Figure
1.4
Figure 1.4 is the closed loop step response of
G(s) after compensation. The specifications asked for an percent overshoot less
than or equal to 10%, and settling time less than or equal to 2 seconds. After
compensation the percent overshoot that I got is 5.19%, and a the settling time is 1.39 seconds.
Design Process
>> g=tf([100],[1,10,0,0])
Transfer function:
100
------------
s^3 + 10 s^2
>> k=tf([1,0],[1,11])
Transfer function:
s
------
s + 11
>> kg=k*g*30*.8
Transfer function:
2400 s
---------------------------
s^4 + 21 s^3 + 110 s^2
>> k1=tf([3.68,3.68*5.476],[1,20.16])
Transfer function:
3.68 s + 20.15
--------------
s + 20.16
>> kg1=kg*k1
Transfer function:
8832 s^2 + 4.836e004 s
--------------------------------------------------
s^5 + 41.16 s^4 + 533.4 s^3 + 2218 s^2
>> kg11=kg1*.349
Transfer function:
3082 s^2 + 1.688e004 s
---------------------------------------------------
s^5 + 41.16 s^4 + 533.4 s^3 + 2218 s^2
>> k2=tf([1,11],[1,64])
Transfer function:
s + 11
------
s + 64
>> k3=tf([1,10],[1,85])
Transfer function:
s + 10
------
s + 85
>> kg111=kg11*k2*k3
Transfer function:
3082 s^4 + 8.161e004 s^3 + 6.935e005
s^2 + 1.857e006 s
-------------------------------------------------------------------------------------------------------------
s^7 + 190.2 s^6 + 1.211e004 s^5 + 3.056e005 s^4 + 3.232e006 s^3 + 1.206e007 s^2
>> kg2=kg111*6.94
Transfer function:
2.139e004 s^4 + 5.664e005 s^3 + 4.813e006 s^2
+ 1.289e007 s
-------------------------------------------------------------------------------------------------------------
s^7 + 190.2 s^6 + 1.211e004 s^5 + 3.056e005 s^4 + 3.232e006 s^3 + 1.206e007 s^2
>> k4=tf([1,5.476],[1])
Transfer function:
s + 5.476
>> kg3=kg2*k4
Transfer function:
2.139e004 s^5 + 6.835e005 s^4 + 7.914e006 s^3 + 3.924e007 s^2 + 7.056e007 s
-------------------------------------------------------------------------------------------------------------
s^7 + 190.2 s^6 + 1.211e004 s^5 + 3.056e005 s^4 + 3.232e006 s^3 + 1.206e007 s^2
>> kg4=kg3*.233
Transfer function:
4984 s^5 + 1.593e005 s^4 + 1.844e006 s^3 + 9.143e006 s^2 + 1.644e007 s
-------------------------------------------------------------------------------------------------------------
s^7 + 190.2 s^6 + 1.211e004 s^5 + 3.056e005 s^4 + 3.232e006 s^3 + 1.206e007 s^2
>> k5=tf([1],[1,80])
Transfer function:
1
------
s + 80
>> kg5=kg4*k5
Transfer function:
4984 s^5 + 1.593e005 s^4 + 1.844e006 s^3 + 9.143e006 s^2 + 1.644e007
s
------------------------------------------------------------------------------------------------------------------------------
s^8 + 270.2 s^7 + 27319 s^6 + 1.274e006 s^5 + 2.768e007 s^4 + 2.706e008 s^3 + 9.651e008 s^2
>> kg6=kg5*311
Transfer function:
1.55e006 s^5 + 4.953e007 s^4 + 5.735e008 s^3 + 2.844e009 s^2 + 5.113e009
s
------------------------------------------------------------------------------------------------------------------------------
s^8 + 270.2 s^7 + 27319 s^6 + 1.274e006 s^5 + 2.768e007 s^4 + 2.706e008 s^3 + 9.651e008 s^2
>> figure(10)
>> bode(kg6)
margin(kg6)
figure(11)
t=minreal(kg6/(1+kg6))
Transfer function:
1.55e006 s^4 + 4.798e007 s^3 + 5.41e008 s^2 + 2.627e009 s + 4.648e009
--------------------------------------------------------------------------------------------------------------------------------------------
s^7 + 269.2 s^6 + 2.706e004 s^5 + 2.8e006 s^4 + 7.465e007 s^3 + 7.95e008 s^2 + 3.505e009 s + 4.648e009
>> step(t)
>> figure(3)
>> rlocus(kg6)
>> pole(kg6)
ans =
0
0
-85.0000
-80.0000
-64.0000
-20.1600
-11.0000
-10.0000
>> zero(kg6)
ans =
0
-11.0000
-10.0000
-5.4760
-5.4760
End of Compensation
Example
