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Three Extra Review Slides

# Three Extra Review Slides - 2 1 2 1 2 1 2 1 1 b b z b z c...

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1 2 0 1 2 [ ] [ 1] [ 2] [ ] [ 1] [ 2] y n a y n a y n b x n b x n b x n + - + - = + - + - Assume that the input is causal Assume you are given ICs: y [-1] & y [-2] 1 2 1 0 1 2 1 2 2 1 2 1 2 1 2 1 2 [ 1] [ 1] [ 2] ( ) ( ) 1 1 b b z b z a y a y z a y Y z X z a z a z a z a z - - - - - - - + + - - - - - - = + + + + + Full Solution for Second-order case Solution: H ( z ) Part of sol’n driven by IC “Zero-Input Sol’n” Part of sol’n driven by input “Zero-State Sol’n”

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1 2 1 0 1 2 1 2 1 2 1 2 1 2 ( ) ( ) 1 1 b b z b z c dz Y z X z a z a z a z a z - - - - - - - + + + = + + + + + Full Solution for Second-order case: Poles @ zero angle X X Im{z} “Zero-Input Sol’n” “Zero-State Sol’n” ( ) ( ) ( ) Az Bz Cz Dz E z Y z z a z b z a z b F z = + + + + - - - - “Zero-Input Sol’n” “Zero-State Sol’n” Convert to “z-form”… Do “store z” trick… then do PFE… then un-do “store z trick”: ae j0 = a, a 0 Transient Part Steady State Part [ ] [ ] [ ] [ ] [ ] [ ] n n n n ss y n Aa u n Bb u n Ca u n Db u n y n = + + + + “Zero-Input Sol’n” “Zero-State Sol’n” Do IZT via Table: Transient Part Steady State Part be j0 = b, b 0 Stable System : Poles inside Unit Circle ZI & Transient Decay n Decay, No Oscillation
1 2 1 0 1 2 1 2 1
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Unformatted text preview: 2 1 2 1 2 1 2 ( ) ( ) 1 1 b b z b z c dz Y z X z a z a z a z a z-------+ + + = + + + + + Full Solution for Second-order case: Poles @ non-zero angle X X Im{z} “Zero-Input Sol’n” “Zero-State Sol’n” * * ( ) ( ) ( ) o o o o j j j j Az A z Cz C z E z Y z z ae z ae z ae z ae F z Ω- Ω Ω- Ω = + + + +----“Zero-Input Sol’n” “Zero-State Sol’n” Convert to “z-form”… Do “store z” trick… then do PFE… then un-do “store z trick”: Transient Part Steady State Part [ ] cos( ) [ ] cos( ) [ ] [ ] n n o o ss y n Aa n u n Ca n u n y n φ θ = Ω + + Ω + + “Zero-Input Sol’n” “Zero-State Sol’n” Do IZT via Table then combine w/ Euler: Transient Part Steady State Part Stable System : Poles inside Unit Circle ZI & Transient Decay Ω – Ω , o j ae a- Ω , o j ae a Ω Decaying Oscillation...
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