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Unformatted text preview: 13.2 The control system shown in fig P 13.2 contains threemode controller.
(a) For the closed loop, develop formulas for the natural period of oscillation
τ and the damping factor ξ in terms of the parameters K, τ D , τ I and τ 1 .
(b) Calculate ξ when K is 0.5 and when K is 2.
(c ) Do ξ & τ approach limiting values as K increases, and if so, what are
these values?
(d ) Determine the offset for a unit step change in load if K is 2.
(e ) Sktech the response curve (C vs t) for a unitstep change IN LOAD
WHEN k is 0.5 and when K is 2.
(f) In both cases of part (e) determine the max value of C and the time at
which it occurs. a) C
=
R 1
1 k 1 + τ D s +
τ 1s + 1 τIs 1+ k
1
1 + τ D s + τ 1s + 1 τIs 1
(τ 1 s + 1)
C
=
U 1 k 1 + τ D s + 1 1+ τ 1s + 1 τ I s τIs
= k
τ Iτ I 2 k + 1 s + τ I τ D +
τ I s + 1
k
k 1 k 1 + τ D s + τIs
C =
R 1 τ 1 s + 1 + k 1 + τ D s + τIs ( ) k τ Dτ I s 2 + τ I s + 1
C
=
R (kτ I τ D + τ 1τ I ) s 2 + (k + 1)τ I s + k τ2 = τ I (kτ D + τ I ) =2× =ξ = =T = k ;2τξ = τ I (kτ D + τ I )
k ξ= (k + 1)τ I
k (k + 1)τ I
k τI
(k + 1)
2
k (kτ D + τ 1 )
2π × 2τπ
1− ξ 2 = τ I (kτ D + τ 1 )
k 4k (kτ D + τ 1 ) − (k + 1) 2 τ D
2 k (kτ D + τ 1 ) T= 4π (kτ D + τ 1 ) τ
4k k D τ
I τ1
+ τ I − (k + 1) 2 B) τ D = τ I =1; τ 1 .=2
For k = 0.5 ; ξ =0.75
For k = 2 ; ξ =1.5 1
= 0.671
0.5(2.5) 1
= 0.530
2×3
2 1
1 + τ I
2
1 (k + 1) τ I
1 k
C) = ξ =
=
τ
2 k (kτ D + τ 1 ) 2 τ D + I k ...
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This note was uploaded on 11/13/2011 for the course COP 4355 taught by Professor Koslov during the Spring '10 term at University of Florida.
 Spring '10
 Koslov

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