compensator7 - 29 29 29 3 2 1 8 15 5 50 C KG s G s s s K s...

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Compensators Q UESTION 7 For the closed–loop system in Figure 7 with ( 29 1 = s G C and ( 29 ( 29 ( 29 ( 29 5 10 3 5 s G s s s s + = + + , complete the following: a. Sketch the Root Locus Diagram. Qualitatively describe the closed–loop system stability and transient characteristics for 0 < K < ∞. b. For K = 12 and r ( t ) = 0.8 t , calculate the steady–state error. c. For K = 12 and r ( t ) = 0.8 t , simulate the closed–loop system and plot y ( t ), e ( t ), and u ( t ). Turn in your Matlab code. G(s) KG C (s) U(s) R(s) + - Y(s) E(s) Figure 7 There is m = 1 finite open–loop zero at –10 There are n = 3 finite open–loop poles at 0, –3, and –5 There are n = 3 branches and n m = 2 branches go to infinity The Root Locus is on the real axis between 0 and –3 and between –5 and –10 The asymptote real–axis intercept is [ ] [ ] 0 3 5 10 1 a n m σ - - - - = = - The asymptote angles are [ ] 0 2 1 180 0, 1, 2, a i i n m θ + = = ± ± - K There are n m = 2 distinct asymptotes with angles 0 0 90 , 90 a = -
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Compensators The closed–loop characteristic equation is
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Unformatted text preview: ( 29 ( 29 ( 29 3 2 1 8 15 5 50 C KG s G s s s K s K + = → + + + + = Substituting s = jω into the closed–loop characteristic equation and rearranging 2 2 50 8 15 5 K K j ϖ -+ +-= Setting the real and imaginary parts equal to zero and solving simultaneously, K = 12. Therefore, the closed–loop system is stable for 0 < K < 12. The closed–loop system is underdamped for all K . The 2% settling time decreases as K increases and approaches ∞ as K approaches 12. Since the closed–loop system is marginally stable for K = 12, the steady–state error cannot be calculated. The Root Locus Diagram and simulation plots are given below. 5 2 4 6 time (s) r y 5-0.2 0.2 time (s) e(t) 5-1 1 2 time (s) u(t)-15-10-5 5-50 50 2...
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This note was uploaded on 02/01/2012 for the course MECH ENG 279 taught by Professor Landers during the Spring '11 term at Missouri S&T.

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compensator7 - 29 29 29 3 2 1 8 15 5 50 C KG s G s s s K s...

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