"
NASA
Technical Memorandum
89427
Estimability
and
Regulability
of
Linear Systems
Yoram Baram and
Thomas
Kailath
(EiASA-TR-89427)
€SIII'!ABILITY
AbC
~87-2~7~0
REGULABIIITY
CE
&LINEA%
SYSBEP5
(NASA)
31
p
CSCL
128
Unclas
February
1987
.
National Aeronautics and
Space Administration
~~
~-
NASA
Technical Memorandum
89427
Linear Systems
Yoram Baram, Ames Research Center, Moffett Field, California
Thomas Kailath, Department of Engineering, Stanford University, Stanford, California
February
1987
National Aeronautics and
Space Administration
Ames
Research
Center
Moffett
Field, California
94035
ESTIMABILITY AND REGULABILITY OF LINEAR SYSTEMS
Yoram Baram* and Thomas Kailath**
Abstract
A
linear state-space system will be said to be estimable
if
in estimating
its
state from its output the posterior error covariance matrix
is
strictly
smaller than the prior covariance matrix.
the quadratic cost
of
state feedback control is strictly smaller than the cost
when no feedback
is
used.
different from the well known observability and controllability properties
of
linear systems.
regulability are derived for time variant and time invariant systems,
in
discrete and continuous time.
It will be said to be regulable
if
These properties, which are shown to be dual, are
Necessary and sufficient conditions for estimability and
*Y.
Baram
is
with the NASA Ames Research Center, Moffett Field, CA
94035,
on sabbatical leave from the Department
of
Electrical Engineering,
Technion-Israel Institute
of
Technology, Haifa
32000
Israel.
**T.
Kailath
is
with the Information Systems Laboratory, Department
His
of
Electrical Engineering, Stanford University, Stanford, CA
94305.
work was supported in part by
the
National Science Foundation under
Grant DCI-84-21315-A1.
1
1.
Introduction
The benefit of using observation or feedback signals in state estimation
or regulation
of
dynamical systems
is
normally manifested by the reduction
of
certain cost functions
with
respect to their values when no such signals are
used.
yield mean-square error reduction in state estimation and quadratic cost
reduction in state feedback control. These properties, which we call
estirna-
bility
and
regulability,
are different from the properties
of
observability
and controllability, which were introduced
by
Kalman
[l]
and are widely recog-
nized as key structural properties in linear estimation and control.
In this paper we introduce those properties of linear systems that
We shall say that a stochastic linear system
is
estimable
if,
in estimat-
ing
its state from
its
output,
the posterior error covariance
matrix
is
strictly smaller than the prior state covariance matrix.
equivalent to the condition that no direction in the state space at any time
is orthogonal to all the past observations.
to the nonsingularity of
a
certain Gramian matrix.
is independent of that of observability in the sense that one does not
imply
the other.
can be reduced to
a
lower-order estimator of the state process.
state-space system driven
by
white noise
is
shown
to
be
a
minimal order
realization of its output process
if
and only
if
it
is
observable and esti-
mable.
in
discrete and continuous time.
condition leads directly to conditions derived
by
Baram and Shaked
[2],
131,
for
minimality of
the
Kalman filter.
This condition
is
It
is also shown to be equivalent
The notion
of
estimability
When
a
system
is
not estimable, the corresponding Kalman filter
A
linear
These results apply to time-variant and time-invariant linear systems
In the stationary case, the estimability
2
We shall say that
a
linear system
is
regulable
if,
for any nonzero ini-
tial condition, the quadratic cost in applying optimal control
is
strictly
smaller than the cost when
no
control
is
applied.
system is not regulable, the feedback signal may be eliminated for some non-
zero initial condition without increasing the cost.
conditions for regulability are obtained for time-variant and time-invariant
systems in discrete and continuous time.
As
might be expected, regulability
and estimability are shown to be dual properties.
This means that when a
Necessary and sufficient
2.
Estimability
Consider the system
=
F
x
+
Gkwk
k+
1
kk
X
n
xk
E
R
=Hx
+v
k
=
0,1,2,
...
'k kk k
where
E{x0wE}
=
0,
E{w
v*}
=
0,
E{w
w*}
=
Qk6k,J,
E{v
v*}
=
R
6
denotes expectation,
*
denotes Hermitian transpose, and
6
Kronecker delta. Let Xk denote the linear least-squares estimate of Xk
E{xo}
=
0,
E{x
x*}
=
no,
{wk)
and
{vk)
are zero-mean sequences with
00
Also,
E{*}
kJ k.l kJ k k,J'
denotes the
k,
.I
n
given (yk-19yk-2,
1
and
let
3