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b) Can a P controller C(s) = Kp stabilize the plant G(s)? If so, find the values of Kp that are necessary and sufficient. c) Show using the Final Value Theorem that the system with the P controller from (b) can track a unit-step reference r(t) = 1 with zero steady-state error lim-+00 e(t) =0. %3D %3D %3D

b) Can a P controller C(s) = Kp stabilize the plant G(s)? If so, find the values of Kp that are necessary and sufficient. c) Show using the Final Value Theorem that the system with the P controller from (b) can track a unit-step reference r(t) = 1 with zero steady-state error lim-+00 e(t) =0. %3D %3D %3D

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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Only parts B and C. No matlab

316 and controller C(s), as shown in the 3. Consider a unity-feedback control system with plant G(s) following figure. Reference Error Controller Plant r(t) e(t) u(t) y(1) C(s) G(s) (a) Determine the poles, zeros, order, type, relative degree, and de gain of the plant G(s) and show using the Routh-Hurwitz criterion that G(s) is not stable. = Kp stabilize the plant G(s)? If so, find the values of Kp that are (b) Can a P controller C(s) necessary and sufficient. (c) Show using the Final Value Theorem that the system with the P controller from (b) can track a unit-step reference r(t) = 1 with zero steady-state error limo e(t) = 0. (d) Show that it however cannot track a unit-ramp reference r(t) =t with zero steady-state error. Can the error be made arbitrarily small with Kp without losing stability? (e) Can a PI controller C(s) = Kp + AL stabilize the plant G(8) and, at the same time, yield zero steady-state error to both unit-step and unit-ramp references? If so, find the values of Kp and Ki that are necessary and sufficient. Reconsider the unity-feedback control system shown in Problem 3 and let the controller C(s) and plant G(s) be defined as follows. (a) Let C(s) = Kp and G(s) = : Find the value of Kp that yields 10% of overshoot in y(t) when r(t) is a unit-step input. Hint: 2nd-order system. (b) Let C(s) = Kp and G(s) = S+21ASL5): Sketch the root locus of the open-loop transfer function C(s)G(s) by hand and using MATLAB's rlocus. Derive a condition that Kp must satisfy in order for the closed-loop system to be asymptotically stable. (s+3) %3D
Transcribed Image Text:316 and controller C(s), as shown in the 3. Consider a unity-feedback control system with plant G(s) following figure. Reference Error Controller Plant r(t) e(t) u(t) y(1) C(s) G(s) (a) Determine the poles, zeros, order, type, relative degree, and de gain of the plant G(s) and show using the Routh-Hurwitz criterion that G(s) is not stable. = Kp stabilize the plant G(s)? If so, find the values of Kp that are (b) Can a P controller C(s) necessary and sufficient. (c) Show using the Final Value Theorem that the system with the P controller from (b) can track a unit-step reference r(t) = 1 with zero steady-state error limo e(t) = 0. (d) Show that it however cannot track a unit-ramp reference r(t) =t with zero steady-state error. Can the error be made arbitrarily small with Kp without losing stability? (e) Can a PI controller C(s) = Kp + AL stabilize the plant G(8) and, at the same time, yield zero steady-state error to both unit-step and unit-ramp references? If so, find the values of Kp and Ki that are necessary and sufficient. Reconsider the unity-feedback control system shown in Problem 3 and let the controller C(s) and plant G(s) be defined as follows. (a) Let C(s) = Kp and G(s) = : Find the value of Kp that yields 10% of overshoot in y(t) when r(t) is a unit-step input. Hint: 2nd-order system. (b) Let C(s) = Kp and G(s) = S+21ASL5): Sketch the root locus of the open-loop transfer function C(s)G(s) by hand and using MATLAB's rlocus. Derive a condition that Kp must satisfy in order for the closed-loop system to be asymptotically stable. (s+3) %3D
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