Vai's Uni Notes

Internal Stability

6 min read

Stability and convergence of the homogeneous (“internal”) solutions 𝐱(𝑡)=Φ(𝑡,𝑡0)𝐱0, stated with norms.

Motivation

  • Stability and convergence of homogeneous solutions
    • 𝐱(𝑡)=Φ(𝑡,𝑡0)𝐱0
  • In simple LTI/SISO, the condition is
    • 𝑦ℎ(𝑡)→0 for all ICs ⇔ all poles in LHP
  • To generalize such ideas to vector functions, we’ll use vector norms
  • Use these notions to describe stability / boundedness / convergence, first of the state eq’n, then of input/output behavior
    • Example: ∃𝐵<∞ such that ‖𝐱(𝑡)‖≤𝐵<∞ ∀𝑡≥𝑡0
    • or ‖𝐱(𝑡)‖→0 as 𝑡→∞

Stability in the Sense of Lyapunov

  • More formally, a system is stable i.s.L. (“in the sense of Lyapunov”) if ∃𝛾<∞ such that
    • ‖𝐱(𝑡)‖≤𝛾‖𝐱0‖∀𝑡≥𝑡0
  • Note
    • ‖𝐱(𝑡)‖=‖Φ(𝑡,𝑡0)𝐱0‖≤‖Φ(𝑡,𝑡0)‖𝑖‖𝐱0‖
    • ⇒ the system is stable i.s.L. if
    • ‖Φ(𝑡,𝑡0)‖𝑖≤𝛾<∞∀𝑡
    • i.e. ‖Φ(𝑡,𝑡0)‖𝑖 is bounded
  • The system is asymptotically stable if ‖𝐱(𝑡)‖→0 as 𝑡→∞
    • ⇒‖Φ(𝑡,𝑡0)‖𝑖→0 as 𝑡→∞
  • How to deduce whether Φ has these properties if we don’t know exactly what Φ looks like? → Lyapunov Functions

Formal Definitions

Stability and Uniformity

  • Stable (i.s.L.): if ∃𝛾(𝑡0)<∞ so that
    • ‖𝐱(𝑡)‖≤𝛾(𝑡0)‖𝐱0‖∀𝑡≥𝑡0
  • If we can find a single 𝛾, independent of 𝑡0, that works for all 𝑡0, the stability is said to be “uniform”

Example where γ depends on t₀

  • ̇𝑥(𝑡)=𝑎(𝑡)𝑥(𝑡)𝑎(𝑡)=4𝑡sin2𝑡−2𝑡
  • With 𝑡0=2𝑘𝜋 (𝑘=0,1,2,…), the solution for 𝑥(𝑡) reaches a max at 𝑡=𝑡0+𝜋, for which
    • |𝑥(𝑡0+𝜋)||𝑥0|=exp[(2𝑡0+𝜋)(4−𝜋)]⏟𝛾(𝑡0)
  • 𝛾 grows without bound as 𝑡0 increases, so this stability cannot be uniform

Possible slip in notes

  • The coefficient reads 𝑎(𝑡)=4𝑡sin2𝑡−2𝑡, but the stated ratio exp[(2𝑡0+𝜋)(4−𝜋)] is what ∫𝑡0+𝜋𝑡0𝑎(𝜏)d𝜏 gives for 𝑎(𝑡)=4𝑡sin𝑡−2𝑡, so the "2" inside the sine may be a slip

Asymptotic Stability

  • Asymptotic stability:
    1. stable i.s.L., and
    2. ‖𝐱(𝑡)‖→0 as 𝑡→∞
  • Why require condition 1 as well? Sketch: |𝑥(𝑡)| starting from 𝑥0 at 𝑡0 could blow up, 𝑥(𝑡)→∞ as 𝑡→𝑡esc (finite escape time), and then still decay to 0 afterward; condition 2 alone would not rule this out
  • Condition 2 can be expressed as: for any 𝜀>0, ∃𝑇(𝜀,𝑡0) so that
    • ‖𝐱(𝑡)‖≤𝜀‖𝐱0‖∀𝑡≥𝑡0+𝑇
  • If a single 𝑇 can be found that works for every 𝑡0, the system is said to be uniformly asymptotically stable (UAS)
    • Note: 𝑇 will still depend on 𝜀!

Asymptotically Stable but Not Uniformly

  • Linear system with fundamental solution 𝜓(𝑡)=11+𝑡 and STM
    • Φ(𝑡,𝑡0)=1+𝑡01+𝑡
  • so
    • 𝑥(𝑡)=[1+𝑡01+𝑡]𝑥0
  • The system is A.S. Is it UAS?
    • |𝑥(𝑡)||𝑥0|=1+𝑡01+𝑡
  • Suppose we take 𝜀=12. Then we need to find 𝑡 such that
    • 12≥1+𝑡01+𝑡
    • If 𝑡0=0⇒𝑡≥1=1+𝑡0⇒𝑇(12,0)=1
    • If 𝑡0=10: 1+101+𝑡=1+𝑡01+𝑡≤12⇒𝑡≥21=𝑡0+11, so 𝑇(12,10)=11
  • Not uniformly asymptotically stable
  • Generally here 𝑇 must satisfy
    • 1+𝑡01+𝑡0+𝑇≤𝜀
    • (𝑇 depends on both 𝜀 and 𝑡0!)

Exponential Stability

  • Exponentially stable: if ∃𝛾(𝑡0)<∞ and 𝜆(𝑡0)>0 such that
    • ‖𝐱(𝑡)‖≤𝛾𝑒−𝜆(𝑡−𝑡0)‖𝐱0‖∀𝑡≥𝑡0
    • Sketch: ‖𝐱(𝑡)‖ may wiggle up and down, but it stays under a decaying envelope ∼𝑒−𝜆(𝑡−𝑡0)
  • ES is uniform (UES) if both 𝛾 and 𝜆 are constants, independent of 𝑡0

UAS Implies UES for Linear Systems

  • Note: ES → AS, but the reverse is not generally true

Theorem

  • For linear systems it is true that UAS ⇒ UES
  • Proof: recall
    • 𝐱(𝑡)=Φ(𝑡,𝑡0)𝐱0⇒‖𝐱(𝑡)‖≤‖Φ(𝑡,𝑡0)‖𝑖‖𝐱0‖
  • UAS ⇒∃𝑇(𝜀)<∞ so that
    • ‖Φ(𝑡0+𝑇,𝑡0)‖𝑖≤𝜀for𝑎𝑛𝑦𝑡0
  • Chop time into intervals of length 𝑇: [𝑡0,𝑡0+𝑇], [𝑡0+𝑇,𝑡0+2𝑇], etc.
    1. For 𝑡∈[𝑡0,𝑡0+𝑇]:
      • ‖𝐱(𝑡)‖≤𝛾‖𝐱0‖
    2. For 𝑡∈[𝑡0+𝑇,𝑡0+2𝑇] (using the composition property):
      • 𝐱(𝑡)=Φ(𝑡,𝑡0)𝐱0=Φ(𝑡,𝑡0+𝑇)Φ(𝑡0+𝑇,𝑡0)𝐱0
      • ‖𝐱(𝑡)‖≤‖Φ(𝑡,𝑡0+𝑇)‖⏟𝛾‖Φ(𝑡0+𝑇,𝑡0)‖⏟𝜀‖𝐱0‖⇒‖𝐱(𝑡)‖≤𝛾𝜀‖𝐱0‖
    3. For 𝑡∈[𝑡0+2𝑇,𝑡0+3𝑇]:
      • 𝐱(𝑡)=Φ(𝑡,𝑡0+2𝑇)Φ(𝑡0+2𝑇,𝑡0+𝑇)Φ(𝑡0+𝑇,𝑡0)𝐱0
      • ‖𝐱(𝑡)‖≤𝛾𝜀⋅𝜀⋅‖𝐱0‖≤𝛾𝜀2‖𝐱0‖
    4. For 𝑡∈[𝑡0+𝑘𝑇,𝑡0+(𝑘+1)𝑇]:
      • ‖𝐱(𝑡)‖≤𝛾𝜀𝑘‖𝐱0‖
      • →0 as 𝑘→∞ if 𝜀<1
  • Say 𝜀=12 for illustration:
  • Normalized response ||x(t)||/||x0|| bounded by gamma, gamma/2, gamma/4, gamma/8 on successive intervals of length T, with the exponential envelope 2 gamma exp((ln(epsilon)/T)(t - t0)) through the corners (from Prof. Sanner's notes)
    • The staircase bounds 𝛾,𝛾𝜀=𝛾/2,𝛾𝜀2=𝛾/4,𝛾𝜀3=𝛾/8 lie under the exponential envelope 2𝛾𝑒(ln(𝜀)/𝑇)(𝑡−𝑡0), i.e. −𝜆=ln(𝜀)/𝑇
  • Thus UAS (equal contraction, 𝜀, on equal time intervals, 𝑇) ⇒

UAS ⇒ UES

  • ‖𝐱(𝑡)‖≤𝛾1𝑒−𝜆(𝑡−𝑡0)‖𝐱0‖with𝜆=−ln(𝜀)𝑇⇒UES
  • and 𝜀,𝑇(𝜀) describe the asymptotic stability properties of the system

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