Unformatted text preview: 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: 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 Class activity: Find A and B. A=
⎡z ⎤
dx
Define x = ⎢ 1 ⎥ . Then = Ax + Bu , where dt
⎢ z2 ⎥
B=
⎣
⎦ 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 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.) dx
= Ax + Bu
dt
y = Cx + Du This is all the same as before, but need to use matrix operations for inverse. ( sI − A) X ( s) = BU ( s) G ( s) = Y ( s)
=
U ( s) So ( sI − A) must be invertible if G(s) is to be bounded. This is the same criterion as computing the eigenvalues above. Recap on second
order systems: A. Oscillations • ζ > 0 , overdamped, no natural oscillation • ζ = 1 , critically damped, no natural oscillation, repeated roots • 0 < ζ ≤ 1 , underdamped, sines and cosines in the solution B. Stability: Determined by Re(s). If Re(s) > 0 for any root s, then unstable. Else stable. Class activity: Step response of second order system. Inspect the solution from Line 22 of Table 3.1. Discuss what happens as (divide class up by rows) ζ →0
ζ →1
τ →0
τ →∞ Qualitative features of an underdamped process. Can use these as a graphical approach to empirical modeling....
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 Spring '08
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