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Unformatted text preview: ME 132 Fall 2009 Solutions to Homework 4 1. ( 3.3, Problem 13) (a) The steady state gains are u y = G = cb 2 a d y = H = cb 1 a (b) By plugging in the controller and output equations, we see that x = ax + b 1 d + b 2 [( K 1 + K 2 ) r K 2 y K 2 n ] = ( a b 2 K 2 c ) x + b 2 ( K 1 + K 2 ) r + b 1 d b 2 K 2 n so the constants for the closed loop dynamics are A = a b 2 K 2 c B 1 = b 2 ( K 1 + K 2 ) B 2 = b 1 B 3 = b 2 K 2 For the output equations, the constants can be found by inspection: C 1 = c D 11 = D 12 = D 13 = 0 C 2 = cK 2 D 21 = K 1 + K 2 D 22 = 0 D 23 = K 2 (c) The system is stable if and only if A < 0, or a b 2 K 2 c < The closedloop steady state gain from r y is r y = C 1 B 1 A + D 11 = cb 2 ( K 1 + K 2 ) b 2 K 2 c a The closedloop steady state gain from d y is d y = C 1 B 2 A + D 12 = cb 1 b 2 K 2 c a 1 (d) i. All of the closedloop steady state gains are given below: r y = C 1 B 1 A + D 11 = cb 2 K 1 a = cb 2 1 G a = 1 d y = C 1 B 2 A + D 12 = cb 1 a n y = C 1 B 3 A + D 13 = cb 2 K 2 a = 0 r u = C 2 B 1 A + D 21 = cK 2 b 2 ( K 1 + K 2 ) a + K 1 + K 2 = 1 G d u = C 2 B 2 A + D 22 = cK 2 b 1 a = 0 n u = C 2 B 3 A + D 23 = cb 2 K 2 2 a K 2 = 0 ii. The steady state gain from r y is 1. This means that, in the absence of disturbances and noise, the controller achieves perfect tracking of the reference signal in the steady state. iii. Treating K 1 as a fixed number and recalling the definition of sensitivity, S r y b 2 := r y b 2 b 2 r y = b 2 parenleftbigg cb 2 K 1 a parenrightbigg b 2 r y = cK 1 b 2 a a cK 1 b 2 = 1 Sensitivity ranges from 0 (least sensitive) to 1 (most sensitive). This result implies that the feedforward controllers performance in tracking r is ex tremely sensitive to parameter uncertainty in the plant. This is easily seen from letting K 1 = a c b 2 for b 2 negationslash = b 2 , which causes r y negationslash = 1. iv. The openloop steady state gain from d y is the same as the closedloop steady state gain from d y . This means that there is no disturbance rejection for the feedforward controller. Intuitively, this makes perfect sense because the controller has no measurements of the output, so there is no way of detecting an unknown disturbance. v. The steady state gain from d u is 0. The fact that the controller is not compensating for the disturbance makes the result in the previous part un surprising....
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 Spring '08
 Tomizuka

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