ECE 486 Assignment # 9
http://courses.ece.uiuc.edu/ece486/
Issued: April 24 Due: May 1, 2009
Problems:
23 A delay of length Tcan be expressed as an LTI system with transfer function e−Ts .
Using the approximation ex≈1 + x,x∼0, we obtain the low-frequency approxima-
tion,
e−T s =e−T s/2
eT s/2≈1−T s/2
1 + T s/2
Consider the DC motor model with transfer function [(s+ 1)s]−1, and a one second
delay. The overall transfer function is approximated by
Gp(s) = 2−s
2 + s
1
s(s+ 1) (∗)
(a) Explain using a root locus plot that a stabilizing PI compensator can be devised.
Note that only a 0◦-locus will do.
Also, explain why you might use a PI compensator for a type I plant.
(b) Explain using a Bode plot that a stabilizing PI compensator can be devised.
(c) Obtain a four dimensional state space model in which one state is the integral,
xI(t) = Zt
0
y(s)−r(s)ds
(d) Using Matlab, obtain a state-feedback gain in the controller u=−KAxAthat
places the closed loop poles in the left half plane, at numerical positions consistent
with a closed loop bandwidth of your choice.
Note: You will have a great deal of trouble if ωBW ≫1.
(e) Using Matlab, obtain an observer to estimate x(t) given the input and output.
Explain your choice of observer poles.
(f) Write down a block diagram description of the plant and your compensator.
(g) If you are curious, and have the time, try a step response using simulink (it won’t
take you long). If you chose ωBW ≫1 you can expect some strange behavior. For
closed loop poles with small natural frequency the step response will be relatively
calm.