Vai's Uni Notes

Local Stability

2 min read

“Local” stability of linear systems: what the Lyapunov conclusions for 𝐴(𝑡) say about the original nonlinear system once the linearization remainder is put back in.

The Remainder Term

  • Recall that the homogeneous dynamics actually satisfy
    • ̇𝐱(𝑡)=𝐴(𝑡)𝐱(𝑡)+𝚪(𝐱(𝑡),𝑡)
    • where 𝚪(𝐱(𝑡),𝑡) is the truncation of the Taylor series expansion of the original nonlinear state equation function 𝐟 (see Linearized Deviation Model)
  • Can the presence of this term radically change our conclusions of stability based on 𝐴(𝑡)?
    • Generally no, as long as we can ensure ‖𝐱(𝑡)‖ remains sufficiently small
    • But then this means that our stability conclusions are necessarily limited to such (sufficiently small) 𝐱

Local vs. Global

  • Thus these conclusions are local (holding only for a restricted set of 𝐱), and not global (holding for arbitrary 𝐱)
  • But how small must this local set be? In particular, what is the “region of convergence” within which our stability analysis will be unaffected by 𝚪(𝐱,𝑡)?
  • Lyapunov gives us the tools to quantify this region

Bounding the Added Term

  • Including the remainder term, 𝑉=𝐱𝖳𝑃𝐱 now has derivative
    • ̇𝑉=−𝐱𝖳𝑄𝐱+2𝐱𝖳𝑃𝚪
    • where again
    • −𝑄=̇𝑃+𝐴𝖳𝑃+𝑃𝐴
  • If 𝑄(𝑡) is uniformly PD (so 𝐴(𝑡) is UES), then as before
    • ̇𝑉≤−𝜈‖𝐱‖22+2𝐱𝖳𝑃𝚪
    • and we need to quantify the impact of the added term
  • We can use a worst-case bound:
    • 𝐱𝖳𝑃𝚪≤|𝐱𝖳𝑃𝚪|≤‖𝐱‖‖𝑃‖𝑖‖𝚪‖
  • Let
    • 𝛾𝑝=max𝑡‖𝑃(𝑡)‖𝑖<∞
    • (since 𝑃(𝑡) is bounded)
  • Then

Derivative Bound with the Remainder

  • ̇𝑉≤−𝜈‖𝐱‖22+2𝛾𝑝‖𝐱‖‖𝚪‖
  • So we have a (negative) quadratic term in ‖𝐱‖, together with a (positive) linear term in ‖𝐱‖
  • When does the negative “overpower” the positive?

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