system dynamics and control assignment 7

where, �㕟:the system input;�㕐:the system output Find the steady-state error in velocity for an input of �㕡Ø�㕢�㕡 to a unity feedback system with a... is to be designed to meet the followi

SYSTEM DYNAMICS AND CONTROL

ASSIGNMENT 7

STUDENT REPORT

Contents

7.1 3

7.2 3

7.3 4

7.4 4

7.5 6

7.6 7

7.7 8

7.8 8

7.9 9

7.10 9

7.11 10

7.12 10

7.13 11

7.14 12

7.15 12

7.16 13

7.19 14

7.20 14

7.21 15

7.22 15

7.23 16

7.27 17

7.28 18

7.29 18

7.30 19

7.31 20

7.32 21

7.33 22

7.34 23

7.35 23

7.44 24

7.50 25

7.55 26

25609.02= 0.04 42�㕡 �㕢(�㕡): �㕒 ØØ(∞) = 84

a.the steady-state error;

b.the steady-state error if the input becomes �㕟(�㕡) = �㕡�㕢(�㕡)

a,From the figue:

The steady-state error

�㕒(∞) = 5 − 3 = 2

b,

7.4

For the system shown in the figure, what steady state error can be expected for the following test

�(�) 1

�

� 4�

�(�)2

3�+ 3

1 − 3�

�(1 − 4�)

4�

�(�)2(1 − 3�)

�(1 − 4�)(3�+ 3)

�(�)

4�

�(�)2(1 − 3�)

lim

150.1818= 82.5 15�㕡 �㕢(�㕡): �㕒 ØØ(∞) = 30

70

0 = ∞ 81�㕡 �㕢(�㕡): �㕒 ØØ(∞) = 162

where, �㕟:the system input;�㕐:the system output

Find the steady-state error in velocity for an input of �㕡Ø�㕢(�㕡) to a unity feedback system with a forward TF of

a.What is the expected percent overshoot for a unit step input?

b.What is the settling time for a unit step input?

c.What is the steady-state error for an input of 5�㕢(�㕡)?

d.What is the steady-state error for an input of 5�㕡�㕢(�㕡)?

e.What is the steady-state error for an input of 5�㕡 �㕢(�㕡)?

5500075

50�㕡�㕢(�㕡): �㕒 ØØ(∞) =50

�㔾Ø=50

0 = 0.075 50�㕡 �㕢(�㕡): �㕒 ØØ(∞) =100

�

�(�) 100

�(�+ 5)

�100

�( �)10

a.type 0 TF → �㔺2 must be zero

b.type 1 TF → �㔺2 must be type 0

c.type 2 TF → �㔺2 must be type 1

Given the system of the figure, design the value of �㔾 so that for an input of 100�㕡�㕢(�㕡), there will

a 0.01 error in the steady state

�(�) �

�(�+ 1)10�

b.What error can be expected for an input of 12�㕢(�㕡)?

1 +�㔾(�㕠 + �㕎)(�㕠 + �㗽)

�㕎 = 12 − 3 10√

7.28

A second-order, unity feedback system is to follow a ramp input with the following specifications: the steady-state output position shall differ from the input position by 0.01 of the input velocity; the natural frequency of the closed-loop system shall be 10�㕟�㕎�㕑/�㕠 Find the following

a,The system type

Because the steady-state output position shall differ from the input position by 0.01 of the input velocity ⟹ The system type is type 1

b.The exact expression for the forward-path TF

is to be designed to meet the following requirements: The steady-state position error for a unit ramp input equals 1/10; the closed-loop poles will be located at −1 ± �㕗1 Find �㔾 �㗼, , and �㗽 in order meet the specifications

find the following

a.�㔾 and �㕎 to yield �㔾Ø= 1000 and a 20% overshoot

The system type is type 1

c.The steady-state error for an input of 5�㕢(�㕡)

Since system is Type 1, �㕒 = 0

d.The steady-state error for an input of 5�㕡�㕢(�㕡)

5�㕡�㕢(�㕡): �㕒 ØØ(∞) = 5

lim

51

3 × 1

= 15

e.Discuss the validity of your answers to Parts c and d

The system type is Type 1

c.The steady-state error for an input of 5�㕢(�㕡)

Since system is Type 1, �㕒 = 0

d.The steady-state error for an input of 5�㕡�㕢(�㕡)

5�㕡�㕢(�㕡): �㕒 (∞) = 5

lim

52032

= 8 e.Discuss the validity of your answers to Parts c and d

( �+ 12 )( �+ 32) (�+ 38)

13

Use matlab to find the following

a.The system type

find the following

a.The system type

�2( �+ 2)

− 1 + �

�(�)

�(�) �+ 1

�3+ 2�2+ (�− 1)�+ �− 1

�(�)

The system type is Type 0

b.The value of �㔾 to yield 0.1% error in the steady state

�㔸(�㕠) = �㕅(�㕠)[1 − �㔶(�㕠�㔼 − �㔴)�㕠Ø+ 20�㕠 + 111�㕠 + 164ØØ�㔵] = �㕅(�㕠) =�㕠Ø+ 19�The steady-state error

�㕒(∞) = lim

''�㕠�㕅(�㕠)�㕠Ø+ 20�㕠 + 111�㕠 + 164�㕠Ø+ 19�㕠 + 108�㕠 + 180For a unit step, �㕅(�㕠) = 1/�㕠, and �㕒(∞) = 45/41 = 1.0976 For a unit ramp, �㕅(�㕠) = 1/�㕠

= ∞ Notice that the system behaves like a Type 0 system

Input substitution methods

For a unit step input, the steady-state error

�㕒(∞) = lim

''�㕠�㕅(�㕠)�㕠Ø+ 9�㕠 + 5�㕠 + 7�㕠Ø+ 9�㕠 + 5�㕠For a unit step, �㕅(�㕠) = 1/�㕠, and �㕒(∞) = 0 For a unit ramp, �㕅(�㕠) = 1/�㕠 , and �㕒(∞that the system behaves like a Type 1 system

Input substitution methods

For a unit step input, the steady-state error

�㕒(∞) = lim

''�㕠�㕅(�㕠)�㕠Ø+ 14�㕠 + 33�㕠 − 29�㕠Ø+ 2�㕠 + 29�㕠 − 204For a unit step, �㕅(�㕠) = 1/�㕠, and �㕒(∞) = −7.0345 For a unit ramp, �㕅(�㕠) = 1/�㕠 , andNotice that the system behaves like a Type 0 system

Input substitution methods

For a unit step input, the steady-state error