Control ENG HW_Part_52

Control ENG HW_Part_52 - 13.2 The control system shown in...

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Unformatted text preview: 13.2 The control system shown in fig P 13.2 contains three-mode 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 unit-step 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.

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Control ENG HW_Part_52 - 13.2 The control system shown in...

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