Vai's Uni Notes

Lyapunov Equation

7 min read

Stability tests for ̇𝐱=𝐴(𝑡)𝐱 from a time-varying quadratic Lyapunov function 𝑉(𝐱,𝑡)=𝐱𝖳𝑃(𝑡)𝐱, and the converse result that such a 𝑃(𝑡) always exists.

The Lyapunov Equation

  • Suppose the matrix 𝑃(𝑡) is uniformly bounded and uniformly positive definite, i.e.
    • 𝜌=max𝑡𝜆max(𝑃(𝑡))<∞𝜂=min𝑡𝜆min(𝑃(𝑡))>0
  • Define 𝑄(𝑡) by

Lyapunov Equation

  • −𝑄(𝑡)=̇𝑃(𝑡)+𝐴𝖳(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)

Stability Theorem

Theorem

  • With 𝑃(𝑡) uniformly bounded and uniformly PD, and 𝑄(𝑡) from the Lyapunov eq’n:
    1. If 𝑄(𝑡) is positive semi-definite, then the system is stable i.s.L.
    2. If 𝑄(𝑡) is uniformly positive definite, then the system is exponentially stable

NOTE

  • If the tests fail, the result is inconclusive → a different choice of 𝑃(𝑡) may yield different results
  • Is it true that every stable/ES system will have (𝑃,𝑄) satisfying the required conditions? Yes! In fact, infinitely many 𝑃,𝑄 pairs can be found. Indeed
    • 𝑃(𝑡)≝∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏
    • (Φ: STM for 𝐴(𝑡)) satisfies the Lyapunov eq’n (see Converse Theorem)

Proof

  • Let
    • 𝑉(𝐱,𝑡)=𝐱𝖳𝑃(𝑡)𝐱
  • ̇𝑉=dd𝑡𝑉(𝐱(𝑡),𝑡)=̇𝐱𝖳𝑃𝐱+𝐱𝖳̇𝑃𝐱+𝐱𝖳𝑃̇𝐱=(𝐴𝐱)𝖳𝑃𝐱+𝐱𝖳̇𝑃𝐱+𝐱𝖳𝑃(𝐴𝐱)=𝐱𝖳[̇𝑃+𝐴𝖳𝑃+𝑃𝐴]𝐱=−𝐱𝖳𝑄(𝑡)𝐱
  1. 𝑄 positive semi-definite ⇒̇𝑉≤0, so (using Rayleigh-Ritz on both ends)
    • 𝜂‖𝐱(𝑡)‖22≤𝐱𝖳(𝑡)𝑃(𝑡)𝐱(𝑡)=𝑉(𝐱(𝑡),𝑡)≤𝑉(𝐱0,𝑡0)=𝐱𝖳0𝑃(𝑡0)𝐱0≤𝜌‖𝐱0‖22
    • Thus
    • 𝜂‖𝐱(𝑡)‖22≤𝜌‖𝐱0‖22⇒‖𝐱(𝑡)‖2≤√𝜌𝜂⏟𝛾‖𝐱0‖2
    • = the definition of i.s.L. stability
  2. 𝑄 uniformly positive definite
    • ⇒𝜈=min𝑡𝜆min(𝑄(𝑡))>0
    • Then
    • ̇𝑉=−𝐱𝖳𝑄(𝑡)𝐱≤−𝜈‖𝐱(𝑡)‖22
    • To make further progress we need an important lemma:

Lemma ("Comparison Lemma")

  • Suppose ̇𝑤(𝑡)=𝑓(𝑤(𝑡),𝑡), 𝑤(𝑡0)=𝑤0 has known solution 𝑤(𝑡)
  • Then the differential inequality
    • ̇𝑣(𝑡)≤𝑓(𝑣(𝑡),𝑡)𝑣(𝑡0)=𝑣0≤𝑤0
  • has solution 𝑣(𝑡) satisfying
    • 𝑣(𝑡)≤𝑤(𝑡)∀𝑡≥𝑡0
  • Continuing 2.:
    • ̇𝑉=−𝐱𝖳𝑄(𝑡)𝐱≤−𝜈‖𝐱‖22
    • Note that 𝐱𝖳𝑃𝐱≤𝜌‖𝐱‖22 (=𝑉)
    • ⇒−‖𝐱‖22≤(−1𝜌)𝐱𝖳𝑃𝐱
    • Hence
    • ̇𝑉≤−𝜈‖𝐱‖22≤−(𝜈𝜌)𝐱𝖳𝑃𝐱(=𝑉)
    • or
    • ̇𝑉≤−(𝜈𝜌)𝑉
    • By the comparison lemma (compare with ̇𝑤=−(𝜈𝜌)𝑤):
    • 𝑉(𝑡)≤𝑉(𝑡0)𝑒−𝜆1(𝑡−𝑡0)with𝜆1=𝜈𝜌
  • So
    • 𝜂‖𝐱(𝑡)‖22≤𝐱𝖳(𝑡)𝑃(𝑡)𝐱(𝑡)=𝑉(𝑡)≤𝑉(𝑡0)𝑒−𝜆1(𝑡−𝑡0)=𝐱𝖳0𝑃(𝑡0)𝐱0𝑒−𝜆1(𝑡−𝑡0)≤𝜌‖𝐱0‖22𝑒−𝜆1(𝑡−𝑡0)
    • ⇒𝜂‖𝐱(𝑡)‖22≤𝜌‖𝐱0‖22𝑒−𝜆1(𝑡−𝑡0)

Exponential Stability Bound

  • ‖𝐱(𝑡)‖2≤√𝜌𝜂𝑒−𝜆(𝑡−𝑡0)‖𝐱0‖2with𝜆=𝜈2𝜌
  • Note 𝜆 is determined by the ratio of the min eigenvalue of 𝑄 (𝜈) to the max eigenvalue of 𝑃 (𝜌)

Aside

  • Suppose ̇𝑉=𝛼(𝑡)𝑉(𝑡). If 𝛼(𝑡)>0 ∀𝑡, does 𝑉(𝑡)→∞ necessarily?

Converse Theorem

Theorem

  • If 𝐴(𝑡) is UES, then for any bounded, uniformly PD 𝑄(𝑡), there exists a bounded, uniformly PD 𝑃(𝑡) such that
  • ̇𝑃(𝑡)+𝐴𝖳(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)=−𝑄(𝑡)
  • Pf: by construction. Define
    • 𝑃(𝑡)=∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏

a) P(t) Is Uniformly Positive Definite

  • 𝐱𝖳𝑃(𝑡)𝐱=∫∞𝑡(Φ𝐱)𝖳𝑄(𝜏)(Φ𝐱)d𝜏
  • Define 𝐳=Φ(𝜏,𝑡)𝐱
  • Note: since Φ(𝜏,𝑡) is nonsingular for any 𝑡,𝜏, 𝐳=Φ(𝜏,𝑡)𝐱≠𝟎 for any 𝐱≠𝟎
  • Then, for any nonzero 𝐱:
    • 𝐱𝖳𝑃(𝑡)𝐱=∫∞𝑡𝐳𝖳𝑄(𝜏)𝐳d𝜏≥∫∞𝑡𝜈‖𝐳‖22d𝜏>0
    • since ‖𝐳‖>0, and 𝜈=min𝑡𝜆min(𝑄(𝑡))>0

b) P(t), Q(t) Satisfy the Lyapunov Equation

  • Note that
    • ̇𝑃(𝑡)=𝜕𝜕𝑡∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏
  • Using Leibniz’ rule:
    • ̇𝑃=−[Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)]𝜏=𝑡+∫∞𝑡𝜕𝜕𝑡[Φ(𝜏,𝑡)𝖳]𝑄(𝜏)Φ(𝜏,𝑡)d𝜏+∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)[𝜕𝜕𝑡Φ(𝜏,𝑡)]d𝜏
  • Recall
    • 𝜕𝜕𝑡Φ(𝜏,𝑡)=−Φ(𝜏,𝑡)𝐴(𝑡)
    • (differentiation w.r.t. the 2nd time argument of the STM: property 4b)
    • and Φ(𝑡,𝑡)=𝐼 for any 𝑡
  • Then
    • ̇𝑃=−𝐼𝑄(𝑡)𝐼−𝐴𝖳(𝑡)∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏⏟𝑃(𝑡)−∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏⏟𝑃(𝑡)𝐴(𝑡)
    • ⇒̇𝑃(𝑡)=−𝑄(𝑡)−𝐴(𝑡)𝖳𝑃(𝑡)−𝑃(𝑡)𝐴(𝑡)
    • or
    • ̇𝑃(𝑡)+𝐴𝖳(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)=−𝑄(𝑡)✓

c) P(t) Is Bounded for Any t

  • 𝑃(𝑡)=∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏
  • ⇒‖𝑃(𝑡)‖𝑖≤∫∞𝑡‖Φ(𝜏,𝑡)‖2𝑖‖𝑄(𝜏)‖𝑖d𝜏
  • 𝑄(𝑡) bounded ⇒‖𝑄(𝜏)‖𝑖≤𝛾𝑄<∞ ∀𝜏, so
    • ‖𝑃(𝑡)‖𝑖≤𝛾𝑄∫∞𝑡‖Φ(𝜏,𝑡)‖2𝑖d𝜏
  • If 𝐴(𝑡) is UES, then
    • ‖Φ(𝜏,𝑡)‖≤𝑐𝑒−𝜆(𝜏−𝑡)for some𝜆>0,and some𝑐<∞
  • ⇒‖𝑃(𝑡)‖𝑖≤𝛾𝑄𝑐2𝑒2𝜆𝑡∫∞𝑡𝑒−2𝜆𝜏d𝜏
  • So
    • ‖𝑃(𝑡)‖𝑖≤−𝛾𝑄𝑐22𝜆𝑒2𝜆𝑡[𝑒−2𝜆𝜏]𝜏=∞𝜏=𝑡=−𝛾𝑄𝑐22𝜆𝑒2𝜆𝑡[0−𝑒−2𝜆𝑡]
  • Thus
    • ‖𝑃(𝑡)‖𝑖≤𝛾𝑄𝑐22𝜆<∞∀𝑡

Consequences

  • Thus for UES 𝐴(𝑡) (and, with a slight modification of the above argument, for i.s.L. 𝐴(𝑡)) we are guaranteed the existence of 𝑃(𝑡),𝑄(𝑡) pairs that will work to show stability
  • In fact, if there exists one 𝑃(𝑡),𝑄(𝑡) pair that works, there exist infinitely many other pairs that will work

Side note

  • If we view the Lyapunov eq’n as an arbitrary matrix DE:
    • ̇𝑃(𝑡)+𝐴𝖳(𝑡)𝑃(𝑡)+𝑃(𝑡)𝐴(𝑡)=−𝑄(𝑡)
  • the corresponding solution is
    • 𝑃(𝑡)=∫∞𝑡Φ𝖳(𝜏,𝑡)𝑄(𝜏)Φ(𝜏,𝑡)d𝜏
    • (Φ: STM for 𝐴(𝑡))

LTI Systems

  • For LTI systems, we can take 𝑃(𝑡)=const, and ̇𝑃(𝑡)=0, so the Lyapunov eq’n becomes

Algebraic Lyapunov Equation (LTI)

  • 𝐴𝖳𝑃+𝑃𝐴=−𝑄
  • And the above theorem becomes: if 𝐴 is Hurwitz, then for any positive definite 𝑄, there exists a positive definite 𝑃 such that 𝐴𝖳𝑃+𝑃𝐴=−𝑄

Careful with the wording!

  • The above does not imply that for any 𝑃, the corresponding 𝑄 is positive definite!
  • For example, 𝑃=𝐼 (𝑉(𝐱)=‖𝐱‖22) might not yield a positive definite 𝑄 even when 𝐴 is stable
  • However, there are infinitely many other choices 𝑃≠𝐼 for which the corresponding 𝑄 would be positive definite
  • Solving for 𝑃 given 𝑄 and 𝐴 for LTI systems is a well studied problem in linear algebra
    • Matlab’s lyap function can solve for 𝑃 given 𝑄 and 𝐴

Inconclusive Results

  • Now what if a particular 𝑃,𝑄 gives inconclusive results? i.e. you can find 𝑉(𝐱,𝑡)=𝐱𝖳𝑃(𝑡)𝐱 for which
    • ̇𝑉=−𝛼(𝑡)‖𝐱(𝑡)‖22
    • but 𝛼(𝑡) is not ≥0 generally (maybe even 𝛼(𝑡)<0!)
  • Here you can:
    1. Try different 𝑃(𝑡),𝑄(𝑡)
    2. Try to investigate further, i.e.
      • ̇𝑉≤−𝛼(𝑡)‖𝐱(𝑡)‖22⇒̇𝑉≤(−𝛼(𝑡)𝜌)𝑉
      • (as in the proof of UES, where again 𝜌=max𝑡𝜆max(𝑃(𝑡)))
      • compare with ̇𝑤=−𝛼(𝑡)𝜌𝑤⇒𝑤(𝑡)=𝑤(𝑡0)𝑒−1𝜌∫𝑡𝑡0𝛼(𝜏)d𝜏
      • Then from the comparison lemma:
      • 𝑉(𝐱(𝑡),𝑡)≤𝑉(𝐱0,𝑡0)𝑒−1𝜌∫𝑡𝑡0𝛼(𝜏)d𝜏
      • and we can investigate the behavior of the integral to see if we can ensure boundedness or convergence
  • What about the nonlinear terms neglected by linearization? → Local Stability

Graph View

Backlinks