, analysis/synthesis, and some related problems in multidimensional systems

It is shown that for a given complex matrix M, and a purely complex uncertainty structure /spl Delta/, the problem of checking whether the inequality /spl mu//sub /spl Delta//(M)<1 holds, is /spl Nscr//spl Pscr/-hard. It is also shown that, the problem of checking whether the frequency domain /spl mu/-norm, /spl par/M(s)/spl par//sub /spl mu//, of an LTI system, M(s), is less than 1, and the problem of checking whether the best achievable /spl mu/-norm, inf/sub Q/spl epsiv//spl Hscr//spl infin///spl par//spl Fscr/(T,Q)/spl par//sub /spl mu//, of an LFT, /spl Fscr/(T,Q), is less than one, are both /spl Nscr//spl Pscr/-hard problems, namely purely complex /spl mu/ computation, analysis/synthesis are all /spl Nscr//spl Pscr/-hard. Although general /spl Hscr//sup /spl infin// norm computation, analysis/synthesis have a well established theory for LTI systems, there is no known nonconservative polynomial time procedure for purely complex /spl mu/ computation, analysis/synthesis problems. The results obtained imply that it is rather unlikely to find nonconservative polynomial time procedures for the purely complex /spl mu/ computation, analysis/synthesis problem, contrary to the standard /spl Hscr//sup /spl infin// problems. As independent results, it is also shown that the problem of checking the stability and the problem of computing the /spl Hscr//sup /spl infin// norm, are both /spl Nscr//spl Pscr/-hard problems for multidimensional systems. These results imply that it is rather unlikely to find a simple analogue of the Schur-Cohn test for checking the stability and an efficient generalization of bisection method for computing the /spl Hscr//sup /spl infin// norm, in the context of multidimensional systems.

[1]  J. Shanks,et al.  Stability criterion for N -dimensional digital filters , 1973 .

[2]  A. Packard,et al.  Continuity properties of the real/complex structured singular value , 1993, IEEE Trans. Autom. Control..

[3]  E. Jury Stability of multidimensional scalar and matrix polynomials , 1978, Proceedings of the IEEE.

[4]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[5]  Pramod P. Khargonekar,et al.  Robustness margin need not be a continuous function of the problem data , 1990 .

[6]  Brian D. O. Anderson,et al.  Stability of multidimensional digital filters , 1974 .

[7]  Andrew Packard,et al.  The complex structured singular value , 1993, Autom..

[8]  Michael G. Strintzis,et al.  Tests of stability of multidimensional filters , 1977 .

[9]  Stephen P. Boyd,et al.  A bisection method for computing the H∞ norm of a transfer matrix and related problems , 1989, Math. Control. Signals Syst..

[10]  Petros G. Voulgaris,et al.  On robust performance in , 1998 .

[11]  Svatopluk Poljak,et al.  Checking robust nonsingularity is NP-hard , 1993, Math. Control. Signals Syst..

[12]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[13]  Ezra Zeheb,et al.  Zero sets of multiparameter functions and stability of multidimensional systems , 1981 .

[14]  N. Bose,et al.  Algorithm for stability test of multidimensional filters , 1974 .

[15]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[16]  J. Doyle,et al.  Review of LFTs, LMIs, and mu , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[17]  Ezra Zeheb,et al.  N-dimensional stability margins computation and a variable transformation , 1982 .

[18]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[19]  N.K. Bose,et al.  Problems and progress in multidimensional systems theory , 1977, Proceedings of the IEEE.

[20]  R. Saeks,et al.  Multivariable Nyquist theory , 1977 .

[21]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[22]  Kameshwar Poolla,et al.  Robust performance against slowly-varying structured perturbations , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[23]  M. Morari,et al.  Computational complexity of μ calculation , 1994, IEEE Trans. Autom. Control..

[24]  A. Tarski A Decision Method for Elementary Algebra and Geometry , 2023 .

[25]  K. Poolla,et al.  Robust performance against time-varying structured perturbations , 1995, IEEE Trans. Autom. Control..

[26]  J. Shamma Robust stability with time-varying structured uncertainty , 1994, IEEE Trans. Autom. Control..

[27]  J. Doyle,et al.  mu analysis with real parametric uncertainty , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[28]  N. Jacobson,et al.  Basic Algebra I , 1976 .