(a) Verify that the closed-loop transfer function T (s) = C(s)/R(s) in terms of parameters α andβ, when expressed as a ratio of two polynomials in s, is T (s) = β/ s2 + (α + 2)s + 2α + β(b) The continuous-time, linear, time-invariant feedback control system pictured in Figure 1 consists of a(proportional) controller K(s) and a plant P(s) modeled by, respectively, the transfer functions1K(s) = Band P(s) =(s+2)(s+a)with control parameters a and 3 denoting adjustable real-valued gains.R(s)K(s)P(s)C'(s)Figure 1: A Continuous-Time, Linear, Time-Invariant Feedback Control System Verify that the closed-loop transfer function T(s) = C(s)/R(s) in terms of parameters a andpressed as a ratio of two polynomials in s, isT(s) =38² + (x+2)s +2a+B

Given that: The block diagram is shown below. A continuous time, linear, time-invariant feedback…

Answered: Verify that the closed-loop transfer…

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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(a) Verify that the closed-loop transfer function T (s) = C(s)/R(s) in terms of parameters α and
β, when expressed as a ratio of two polynomials in s, is
T (s) = β/ s2 + (α + 2)s + 2α + β

(b)

The continuous-time, linear, time-invariant feedback control system pictured in Figure 1 consists of a
(proportional) controller K(s) and a plant P(s) modeled by, respectively, the transfer functions
1
K(s) = B
and P(s) =
(s+2)(s+a)
with control parameters a and 3 denoting adjustable real-valued gains.
R(s)
K(s)
P(s)
C'(s)
Figure 1: A Continuous-Time, Linear, Time-Invariant Feedback Control System

Verify that the closed-loop transfer function T(s) = C(s)/R(s) in terms of parameters a and
pressed as a ratio of two polynomials in s, is
T(s) =
3
8² + (x+2)s +2a+B

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(b) Pick a pair of values for parameters α and β such that the closed-loop system is stable

(c) Pick a pair of values for parameters α and β such that the closed-loop step response is underdamped, exhibiting a settling time of 4/5 seconds and a peak time of π/7 seconds.

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