Select one basis for each and put them together as

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Select one basis for each, and put them together as the basis for C n ~ Could be tricky One can then obtain the desired canonical form through similar transformation ~ Not an easy task!
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11/9/15, Lecture 9 ECE5101, Fall 2015, © P. B. Luh 19 Example. For the following system, find its canonical decomposition into four subsystems [ ] x 1 0 1 0 y ; u 0 0 1 2 x 0 1 0 0 1 2 0 0 2 2 1 0 2 2 1 1 x = Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ + Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = ! [ ] B A B A AB B C 3 2 = Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 0 0 0 0 0 0 0 0 1 1 1 1 5 4 3 2 Controllable subspace: = Range (C) S c Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 0 0 1 0 , 0 0 0 1 Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 0 0 1 3 , 0 0 1 2 ~ Subspace spanned by
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11/9/15, Lecture 9 ECE5101, Fall 2015, © P. B. Luh 20 Uncontrollable subspace: c S 1 0 0 0 , 0 1 0 0 Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = = Subspace orthogonal to S c Observable subspace: Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 2 1 1 0 , 1 0 1 0 = Range (O * ) S o Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 3 2 CA CA CA C O Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 4 3 1 0 3 2 1 0 2 1 1 0 1 0 1 0
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11/9/15, Lecture 9 ECE5101, Fall 2015, © P. B. Luh 21 Unobservable subspace o S 1 1 1 0 , 0 0 0 1 Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = = Subspace orthogonal to S o ; 2 1 1 0 , 1 0 1 0 S o Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ = 1 1 1 0 , 0 0 0 1 S o ; 0 0 1 0 , 0 0 0 1 S c Ϊ Ϊ έ Ϊ Ϊ ά Ϋ Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ Υ Υ Υ Υ Φ Τ ΢ ΢ ΢ ΢ Σ Ρ Υ Υ Υ Υ Φ Τ
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  • Fall '15
  • Richard Osborne
  • LTI system theory, P. B. Luh

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