Stability of interconnected systems having slope-bounded nonlinearities

  • Michael G. Safonov
Session 5 Stability II
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 62)


Improved stability criteria are obtained for systems having multiple nonlinearities. The key result (lemma 2) identifies a class of frequency dependent scaling factors d(s) such that, for any time-invariant slope-bounded nonlinearity f(x), the "scaled" operator dfd−1 is in the same L2 conic sector as f(x). Previous results admit only constant scaling factors d.


Feedback System Stability Margin Small Gain Theorem Diagonal Scaling Uncertain Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N.R. Sandell, P. Varaiya, M. Athans, and M.G. Safonov, "Survey of Decentralized Control Methods for Large Scale Systems", IEEE Trans. on Automatic Control, AC-23, pp.108–128, 1978.Google Scholar
  2. [2]
    M. Araki, "Stability of Large-Scale Nonlinear Systems — Quadratic-Order Theory of Composite System Method Using M-Matrices", IEEE Trans, on Automatic Control, AC-23, pp.129–142, 1978.Google Scholar
  3. [3]
    A. Michel, "On the Status of Stability of Interconnected Systems", IEEE Trans. on Circuits and Systems, CAS-30, pp.326–340, 1983. (Also published in IEEE Trans. on Automatic Control, AC-28, June, 1983 and in IEEE Trans. on Systems Man and Cybernetics, BMC-13, July/August 1983.)Google Scholar
  4. [4]
    M.G. Safonov, "Robustness and Stability Aspects of Stochastic Multivariable Feedback System Design", Ph.D. Dissertation, Mass. Inst. of Technology, Cambridge, MA, Sept. 1977; also M.G. Safonov, Stability and Robustness of Multivariable Feedback Systems, MIT Press, Cambridge, MA, 1980.Google Scholar
  5. [5]
    M.G. Safonov and M. Athans, "A Multiloop Generalization of the Circle Stability Criterion for Stability Margin Analysis", IEEE Trans. on Automatic Control. AC-26, pp. 415–422, 1981.Google Scholar
  6. [6]
    J.C. Doyle, "Robustness of Multiloop Linear Feedback Systems", in Proc. 1978 IEEE Conf. on Decision and Control, San Diego, CA, Jan. 10–12, 1979.Google Scholar
  7. [7]
    J.C. Doyle and G. Stein, "Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis", IEEE Trans. on Automatic Control, AC-26, pp. 4–16, 1981.Google Scholar
  8. [8]
    I. Postlethwaite, J.M. Edmunds and A.G.J. MacFarlane, "Principal Gains and Principal Phases in the Analysis of Linear Multivariable Feedback Systems", Ibid., pp. 32–46.Google Scholar
  9. [9]
    M.G. Safonov, A.J. Laub and G.L. Hartmann, "Feedback Properties of Multivariable Systems: The Role and Use of the Return Difference Matrix", Ibid., pp.47–65.Google Scholar
  10. [10]
    N.A. Lehtomaki, N.R. Sandell and M. Athans, "Robustness Results in Linear-Quadratic Gaussian Based Multivariable Control Designs, Ibid., pp 75–92.Google Scholar
  11. [11]
    M.G. Safonov, "Propagation of Conic Model Uncertainty in Hierarchical Systems" IEEE Trans. on Circuits and Systems, CAS-30, pp. 388–396, 1983. (Also published in IEEE Trans. on Automatic Control AC-28, June 1983 and in IEEE Trans. on Systems Man and Cybernetics, SMC-13, July/ August 1983.)Google Scholar
  12. [12]
    D.J.N. Limebeer and Y.S. Hung, "Robust Stability on Inter-connected Systems", Ibid., pp.397–403.Google Scholar
  13. [13]
    M.G. Safonov, "Stability Margins of Diagonally Perturbed Multivariable Feedback Systems", IEEE Proc., 129, Pt.D., pp. 251–256, 1982.Google Scholar
  14. [14]
    J.C. Doyle, "Analysis of Feedback Systems with Structured Uncertainties", Ibid., pp. 242–250.Google Scholar
  15. [15]
    J.C. Doyle, J.E. Wall and G. Stein, "Performance Robustness Analysis for structured Uncertainty", in Proc. IEEE Conf. on Decision and Control, Orlando, FL, December, 1982.Google Scholar
  16. [16]
    M.G. Safonov and J.C. Doyle, "Optimal Scaling for Multivariable Stability Margin Singular Value Computation", in Proc. MECO/EES Symposium, Athens, Greece, August 29-September 2, 1983.Google Scholar
  17. [17]
    M.F. Barratt, "Conservatism with Robustness Tests for Linear Feedback Control Systems", Ph.D. Thesis, University of Minnesota, June 1980; report 80SRC35, Honeywell Systems and Research Center, Minneapolis, MN.Google Scholar
  18. [18]
    K.S. Narendra and J.H. Taylor, "Frequency Domain Criteria for Absolute Stability, Academic Press, NY, 1973.Google Scholar
  19. [19]
    G. Zames, "On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems — Part I: Conditions Using Concepts of Loop Gain, Conicity, and Positivity", IEEE Trans. on Automatic Control, AC-11, pp.228–238, 1966.Google Scholar
  20. [20]
    C.A. Desoer and M. Vidyassagar, "Feedback Systems: Input-Output Properties, Academic Press, NY, 1975.Google Scholar
  21. [21]
    J.C. Willems "The Analysis of Feedback Systems, MIT Press, Cambridge, MA, 1971.Google Scholar
  22. [22]
    E. Polak and D.Q. Mayne, "An Algorithm for Optimization Problems with Functional Inequality Constraints", IEEE Trans. on Automatic Control, AC-21, pp.184–193, 1976.Google Scholar
  23. [23]
    G. Zames and P. Falb, "Stability Conditions for Systems with Monotone and Slope-Restricted Nonlinearities," SIAM J. Control, vol.6, pp.89–108, 1968.Google Scholar
  24. [24]
    P. Falb and G. Zames, "Multipliers with Real Poles and Zeros: An Application of a Theorem on Stability Conditions," IEEE Trans. on Automatic Control, Vol.AC-13, pp.125–126, 1968.Google Scholar
  25. [25]
    R.W. Brockett and J.L. Willems, "Frequency-Domain Stability Criteria — Parts I and II," IEEE Trans. on Automatic Control, Vol. Ac-10, pp.255–261 and pp.407–413.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Michael G. Safonov
    • 1
  1. 1.Department of Electrical Engineering-SystemsUniversity of Southern CaliforniaLos AngelesU.S.A.

Personalised recommendations