Vai's Uni Notes

Vector and Matrix Norms

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The “size” measures used to generalize LTI/SISO pole conditions to stability of vector state equations.

Vector Norms

  • To generalize ideas like ”𝑦ℎ(𝑡)→0 for all ICs ⇔ all poles in LHP” to vector functions, we’ll use vector norms:
    • ‖𝐱‖ ∼ a measure of the geometric length of a vector
  • Many different metrics are possible; all must satisfy:
    • ‖𝐱‖≥0
    • ‖𝐱‖=0⇔𝐱=𝟎
    • ‖𝛼𝐱‖=|𝛼|‖𝐱‖
    • ‖𝐱+𝐲‖≤‖𝐱‖+‖𝐲‖ (triangle inequality)

Common Norms

  • Euclidean / 2-norm:
    • ‖𝐱‖2=[∑𝑛𝑘=1𝑥2𝑘]1/2
  • Also common are:
    • 1-norm:
      • ‖𝐱‖1=∑𝑛𝑘=1|𝑥𝑘|
    • ∞-norm:
      • ‖𝐱‖∞=max1≤𝑘≤𝑛|𝑥𝑘|
    • p-norm:
      • ‖𝐱‖𝑝=[∑𝑛𝑘=1|𝑥𝑘|𝑝]1/𝑝𝑝integer≥1

Norm Equivalence Theorem

  • For every choice of 𝑝,𝑞 (integers ≥1) there exists 𝑐𝑝𝑞>0 and 𝑐𝑝𝑞<∞ so that
  • 𝑐𝑝𝑞‖𝐱‖𝑞≤‖𝐱‖𝑝≤𝑐𝑝𝑞‖𝐱‖𝑞

Matrix Induced Norms

  • Matrix induced norms:
    • ‖𝐴‖𝑖≝max‖𝐱‖≠0‖𝐴𝐱‖‖𝐱‖
  • ⇒‖𝐴𝐱‖≤‖𝐴‖𝑖‖𝐱‖

Computing Induced Norms

  • Calculation of ‖𝐴‖𝑖 depends on the choice of vector norm:
    • ‖𝐴‖𝑖,2=√𝜆max(𝐴𝖳𝐴)‖𝐴‖𝑖,1=max𝑗∑𝑛𝑖=1|𝑎𝑖𝑗|(max abs. column sum)‖𝐴‖𝑖,∞=max𝑖∑𝑛𝑗=1|𝑎𝑖𝑗|(max abs. row sum)

Properties

  • Any induced norm satisfies:
    • ‖𝐴+𝐵‖𝑖≤‖𝐴‖𝑖+‖𝐵‖𝑖
    • ‖𝐴𝐵‖𝑖≤‖𝐴‖𝑖‖𝐵‖𝑖
  • Use these notions to describe stability / boundedness / convergence, first of the state eq’n (Internal Stability), then of input/output behavior

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