1
TECHNICAL REPORT
67-2
W.
M.
WONHAM
FEBRUARY,
1967
ON
POLE ASSIGNMENT IN MULTI-INPUT
CONTROLLABLE LINEAR SYSTEMS
ON
POU
ASSIGNMENT
IN
MULTI-INPUT
CONTROLLABU
LINEAR
SYSTEMS
*
by
W.
M.
Wonham
*
This research
was
supported in part
by
the
Air
Force Office of
Scientific Research,
Office of Aerospace Research, United States
Air
Force, under AFOSR Grant
No.
AF-AFoSR-~~~-~~,
in part
by
the
National Science Foundation under Grant No.
GK-967,
and in part
by
National Aeronautics and Space Administration under Grant
No.
NGR-
40-002-015.
ABSTRACT
OF
ON
POLE
ASSIGNMENT IN MULTI-INPUT
CONTROLLABU LINEAR SYSTEMS
W.
M.
Wonham
It
is
shown that controllability of an open-loop system
is
equivalent to the possibility
of
assigning an arbitrary
set
of
poles to the transfer matrix
of
the closed
loop system,
formed
by
means
of
suitable linear feedback of the state.
ON
F'Om
ASSIGNMENT
IN
MULTI-INPUT
CONTROLLAEKLE
LINEXR
SYSTEMS
W.
M.
Wonham
INTRODUCTION
Consider the system
Here and in the following,
all
vectors and matrices have real-
valued elements and
all
matrices are constants. In
(l),
A,B
are
matrices of dimension respectively n
X
n and n
X
m;
x
is
the
state, an n-vector; and
u
is
an m-vector.
As
usual,
u
denotes
an external input.
Let
us
"close the
loop"
by setting
u=cx+v,
for some
m
X
n matrix
C
and
new external input
v.
Then
(1)
becomes
dx(t)/dt
=
(A+BC)x(t)
+
Bv(t)
. (2)
In applications
it
is
often desirable to choose
C
so
that the
2
matrix
A+BC
has special properties: for example, stability.
Intuitively
it
is
clear that the possibility of such
a
choice
depends
on
the controllability, in an appropriate sense,of the
state
x
with respect to
u.
In this note
we
single out the property
of "pole assignability", and show that
it
is
equivalent to con-
trollability of
(1)
in the usual sense.
To
be precise, let
A
=
(Al,
...,
An)
be an arbitrary set of n complex numbers hi, such that any hi
with
Im
Ai
f
0
appears in
A
in
a
conjugate pair.
We
recall
that the pair
(A,B)
the n
X
mn
matrix
is
(completely) controllable
if
and only
if
K
=
[
B,
AB,.
.
.
,A~-~B~
is
of full rank n.
The result we wish to prove
is
the following.
THEOREM
The pair
(A,B)
is
controllable
if
and only
if,
for every
choice of the set
A,
there
is
a
matrix
C
such that
A
+
BC
has
A
for
its
set
of
eigenvalues.
-
In other words, controllability
is
equivalent to the