ECE 486 Assignment # 4
http://courses.ece.uiuc.edu/ece486/
Issued: February 13 Due: February 20, 2009
Problems:
9 Consider the plant described by the state space model
˙x=0 1
2−2x+0
1u y = [1 α]x
where αis a constant. Consider the PD controller of the form u=−Kpy−Kd˙y+Krr,
where ris a reference input.
(a) Consider α= 0. Find {Kp, Kd, Kr}such that the resulting closed loop system with
input rand output yis BIBO stable, and the following specifications are met:
overshoot to a step input is no more than 25%; settling time is approximately
1 sec.; DC gain from rto yis unity.
(b) Find the transfer function H=Y/R for general α. Identify the closed-loop poles
and zeros in a pole-zero plot.
(c) This final part can only be done using Matlab: Obtain step response plots of the
closed loop system for each of the five values α= 0, 0.1, −0.1, 1, and −1.
Is the final value unity as expected? What about the other specifications?
10 Sketch by hand the root locus plots of the following systems in the unity feedback
configuration:
(a) G(s) = s2+ 4s+ 3
s2+ 6s+ 8 (b) G(s) = s2+ 3s+ 2
s2+ 7s+ 12
(c) G(s) = 1
s2+s+ 4 (d) G(s) = s+ 1
s2+s+ 4
11 Consider the plant with transfer function Gp(s) = [s(s+ 1)]−1, which is placed in the
unity feedback configuration with compensator Gc(s) = KP+Kw(s2+ω2)−1.
(a) Provided the system is stable and Kw>0, show that the steady-state error is
zero for the sinusoidal reference input r(t) = sin(ωt).
Now, assuming ω= 1....
(b) With Kw= 0, find the range of KPfor which the closed loop system is stable.
Fix a value KP>0 for which stability holds, and indicate your selection on a
root locus plot.
(c) Fix the value KP>0 selected in (b), and obtain a root locus plot with respect to
Kwusing Matlab. Can you find a value Kw>0 for which the closed loop system
is stable? Discuss how you might modify your design based on your conclusions.