This preview shows page 1. Sign up to view the full content.
Unformatted text preview: lity of the system is determined by the signs on ζ ,τ • “Critically damped” If ζ 2 < 1 , these roots are complex, and they will be complex conjugates of each other. • The stability of the system is determined by the signs on ζ ,τ Re( s) = •
• −ζ
τ Imaginary part leads to oscillations “Underdamped” If τ > 0,ζ > 0 , then the second
order linear system is stable. Example: Impulse response for 0 < ζ ≤ 1 : Like in Lab 1, dump in the water quickly. K
U ( s)
τ 2 s 2 + 2ζτ s + 1
U ( s) = 1
K
X ( s) = 2 2
τ s + 2ζτ s + 1
X ( s) = Using Line 19 of Table 3.1, x (t ) = ⎛
t⎞
e−ζ t /τ sin ⎜ 1 − ζ 2 ⎟ τ⎠
⎝
τ 1− ζ
K 2 Point out features of exponential growth or decay of a sinusoid. Now consider the state space version: d 2 x 2ζ dx 1
K
+
+ 2 x = 2 u(t ), x (0) = 0, x '(0) = 0 2
τ dt τ
dt
τ First we must rewrite this second
order ODE as a system of first
order ODEs. Need to define new variables z1 , z2 . Goal, get rid of old x, use new z1 , z2 . z1 ≡ x , z2 ≡ dx
dt dz1
= z2
dt
dz2
2ζ
1
K
= − z2 − 2 z1 + 2 u
dt...
View
Full
Document
This note was uploaded on 04/09/2014 for the course CHBE 4400 taught by Professor Staff during the Spring '08 term at Georgia Tech.
 Spring '08
 Staff

Click to edit the document details