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Answer Mid 1

The document contains a series of questions and solutions related to control systems, specifically focusing on PI controllers, proportional controllers, and feedback control systems. It includes calculations for controller gain, integral time, closed loop responses, offsets for unit step changes, and characteristics of second order processes. Each question is followed by detailed mathematical solutions and expressions derived from the given transfer functions.

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ثامر عدنان عباس
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19 views8 pages

Answer Mid 1

The document contains a series of questions and solutions related to control systems, specifically focusing on PI controllers, proportional controllers, and feedback control systems. It includes calculations for controller gain, integral time, closed loop responses, offsets for unit step changes, and characteristics of second order processes. Each question is followed by detailed mathematical solutions and expressions derived from the given transfer functions.

Uploaded by

ثامر عدنان عباس
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Q1: If the measured input to a PI controller is a step change (𝐘𝐦(𝐬) = 𝟐/𝐬)

and the controller output changes initially as shown in the figure below, what are
the values of the controller gain and integral time?

Answer
A
PI controller with step change in error E(s) 
s
1 A
P( s)  K C (1  )
Is s E(t) p(t)
K A 𝐾𝐶 𝐴
 P(t )  K C A  C t A 𝜏𝐼
I

𝐹𝑟𝑜𝑚 𝑎 𝑝𝑙𝑜𝑡 𝐾𝑐𝐴 = 6


𝐴=2
0 ps
𝐾𝑐 = 3 0 t 0 t
𝐾𝑐𝐴 Response of a PI controller (lineaer)
= 1.2
𝜏𝐼
𝐾𝑐𝐴 6
𝜏𝐼 = = =5
1.2 1.2

1
Q2: The block diagram of a system with a proportional controller is shown below:
Unit step input is introduced in the set point. Find:
1. The expression for ψ and τ for
closed loop response
2. The value of Kc to provide a
critically damped response for
τp = 8, and τm = 1.

Solution:

2
3
4
Q3: A feedback control system has the following transfer functions;
Gp(s) = 5/[(2s + 1) (s + 1)], Gm (s) = 1/(0.5s + 1), GV(s) = 0.2/(0.1s +
1),
GC (s) = KC and Gd (s) =1
(a) Obtain the overall closed loop transfer function.
(b) Find the offset for a unit step change in set point if Kc=3.
(c) Find the offset for a unit step change in load if Kc=3.
Solution

5
6
7
𝟓
Q4: The following second order process Gp(s) = is controlled by a
(𝐬+𝟏)(𝟐𝐬+𝟏)

proportional controller of gain (Kc = 1.6).


For a unit step input in setpoint and unity transfer functions for valve and measuring
elements (GV = Gm = 1), find the: (1) damping coefficient, (2) overshoot, (3) period
of oscillation, (4) Maximum value of response (ymax), (5) offset.
Answer
𝑌(𝑠) 𝐺𝐶 𝐺𝑉 𝐺𝑃
=
𝑋(𝑠) 1 + 𝐺𝐶 𝐺𝑉 𝐺𝑃 𝐺𝑚
5
𝑌(𝑠) 1.6 ×
(s + 1)(2s + 1)
=
𝑋(𝑠) 1 + 1.6 × 5
(s + 1)(2s + 1)
8
𝑌(𝑠) (2s 2 + 3s + 1)
=
𝑋(𝑠) 1 + 8
2
(2s + 3s + 1)
𝑌(𝑠) 8
= 2
𝑋(𝑠) 2s + 3s + 1 + 8
8
𝑌(𝑠) 9
=
𝑋(𝑠) 2 s
s2 + + 1
9 3

2
𝜏 = √ =0.4714
9
ξ=0.35355
 
Overshoot  exp  0.305
1 2 )

2
T  3.1664
1 2

UV=8/9=0.888889
Y(max)=UV*(1+OS)=1.16
Offset=1-UV=0.111111

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