Vai's Uni Notes

Floquet Theory

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For periodic 𝐴(𝑡), the time-varying transformation 𝑃(𝑡)=Φ(𝑡,𝑡0)𝑒−𝑅𝑡 is itself periodic, so it is a Lyapunov transformation onto an LTI system.

Linear Time-Periodic Systems

  • “Floquet” theory is for LTP (“linear time periodic”) systems: systems for which
    • ∃𝑇<∞such that𝐴(𝑡+𝑇)=𝐴(𝑡)∀𝑡≥𝑡0
  • For such systems, 𝑃(𝑡) itself is actually also periodic, and hence a Lyapunov transformation

Monodromy Matrix

  • Let
    • 𝑄=Φ(𝑡0+𝑇,𝑡0)"monodromy matrix"
  • Define 𝑅 by
    • 𝑒𝑅𝑇=𝑄⇒𝑅=(1𝑇)log𝑄
    • (matrix logarithm; Matlab logm)
  • Define
    • 𝑃(𝑡)=Φ(𝑡,𝑡0)𝑒−𝑅𝑡

P(t) Is Periodic

  • Claim: 𝑃(𝑡+𝑇)=𝑃(𝑡) ∀𝑡≥𝑡0
  • 𝑃(𝑡+𝑇)=Φ(𝑡+𝑇,𝑡0)𝑒−𝑅(𝑡+𝑇)=Φ(𝑡+𝑇,𝑡0)𝑒−𝑅𝑇⏟𝑄−1𝑒−𝑅𝑡
  • 𝑒−𝑅𝑇=[𝑒𝑅𝑇]−1=𝑄−1=[Φ(𝑡0+𝑇,𝑡0)]−1=Φ(𝑡0,𝑡0+𝑇)
  • So now (by the composition property)
    • 𝑃(𝑡+𝑇)=Φ(𝑡+𝑇,𝑡0)Φ(𝑡0,𝑡0+𝑇)𝑒−𝑅𝑡=Φ(𝑡+𝑇,𝑡0+𝑇)𝑒−𝑅𝑡≟Φ(𝑡,𝑡0)𝑒−𝑅𝑡=𝑃(𝑡)
    • only if Φ(𝑡,𝑡0)=Φ(𝑡+𝑇,𝑡0+𝑇). Is this true?
  • Let
    • Φ0(𝑡)=Φ(𝑡,𝑡0)Φ𝑇(𝑡)=Φ(𝑡+𝑇,𝑡0+𝑇)
  • ̇Φ0(𝑡)=𝜕𝜕𝑡Φ(𝑡,𝑡0)=𝐴(𝑡)Φ0(𝑡),Φ0(𝑡0)=𝐼̇Φ𝑇(𝑡)=𝜕𝜕𝑡Φ(𝑡+𝑇,𝑡0+𝑇)=𝐴(𝑡+𝑇)Φ𝑇(𝑡)=𝐴(𝑡)Φ𝑇(𝑡),Φ𝑇(𝑡0)=𝐼
  • By uniqueness of solutions for linear DEs, Φ0(𝑡)=Φ𝑇(𝑡); hence

Periodicity of the STM

  • Φ(𝑡+𝑇,𝑡0+𝑇)=Φ(𝑡,𝑡0)
  • true for time-periodic systems
  • and 𝑃(𝑡+𝑇)=𝑃(𝑡) (for this 𝑃(𝑡))

Floquet Transformation

  • So 𝑃(𝑡) provides a Lyapunov transformation turning an LTP system into the LTI system
    • ̇𝐳=𝑅𝐳𝑅=(1𝑇)logm𝑄
  • and

Floquet Transformation

  • Φ(𝑡,𝜏)=𝑃(𝑡)𝑒𝑅(𝑡−𝜏)𝑃−1(𝜏)𝑃(𝑡)periodic
  • The “Floquet transformation” splits the analysis into an LTI part and a periodic part

Graph View

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