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hw4soln_f09

# hw4soln_f09 - ME 132 Fall 2009 Solutions to Homework 4...

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ME 132 Fall 2009 Solutions to Homework 4 1. ( § 3.3, Problem 13) (a) The steady state gains are γ u y = G = cb 2 a γ 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 < 0 The closed-loop steady state gain from r y is γ r y = C 1 B 1 A + D 11 = cb 2 ( K 1 + K 2 ) b 2 K 2 c a The closed-loop steady state gain from d y is γ d y = C 1 B 2 A + D 12 = cb 1 b 2 K 2 c a 1

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(d) i. All of the closed-loop steady state gains are given below: γ r y = C 1 B 1 A + D 11 = cb 2 K 1 a = cb 2 1 G a = 1 γ d y = C 1 B 2 A + D 12 = cb 1 a γ n y = C 1 B 3 A + D 13 = cb 2 K 2 a = 0 γ r u = C 2 B 1 A + D 21 = cK 2 b 2 ( K 1 + K 2 ) a + K 1 + K 2 = 1 G γ d u = C 2 B 2 A + D 22 = cK 2 b 1 a = 0 γ 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 γ r y b 2 := ∂γ r y ∂b 2 b 2 γ r y = ∂b 2 parenleftbigg cb 2 K 1 a parenrightbigg b 2 γ 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 “feed-forward” controller’s 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 γ r y negationslash = 1. iv. The open-loop steady state gain from d y is the same as the closed-loop steady state gain from d y . This means that there is no disturbance rejection for the feed-forward 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. vi. The steady state gain from n to both y and u are 0. This makes sense because the noise is modeled as entering the system through the sensor, which is inconsequential because K 2 = 0.
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hw4soln_f09 - ME 132 Fall 2009 Solutions to Homework 4...

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