Mathematics of Control , Signals , and Systems Complexity Issues in Robust Stability of Linear Delay-Differential Systems *

In this paper the following stability problems concerning linear delay-differential systems are shown to beNP-hard: (i) asymptotic stability independent of delay, and (ii) robust asymptotic stability, when each delay is known to lie in an interval. The main results are based on theNP-hardness of complex bilinear programming over the polydisk n which also shows that the purely complexμ computation, analysis/synthesis problems areNP-hard even if there are no repeated blocks. Another side result of the paper is that checking robust nonsingularity, robust Hurwitz stability, and robust Schur stability of a disk matrix areNP-hard problems.

[1]  E. Kamen.  Correction to "Linear systems with commensurate time delays: Stability and stabilization independent of delay" , 1983 .

[2]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

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

[4]  A. Nemirovskii.  Several NP-hard problems arising in robust stability analysis , 1993 .

[5]  B. Ross Barmish,et al.  Robust stability of perturbed systems with time delays, , 1989, Autom..

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

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

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

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

[10]  C. Nett,et al.  A new method for computing delay margins for stability of linear delay systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[11]  A. Johnson,et al.  Delay-independent robust stability of uncertain linear systems , 1995 .

[12]  A. Olbrot,et al.  Robust stability for time-delay systems: the edge theorem and graphical tests , 1989 .

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

[14]  Ezra Zeheb,et al.  Simplified analytic stability test for systems with commensurate time delays , 1984 .

[15]  E. Kamen.  On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations , 1980 .

[16]  Jie Chen,et al.  Frequency sweeping tests for stability independent of delay , 1995, IEEE Trans. Autom. Control..

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

[18]  A. Hmamed.  Further results on the robust stability of uncertain time-delay systems , 1991 .

[19]  Christopher L. DeMarco,et al.  The computational complexity of approximating the minimal perturbation scaling to achieve instability in an interval matrix , 1994, Math. Control. Signals Syst..

[20]  O. Toker,et al.  Complexity Issues in System Theory and Solution Procedures for Certain Robust Control Problems , 1995 .

[21]  Guoxiang Gu,et al.  On the Stability Testing of Time Delay Systems , 1988, 1988 American Control Conference.

[22]  E. W. Kamen,et al.  Linear systems with commensurate time delays: stability and stabilization independent of delay , 1982 .

[23]  Erik I. Verriest,et al.  Frequency domain robust stability criteria for linear delay systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[24]  E. Kamen,et al.  Pointwise stability and feedback control of linear systems with noncommensurate time delays , 1984 .

[25]  H. Ozbay,et al.  On the /spl Nscr//spl Pscr/-hardness of the purely complex /spl mu/ computation, analysis/synthesis, and some related problems in multidimensional systems , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[26]  S. Żak,et al.  On stability independent of delay for linear systems , 1982 .