On the Estimation of Asymptotic Stability Regions: State of the Art and New Proposals

IEEE Transactions on Automatic Control

Volumn 30, Issue 8, 1985, Pages 747-755

  Genesio, Roberto ^a   Tartaglia, Michele ^b   Vicino, Antonio ^b  

^a UNIVERSITY OF FLORENCE   (Italy)

^b POLITECNICO DI TORINO   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

SYSTEM STABILITY;

ESTIMATION OF ASYMPTOTIC STABILITY REGION; TRAJECTORY REVERSING METHOD;

CONTROL SYSTEMS, NONLINEAR;

EID: 0022105529     PISSN: 00189286     EISSN: 15582523     Source Type: Journal    
DOI: 10.1109/TAC.1985.1104057     Document Type: Article

References (95)

1. J. P. La Salle and S. Lefschetz, Stability by Lyapunov's Direct Method. New York: Academic, 1961.
2. H. A. Antosiewicz, “A survey of Lyapunov's second method,” in Contributions to the Theory of Nonlinear Oscillations, vol. 4. Princeton, NJ: Princeton University Press, 1958, pp. 141–166.
3. A. M. Letov, Stability in Nonlinear Control Systems. Princeton, NJ: Princeton University Press, 1961.
4. W. Hahn, Theory and Applications of Lyapunov's Direct Method. Englewood Cliffs, NJ: Prentice-Hall, 1963.
5. V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems. Israel: Jerusalem Academic Press, 1962.
6. V. I. Zubov, Methods of A.M. Lyapunov and Their Application. The Netherlands: Noordhoff, 1964.
7. W. Hahn, Stability of Motion. Berlin: Springer-Verlag, 1967.
8. S. G. Margolis and W. G. Vogt, “Control engineering applications of V.I. Zubov's construction procedure for Lyapunov functions,” IEEE Trans. Automat. Contr., vol. AC-8, pp. 104–113, Apr. 1963.
9. J. R. Hewit and C. Storey, “Optimization of the Zubov and Ingwerson methods for constructing Lyapunov functions,” Electron. Lett., vol. 3, pp. 211–213, May 1967.
10. J. R. Hewit and C. Storey, “Comparison of numerical methods in stability analysis,” Int. J. Contr., vol. 10, pp. 687–701, 1969.
11. F. Fallside, M. R. Patel, and M. Etherthon, “Discussion of ‘Control engineering application of V. I. Zubov's construction procedure for Lyapunov functions'” and “Author's comment,” IEEE Trans. Automat. Contr., vol. AC-10, pp. 220–222, Apr. 1965.
12. J. J. Rodden, “Numerical applications of Lyapunov stability theory,” Preprints JACC, Stanford, CA, 261–268.
13. Y. Yu and K. Vongsuriya, “Nonlinear power system stability study by Lyapunov function and Zubov's method,” IEEE Trans. Power App., Syst., vol. PAS-86, pp. 1480–1485, Dec. 1967.
14. A. K. De Sarkar and N. D. Rao, “Zubov's method and transient-stability problems of power systems,” Proc. Inst. Elec. Eng., vol. 118, pp. 1035–1040, Aug. 1971.
15. A. Vannelli and M. Vidyasagar, “Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems,” Preprints 8th IFAC World Congr., Kyoto, Japan, Aug. 1981, paper 5.1.
16. N. E. Kirin, R. A. Nelepin, and V. N. Baidaev, “Construction of the attraction region by Zubov's method,” J. Differential Equations, vol. 17, 871–880, 1982.
17. G. Burnand and G. Sarlos, “Determination of the domain of stability,” J. Math. Anal. Appl., vol. 23, pp. 714–722, 1968.
18. J. Kormanik and C. C, Li, “Decision surface estimate of nonlinear system stability domain by Lie series method,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 666–669, Oct. 1972.
19. I. Troch, “The evaluation of the domain of attraction of nonlinear control systems with hybrid computing systems,” in Proc. 5th IFAC World Congr. Paris, France, 1972, paper 32.1.
20. R. K. Bansal and R. Subramanian, “Stability analysis of power systems using Lie series and pattern-recognition techniques,” in Proc. Inst. Elec. Eng., vol. 121, pp. 623–629, July 1974.
21. H. Miyagi and T. Taniguchi, “Application of the Lagrange-Charpit method to analyze the power system's stability,” Int. J. Contr., vol. 32, pp. 371–379, 1980.
22. H. Miyagi and T. Taniguchi,, “Lagrange-Charpit method and stability problem of power systems,” Proc, Inst. Elec. Eng. vol. 128, pp. 117–122, May 1981.
23. M. Abu Hassan and C. Storey, “Numerical determination of domains of attraction for electrical power systems using the method of Zubov,” Int. J. Contr., vol. 34, 371–381, 1981.
24. G. P. Szego, “On a new partial differential equation for the stability analysis of time invariant control systems,” SIAM J. Contr., vol. 1, pp. 63–75, 1963.
25. G. P. Szego, “New methods for constructing Lyapunov functions for time-invariant control systems,” in Proc. 2nd IFAC World Congr., Basel, 1963, pp. 584–589.
26. G. P. Szego and G. K. Geiss, “A remark on ‘A new partial differential equation for the stability analysis of time invariant control system,”' SIAM J. Contr., vol. 1, pp. 369–376, 1963.
27. F. S. Prabhakara, A. H. El-Abiad, and A. J. Koivo, “Application of generalized Zubov's method to power system stability,” Int. J. Contr., vol. 20, pp. 203–212, 1974.
28. J. P. La Salle, “Some extensions of Lyapunov's second method,” IRE Trans. Circuits Theory, vol. CT-7, pp. 520–527, 1960.
29. D. G. Shultz, “The generalization of Lyapunov functions,” in Advances in Control Systems, C. T. Leondes, Ed. New York: Academic, 1965, pp. 1–64.
30. D. R. Ingwerson, “A modified Lyapunov method for nonlinear stability analysis,” IRE Trans. Automat. Contr., vol. AC-6, pp. 199–210, May 1961.
31. G. P. Szego, “A contribution to Lyapunov's second method: Nonlinear autonomous systems,” in International Symposium in Nonlinear Differential Equations and Non-linear Mechanics, J. P. La Salle and S. Lefschetz, Eds. New York: Academic, 1963, 421–430.
32. J. R. Hewit and C Storey, “Numerical application of Szego's method for constructing Lyapunov functions,” IEEE Trans. Automat. Contr. vol. AC-14, pp. 106–108, Feb. 1969.
33. D. G. Schultz and J. E. Gibson, “The variable gradient method for generating Lyapunov functions,” AI EE Trans. Appl. Ind., vol. 81, pp. 203–210, Sept. 1962.
34. C. C. Hang and J. A. Chang, “An algorithm for constructing Lyapunov functions based on the variable gradient method,” IEEE Trans. Automat. Contr., vol. AC-15, pp. 510–512, Aug. 1970.
35. D. D. Perlmutter, Stability of Chemical Reactors. Englewood Cliffs, NJ: Prentice-Hall, 1972.
36. A. J. Berger and L. Lapidus, “Stability of high dimensional nonlinear systems using Krasovskii's theorem,” AIChE J., vol. 15, pp. 171—177, Mar. 1969.
37. S. Weissenberger, “Stability-boundary approximations for relay control systems via a steepest-ascent construction of Lyapunov functions,” Trans. ASME J. Basic Eng., vol. 82, pp. 419–428, June 1966.
38. E. J. Davison and E. M. Kurak, “A computational method for determining quadratic Lyapunov functions for non-linear systems,” Automatica, vol. 7, pp. 627–636, 1971.
39. D. N. Shields and C. Storey, “The'behavior of optimal Lyapunov functions,” Int. J. Contr., vol. 21, pp. 561–573, 1975.
40. A. N. Michel, N. R. Sarabudla, and R. K. Miller, “Stability analysis of complex dynamical systems. Some computational methods,” Cir. Syst. Signal Processing, vol. 1, pp. 171–202, 1982.
41. E. Noldus, “On the stability of systems having several equilibrium states,” Appl. Sci. Res., vol. 21, pp. 218–233, 1969.
42. E. Noldus, A. Galle, and L. Josson, “The computation of stability regions for systems with many singular points,” Int. J. Contr., vol. 17, pp. 641–652, 1973.
43. E. Noldus, J. Spriet, E. Verriest, and A. Van Cauwenberghe, “A new Lyapunov technique for stability analysis of chemical reactors,” Automatica, vol. 10, pp. 675–680, 1974.
44. E. J. Noldus, “New direct Lyapunov-type method for studying synchronization problems,” Automatica, vol. 13, pp. 139–151, 1977.
45. E. J. Noldus, “A new frequency domain approach to certain transient stability problems in power systems analysis,” Int. J. Contr., vol. 26, pp. 33- 1977.
46. L. B. Jocic, “On the attractivity of imbedded systems,” Automatica, vol. 17, 853–860, 1981.
47. A. N. Michel, N. R. Sarabudla, and R. K. Miller, “Some constructive approaches in stability analysis,” in Proc. 1981 Int. Symp. Circuit Syst., Apr. 1981, pp. 465–468.
48. R. K. Brayton and C. H. Tong, “Constructive stability and asymptotic stability of dynamical systems,” IEEE Trans. Circuits Syst., vol. CAS-27, pp. 1121–1130, Nov. 1980.
49. J. L. Willems, “Direct methods for transient stability studies in power system analysis,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 332-341,Aug. 1971.
50. M. Ribbens-Pavella, “Critical survey of transient stability studies of multi-machine power systems by Lyapunov's direct method,” in Proc. 9th Ann. Allerton Conf on Circ. Syst. Theory, Oct. 1971, pp. 751–767.
51. A. A. Fouad, “Stability theory-criteria for transient stability” in Proc. Eng. Foundation Conf. Syst. Eng. for Power, Henniker, NH, 1975, 421–450.
52. M. Gatto and S. Rinaldi, “Stability analysis of predator-prey models via the Lyapunov method,” in Analysis and Computation of Equilibria and Regions of Stability, H. R. Grumm, Ed. Laxen-burg, Austria, 1975, pp. 139–153.
53. J. A. Walker and N. H. McClamroch, “Finite regions of attraction for the problem of Lur'e,” Int. J. Contr., vol. 6, pp. 331–336, 1967.
54. S. Weissenberger, “Application of results from the absolute stability problem to the computation of finite stability domains,” IEEE Trans. Automat. Contr., vol. AC-13, pp. 124–125, Feb. 1968.
55. S. Weissenberger, “Comments on ‘Application of results from the absolute stability problem to the computation of finite stability domains,” IEEE Trans. Automat. Contr., vol. AC-14, 109, 1969.
56. B. D. O. Anderson, “Stability of control systems with multiple nonlinearities,” J. Franklin Inst., vol. 282, pp. 155–160, Sept. 1966.
57. J. L. Willems and J. C. Willems, “The application of Lyapunov methods to the computation of transient stability regions for multimachine power systems,” IEEE Trans. Power App. Syst., vol. PAS-89, pp. 795–801, Mar./June 1970.
58. J. L. Willems, “Optimum Lyapunov functions and transient stability regions for multimachine power systems,” Proc. Inst. Elec. Eng., vol. 117, pp. 573–578, Mar. 1970.
59. W. R. Foster and M. S. Davies, “Estimating the domain of attraction for systems with multiple non-linearities,” Int. J. Contr., vol. 15, pp. 1001–1003, 1972.
60. M. A. Pai and M. Anandomohan, “Transient stability of multimachine systems by Popov's method,” IEEE Summer Power Meet., July 1970, 70-CP-667.
61. M. A. Pai, “Transient stability studies in power systems using Lyapunov-Popov approach,” in Proc. 5th IFAC World Congr., Paris, France, 1972, paper 31.5.
62. K. S. Narendra and C. P. Neumann, “Stability of continuous time dynamical systems with M-feedback nonlinearities,” AIAA J., vol. 5, pp. 2012–2027, Nov. 1967.
63. J. L. Willems, “The computation of finite stability regions by means of open Lyapunov surfaces,” Int. J. Contr., vol. 10, pp. 537–544, 1969.
64. T. L. Chang and M. S. Davies, “Regions of transient stability for power systems involving saliency using the Popov criterion,” Proc. Inst. Elec. Eng., vol. 119, 625–628, May 1972.
65. R. Jha and A. K. Mahalanabis, “Improved technique for computation of power-system transient-stability regions,” Proc. Inst. Elec. Eng., vol. 122, pp. 1402–1404, Dec. 1975.
66. M. A. Pai and C. L. Narayana, “Finite regions of attraction for multinonlinear systems and its application to the power system stability problem,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 716–721, Oct. 1976.
67. N. Kakimoto, Y. Ohsawa, and M. Hayashi, “Transient stability analysis of multimachine power systems with field flux decays via Lyapunov's direct method,” IEEE Trans. Power App. Syst., vol. PAS-99, pp. 1819–1827, Sept./Oct. 1980.
68. W. O. Paradis and D. D. Perlmutter, “Tracking function approach to practical stability and ultimate boundedness,” AIChE J., vol. 12, pp. 13–136, Jan. 1966.
69. J. R. Hewit and C. Storey, “Computer application of the tracking function approach to practical stability,” Electron. Lett., vol. 2, pp. 408–409, 1966.
70. J. R. Hewit and C. Storey, “Practical stability in polar co-ordinates,” Electron. Lett., vol. 3, pp. 558–559, Dec. 1967.
71. J. R. Hewit and C. Storey, “Computer program for obtaining regions of practical stability for second-order, autonomous systems,” Proc. Inst. Elec. Eng., vol. 114, pp. 1347–1350, Sept. 1967.
72. J. F. Leathrum, E. F. Johnson, and L. Lapidus, “A new approach to the stability and control of nonlinear processes,” AIChE J., vol. 10, pp. 16–25, Jan. 1964.
73. R. Luus and L. Lapidus, “An averaging technique for stability analysis,” Chem. Eng. Sci., vol. 21, pp. 159–181, 1966.
74. H. N. Scofield, “An estimate of the stable initial condition region based on the describing function,” IEEE Trans. Automat. Contr., vol. AC-10, pp. 484–485, Oct. 1965.
75. E. F. Infante and L. G. Clark, “A method for the determination of the domain of stability of second-order nonlinear autonomous systems,” J. Appl. Mech., vol. 86, pp. 315–320, June 1964.
76. L. B. Jocic, “Planar regions of attraction,” IEEE Trans. Automat. Contr., vol. AC-27, 708–710, 1982.
77. E. J. Davison and K. C. Cowan, “A computational method for determining the stability region of a second-order non-linear autonomous system,” Int. J. Contr., vol. 9, pp. 349–357, 1969.
78. J. Texter, “Numerical algorithm for implementing Zubov's construction in two-dimensional systems,” IEEE Trans. Automat. Contr., vol. AC-19, 62–63, Feb. 1974.
79. H. Yee, “Region of transient stability in state space for synchronous generator,” Proc. Inst. Elec. Eng., vol. 122, pp. 739–744, July 1975.
80. H. Yee and B. D. Spalding, “Transient stability analysis of multimachine power systems by the method of hyperplanes,” IEEE Trans. Power App. Syst., vol. PAS-96, pp. 276–284, Jan./Feb. 1977.
81. H. Yee and M. J. Muir, “Transient stability region of synchronous generator with saturable exciter,” Proc. Inst. Elec. Eng., vol. 125, pp. 943–947, Oct. 1978.
82. H. Yee and M. J. Muir, “Transient stability of multimachine systems with saturable exciter,” Proc. Inst. Elec. Eng., vol. 127, pp. 9–14, Jan. 1980.
83. K. A. Loparo and G. L. Blankenship, “Estimating the domain of attraction of nonlinear feedback systems,” IEEE Trans. Automat. Contr., vol. AC-23, pp. 602–608, Aug. 1978.
84. R. Genesio and A. Vicino, “New techniques for constructing asymptotic stability regions for nonlinear systems,” IEEE Trans. Circuits Syst., vol. CAS-31, pp. 574–581, June 1984.
85. G. Sansone and R. Conti, Non-linear Differential Equations. New York: Macmillan, 1964.
86. N. P. Erugin, “On the continuation of solutions of differential equations,” Prikl. Mat. Mekh., vol. 15, pp. 55–58, 1951.
87. N. P. Erugin, “Certain general questions in the theory of the stability of motion,” Prikl. Mat. Mekh., vol. 15, pp. 227–236, 1951.
88. V. Guillemin and A. Pollack, Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974.
89. J. W. Milnor, “Differential topology,” in Lectures in Modern Mathematics, vol. II. T. L. Saaty, Ed. New York: Wiley, 1964. pp. 165–183.
90. V. I. Arnold, Equations differentielle Ordinairies. Moscow: Mir, 1979.
91. F. W. Wilson, “The structure of the level surfaces of a Lyapunov function,” J. Differential Equations, vol. 3, pp. 323–329, 1967.
92. N. P. Bhatia and G. P. Szego, Stability Theory of Dynamical Systems. New York: Springer-Verlag, 1970.
93. J. Lagasse and C. Mira, “Study of recurrence relationships and their applications by the Laboratoire d'Automatique et de ses Applications Spatiales,” in Proc. 5th World Congr., Paris, France, 1972, paper 32.3.
94. R. Genesio, A. Vicino, and M. Tartaglia, “Qualitative analysis of mathematical arc models using Lyapunov theory,” Int. J. Elec. Power Energy Syst., vol. 4, pp. 245–252, 1982.
95. O. Mayr, “Beitrag zur Theorie der Statistischen und der Dynamischen Lichtbogens,” Archiv. F. Electr., vol. 37, pp. 588–608, 1943.