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final04_sol

final04_sol - ECE 147A FEEDBACK CONTROL SYSTEMS THEORY AND...

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ECE 147A FEEDBACK CONTROL SYSTEMS - THEORY AND DESIGN F04 Final Exam Closed book and notes. No calculators. Show all work. Name: Solution Problem 1: /20 Problem 2: /20 Problem 3: /20 Problem 4: /20 Problem 5: /20 Total: /100 Miscellaneous: tan - 1 (1 / 3) 18 degrees, tan - 1 (1) = 45 degrees. tan - 1 (2) 63 degrees, tan - 1 (5) 78 degrees, tan - 1 (10) 84 degrees, tan - 1 (20) 87 degrees, sin(45 ) = 2 / 2 0 . 7, sin(50 ) 0 . 77, sin(55 ) 0 . 82, sin(60 ) = 3 / 2 0 . 87.

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Name: Problem 1 1. For the unity negative feedback configuration, describe the three properties that typically characterize a good choice for the function ω 7→ | C ( ω ) P ( ω ) | and explain why these three properties are important. (a) The function should have large values for ω small. This is important for good disturbance attenuation and reference tracking. (b) The function should have small values for ω large. This is important to keep measurement noise from affecting the feedback system significantly, and it is important for stability robustness with respect to unmodeled dynamics. (c) The function should decrease slowly in the region where its magnitude is near one. This is important to guarantee that the unity feedback configuration will be stable. 2. What are the typical motivations for using a pole at the origin in a feedback controller? A pole at the origin in the feedback controller is used to guarantee that the output of the plant exactly reproduces step references asymptotically, even in the presence of constant disturbances at the plant input and unmodeled or uncertain plant dynamics. 3. The loop gain for a given unity negative feedback system is shown as the solid curve in the figure below. (The dashed curve is a circle of radius one.) (a) Label on the plot the crossover frequency of the open-loop. The crossover frequency corresponds to the frequency at which the solid curve pierces the dashed circle. (b) Estimate the phase margin. The phase margin is given by the angle on the dashed circle between the point ( - 1 , 0) and the point where the solid curve pierces the dashed circle. It is roughly 30 degrees. (c) Estimate the gain margin. It is about 2, since the gain can be increased by 2 before the number of encirclements of ( - 1 , 0) changes. (d) Label on the plot the frequency range over which the feedback system attenuates the effect of input additive disturbances. Explain your answer. To determine this frequency range, we should draw another circle of radius one, this one centered at ( - 1 , 0) . Disturbance attenuation occurs at the frequnecies for which the solid curve is outside of this new circle. This is because disturbance attenuation occurs where | S ( ω ) | < 1 , i.e., | C ( ω ) P ( ω ) + 1 | > 1 , i.e., | C ( ω ) P ( ω ) - ( - 1 , 0) | > 1 . The latter condition is exactly the condition that C ( ω ) P ( ω ) is outside of this new circle.
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final04_sol - ECE 147A FEEDBACK CONTROL SYSTEMS THEORY AND...

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