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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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Consider the control system shown in Figure B1, where the feedback controller is a positive proportional gain. In the diagram: r is the reference signal, y is the plant’s output, d1 is a disturbance acting on the plant’s input, d2 is a disturbance acting on the plant’s output, and e is the tracking error. Figure B1 i) Showthattheclosed-loopcharacteristicequationis:s2+10s+24+4k=0. Let r = 0, and let: • e1• denote the steady-state tracking error when d1 is a unit step, and d2 = 0; • e2• denote the steady-state tracking error when d2 is a unit step, and d1 = 0; ii) find an expression for e1• as a function of k. iii) find an expression for e2• as a function of k. iv) find the minimum value of k such that both |e1•| and |e2•| are less than or equal to 0.1 v) find the the damping ratio (z) of the closed-loop poles when k has the value obtained in iv), does it correspond to an oscillatory behaviour?

Control
ENGD3113 2021/2022
Revision
Dr. Zehor Belkhatir
Problem 1
Consider the control system shown in Figure BI, where the feedback controller is
a positive proportional gain. In the diagram: r is the reference signal, y is the plant's
output, di is a disturbance acting on the plant's input, d; is a disturbance acting on the
plant's output, and e is the tracking error.
(s+ 6(s+ 4)
Figure BI
i) Show that the closed-loop characteristic equation is: +10s+24+4k = 0.
Let r0, and let:
• e denote the steady-state tracking error when di is a unit step, and d; =0;
• 2 denote the steady-state tracking error when dz is a unit step, and di = 0;
ii) find an expression for el as a function of k.
iii) find an expression for e as a function of k.
iv) find the minimum value of k such that both le| and 2 are less than or equal to
0.1
v) find the the damping ratio (5) of the closed-loop poles when k has the value
obtained in iv), does it correspond to an oscillatory behaviour?
Page I
Control
ENGD3113 2021/2022
Problem 2
onsider the feedback system in Figure B2
C(s)
G(s)
Figure B2
where
C1) =.
G(6) = +2)(s+3)
and k is a positive gain.
- Plot the root locus for this particular closed-loop system, and describe how
the stability of the system and the transient responses are likely to vary for
different ranges of the controller's gain k.

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