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Unformatted text preview: 2.14/2.140 Problem Set 4 Assigned: Thurs. March 8, 2007 Due: Thurs. March 15, 2007, in class Reading: Nise Chapter 8; Notes Chapter 3 on root locus. Reading for 2.140 students: The following problems are assigned to both 2.14 and 2.140 students. Problem 1 Nise Ch. 8, Problem 2 Problem 2 The four plots below show the pole and zero locations of the loop transmission of a feedback system, informally called the open-loop poles and zeros. Each of these loops also have a variable gain K > 0, which is used to move the poles along the root locus branches. For the four systems shown below, sketch the approximate shape of the root locus plot for K > 0. Note that you will need to pay particular attention to the angle criteria in the vicinity of the complex poles and zeros. If the complex pairs are lightly damped, which of these systems presents a danger of instability as the loop gain is varied? This analysis has practical relevance for the situation where a notch filter is used to help stabilize a system with a lightly-damped pair of poles. 1 Problem 3 The block diagram for a feedback loop has a forward path transfer function G ( s ) = Ka ( s ) /b ( s ), and a feedback path transfer function H ( s ) = c ( s ) /d ( s ) as shown below. Prove that the closed-loop zeros are located at: 1) the zeros of the forward path and 2) the poles of the feedback path, independent of the loop gain K . Problem 4 This problem considers the system a) Derive the transfer function X ( s ) in terms of c and k. For this part of the question, ignore F d . F ( s ) What is the time constant, , of the system?...
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- Spring '07