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Unformatted text preview: τ
τ
τ
z1 (0) = 0, z2 (0) = 0 Class activity: Find A and B. ⎡z ⎤
1
⎥ . Then dx = Ax + Bu , where dt
⎢ z2 ⎥
⎣
⎦ Define x = ⎢ ⎡0
⎢
A= ⎢
1
−
⎢ τ2
⎣ 1
2ζ
−
τ ⎡0⎤
⎢
⎥
B=⎢ K ⎥
⎢ τ2 ⎥
⎣
⎦ ⎤
⎥
⎥
⎥
⎦ Now we can check the stability of the system by computing the eigenvalues of A. Here we use s for the eigenvalue, although λ could be used instead.  sI − A = 0
⎡
1⎤
s
⎡1 0⎤ ⎢ 0
⎥
s⎢
1
2ζ ⎥ = 1
⎥−⎢
⎣ 0 1 ⎦ ⎢ −τ2 − τ ⎥
τ2
⎣
⎦
⎛
2ζ ⎞ 1
2ζ
1
s ⎜ s + ⎟ + 2 = s2 +
s+ 2 = 0
τ⎠ τ
τ
⎝
τ −1
2ζ
s+
τ =0 τ 2 s 2 + 2ζτ s + 1 = 0 Note: This is the same polynomial that we had before, so we can use the same quadratic formula and arguments as above to determine the stability. Can use Laplace to solve the state
space system. (Or can use Runge
Kutta as we did before in Simulink.) Will need this for Lab 4. dx
= Ax + Bu
dt
y = Cx + Du
sX ( s) = AX ( s)...
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This note was uploaded on 04/09/2014 for the course CHBE 4400 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff

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