198
Views
188
CrossRef citations to date
0
Altmetric
Original Articles

Structured and simultaneous Lyapunov functions for system stability problems

&
Pages 2215-2240 | Received 23 May 1988, Published online: 01 Apr 2008

References

  • Anderson , B. D. O. 1967 . A system theory criterion for positive real matrices . SIAM Journal of Control , 5 : 171 – 182 .
  • Araki , M. 1976 . Input-output stability of composite feedback systems . I.E.E.E. Transactions on automatic Control , 21 : 254 – 259 .
  • Arrow , K. J. , Hurwicz , L. and Uzawa , H. 1958 . Studies in Linear and Nonlinear Programming , Stanford : Stanford University Press .
  • Barker , G. P. , Berman , A. and Plemmons , R. J. 1978 . Positive diagonal solutions to the Lyapunov equations . Linear and Multilinear Algebra , 5 : 249 – 256 .
  • Brayton , R. K. and Tong , C. H. 1979 , Stability of dynamical systems: a constructive approach . I.E.E.E. Transactions on Circuits and Systems , 26 , 224 – 234 ; 1980, Ibid., 27, 1121–1130 .
  • Brockett , R. W. and Willems , J. L. 1965 , Frequency domain stability criteria . I.E.E.E. Transactions on automatic Control , 10 , 255 – 261 , 401 – 413 .
  • Clarke , F. H. 1983 . Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series , New York : Wiley .
  • Demyanov , V. F. and Vasilev , L. V. 1985 . Nondifferentiable Optimization. Optimization Software , New York : Springer-Verlag .
  • Desoer , C. A. and Vidyasagar , M. 1975 . Feedback Systems: Input-Output Properties , New York : Academic Press .
  • Doyle , J. C. 1982 , Analysis of feedback systems with structured uncertainties , Proceedings of the Institution of Electrical Engineers , 129 (6) .
  • Fiedler , M. , and Ptak , V. 1962 , On matrices with non-positive off-diagonal elements and positive principal minors . Czechoslovak Mathematical Journal , 87 , 382 – 400 ; 1966, Some generalizations of positive definiteness and monotonicity. Numerische Mathematik, 9, 163–172 .
  • Horisberger , H. P. and Belanger , P. R. 1976 . Regulators for linear, time invariant plants with uncertain parameters . I.E.E.E. Transactions on automatic Control , 21 : 705 – 708 .
  • Hui , Hu . 1987 . An algorithm for rescaling a matrix positive definite . Linear Algebra and Its Applications , 96 : 131 – 147 .
  • Kamenetskii , V. A. 1983 , Absolute stability and absolute instability of control systems with several nonlinear nonstationary elements . Automation and Remote Control (a trans- lation of Automatika i Telemekhnika), (12) , 1543 – 1552 .
  • Kamenetskii , V. A. and Pyatnitskii , E. S. 1987 , Gradient method of constructing Lyapunov functions in problems of absolute stability . Automation and Remote Control (a trans- lation of Automatika i Telemekhnika) , (1) , 1 – 9 .
  • Kato , T. 1984 . Perturbation Theory for Linear Operators , Second edition , New York : Springer- Verlag .
  • Kelley , J. E. 1960 . The cutting-plane method for solving convex programs . Journal of the Society of Industrial Applications of Mathematics , 8 : 703 – 712 .
  • Khalil , H. K. 1982 . On the existence of positive diagonal P such that PA + ATP <0 . I.E.E.E. Transactions on automatic Control , 27 : 181 – 184 .
  • Khalil , H. K. and Kokotovic , P. V. 1979 . D-stability and multiparameter singular perturbation . SIAM Journal of Control Optimization , 17 : 56 – 65 .
  • Moylan , P. J. and Hill , D. J. 1978 . Stability criteria for large-scale systems . I.E.E.E. Transactions on automatic Control , 23 : 143 – 149 .
  • Overton , M. L. 1987 , On minimizing the maximum eigenvalue of a symmetric matrix . Technical Report, Centre for Mathematical Analysis, Australian National University. Linear Algebra in Signals, Systems and Control, to be published .
  • Overton , M. L. , and Wormley , R. S. , 1987 , On minimizing the spectral radius of a nonsymmetric matrix function—optimality conditions and duality theory . Technical Report, Centre for Mathematical Analysis, Australian National University .
  • Parlett , B. N. 1980 . The Symmetric Eigenvalue Problem , New York : Prentice Hall .
  • Polak , E. and Wardi , Y. 1983 . A nondifferentiable optimization algorithm for the design of control systems subject to singular value inequalities over a frequency range . Automatica , 18 : 267 – 283 .
  • Pyatnitskii , E. S. and Skorodinskii , V. I. 1983 , Numerical method of construction of Lyapunov functions and absolute stability criteria in the form of numerical procedures . Automation and Remote Control (a translation of Automatika i Telemekhnika), (11) , 1427 – 1437 .
  • Rockefellar , R. T. 1972 . Convex Analysis , Second edition , Princeton : Princeton University Press .
  • Safonov , M. G. and Wyetzner , G. 1987 . Computer-aided stability criterion renders Popov criterion obsolete . I.E.E.E. Transactions on automatic Control , 32 : 1128 – 1131 .
  • Shor , N. Z. 1985 . Minimization Methods for Non-differentiable Functions. Springer series in computation mathematics , Berlin : Springer-Verlag .
  • Willems , J. C. 1971a , The construction of Lyapunov functions for input-output stable systems . SIAM Journal of Control , 9 , 105 – 134 ; 1971b, Least squares stationary optimal control and the algebraic Riccati equation. I.E.E.E. Transactions on automatic Control, 16, 621–634; 1972, Dissipative dynamical systems I: General theory. II: Linear systems with quadratic supply rates. Archives for Rational Mechanics and Analysis, 45, 321–343 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.