The University of Texas at Austin
Department of Electrical and Computer Engineering
EE362K: Introduction to Automatic Control—Fall 2009
Problem Set Six Solution
C. Caramanis
Due: Friday, October 30, 2009.
This problem set focuses on the new concepts introduced in the last three classes: reachability
and state feedback, and integral action, and also observability and state estimation. As usual, we
also work in some exercise with important concepts from linear algebra.
1. Integral Action:
(a) Consider the system:
A
=
1
−
6
40
,B
=
0
1
,
C
=[10
]
Compute a feedback control policy,
u
=
−
Kx
, so that the closed loop eigenvalues are
both at
−
1.
(b) Let the reference signal
r
be equal to 1. Compute the feedforward term so that the
steady state output is equal to the reference signal,
r
.
(c) Plot the trajectory of the output (i.e., plot
y
(
t
)) to verify that you have chosen
K
and
k
r
appropriately so that the system is stable, and so that the steady state output is equal
to
r
.
(d) As in the previous exercise, compute a perturbation matrix, and choosing a value of
p
so that the closed loop system is still stable, plot the behavior of the perturbed system.
(e) Now back to the nominal system: suppose we have a constant disturbance, so that the
system dynamics are now:
˙
x
=
Ax
+
Bu
+
Fd,
where
d
is an unknown but constant disturbance, and
F
is the 2
×
1matrix(1
,
1)
>
.P
ick
a constant value for
d
, and plot the output trajectory for the same values of
K
and
k
r
.
(f) Write down the closed loop augmented system, where you add a state
z
that is the
integral of the error,
y
−
r
, and where you use feedback control with integral action:
u
=
−
−
k
i
z.
In particular, write down the augmented system matrix (this should be a 3
×
3matr
ix)
.
(g) Find values for
K
and
k
i
so that the augmented closed loop system matrix is stable.
(You can use the same
K
you computed above, if you like).
(h) Plot the output trajectory (
y
(
t
)) for di±erent values of the feedforward gain,
k
r
.
1