New relationships between input-output and Lyapunov stability (original) (raw)

Abstract

Theorem 2. I] such that G : = F '0, is a gcrd of (&E, Np, Dc), implying Manuscnpt reccived June 2. 1981: revhed August 18. 1981. ?vi M. Fahmy i b with the Department of Electrical Engineering and Electronics, University of Liverpool. Liverpool L69 3BX. England. on leave from the Department of Electrical Engineering. Asiut University. Auiut. Eg>~t. of Liverpool. 1.iverpool L69 3BX. England. J OReilly i5 with the Department of Electrical Engineenng and Electronlcs. Unlvcrrit)

Key takeaways

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  1. The text establishes a relationship between input-output stability and Lyapunov stability for autonomous systems.
  2. Small-signal L1-stability is defined and shown to be linked to Lyapunov stability.
  3. Conditions for small-signal L1-stability involve finite constants and unique solutions under specific initial conditions.
  4. Theorem 2 states that local conditions ensure the equilibrium point is attractive if small-signal L1-stability holds.
  5. Exploration of input-output and state-space theories can facilitate problem-solving across both frameworks.

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References (10)

  1. K. S. Narendra and I. H. Taylor. Frequena Domarn Crrrerru /or Ahsolure Szuhrhfr..
  2. J. C. Willems, The Anu/ws of Feedbuck Svsrenrr. Cambridge. M4:M.I.T. Press.
  3. C. A. Desoer and M. Vidyasagar. Feedbuck S.psrems: Inpur-Ourpar Properrres. Ken,
  4. A. N. Hichel and R. K. Miller. Quulirurrce Anu/vsm of Lorge Scule Dvnunmul [SI D. D. SilJak. Lurge-Sr-ole Dwunnc S\mrenu. New York: North-Holland. 1978
  5. M. VidyTagar. Inpur-Ourprrr Anu/ws ?/ Lurge-Scule Inrercomecred Sssrena. New
  6. W . Hahn. Sfuhih!). o/ Morron. Berlin: Springer-Verlag. 1967.
  7. J. C . Willems. "The generation of Lyapunov functions for input-output stable systems." SIAM J . Conrr.. vol. 9. pp. 105-133. Feb. 1971
  8. D. J. Hill and P. J. Moylan. "Connections between finite gain and asymptotic
  9. W Walter, Diflerenlrul und Itnegrul Inequubrres. New York: Springer-Vcrlag. 1970. stability." I E E E 7 r u m Aulonlur. Cosrr.. vol. AC-25. pp. 931-936, Oct. 1980. New York: Academic. 1973. 1971. York: Academic. 1975. Svsrmu. New York: Academic, 1977. York: Spnnger-Verlag. I98 I . [ I I ] C . A. Desoer and M. Y. \Vu. "Stahllity of linear time-invariant systems." IEEE Trurls. Circrrrr 77leon. vol. CT-15. pp. 245-250. k t . 196X.
  10. I?] P. J. Moylan. A. Vannelli. and M. Vidyasagar. "On the stability and well-posednesr of interconnected nonlinear dynamid sybtems." IEEE Truns. Cfrmrs S u r . . $01. CAS-27. pp. 1097-1 102. Nov. 1980

FAQs

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What new concept does the paper introduce related to stability?add

The paper defines 'small-signal L∞-stability' and links it to Lyapunov stability. This is significant for analyzing relationships between input-output and Lyapunov stability across various systems.

How does Lyapunov stability relate to input-output stability?add

The research reveals that many stability criteria are more comprehensible in the input-output framework. Furthermore, key findings suggest that establishing input-output stability can utilize results from Lyapunov stability, promoting methodological synergy.

What are the sufficient conditions for loop stability discussed?add

The findings indicate that under specific conditions, including a compensator's existence, loop stability is achievable. In particular, the study illustrates that if G is a gcrd of given matrices, stability conditions are satisfied.

What makes the methodology in this paper noteworthy?add

The methodology combines local conditions of exponential stability with system characteristics to derive small-signal L∞-stability. This approach enhances the applicability of results to both stable and unstable systems.

What implications does the research have for large-scale systems?add

The study posits that the properties of local observability and stability are crucial for large-scale systems. Consequently, it highlights how feedback mechanisms can preserve certain stability characteristics in complex interconnections.