Vai's Uni Notes

Lyapunov Functions

4 min read

“Geometric” (Lyapunov) stability analysis: deduce stability from how a size measure of 𝐱(𝑡) changes, without knowing Φ(𝑡,𝑡0).

Rate of Change of the Norm

  • How do we deduce whether Φ has the stability properties if we don’t know exactly what Φ looks like?
  • We could equivalently think in terms of the change in ‖𝐱(𝑡)‖ over time, i.e.
    • dd𝑡‖𝐱(𝑡)‖≤0⇒stable i.s.L.
    • (in fact with 𝛾=1, so perhaps more restrictive than necessary…)
  • or
    • dd𝑡‖𝐱(𝑡)‖22≤0
  • dd𝑡(𝐱𝖳𝐱)=(dd𝑡𝐱𝖳)𝐱+𝐱𝖳(dd𝑡𝐱)
  • With the homogeneous dynamics ̇𝐱=𝐴𝐱:
    • dd𝑡‖𝐱‖22=(𝐴𝐱)𝖳𝐱+𝐱𝖳(𝐴𝐱)=𝐱𝖳[𝐴𝖳+𝐴]𝐱(quadratic form)≤0??- must investigate!

A First Stability Test

  • “Geometric” stability analysis (Lyapunov): examine the behavior of dd𝑡‖𝐱(𝑡)‖ via different metrics
    • [Initially homogeneous / “internal” stability]
  • For example:
    • dd𝑡‖𝐱(𝑡)‖22=𝐱𝖳(𝑡)[𝐴𝖳(𝑡)+𝐴(𝑡)]𝐱(𝑡)=𝐱𝖳(𝑡)𝑀(𝑡)𝐱(𝑡)
    • which is a quadratic form
  • With 𝑀(𝑡)=𝐴𝖳(𝑡)+𝐴(𝑡) (symmetric), the Rayleigh-Ritz inequality gives
    • dd𝑡‖𝐱‖22≤𝛼(𝑡)‖𝐱‖22𝛼(𝑡)=max𝑘𝜆𝑘[𝐴𝖳(𝑡)+𝐴(𝑡)]
  • ⇒ if max𝑡𝛼(𝑡)≤0 ⇒ the system is stable i.s.L.

Theorem

  • If
    • max𝑡𝜆max[𝐴𝖳(𝑡)+𝐴(𝑡)]≤0
  • then the system is stable i.s.L.

Careful

  • This does not mean that 𝛼(𝑡)>0 ∀𝑡 implies instability!
  • It might, or might not ⇒ requires further investigation

Re-Interpreting the Result

  • (Previous result, re-interpreted.) We have shown
    • dd𝑡‖𝐱‖22=−𝐱𝖳𝑄(𝑡)𝐱
    • where
    • 𝑄(𝑡)=−[𝐴𝖳(𝑡)+𝐴(𝑡)]=−2𝐴sym(𝑡)(recall𝐴sym=12(𝐴+𝐴𝖳))
  • So that dd𝑡‖𝐱‖22≤0 if
    • min𝑡𝜆min(𝑄(𝑡))≥0
    • i.e. if 𝑄(𝑡) is positive semi-definite. Equivalently
    • max𝑡𝜆max[𝐴𝖳(𝑡)+𝐴(𝑡)]≤0

Too restrictive

  • dd𝑡‖𝐱‖22≤0 is too restrictive to fully characterize stability
  • Sketch of ‖𝐱(𝑡)‖22 vs. 𝑡, both starting from ‖𝐱0‖22:
    • a blue curve that only decreases (dd𝑡‖𝐱‖22≤0)
    • a red curve that first rises well above ‖𝐱0‖22, with bumps, and then decays: the red one is also stable, but doesn’t satisfy dd𝑡‖𝐱‖22≤0 ∀𝑡≥𝑡0
  • ⇒ reinterpret the argument more geometrically

Level Sets and Invariant Sets

  • Let
    • 𝑉(𝐱)=‖𝐱‖22
  • Note that:
    1. The “level sets” 𝑉(𝐱)=𝑐 for 𝑐>0 define a nested family of closed contours
    2. Rate of change along solutions:
      • dd𝑡‖𝐱‖22=dd𝑡𝑉(𝐱)=𝜕𝑉𝜕𝐱⏟outward normalvector to contour lines⋅̇𝐱⏟direction statevector is moving
  • ̇𝑉≤0:
    • ⇒𝜕𝑉𝜕𝐱⋅̇𝐱≤0
    • ⇒̇𝐱 points into the contour (or at worst along the boundary)
  • Top: circular contours of constant V(x) = c for c = 1, 2, 4 in the x1-x2 plane. Bottom: on a contour V(x) = c, the outward normal dV/dx and the velocity x-dot; where x-dot points inward the state moves to the interior (from Prof. Sanner's notes)
  • 𝜕𝑉𝜕𝐱⋅̇𝐱≤0 ∀𝐱 ⇒ motion at any point on any contour is directed to the interior of that contour

Invariant Sets

  • This means that the interior of each contour,
    • 𝑆(𝑐)={𝐱|𝑉(𝐱)≤𝑐}
    • defines an “invariant set”, i.e. if ∃𝑡1 such that 𝐱(𝑡1)∈𝑆(𝑐), then necessarily 𝐱(𝑡)∈𝑆(𝑐) ∀𝑡≥𝑡1
  • 𝜕𝑉𝜕𝐱⋅̇𝐱≤0 ∀𝐱 means each contour is an invariant set, and the state can never “escape” the contour it starts inside ⇒ stability i.s.L.

NOTE

  • Nothing in these arguments requires circular contours! Any nested set of closed contours works
  • Nested closed, non-circular contours V(x) = c with V(x) not equal to the squared 2-norm (from Prof. Sanner's notes)

Quadratic Lyapunov Functions

  • For linear systems, we need only consider 𝑉(𝐱) for which the level sets 𝑉(𝐱)=𝑐 are elliptical, which can be described using the quadratic form
    • 𝑉(𝐱)=𝐱𝖳𝑃𝐱𝑃positive definite
    • (nested ellipses 𝐱𝖳𝑃𝐱=𝑐)
  • Terminology:
    • A general 𝑉(𝐱) satisfying the nested contour requirement above is known as a “Lyapunov function”
    • A quadratic form for 𝑉(𝐱) is known as a quadratic Lyapunov function (QLF)
  • Note that our original 𝑉(𝐱)=‖𝐱‖22=𝐱𝖳𝑃𝐱 with 𝑃=𝐼
  • Allowing time-varying 𝑃(𝑡) leads to the Lyapunov Equation

Graph View

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