Difference Equations and Stability

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Difference Equations and Stability
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Example Second-Order Equation

• yn2 - 0.6 yn1 - 0.16 yn 5 xn2
  withy-1 0 and y-2 6.25 and xn 4-n
  un
• Zero-input response
• Characteristic polynomial g2 - 0.6 g - 0.16 (g
  0.2) (g - 0.8)
• Characteristic equation
  (g 0.2) (g - 0.8) 0
• Characteristic roots
  g1 -0.2 and g2 0.8
• Solution
  y0n C1 (-0.2)n C2 (0.8)n
• Zero-state response

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Example Impulse Response

• hn2 - 0.6 hn1 - 0.16 hn 5 dn2with
  h-1 h-2 0 because of causality
• General form of impulse response
• hn (bN/aN) dn y0n un
• Since aN -0.16 and bN 0,
• hn y0n un C1 (-0.2)n C2 (0.8)n un
• Discrete-time version of slides 5-9 and 5-10

balances impulsive events at origin
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Example Impulse Response

• Need two values of hn to solve for C1 and C2
• h0 - 0.6 h-1 - 0.16 h-2 5 d0 ? h0 5
• h1 - 0.6 h0 - 0.16 h-1 5 d1 ? h1 3
• Solving for C1 and C2
• h0 C1 C2 5
• h1 -0.2 C1 0.8 C2 3 ? C1
  1, C2 4
• hn (-0.2)n 4 (0.8)n un
• Usefulresult

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Example Solution

• Zero-state response solution
• ysn hn xn (-0.2)n 4(0.8)n un
  (4-n un)
• ysn -1.26 (4)-n 0.444 (-0.2)n 5.81
  (0.8)n un
• Total response yn y0n ysn
• yn C1(-0.2)n C2(0.8)n -1.26
  (4)-n 0.444 (-0.2)n 5.81 (0.8)n un
• With y-1 0 and y-2 6.25
• y-1 C1 (-5) C2(1.25) 0
• y-2 C1(25) C2(25/16) 6.25
• Solution C1 0.2, C2 0.8

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Repeated Roots

• For r repeated roots of Q(g) 0
• y0n (C1 C2 n Cr nr-1) gn
• Similar to continuous-time case

Continuous Time DiscreteTime Case
non-repeated roots
repeated roots
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Stability of Zero-Input Response

• Asymptotically stable if andonly if all
  characteristic rootsare inside unit circle.
• Unstable if and only if one orboth of these
  conditions exist
• At least one root outside unit circle
• Repeated roots on unit circle
• Marginally stable if and only if no roots are
  outside unit circle and no repeated roots are on
  unit circle

Discrete-Time Systems
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Stability of Zero-Input Response
Continuous-Time Systems
Discrete-Time Systems
Marginally stable non-repeated characteristic
roots on the unit circle (discrete-time systems)
or imaginary axis (continuous-time systems)