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Problem 1. For the feedback control system given, where R(s) C(s) G.(s) G,(s) 1 G.(s) = K(s+ 10) and Gp(s) = s+ 1)(s+ 2)(s + 6) (a) Sketch the root locus. When sketching the root locus, if necessary, make use of the asymptotes finding ơa and 0a that are the intersecting point and angles with the real axis, respectively, with the following formula, E finite poles-E finite zeros #finite poles-#finite pzeros and Oa = (2k+1)n #finite poles-#finite pzeros' ,where k = 0,±1,±2, ... %3D (b) If applies, what are the values of the gain and poles at the imaginary axis crossings? (c) Write the range of the gain making the system stable?

Problem 1. For the feedback control system given, where R(s) C(s) G.(s) G,(s) 1 G.(s) = K(s+ 10) and Gp(s) = s+ 1)(s+ 2)(s + 6) (a) Sketch the root locus. When sketching the root locus, if necessary, make use of the asymptotes finding ơa and 0a that are the intersecting point and angles with the real axis, respectively, with the following formula, E finite poles-E finite zeros #finite poles-#finite pzeros and Oa = (2k+1)n #finite poles-#finite pzeros' ,where k = 0,±1,±2, ... %3D (b) If applies, what are the values of the gain and poles at the imaginary axis crossings? (c) Write the range of the gain making the system stable?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Problem 1. For the feedback control system given, where R(s) C(s) Gc(s) G,(s) 1 G.(s) = K(s+ 10) and Gp(s) (s + 1)(s + 2)(s + 6) (a) Sketch the root locus. When sketching the root locus, if necessary, make use of the asymptotes finding ơa and 0a that are the intersecting point and angles with the real axis, respectively, with the following formula, E finite poles-E finite zeros O, = #finite poles-#finite pzeros (2k+1)n #finite poles-# finite pzeros' and 0a = where k = 0,±1,±2, ... (b) If applies, what are the values of the gain and poles at the imaginary axis crossings? (c) Write the range of the gain making the system stable? (d) It is determined that the dominant poles are at s12 = -1±j3.5 when the gain is K = 8. Find the location of the third pole for this value of gain. (e) What is the system type? (1) Find the steady-state errors for the same gain, K = 8, when r(t) = 0.6u(t) step and 0.6tu(t) ramp inputs are applied, respectively. (g) In order to improve the steady state response of the system add a PI controller having the same gain and a zero of s = -0.2. Now find the steady-state errors for the same inputs.
Transcribed Image Text:Problem 1. For the feedback control system given, where R(s) C(s) Gc(s) G,(s) 1 G.(s) = K(s+ 10) and Gp(s) (s + 1)(s + 2)(s + 6) (a) Sketch the root locus. When sketching the root locus, if necessary, make use of the asymptotes finding ơa and 0a that are the intersecting point and angles with the real axis, respectively, with the following formula, E finite poles-E finite zeros O, = #finite poles-#finite pzeros (2k+1)n #finite poles-# finite pzeros' and 0a = where k = 0,±1,±2, ... (b) If applies, what are the values of the gain and poles at the imaginary axis crossings? (c) Write the range of the gain making the system stable? (d) It is determined that the dominant poles are at s12 = -1±j3.5 when the gain is K = 8. Find the location of the third pole for this value of gain. (e) What is the system type? (1) Find the steady-state errors for the same gain, K = 8, when r(t) = 0.6u(t) step and 0.6tu(t) ramp inputs are applied, respectively. (g) In order to improve the steady state response of the system add a PI controller having the same gain and a zero of s = -0.2. Now find the steady-state errors for the same inputs.
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