ECE 147A FEEDBACK CONTROL SYSTEMS  THEORY AND DESIGN F09
Homework 8 Solutions
Problem 1
Consider a plant modeled by
P
(
s
) =
10
s
+ 5
. Design a controller that makes the
transfer function from reference to tracking error of Type 1, provides no worse that 25% over
shoot to step references, gives a factor of 20 improvement in plant input disturbance attenuation
(compared to using no control at all) over the frequency range up to 1 rad/sec, and provides
attenuation of measurement noise in the frequency range above 1000 rad/sec by a factor of at
least 1000.
Explain your design using Bode plots, phase margins, etc. You may use matlab to compute
these.
Test your controller using the simulink file given with this homework assignment at the
course web page. In the file, enter the poles, zeros and gain of your controller into the “con
troller” block. (The template shows three poles and three zeros in the controller, but you can
use as many poles and zeros as you wish.)
The simulation, which covers 100 seconds, turns
the controller on at time
t
= 30, has a step up at
t
= 45, a step down at
t
= 55 and turns the
controller back off at
t
= 75. By plotting “
tim
” vs. “
clean_out
” in the workspace, determine
the overshoot percentage for the step up and the step down. Also estimate the steadystate
average output magnitude due to the input disturbances compared to the steadystate average
output magnitude without control.
Run at least three simulations using a different “seed”
in the “input disturbance generator” block. (Changing the seed changes the “random” input
sequence generated by simulink.)
Extra credit:
See how small you can make the steadystate average magnitude of the output
in the simulation when the control is active.
As with any engineering design problem, this control design problem begins with identify
ing the spcifications and developing a design methodolgy. First, we interpret the specifications
and then proceed to form a method for satisfying those specifications. We are given a transfer
function of a plant:
P
(
s
) =
10
s
+ 5
and we are required to satify the following specifications:
1. Type I tracking error
2. Overshoot
≤
25%
3. Factor of 20 improvement in plant disturbance input attenuation up to 1 rad/sec
4. Attenuation of measurement noise above 1000 rad/sec by a factor of 1000
Now, because these specifications were given with respect to the frequency domain, Bode
design would be the most helpful here. Intpreting our specifications in terms of the frequency
domain, we can see what needs to be accomplished by the controller in this sense.
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1. Type I tracking error
⇒
Controller must have 1 integrator.
2. Overshoot
≤
25%
⇒
Phase margin of approximately
45
◦
.
3. Factor of 20 improvement in plant disturbance input attenuation up to 1 rad/sec
⇒
Controller gain must bring low frequency magnitude (below 1 rad/sec)
20 log 20
≈
26
dB above the magnitude of the plant (without control).
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 Spring '10
 VolkanRodoplu
 Decibel, Bode Diagram, Closed loop, measurement noise

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