Vai's Uni Notes

Lyapunov Transformations

3 min read

Time-varying coordinate transformations 𝐱=𝑃(𝑡)𝐳 that can make any 𝐴(𝑡) constant, and the extra conditions needed so that they preserve stability properties.

Any A(t) Can Be Made Constant

  • Recall
    • 𝐴𝑝(𝑡)=𝑃−1(𝑡)[𝐴(𝑡)𝑃(𝑡)−̇𝑃(𝑡)]
  • Claim: for any 𝐴(𝑡), and any constant 𝑅, there exists 𝑃(𝑡) so that 𝐴𝑝(𝑡)=𝑅
    • i.e. (looking only at the homogeneous dynamics)
    • ̇𝐱(𝑡)=𝐴(𝑡)𝐱(𝑡)⟺𝑃(𝑡)̇𝐳(𝑡)=𝑅𝐳(𝑡)
  • Let Ψ(𝑡) be any fundamental matrix for 𝐴(𝑡), and let
    • 𝑃(𝑡)=Ψ(𝑡)𝑒−𝑅𝑡
  • Note
    • det𝑃(𝑡)=detΨ(𝑡)⏟≠0for any finite𝑡det𝑒−𝑅𝑡⏟≠0for any finite𝑡≠0
  • Then
    • ̇𝑃=[dd𝑡Ψ(𝑡)]⏟𝐴(𝑡)Ψ(𝑡)𝑒−𝑅𝑡+Ψ(𝑡)[dd𝑡𝑒−𝑅𝑡]⏟𝑒−𝑅𝑡(−𝑅)
    • 𝐴𝑝=[Ψ(𝑡)𝑒−𝑅𝑡]−1[𝐴(𝑡)Ψ(𝑡)𝑒−𝑅𝑡−𝐴(𝑡)Ψ(𝑡)𝑒−𝑅𝑡+Ψ(𝑡)𝑒−𝑅𝑡𝑅]=𝑅✓

Turning an Unstable System into a Stable One?

  • ̇𝑥=𝑥⇒𝐴=1
  • Let 𝑅=−1=𝐴𝑝⇒̇𝑧=−𝑧
  • Ψ(𝑡)=𝑒𝑡, 𝑒−𝑅𝑡=𝑒𝑡
  • 𝑃(𝑡)=Ψ(𝑡)𝑒−𝑅𝑡=𝑒2𝑡̇𝑃=2𝑒2𝑡
  • 𝐴𝑝=𝑃−1[𝐴𝑃−̇𝑃]=𝑒−2𝑡[(1)𝑒2𝑡−2𝑒2𝑡]=−1=𝑅✓
  • ̇𝑧=−𝑧⇒𝑧(𝑡)=𝑧0𝑒−𝑡
  • 𝑥(𝑡)=𝑥0𝑒𝑡=𝑃(𝑡)𝑧(𝑡)=𝑒2𝑡[𝑧0𝑒−𝑡]=𝑧0𝑒𝑡✓
  • Consistent, but it seems “weird” that we can turn an unstable system into a stable one!

STM from the Transformation

  • We have shown 𝑃(𝑡)=Ψ(𝑡)𝑒−𝑅𝑡, where Ψ(𝑡) is any fundamental matrix for 𝐴(𝑡) and 𝑅 is any constant matrix, transforms
    • ̇𝐱(𝑡)=𝐴(𝑡)𝐱(𝑡)⇒̇𝐳(𝑡)=𝑅𝐳(𝑡)
  • so hence (by the STM under a coordinate change)
    • Φ(𝑡,𝜏)⏟STM for𝐱=𝑃(𝑡)𝑒𝑅(𝑡−𝜏)⏟STM for𝐳𝑃−1(𝜏)
  • Note if 𝑅=diag{𝑟𝑖} then 𝑒𝑅𝑡=diag{𝑒𝑟𝑖𝑡}

Lyapunov Transformation

  • Whether this gives useful insight depends on the required 𝑃(𝑡)
  • A “Lyapunov transformation” preserves properties like convergence or divergence
  • Generally we want to ensure (in addition to nonsingularity) that
    • 𝑃(𝑡)↛0
    • 𝑃−1(𝑡)↛0
    • as 𝑡→∞
  • Stated with determinants: 𝑃(𝑡) must also satisfy: ∃𝜂>0 so that for all 𝑡
    • |det𝑃(𝑡)|≥𝜂>0|det𝑃−1(𝑡)|≥𝜂>0
    • (rules out the “weird” behavior of the previous example!)
  • A transformation with these additional restrictions is known as a Lyapunov transformation
  • The challenge is to ensure 𝑃(𝑡) has this property without solving for Ψ(𝑡) directly
    • For linear time-periodic systems, a basic “trick” can be applied while ensuring that 𝑃(𝑡) is a Lyapunov transformation: Floquet Theory

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