Stability of Time Varying Linear Systems

  • Shashi K. Shrivastava
  • S. Pradeep
Conference paper
Part of the Astrophysics and Space Science Library book series (ASSL, volume 127)


The concept of stability is one of the most fundamental in the study of natural sciences. The complexity of the stability problem depends on the degree of mathematical idealization of the system. In general, time invariant systems are much easier to handle than time varying systems. Here, first a bird’s eye view is presented on the important methods in constant parameter linear and nonlinear systems. Following this is a quick review of stability analysis of time varying systems touching upon periodic systems, second order systems, arbitrarily time varying systems, the absolute stability problem and the functional analytic approach. The paper also includes a summary of some important results of the authors’ work on the stability of second order multidimensional linear arbitrarily time varying systems. The theorems derived generalize some existing theorems. Application of one of them to the damped Mathieu equation yields better results than those existing in literature for some values in the parameter space. The asymptotic behaviour and boundedness of almost constant coefficient systems have also been studied and some new theorems derived.


Time Invariant System Circle Criterion Time Vary Linear System Multiplier Criterion Large Space Structure 
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.


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  1. Aizerman, M.A. : 1947, ‘On the effect of nonlinear functions of several variables on the stability of automatic control systems’, Avtom, i Telemekh. ,VIII.Google Scholar
  2. Aizerman, M.A.: 1949, ‘On a problem relating to the ‘global’ stability of dynamical systems’, Usp. Matem. Nauk, IV.Google Scholar
  3. Aizerman, M.A., and F.R. Gantmacher: 1964, Absolute Stability of Regulator Systems, Holden-Day Inc., San Francisco.Google Scholar
  4. Anderson, B.D.O.: 1966, ‘Stability of control systems with multiple nonlinearities’, J.Franklin Institute, 282, 155–160.zbMATHCrossRefGoogle Scholar
  5. Antosiewicz, H.A. : 1958, ‘A survey of Liapunov’s direct method; Annals of mathematical studies, 41, Contributions to the Theory of Nonlinear Oscillations, IV, Princeton Univ. Press, Princeton, pp. 141–166.Google Scholar
  6. Arnold, V.I.: 1973, Ordinary Differential Equations, MIT Press.Google Scholar
  7. Arscott, F.: 1964, Periodic Differential Equations, Pergamon Press, New York.zbMATHGoogle Scholar
  8. Barbashin, E.A.: 1970, Introduction to the Theory of Stability, Wolters-Noordhoff, Groningen, The Netherlands.zbMATHGoogle Scholar
  9. Bellman, R.: 1943, ‘The stability of solutions of linear differential equations’, Duke Math. J., 10, pp. 643–647.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Bellman, R.: 1953, Stability Theory of Differential Equations, McGraw Hill, New York.zbMATHGoogle Scholar
  11. Birkoff, G.D.: 1927, Dynamical Systems, Amer. Math. Soc. Colloquium Publications, 9, New York.Google Scholar
  12. Black, H.S.: 1934, ‘Stabilized feedback amplifiers’, Bell System Tech. J.,13, pp. 1–18.Google Scholar
  13. Blaquiere, A.: 1966, Nonlinear System Analysis, Academic Press, New York.Google Scholar
  14. Bode, H.W.: 1940, ‘Relation between attenuation and phase in feedback amplifier design’, Bell System Tech.J., 19, pp. 421–450.Google Scholar
  15. Brockett, R.W., and J.W. Willems: 1965, ‘Frequency domain stability criteria’, Part I and II, Proc. Joint Automatic Control Conference, Troy, New York, pp. 735–747.Google Scholar
  16. Cesari,L.: 1963, Asymptotic Behaviour of Stability Problems in Ordinary Differential Equations, Springer-Verlag, New York.Google Scholar
  17. Chetayev, N.G.: 1961, The Stability of Motion, Pergamon Press, New York.Google Scholar
  18. Cho, Yo-Sung, and K.S, Narendra: 1968, ‘An off-axis circle criterion for the stability of feedback systems with a monotone nonlinearity’, IEEE Trans. Auto. Control, AC-13, pp 413–416.MathSciNetGoogle Scholar
  19. Coddington, E.A., and N. Levinson: 1955, Theory of Ordinary Differential Equations, McGraw Hill, New York.zbMATHGoogle Scholar
  20. Coppel,W.A.: 1965, Stability and Asymptotic Behaviour of Differential Equations, D.C. Heath and Company, Boston.Google Scholar
  21. Cunningham, W.J.: 1958, Introduction to Nonlinear Analysis, McGraw Hill, New York.zbMATHGoogle Scholar
  22. Desoer, C.A.: 1965, ‘A generalization of the Popov criterion’, IEEE Trans. Auto. Control, AC-10, pp. 182–185.CrossRefMathSciNetGoogle Scholar
  23. Desoer, C.A., and M. Vidyasagar: 1975, Feedback Systems : Input-Output Properties, Academic Press, New York.zbMATHGoogle Scholar
  24. Erugin, N.P.: 1966, Linear Systems of Ordinary Differential Equations with Periodic and Quasi-periodic Coefficients, Academic Press, New York.zbMATHGoogle Scholar
  25. Evans, W.R.: 1948, ‘Graphical analysis of control systems’, Trans.IEEE, 67, pp. 547–551.Google Scholar
  26. Evans, W.R.: 1950, ‘Control system synthesis by root locus method’, Trans. IEEE, 69. pp. 1–4.Google Scholar
  27. Floquet, G.: 1883, ‘ Sur les equations differentiales lineaires’, Ann. L’Ecole Normale Super, 12, pp. 47–89.MathSciNetGoogle Scholar
  28. Gunderson, H., H. Rigas, and F.S. Van Vleck: 1974, ‘A technique for determining the stability regions for the damped Mathieu equation’, SUM J. of Appl. Math., 26, 345–349.zbMATHCrossRefGoogle Scholar
  29. Hahn, W.: 1963, Theory and Application of Liapunov’s Direct Method, Prentice Hall, Englewood Cliffs.zbMATHGoogle Scholar
  30. Hahn,W.: 1967, Stability of Motion, Springer Verlag, Berlin.zbMATHGoogle Scholar
  31. Hartman, P.: 1964, Ordinary Differential Equations, Wiley, New York.zbMATHGoogle Scholar
  32. Hill, G.W.: 1886, ‘On the part of the lunar perigee which is a function of the mean motions of the Sun and Moon’, Acta Math., 8, pp.1–36.zbMATHCrossRefMathSciNetGoogle Scholar
  33. Huseyin, K.: 1978, Vibrations and Stability of Multiple Parameter Systems, Sijthoff and Noordhoff, The Netherlands.zbMATHGoogle Scholar
  34. Kalman, R.E.: 1963, ‘Liapunov functions for the problem of Lur’e in automatic control’, Proc. Natl. Acad. Sci., 49, 201–205.ADSzbMATHCrossRefMathSciNetGoogle Scholar
  35. Kalman, R.E., and J.E. Bertram: 1960, ‘Control system design via the “Second Method” of Liapunov’, Trans. ASME, J. Basic Engineering, 82, pp. 371–400.CrossRefMathSciNetGoogle Scholar
  36. Krasovskii, N.N.: 1963, Problems of the Theory of Stability of Motion, Stanford University Press, Stanford, Ca.Google Scholar
  37. Kreider, D.L., R.G. Kuller, and D.R. Ostberg: 1968, Elementary Differential Equations, Addison Wesley, Cambridge, Ma., 274–277.zbMATHGoogle Scholar
  38. Lagrange, J.L.: 1788, Mecanique Analytique, Desaint, Paris.Google Scholar
  39. La Salle, J.P., and S. Lefschetz: 1961, Stability by Liapunov’s Direct Method with Applications, Academic Press, New York.zbMATHGoogle Scholar
  40. Lefschetz, S.: 1962, Differential Equations:Geometric Theory, Wiley, New York.Google Scholar
  41. Leipholz, H.: 1970, Stability Theory, Academic Press, New York.zbMATHGoogle Scholar
  42. Liapunov, A.M.: 1949,’Problem general de la stabilite du mouvement’, Annals of Mathematical Studies 17, Princeton University Press, Princeton.Google Scholar
  43. Lips, K.W., and V.J. Modi: 1978, ‘Transient attitude dynamics of satellites with deploying flexible appendages’, Acta Astronautica, 5, pp. 797–815.zbMATHCrossRefGoogle Scholar
  44. Lur’e, A.I., and V.N. Postnikov: 1944, ‘On the theory of stability of control systems’, Prikl. Mat. i. Mehk., VIII.Google Scholar
  45. Magnus, W., and S. Winkler: 1966, Hill’s Equation, Wiley, New York.zbMATHGoogle Scholar
  46. Malkin, I.G.: 1952,Theory of Stability of Motion, Translation series AEC-tr-3352, Physics and Mathematics, United States Atomic Energy Commission.Google Scholar
  47. McLachlan, N.W.: 1964, Theory and Application of Mathieu Functions, Dover, New York.zbMATHGoogle Scholar
  48. Meissner, E.: 1918, ‘Uber schuttelerscheinungen im system mit periodisch veranderlicher Elastizitat’, Schweiz. Bauztq., 72, 95–98.Google Scholar
  49. Minorsky, N.: 1962, Nonlinear Oscillations, Van Nostrand, Princeton.zbMATHGoogle Scholar
  50. Modi, V.J.: 1974, ‘Attitude dynamics of satellites with flexible appendages — a brief review’, Journal of Spacecraft and Rockets, 11, 743–751.ADSCrossRefGoogle Scholar
  51. Narendra, K.S., and R.M. Goldwyn: 1964, ‘A geometrical criterion for the stability of certain nonlinear nonautonomous systems’, IEEE Trans. Circuit Theory, CT-11(3), 406–407.Google Scholar
  52. Narendra, K.S., and C.P. Neuman: 1966, ‘Stability of a class of differential equations with a single monotone nonlinearity’, SIAM J.Control, 4, 295–308.zbMATHCrossRefMathSciNetGoogle Scholar
  53. Narendra, K.S. and J.H. Taylor: 1973, Frequency Domain Criteria for Absolute Stability, Academic Press, New York.zbMATHGoogle Scholar
  54. Nemytskii, V.V., and V.V. Stepanov: 1960, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton.zbMATHGoogle Scholar
  55. Nyquist, H.: 1932, ‘Regeneration theory’, Bell System Tech. J. 11, 126–147.zbMATHGoogle Scholar
  56. Pipes, L.A.: 1953, ‘Matrix solutions of equations of the Mathieu-Hill type’, J. Applied Physics, 24, 902–910.ADSzbMATHCrossRefMathSciNetGoogle Scholar
  57. Pliss, V.A.: 1958, Certain Problems in the Theory of Stability in the Whole, published by LGU.Google Scholar
  58. Poincare, H.: 1957, Les Methodes Nouvelles de la Mechanique Celeste, Dover, New York.Google Scholar
  59. Popov, V.M.: 1961, ‘Absolute stability of nonlinear systems of automatic control’, Automat. Remote Control, 22, 857–875.zbMATHMathSciNetGoogle Scholar
  60. Pradeep, S., and S.K. Shrivastava: 1986a, ‘On the asymptotic behaviour and boundedness of systems with time varying coefficients’, submitted for publication to the Journal of Applied Mechanics Google Scholar
  61. Pradeep, S., and S.K. Shrivastava: 1986b, ‘Stability regions of the damped Mathieu equation’, paper under preparation.Google Scholar
  62. Richards, J.A.: 1976, ‘Stability diagram approximation for the lossy Mathieu equation’, SIAM J. of Appl. Math. 30, 240–247.zbMATHCrossRefGoogle Scholar
  63. Richards, J.A.: 1977, ‘Modelling parametric processes — a tutorial review’, Proc. IEEE, 65, 1549–1557.ADSCrossRefGoogle Scholar
  64. Richards, J.A.: 1983, Analysis of Periodically Time Varying Systems, Springer Verlag, New York.zbMATHGoogle Scholar
  65. Rouche, N., P. Habets, and M. Laloy: 1977, Stability Theory by Liapunov’s Direct Method, Springer Verlag, New York.zbMATHGoogle Scholar
  66. Routh, E.J.: 1975, Stability of motion, Edited by A.T. Fuller, Taylor and Francis Ltd.Google Scholar
  67. Sandberg,I.W.: 1964a, ‘On the L2 boundedness of solutions of nonlinear functional equations’, Bell Systems Tech.J. , 1581–1599.Google Scholar
  68. Sandberg, I.W.: 1964b, ‘A frequency domain condition for the stability of feedback systems containing a single time varying nonlinear element’, Bell System Tech. J., 43, 1601–1608.zbMATHMathSciNetGoogle Scholar
  69. Sandberg, I.W.: 1964c, ‘A condition for the stability of feedback systems containing a single time varying nonlinear element’, Bell System Tech. J., 43, 1815–1817.zbMATHGoogle Scholar
  70. Sandberg, I.W., and V.E. Benes: 1964, ‘On the properties of nonlinear integral equations that arise in the theory of dynamical systems’, Bell System Tech. J., 43, 2839–2853.zbMATHMathSciNetGoogle Scholar
  71. Sandberg, I.W.: 1965a, ‘On the boundedness of solutions of nonlinear integral euqations’, Bell System Tech. J.,44, 439–453.ADSMathSciNetGoogle Scholar
  72. Sandberg, I.W.: 1965b, ‘Some results on the theory of physical systems governed by nonlinear functional equations’, Bell System. Tech. J., 44, 871–898.ADSzbMATHMathSciNetGoogle Scholar
  73. Sansone, G., and R. Conti: 1964, Nonlinear Differential Equations, Pergamon Press, New York.Google Scholar
  74. Shrivastava, S.K.: 1981, ‘Stability theorems for multidimensional linear systems with variable parameters’, J Appl.Mech.,48, 174–176.zbMATHCrossRefMathSciNetGoogle Scholar
  75. Shrivastava, S.K., and V.J. Modi: 1983, ‘Satellite attitude control in the presence of environmental torques — a brief survey’, Journal of Guidances Control and Dynamics,6, 461–471.ADSzbMATHCrossRefGoogle Scholar
  76. Shrivastava, S.K., and S. Pradeep: 1985, ‘On the stability of multidi-mensional linear time varying systems’, Journal of Guidance, Control and Dynamics, 8, 579–583.zbMATHCrossRefGoogle Scholar
  77. Shrivastava, S.K., C. Tschann, and V.J. Modi: 1969, ‘Librational dynamics of earth orbiting satellites’, Proceedings of the 14th Congress of the Indian Society of Theoretical and Applied Mechanics,Kurukshetra, India, 284–306.Google Scholar
  78. Starzinskii, V.M.: 1955, ‘Survey of works on conditions of stability of the trivial solution of a system of linear differential equations with periodic coefficients’, American Mathematical Society Translations, Series 2, 1, 189–238.Google Scholar
  79. Szebehely, V.: 1967, Theory of Orbits, Academic Press, New York.Google Scholar
  80. Taylor, J.H., and K.S. Narendra: 1969, ‘Stability regions for the damped Mathieu equation’, SIAM J. of Appl. Math., 17, 343–352.zbMATHCrossRefMathSciNetGoogle Scholar
  81. Thomson, W. (Lord Kelvin), and P.G. Tait: 1962, Principles of Mechanics and Dynamics, Dover Publications Inc., New York.Google Scholar
  82. Van der Pol, B., and M.J.O. Strutt: 1928, ‘On the stability of Mathieu’s equation’, Philos. Mag., 5, 18–39.Google Scholar
  83. Venkatesh, Y.V.: 1977, Energy Methods in Time-Varying System Stability and Instability Analyses, Springer-Verlag, New York.zbMATHGoogle Scholar
  84. Vidyasagar, M.: 1978, Nonlinear Systems Analysis, Prentice-Hall, Inc., New Jersey.Google Scholar
  85. Willems, J.C.: 1970, The Analysis of Feedback Systems, MIT Press, Cambridge, Mass.Google Scholar
  86. Willems, J.W., and R.W. Brockett: 1968, ‘Some new rearrangement inequalities having application in stability analysis’, IEEE Trans. Auto. Ctrl., AC-13(5), 539–549.CrossRefMathSciNetGoogle Scholar
  87. Yakubovich, V.A.: 1964, ‘Solution of certain matrix inequalities encountered in nonlinear control theory’, Soviet Mathematics, 5, 652–656.zbMATHGoogle Scholar
  88. Yakubovicz, V.A.: 1965, ‘Frequency conditions for the absolute stability and dissipativity of control systems with a single differential nonlinearity’, Soviet Math. Dokl.,6, 98–101.Google Scholar
  89. Yakubovich, V.A. , and V.M. Starzhinskii: 1975, Linear Differential Equations with Periodic’ and Qua si-Periodic Coefficients, Wiley, New York.Google Scholar
  90. Zames, G.: 1964a, ‘On the stability of nonlinear, time varying feedback systems’, Proc. NEC, 20, 725–730.Google Scholar
  91. Zames, G.: 1964b, ‘Contracting trasnformations — a theory of stability and iteration for nonlinear time varying systems’, International Conference on Microwaves, Circuit theory and Information Th., 121–122.Google Scholar
  92. Zames, G.: 1965, ‘Nonlinear time varying feedback systems — conditions for L boundedness derived using conic operators on exponentially weighted spaces’, Proc. Allerton Conf., 460–471.Google Scholar
  93. Zames, G.: 1966, ‘On the input-output stability of time varying non-linear feedback systems’, IEEE Trans. Auto. Control, part I AC-11(2) 228–238, part II, AC-11(3), 465–476.Google Scholar
  94. Zames,G., and P.L. Falb: 1968, ‘Stability conditions for systems with monotone and slope restricted nonlinearities’,SIAM J.Ctrl.,6,89–108.zbMATHCrossRefMathSciNetGoogle Scholar
  95. Zubov,V.I.: 1964, The Methods of Liapunov and Their Applications, Noordhoof, Groningen.Google Scholar

Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • Shashi K. Shrivastava
    • 1
  • S. Pradeep
    • 1
  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia

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