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# ps8 - EE 428 PROBLEM SET 8 DUE 10 December 2007 Reading...

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EE 428 PROBLEM SET 8 DUE: 10 December 2007 Reading assignment: Ch 6, sections 6.1 through 6.4 Laboratory sections meet during the week of December 3. Problem 34: (17 points) The closed-loop system in Figure 1 contains a plant with transfer function G p ( s ) = 100 ( s + 1)( s 2 + 10 s + 100) and uses proportional control with G c ( s ) = K, where K 0. In this problem you will use the Routh-Hurwitz criterion and the concept of gain margin to determine the range of K for which the closed-loop system is stable. Figure 1: Feedback control system using cascade compensation. 1. (3 points) Using the Routh-Hurwitz criterion, determine the largest positive gain K for which the closed-loop system is stable. 2. (6 points) Using MATLAB, generate the Bode magnitude and phase plots of the open-loop transfer function B ( s ) E ( s ) = G c ( s ) G p ( s ) for unity gain feedback, G c ( s ) = K = 1. 3. (2 points) Using the Bode phase plot, determine the frequency at which the phase of the open-loop transfer function is -180 . This point is called the phase crossover frequency and is denoted by ω pc . 4. (2 points) The gain margin GM is the number of decibels which must be added to the magnitude curve in order to make | G c ( ω pc ) G p ( ω pc ) | = 0 dB. With K = 1, determine the gain margin of the system shown in Figure 1. 5. (2 points) Verify your answers in parts (4) and (5) using the MATLAB command margin(num,den) , where num and den are the numerator and denominator polynomials of G c ( s ) G p ( s ), respectively. 6. (2 points) Using the gain margin measured in part (5), determine the largest positive value of the gain K for which the system is closed-loop stable. Compare your result to that obtained using the Routh-Hurwitz criterion.

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Problem 35: (17 points) In this problem you will investigate the effect of a time delay on the stability of a closed-loop system. Time delays occur frequently in industrial applications. For example, in temperature control systems the control engineer must
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