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Torsion Discriminance for Stability of Linear Time-Invariant Systems

Author

Listed:
  • Yuxin Wang

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China)

  • Huafei Sun

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China)

  • Yueqi Cao

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China)

  • Shiqiang Zhang

    (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China)

Abstract

This paper extends the former approaches to describe the stability of n -dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the i th curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Suggested Citation

  • Yuxin Wang & Huafei Sun & Yueqi Cao & Shiqiang Zhang, 2020. "Torsion Discriminance for Stability of Linear Time-Invariant Systems," Mathematics, MDPI, vol. 8(3), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:386-:d:330501
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