Design a Feedback Control System - IT Assignment Help

Assignment Task:

Task:

Common hints for all the questions:
1. Note that you do not need to add far away poles for your controller if it is already causal.
2. Use internal model principle wherever relevant
3. If you have 2 poles varying simultaneously, the similar triangles idea will not work. You would need to go “back to the basics”, i.e., K=-1/CP and then undertake term-by-term cancellation.

Problem#1
Suppose the plant you wish to control is the same as the one in Homework#3, i.e., 2 P s K s p s p s p ( ) /{[5 4 1][ (6 ) 1][5 1]} = + + + . Here p?represents the sum of the digits (all 18 digits) of your S.R. number. The nominal value of the DC gain is 0.3p but the actual value can be anywhere between
0.2 p
to
0.9 p
, i.e.,
0.2 0.9 p K p ? ? .

Design a feedback control system that achieves the following performance and stability specifications: The closed-loop system should show zero steady state error to DC inputs. The dominant poles of the nominal closed-loop system are to be situated at − ? 3 7 p j p and should vary by less than 0.1p when the plant gain varies. Further, the damping factor of all the closed-loop poles should be at least 0.25. Plot the step response of your closed-loop system for different values of the plant gain.
(b) Suppose we express the plant in the standard form used for plotting the root locus, viz., 2 2 P s K p p p s p s p s p ( ) ( .0.8 .6 . 5) /[( 0.8 )( 6 )( 5)] = + + + , where K p = 0.3

. The dominant open- loop pole of your system at s p = −0.8 varies between the limits −0.1p and −2.5 p , redesign your control system to satisfy the same performance and stability specifications as that of part (a).
(c) Suppose, now, that both K and the pole at −0.8 p vary between the limits indicated above, redesignyour control system to satisfy the same performance and stability specifications.

Problem#2
A plant is given by 2 2 ( ) 1 ( 2 ) P s s s = + + ?? ? n n

, where nominally
? = 0.5 , and
1 ?n = but ?n can vary

between the limits 0 2 ? ? ?n
. Design a two-degree of freedom control system satisfying the following three specifications:
(i) It possesses a dominant closed-loop pole nominally at s=-1.
(ii) The variation of the dominant pole due to plant uncertainty is at most 0.01.
(iii) The real parts of the other closed-loop poles are less than or equal to -10 (i.e., since the real parts are negative, the magnitude of the real part is greater than or equal to 10).

Problem#3
A plant has the following transfer function:

2 P s s s s p s p ( ) ( 12 40) / ( )( ) = + + − − , where the nominal value of p=-5+1j but the real part of p can vary from -3 to -7.

Design a two degree of freedom control system such that
(i) the closed-loop system tracks DC signals with zero steady state error.
(ii) the closed-loop system has a single dominant pole at s =−1 and the variation of the pole due to that of the plant is at most 0.1.
What is the steady state error of the closed-loop system in tracking a ramp signal r(t)=t.1(t)?

Problem#4
A plant has the following transfer function: 2 P s K s ( ) / ( 1) = + , where the nominal value of K=1, but K can vary from 0.5 to 4.
Design a two degree of freedom control system such that
(i) the closed-loop system perfectly rejects output disturbances of the form 0 1 d t a a t ( ) sin( ) = + + ? , where a0, a1 and φ are constants.
(ii) The closed-loop system has dominant poles at s j = − ?1 1 and the magnitude of variation of the poles due to variation of plant gain is at most 0.1.

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