Skip to main content

Robust Stability Analysis of Linear State Space Systems

  • Chapter
  • First Online:
Robust Control of Uncertain Dynamic Systems
  • 2812 Accesses

Abstract

In this chapter, which is also one of the main chapters of the book, we address the issue of robust stability analysis for uncertain linear dynamic systems. We first present a rigorous mathematical formulation of the problem, mainly focusing on continuous-time systems. Then, we consider various characterizations of uncertainty discussed in the previous chapter and present corresponding bounds on the perturbations for robust stability.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. Z. Zheng. A method for determining the parameter stability regions of linear control systems. ieeeac, AC-29(2):183–185, 1984.

    Google Scholar 

  2. M. Eslami and D.L. Russell. On stability with large parameter variations stemming from the direct method of lyapunove. AC-25(6):1231–1234, 1980.

    Google Scholar 

  3. S. S. L. Chang and T. K. C. Peng. Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans Auto. Cont., AC-17:474–483, August 1972.

    Article  MathSciNet  Google Scholar 

  4. R.V. Patel, M. Toda, and B. Sridhar. Robustness of linear quadratic state feedback designs in the presence of system uncertainty. ieeeac, AC-22:945–949, 1977.

    Google Scholar 

  5. D. Hinrichsen and A. J. Pritchard. Stability radius of linear systems. Systems and Control Letters, 7:1–10, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  6. J.M. Martin. State space measures for stability robustness. ieeeac, AC-31:509–512, 1987.

    Google Scholar 

  7. R.K. Yedavalli, S.S. Banda, and D.B. Ridgely. Time domain stability robustness measures for linear regulators. JGCD, pages 520–525, 1985.

    Google Scholar 

  8. R. K. Yedavalli. Improved measures of stability-robustness for linear state space models. IEEE AC, 30:577–579, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. K. Yedavalli. Perturbation bounds for robust stability in linear state space models. IJC, 42:1507–1517, 1985.

    MathSciNet  MATH  Google Scholar 

  10. R. V. Patel and M. Toda. Quantitative measures of robustness for multivariable systems. Proc. Joint Amer. Contr. Conf., TP8-A, 1980.

    Google Scholar 

  11. K. Zhou and P. Khargonekar. Stability robustness bounds for linear state space models with structured uncertainty. IEEE Transactions on Automatic Control, AC-32:621–623, July 1987.

    Article  MathSciNet  Google Scholar 

  12. M.Fu and B.R.Barmish. Maximal unidirectional perturbation bounds for stability of polynomials and matrices. Systems and Control Letters, 11:173, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  13. X. Zhang, P. Tsiotras, and A. Lanzon. An approach for computing the exact stability domain for a class of lti parameter dependent systems. International Journal of Control, 79(9):1046–1061, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  14. R.Genesio and A.Tesi. Results on the stability robustness of systems with state space perturbations. Systems and Control Letters, 11:39, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  15. L.El Ghaoui and S. Boyd. Stability robustness of linear systems to real parametric perturbations. Proc. 29th Conference on Decision and Control, pages 46–51, Honolulu 1990.

    Google Scholar 

  16. M. Corless and D. Da. New criteria for robust stability. In Proceedings of the International Workshop on ’Robustness in Identification and Control, Torino, Italy, June 1988.

    Google Scholar 

  17. B.S Chen and C.C Wong. Robust linear controller design: Time domain approach. IEEE Trans Auto. Control, 32(2):161, February 1987.

    Google Scholar 

  18. R.K. Yedavalli and Z. Liang. Reduced conservatism in stability robustness bounds by state transformation. IEEE AC, 31:863–866, 1986.

    Article  MATH  Google Scholar 

  19. D.C. Hyland and D.S. Berstein. The majorant lyapunov equations nonnegative matrix equation for robust stability and performance of large scale systems. IEEE AC, 32:1005–1013, 1987.

    Article  MATH  Google Scholar 

  20. R. K Yedavalli and S.R Kolla. Stability analysis of linear time varying systems. International Journal of Systems Science, 19(9):1853–1858, 1988.

    Google Scholar 

  21. D.D. Siljak. Parameter space methods for robust control design: A guided tour. IEEE Transactions on Automatic Control, AC-34:674, 1989.

    Article  MathSciNet  Google Scholar 

  22. Y. T. Juang, T. S. Kuo, and C. F. Hsu. New approach to time domain analysis for stability robustness of dynamic systems. International Journal of Systems Science, 18(7):1363–1376, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  23. L.Qiu and E.J. Davison. Bounds on the real stability radius. In Robustness of Dynamic Systems with Parameter Variations, 1992.

    Google Scholar 

  24. B.R. Barmish. New tools for robustness of linear systems. Macmillan Publishing company, Riverside, NJ, 1994.

    MATH  Google Scholar 

  25. A.Tesi and A. Vicino. Robust stability of state space models with structured uncertainties. IEEE Trans on Automatic Control, 35:191–195, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Qiu and E.J. Davison. New perturbation bounds for the robust stability of linear state space models. Proceedings of 25th CDC,Athens,Greece, pages 751–755, 1986.

    Google Scholar 

  27. R. K. Yedavalli. Stability robustness measures under dependent uncertainty. Proc. Amer. Control Conf., Atlanta:820–823, June 1988.

    Google Scholar 

  28. L.Qiu and E.J. Davison. The stability robustness determination of state space models with real unstructured perturbations. Mathematics of Control, Signals and Systems, 2:247–267, 1991.

    Google Scholar 

  29. R. K. Yedavalli. Flight control application of new stability robustness bounds for linear uncertain systems. AIAA Journal of Guidance, Control and Dynamics, 16(6):1032, Nov-Dec 1993.

    Google Scholar 

  30. A.T.Fuller. Conditions for a matrix to have only characteristic roots with negative real parts. Journal of Mathematical Analysis and Applications, 23:71–98, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  31. E.I.Jury. Inners and stability of dynamic systems. Wiley, New York, 1974.

    Google Scholar 

  32. N.W Hagood. Cost averaging techniques for robust control of parametrically uncertain systems. Proc. AIAA Guidance, Navigation and Control Conference, AIAA 91-2605, 1991.

    Google Scholar 

  33. L.Saydy, A.L.Tits, and E.H.Abed. Guardian maps and the generalized stability of parameterized families of matrices and polynomials. Mathematics of Control, Signals and Systems, 3:345, 1990.

    Google Scholar 

  34. S. Rern, P. T. Kabamba, and D. S. Bernstein. Guardian map approach to robust stability of linear systems with constant real parameter uncertainty. IEEE Transactions on Automatic Control, 39(1):162–164, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. S Bernstein and W.M Haddad. Robust stability and performance analysis for state space systems via quadratic lyapunov bounds. SIAM Journal of Matrix Analysis and Applications, 11:239–271, 1990.

    Google Scholar 

  36. D. S Bernstein and W.M Haddad. Robust stability and performance analysis for linear dynamic systems. IEEE Trans Auto. Control, 34:751–758, 1989.

    Google Scholar 

  37. J.P Guiver and N.K Bose. Strictly hurwitz property invariance of quartics under coefficient perturbation. IEEE Trans Auto. Control, 28(1):106–107, 1983.

    Google Scholar 

  38. V.L Kharitonov. Asymptotic stability of an equilibrium position of a family of linear differntial equations. Differensialnye Uravneniya, 14:2086–2088, 1978.

    Google Scholar 

  39. B.R. Barmish. Invariance of the strict Hurwitz property for polynomials with perturbed coefficients. AC-29:935–936, 1984.

    Google Scholar 

  40. N.K Bose. A system theoretic approach to stability of sets of polynomials. Contemporary Mathematics, 47:25–35, 1985.

    Google Scholar 

  41. B.D.O.Anderson, E.I. Jury, and M.Mansour. On robust Hurwitz polynomials. AC-32:909–913, 1987.

    Google Scholar 

  42. S.P Bhattacharyya, H Chappellat, and L.H Keel. Robust control: The Parametric approach. Prentice Hall, Englewood cliffs, NJ, 1995.

    Google Scholar 

  43. A.C.Bartlett, C.V.Hollot, and L. Huang. Root locations of an entire polytope of polynomials: it suffices to check the edges. Mathematics of Control,Signals and Systems, 1:61–71, 1988.

    Google Scholar 

  44. B. R Barmish. A generalization of kharitonov’s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations. IEEE Trans Auto. Control, AC-34:157–165, 1989.

    Google Scholar 

  45. J.E. Ackermann, H.Z. Hu, and D. Kaesbauer. Robustness analysis: A case study. AC-35:352–356, 1990.

    Google Scholar 

  46. R. K. Yedavalli. Robust stabilization of linear interval systems. Proc. Amer. Control Conf.,, Minneapolis:1579–1583, June 1987.

    Google Scholar 

  47. B.R.Barmish, M.Fu, and S.Saleh. Stability of a polytope of matrices:counterexamples. IEEE AC, 33(6):569, June 1988.

    Google Scholar 

  48. B.R.Barmish and C.V. Hollot. Counterexample to a recent result on the stability of interval matrices by s. bialas. International Journal of Control, 39:1103, 1984.

    Google Scholar 

  49. K. H Wei and R. K. Yedavalli. Invariance of the strict hurwitz property for uncertain polynomials with dependent coefficients. IEEE Trans On Autom. Control, 32(10):907–909, October 1987.

    Google Scholar 

  50. M.Mansour. Robust stability of interval matrices. Proceedings of the IEEE CDC, 1989.

    Google Scholar 

  51. J.A Heinen. Sufficient conditions for the stability of interval matrices. Int. J of Control, 39(6):1323–1328, 1984.

    Google Scholar 

  52. M.B Argoun. On sufficient conditions for the stability of interval matrices. Int. J of Control, 44(5):1245–1250, 1986.

    Google Scholar 

  53. Z.C Shi and W.B Gao. Stability of interval parameter matrices. Int. J of Control, 45(3):1093–1101, 1987.

    Google Scholar 

  54. R. K. Yedavalli. Stability analysis of interval matrices- another sufficient condition. Int. J of Control, pages 767–772, March 1986.

    Google Scholar 

  55. C.S. Zhou and J.L Deng. The stability of the grey linear system. Int. J of Control, 43(1):313, 1986.

    Google Scholar 

  56. C.L Jiang. Sufficient condition for the asymptotic stability of interval matrices. Int. J of Control, 46(5):1803, 1987.

    Google Scholar 

  57. M Mansour. Sufficient conditions for the asymptotic stability of interval matrices. Int. J of Control, 47(6):1973, 1988.

    Google Scholar 

  58. C.L Jiang. Another sufficient condition for the stability of interval matrices. Int. J of Control, 47(1):181, 1988.

    Google Scholar 

  59. K.Wang, A.N. Michel, and D.Liu. Necessary and sufficient conditions for the hurwitz and schur stability of interval matrices. IEEE AC, 39(6):1251, Jun 1994.

    Google Scholar 

  60. Y. T Juang, T.S Kuo, C. F Hsu, and S.D Wang. Root locus approach to the stability analysis of interval matrices. Int. J of Control, 46(3):817, 1987.

    Google Scholar 

  61. D.B Petkovski. Stability of interval matrices: Improved bounds. Int. J of Control, 48(6):2265, 1988.

    Google Scholar 

  62. J.M Martin and G.A Hewer. Smallest destabilizing perturbations for linear systems. Int. J of Control, 46:1495, 1987.

    Google Scholar 

  63. S.Bialas. A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices. Bulletin of the Polish Academy of Sciences, 33:473, 1985.

    MathSciNet  MATH  Google Scholar 

  64. L.X Xin. Stability of interval matrices. Int. J of Control, 45(1):203–210, 1987.

    Google Scholar 

  65. L.X Xin. Necessary and sufficient conditions for stability of a class of interval matrices. Int. J of Control, 45(1):211–214, 1987.

    Google Scholar 

  66. W.J Wang and T.T Lee. Robust stability for single and large scale perturbed systems. International Journal of System Science, 19(3):405–413, 1988.

    Google Scholar 

  67. S.S Wang and W.G Lin. A new approach to the stability analysis of interval systems. Control Theory Adv Technol, 7(2):271–284, 1991.

    Google Scholar 

  68. M.E. Sezer and D.D Siljak. On stability of interval matrices. IEEE Trans Auto. Control, 39(2):368–371, February 1994.

    Google Scholar 

  69. O Pastravanu and M.H Matcovschi. Comments on “assessing the stability of linear time invariant continuous interval dynamic systems”. IEEE Trans Auto. Control, 56(6):1442–1445, June 2011.

    Google Scholar 

  70. L Kolev and S Petrakieva. Assessing the stability of linear time invariant continuous interval dynamic systems. IEEE Trans Auto. Control, 50(3):393–397, March 2005.

    Article  MathSciNet  Google Scholar 

  71. O Pastravanu and M Voicu. Necessary and sufficient conditions for componentwise stability of interval matrix systems. IEEE Trans Auto. Control, 49(6):1016–1021, June 2004.

    Article  MathSciNet  Google Scholar 

  72. Y.K.Foo and Y.C.Soh. Stability analysis of a family of matrices. IEEE AC, 35(11):1257, Nov 1990.

    Google Scholar 

  73. C.B.Soh. Stability robustness measures of state space models. International Journal of Systems Science, 22(10):1867, 1991.

    Google Scholar 

  74. D.D Siljak. Large Scale Systems: Stability and Structure. North Holland, 1978.

    Google Scholar 

  75. R. K. Yedavalli. Sufficient conditions for the stability of interconnected dynamic systems. Journal of Information and Decision Technologies, 15(2):133–142, 1989.

    MathSciNet  MATH  Google Scholar 

  76. M.A Rotea, M Corless, D Da, and I.R Petersen. Systems with structured uncertainty:relations between quadratic stability and robust stability. IEEE Trans Auto. Cont., 38(5):799–803, May 1993.

    Google Scholar 

  77. A Packard and J.C Doyle. Quadratic stability with real and complex perturbations. IEEE Trans Auto. Control, 35(2):198–201, 1990.

    Google Scholar 

  78. Dorato P. Robust Control. IEEE press, 1987.

    Google Scholar 

  79. P. Dorato and R. K. Yedavalli. Recent advances in robust control. IEEE Press, 1990.

    Google Scholar 

  80. D.C.W Ramos and P.L.D Peres. An lmi condition for the robust stability of uncertain continuous time linear systems. IEEE Trans on Automatic Control, 47(4), April 2002.

    Google Scholar 

  81. F Amato, F Garofalo, L Glielmo, and A Pironti. Robust and quadratic stability via polytopic set covering. International Journal of Robust and Nonlinear Control, 5:745–756, 1995.

    Google Scholar 

  82. T Mori and H Kokame. A parameter dependent lyapunov function for a polytope of matrices. IEEE Trans on Automatic Control, 45:1516–1519, 2000.

    Google Scholar 

  83. F Garofalo, G Celentano, and L Glielmo. Stability robustness of interval matrices via quadratic lyapunov forms. IEEE Trans on Automatic Control, 38:281–284, 1993.

    Google Scholar 

  84. R. K. Yedavalli. Robust stability of linear interval parameter matrix family problem revisited with accurate representation and solution. Proceedings of the American Control Conference, St. Louis(MO):3710–3717, June 2009.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Yedavalli, R.K. (2014). Robust Stability Analysis of Linear State Space Systems. In: Robust Control of Uncertain Dynamic Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9132-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-9132-3_2

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-9131-6

  • Online ISBN: 978-1-4614-9132-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics