March 1983
LIDS-P-1289
Presented at Johns
Hopkins Conference, March 1983
LINEAR ESTIMATION OF TWO-POINT BOUNDARY VALUE PROCESSES
by
C.S. Draper Lab.
M.I.T. Laboratory for Information and Decision Systems
ABSTRACT
We consider estimation of an nxl vector noncausal stochastic process governed by the linear
x(t) = A(t)x(t) + B(t)u(t)
with
boundary
two-point
condition
with two-point boundary condition
0
noncausal estimation problem and following the
methodolocy introduced in [6], we augment the
to
with the
methodology introcess
be estimated
complementary model and invert the combined
system to obtain a differential realization of
the smoother.
2.
LINEAR STOCHASTIC TWO-POINT BOUNDARY VALUE
PROCESS (TPBVP)
T
v = V x(O) + V x(T)
where u is an mxl white input noise with covariance parameter Q(t) and v is an nxl zero mean
Gaussion boundary value with nonsingular covariancev
uncorrelated
m
with u on [OT].
Estimation of processes satisfying models of
this type was first suggested by Krener [1].
1.
Bernard C. Levy
Alan S Willsky
Milton B. Adams
INTRODUCTION
The now classical solutions to linear filtering and smoothing problems for one-dimensional
(l-D), nonstationary, causal processes are dis'
^
cussed in the review paper by Kailath [2].
Notable in the derivations of these solutions is
the use of the Markovian nature of the models for
these 1-D processes.
However, in as much as
stochastic processes in higher dimensions (random
fields) are typically noncausal and consequently
are not Markovian in the usual sense, their estimators cannot be derived through a direct extension of these 1-D derivations. Thus, linear
estimation-problems for noncausal processes require
estipationp roblems for noncausal processes require
new approaches, and a natural foirst step is to
solve the estimation problem for 1-D noncausal
processes.
The General Solution
governed by an nth order linear stochastic
differential equation together with a specified
two-point boundary condition. Accordingly, the
process will be referred to as a linear stochastic two-point boundary value process or TPBVP.
In describing the model we employ the white
noise formalism for representing linear stochastic differential equations. Let u(t) be a mxl
white noise process with covariance parameter
white noise process with covariance parameter
Q(t). Let v be a nxl random vector, independent
of u(t), with covariance matrix
The nxl
of u(t), with covariance matrix E . The nxl
v
boundary value process x(t) is governed on the
boundary value process x(t) is governed on the
O
T
It will be assumed that A and B are continuous on
O,T] and that all random variables are zero-mean.
It is instructive to derive one form of the
. Withthis as motivation, Krener [3] introgeneral solution for (1) as the approach we take
duced anoncausal
class of linear
1- dynamic
in this derivation will be used later.
The form
duced a class of linear noncausal 1-D dynamic
models. In his study of these models, Krener
of the solution which we obtain differs from the
usual Green's function solution (e.g. see [1 ]).
has developed results on controllability, obserLet ¢(t,s) be the transition matrix associated
vability and minimality and has solved a deterministic
linear
control
with
problem.
A(t).
Inaddition,
If x(O) were known, then x(t) could
ministic linear control problem. In addition,
he has posed the fixed-interval linear smoothing
be represented in the variation-of-constants
problem for these systems and has derived interform
gral equations for both the weighting pattern and
O
the error covariance for the optimal smoother.
Working directly with these equations he has had
where
t)
is the solution of la) with
where x (t) is the solution of (la) with x (0) =0:
some success in obtaining a dynamic realization
of the smoother for a special "stationary-cyclic"sbsusds
2b
(t,s)b(s)u(s)ds
x (t)= f
class of these processes [4].
In this paper we solve the estimation problem
for Krener's 1-D noncausal models through a
generalization of the complementary models
approach to linear estimation which was first
introduced by Weinert and Desai [5].
In particular, we present a procedure for finding the
complementary modelwhich is appropriate for the 1-D
Substituting from (2a) at t = T into the boundary
condition (lb), we can write
T
T
v = V x (t) = [V + V T(T,O)]x(O) .
(3a)
For a well-posed problem, there will be a unique
x(O) for a given v and u on [O,T].
Thus
The reason for our choice of subscripts f and b,
well-posedness requires that ihe nxn matrix
0
F = V
T
+ VT (T,O)
be nonsingular.
for x(O) as
(3b)
With F invertible, we can solve
-fo
T
x(O) = F
v - V x (T)
.
- V-1
VTX O t
x (t) = D (t O)x (0)
f(t) = (tO)xf(O)
+ X Ct)
f(t)
(7a)
x
+ xb(t)
(7b)
and
OI(t)
The noncausal nature of the TPBVP x(t) is
clearly displayed if we correlate the value of
x at t = 0 with future values of the input u:
E{x(O)u'(t)} = -F- V
If x(O) and xb(T) were known, then we
could solve for xf(t) and xb(t) as
(3c)
Substituting x(O) into (2a) gives the general
solution for (la,b) as
x(t) =PD(tO)F~v
denoting forward and backward respectively, will
become apparent blow.
(T,t)B(t)Q(t) t [O,T]
(5)
b(t
=(4)(t,T)xb(T)
where xf0(t) is governed by (6b) with xf(0) = 0
where Xf t) is governed by (6b) with Xf(O) = 0
and xx° (t) is governed by (7b) with
(T) = 0.
Following a derivation similar to that used to
obtain the general solution in (4), it can be
shown that
Thus, the n
order model in (1) is not Markovian,
and consequently Kalman filtering and associated
smoothing technqiues are not directly applicable.
xf(t)
f(t,O)
xb(t)
)
x
b
(
xf(t)
- v
where the transformation matrix T(t) is chosen so
that 1) the dynamics of the system model in (1)
become decoupledl:
O
T
[VV +V
+VbF
fb
b(
x(t)
=
[Volb (TJ
(t)
Tl
)
b
0
+ VbOb(O,T)].
bb
(9)
(10)
.
The covariance of x(t)
P (t) = E{x(t)x'(t)}
x
can be derived from (4) as
F
-a
P (t) = P (t) + m(t,0)F
F-1' '(tO ) 1
When there is no risk of confusion we will often
omit explicit reference to the independent variable, i.e. A(t) + A.
T
Covariance of the TPBVP x(t)
0
(6c)
V
As we will see, the two-filter form of the
general solution in (8) is the foundation for the
implementation of the estimator that we develop
later in this paper. The term two-filter is used
to signify that the numerical solution of (8)
requires the integration of a forward process
xf0 and a backward process xb
[xf (T)
T
f[f (T,)
f
The TPBVP x is recovered from (8) by inverting
(6a):
(6b)
[f]
and 2) Af is exponentially stable in the forward
direction and A_ is exponentially stable in the
backward direction.
For time-invariant systems
this is always possible by assigning those modes
associated with eigenvalues greater than or equal
to zero to Af and those less than zero to Ab.
For time-varying dynamics, it may be difficult to
determine the dynamics and boundary conditions
for a transformation T(t) which transforms the
system dynamics into this form. However, for
the systems of interest to us later in this paper
we can overcome this difficulty by invoking results obtained previously for smoothing solutions
for causal processes. The boundary condition for
the transformed process will be written as
j
xb(t)
b
where
Ff
X(0)
(8)
b
1'6'~~b~~~~~t)']
v =[
Iv - vfxf(T)
t
(6a)
[
] rx
[l: [L %J[%J
-fb
1
Below, as an alternative to (4), we present
a second form for the general solution of (1)
which leads to a numerically stable implementation. Consider the equivalent process obtained
by transforming x as
f(L )1 = T(t)x(t)
0
.
(t,T)
=
(11)
TO
[(v + V P
(t,
(t)F-lVTPO(t) -
T'
(T)V
]
(t)VT' F- ''(t,0)
(12a)
where P (t) is the covariance of x (t) satisfying
x
*O
=
P = Ap0
AP + P A' + BQB' ; P(0) =
(12b)
X
X
X
X
with r(t), u(t) and v with covariance matrix
Ib. Define a qx2n matrix W partitioned into
qxn blocks as
q
blocks as
W = [W
An additional expression for P can be derived
from the two-filter form of the general solution
(equation (8)). However, this expression is
somewhat complex, and we will wait until a later
publication to present it in the context of the
study of the estimation error covariance [7].
0
T]
W
.
(17a)
The boundary observation is the qxl random
vector:
Yb = Wxb + rb
(17b)
Define an nx2n matrix V as
Green's Identity
T
V = [V
In establishing the solution to our estimation problem we will make use of Green's Identity
for the differential operator:
L:D(L) + R(L) ; (Lx) (t) = x(t) - A(t)x(t)
(18a)
so that the boundary condition in (lb) can be
written as
(13)
where D(L) is the space of once continuously
differentiable nxl vector functions on [0,T] and
R(L) is the Hilbert space of square integrable
nxl vector functions on (OT]. Let E be a
2nx2n matrix partitioned into nxn blocks with:
V ]
v = Vxb
(18b)
We will assume that the rows of W and the rows
of V are linearly independent (8].
The fixed-interval smoothing problem is to
find the linear minimum variance estimate of
the noncausal TPBVP x(t), t £ [0,T], given the
complete observation set Y:
E =
0
I
(14a)
and define the 2nxl vector
4.
Xb=xO)
(1=4b)
(1)
[x(T)J
t
(L A)(t) = -X(t) - A'(t)X(t)
Y = {Yb, y(t)
.
(14c)
N>
+
L 2[,T0
<x,
L X>
+<xb,
L[O,T]
L
E b> 2
where Xb takes a form similar to xb in (14b).
3.
PROBLEM STATEMENT
The fixed-interval smoothing problem for
the noncausal process x(t) defined earlier is
stated as follows. Let r(t) be a pxl white
noise process uncorrelated with v and u(t) and
with continuous covariance parameter R(t).
Let
C(t) be a pxn matrix whose elements are continuous on [0,T].
The observations of x(t) are
given by the pxl vector stochastic process:
y(t) = C(t)x(t) + r(t)
(16)
In addition, we assume that there may be available a boundary observation Yb defined as follows. Let rb be a qxl random vector uncorrelated
.
(19)
Our approach to the solution of the estimation problem is based on an application of
the method of complementary models introduced
by Weinert and Desai [ ] in deriving the
smoother for causal stochastic processes. A
brief discussion of the basis for this approach
follows. Define the underlying process i as
rv
(20)
r
n
(15)
[0,T]}
SOLUTION TO THE ESTIMATION PROBLEM
Given these definitions, the Green's Identity
for L on the interval [0,T] is obtained directly
by integration by parts, yielding
=
<Lx,
. t
The complementary process Z = {Z, }Z
is
orthogonal to the observations Y fy, yb b }and
when combined with the observations, they span
the space spanned by i. In particular, we can
express the observations Y and the process to
be estimated x as linear mapping of the underlying process
Y[H
I]
(21a)
Yb
A
=M
y
and
x=M
x
5
.
(21b)
4
given by
=Zb
Z
(22a)
= [-I : H*] Z;
M
is
a linear mapping of
where H* is the Hilbert adjoint operator of H
g
z
(1) the orthogonality condition
which satisfies:
Z
LY,
i.e.
]
E
[z'(t),
J
Y
b
=
(23a)
(23b)
g; Y= +NN Z
y
z
(23c)
Y-1
where the partitions of M
are
M
Given M , the final step in determining the
estimator isZto augment M and M as in (23b)
invert.
This step caX be siMplified if we
first determine an internal differential realization for Z in a form similar to that for Y.
Then, following [ 5 ] the internal differential
realizations for Y and Z can be augmented and
inverted in a straightforward manner.
From
(25b) we see that an internal differential
realization for Z implies an internal realization for H*. To obtain such a realization,
start with the inner product identity for
Hilbert adjoints
[]
H*
(26
Ln
Given (23a) and (23c), the linear minimum variance estimate of E given Y is
, the projection
of (23c) into Span (Y). Since
is a linear
mapping of i, the linear minimum variance
estimate of x is
Mx
(H)
(H)
is an arbitrary element in the range
o H, R(H), and
of
is an arbitrary element in
[vU
the domain of H, D(H). Employing
H'
=C
v
(24a)
It is straight-forward to show that the estimation error is the minimum variance estimate
given Z:
v
where
Y
= M M Y
x y
(25b)
H-1
=
z]
EYy
-u
The
[M] Aand
=MC
g
g
Rr
=
Z =
is invertible so that 5 can be written as
=
+
in (21a) and E is the integral operator whose
kernel is the correlation function of the
From the definition of
underlying process i.
the underlying process fr our problem
the underlying process for our problem
Z
0
b
and (2) the complementation condition:
augmented mapping M defined by
r~
(25a)
(27a)
Wxb
and defining
[A
,1`
H*"
fiUX1(27b)
=M
x
x
i
z
y
=M
x
Nz
y
z
Here we have expressed the error as a linear
mapping of the underlying process I, and the
estimation error variance is therefore a linear
function of the variance of E.
For the above construction to be useful we
need to know the mapping M in (22b) and the
inverse of the augmented map M in (23b).
In
[ 8] it is proved that the orthogonality and
complementation conditions are satisfied by M
z
where
H
(26)
<x
.e. H* =
,
O I
can be rewritten as
C*u,> + <X,
+ <x
b'
W*v>
>
b
H
0
(27c)
I
<Lx,
(28a)
5
Applying Green's identity to replace <Lx, X> in
[x(0)T)
r
0
(28a) gives
-
<X, [C*u>
BQB
[A
14 =
tWcl
(T)
(31b)
(T)J
where
is a matrix such that
p-V~l
rV=
T
(0)
L A]> = <Xb, [EEb + V*YIb - W*vX]>
(28b)
Now if W
()
C R
C
]
-A'
(29a)
V0fIT-1V0
'1
0
V-
is invertiable, we can define
VT'
-:O l1V
-WAl?
(P')
=
VXC
(29b)
E
V
and
EXXA
and
W'
0'
-10
v
b
- - - - - -0 - - - -
v
V° '1VT
--
I
(31d)
W
+ WTlb
+w
+ W
Ewl0
:
Then it is shown in [8] that the map
T=
=
H*
(30a)
[
b
th
has an internal differential realization with
interval process X satisfying
L
= C'u,
(30b)
with boundary condition
[d =
L0
=V
r vx
[
Ib
]
rTpi
(30c)
X
FB'X
=ii
II
(30d)
Lb
[WkEb]
LW'bJ WRb
and that this choice of internal realization
satisfies the inner product identity (28b) with
both left and right hand sides identically zero.
Augmenting Z defined by (25b) and- H* given
by (30) with the internal differential realization for Y and inverting as in [ 6] and projecting the solution onto the span of Y (see
(23)) we find [ 8] that an internal differential
realization of the estimator is given by the
2nth order system:
[=]
x
Hamiltonian Diagonalization
0considerations
1.1
=
_ _R
with two-point boundary condition
(31a)
The solution of the 2n
order boundary
value process in (31) could be implemented by
either of the two forms of the general solution
derived earlier. However, by considering the
"time"-invariant case we can anticipate that
there may be numerical instabilities associated
with the first of those methods. In the timeinvariant case the eiqenvalues of the 2nx2n
Hamiltonian matrix Y defined in (la) are symmetric about the imaginary axis [ 9 ], i.e. there
are n eigenvalues in each of the left and right
half planes. Thus, for the time-invariant case,
the right half plane eigenvalues will result in
numerical instabilities for the unidirectional
implementation suggested by (4).
This stability
problem can be avoided in general by transforming
the smoother dynamics into the stable forward/
backward form in (8).
To achieve this second
form, we need a transformation which diagonalizes
the dynamics of P. into two nxn blocks, one stable
in the forward direction and the other backwards
stable.
Since the dynamics of our smoother as represented by tf are identical to those of the
smoother for causal processes as originally derived by Bryson and Frazier [10], any transformation which results in a two-filter smoother
for causal processes will also diagonalize our
smoother. However, choosing a diagonalizing
transformation for our problem requires special
not encountered'in the causal
case. First, because the two-point boundary
condition provides incomplete information for
both the initial and final values of the process,
we will choose a transformation which corresponds to a two-filter solution for causal processes with both filters in information form.
Second, it is important to choose the boundary
conditions properly for the Riccati differential
equations which govern the time-varying elements
of the diagonalizing transformation. Our choice
of diagonalizing transformation and corresponding
boundary conditions makes it possible to formulate a numerically stable two-filter form for
our smoother which is remarkably similar to twofilter smoothers for causal processes.
Define the time-varying transformation T(t)
as the 2nx2n matrix partitioned in nxn blocks
as
T(t)
=
[f
-
(32a)
eb (t)
=
then carrying out the calculation in (33a), it
can be shown that H is diagonalized with diagonal blocks
q
Hf = -[A'
(34e)
- bBQB]
(34f)
(34f)
and
(-A'
b
Thus the dynamics of qf and qb are decoupled and
are given by
f = Hfqf + C'R
a
1
y
(35a)
y
(35b)
and
qb =
Let the transformed process be denoted by
|qf(t)
C'R
bqb
If we assume for time-invariant dynamics
that {A,B} is stabilizable and that {A,C} is
^x(t)
= T(t)
q(t) =
(32b)
q (t(t)
Lb
Also define
'iq = TT
efBQB']
+
(33a)
+ THT
detectable and for time-varying dynamics that
{A,B} is uniformly completely controllable and
{A,C} is uniformly completely reconstructable,
then the invertibility of P in (34b) is guaranteed if both 8 (0) and 8 (Ty are nonnegative
f
b
definite [ 9]. Furthermore, these conditions
guarantee that 6
and eb and their derivatives
are bounded and that Hf and Hfbare forward and
backward stable respectively.
Under the transformation (32a), the boundary
condition (31b) becomes
and
=
G
TG
(33b)
q
so that the dynamics of the transformed process
can be written as]
[0 O' II-1
W
b Yb
H
+
q]
q
Gy
= (t)
_8
t
I
b
b
_ _ _ _ _ _
) f(t
f
_s
Vq = V
_(t)
V
_
q
=V
T
x(
(T)
Of(0) = V-0
-1
b(t)]
,
and if we choose the dynamics for 6f and
(36b)
(
(36c)
.
To simplify the expressions for the boundary
value coefficient matrices in (36b) and (36c),
choose the following nonnegative definite initial
and final conditions for the Riccati equations
(34c) and (34e):
where
V
+ w
+
f
fBQB'6f - C'R- C
b
a
as
EBQB'b
b bb b
(37a)
and
(T)=
V
Then defining e
= 6bA + A'6b-
W
(37a)
(34c)
and
-b
0'
(34b)
T' -1 T
-0f = efA + A'6
(0 q
) b(T)
and
0OP(t)
bt
P (t) = [ef(t) +
s
f
v
+
q Lq
where
If we use the following form for the inverse of
T:
T-
qf(T)
(33c)
.
q
I
1
]T)
[
f(O)
(36a)
wT'
=
qr-i)
0
+
C
(34d)
b C'R ~~~~~
e
T
+
T' -i T
T
.
(37b)
(37b)
as the following nxn matrix:
+
= vT' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'c
lD CWT'b
(38)
vb
it can be shown that the boundary value coefficient matrices can be written as
.O)8P 0fb
(O 1
[ I
qVfcPs()
[ w
qf(t)
_
)
=
(t)Ffb
v0b
bv
]
(
(39a)
0
0cPs (O)qb (0)
and
~~el~P~ (T)
P (T)
c s
[VT
e
s
~The
:
q
V
I
V
(39b)
.
f
.
b
Since the dynamics of qf and qb are decoupled, the only coupling between the two enters
through the boundary condition. By our choice
of initial and final conditions for the Riccati
equations, we have been able to display this
coupling solely as a function of the matrix 0 .
inverting T(t) in (32b) so that we obtain
^(t) = P (t)[qf(t) + qb(t)]
.
(40)
two-filter solution for %and q is formulated
(35a) and (35b) re ecte
overned
b
y
(35a) and (35b) respectivery with boundary
conditions: q° (0) = 0 and $q (T) = 0. Define
Ff
q band
Ffb
and f fb as the 2nx2n matrices
fb a
f (TO
f(T,0)
f
+
b
T
b
(41 )
(41a)
1
I + 0'P
=
(T)(f(TO ) .
e'P (T)
a s(T)(Df(
c s
c
- - - - - - - - - - - - - - - - - - cP
(0)
I +
cPs ()P
(C,T
and
f(t,O)
Ofb(t)=
:
~(38)
0
-L- --
(41b)
SMOOTHING ERROR
From (24b) we see that the smoothing error
is the solution of the augmented and inverted
system projected onto the span of the complementary process Z. By expressing the complementary
process as a linear function of the underlying
process and employing the internal differential
realization of the inverted system it can be shown
[8], that the smoothing error has the same
Hamiltonian dynamics so the smoother (31a) and
that the boundary condition for the smoothing
error can be written in terms of VO and V
in
.
vx
(31d) and (31e). As a consequence, the same
diagonalizing transformation (32a) can be used
to write the error as a linear combination of a
forward stable process and backward stable pro-cess. From this representation it can be shown
[8] that the error covariance can be expressed
8
in terms of Of, b and 6c in (34c), (34d) and
plus the solution of one additional matrix
Riccati equation.
fb
6.
Then the two-filter solution for qCt) is given
by
(41c)
Lq(t)
computational complexity of the noncausal smoother implementation suggested by (41)
is nearly the same as that of the two-filter
smoothers for causal processes such as the
Mayne-Fraser form [11,121.
We note, however,
that before qf and q can be evaluated for any
t S [O,T], bot qf and qb must be computed and
stored along with P and Q
for the entire ins
fb
terval EO,T].
Thus, the required storage exceeds
that of the smoother for causal processes. Indeed, the Mayne-Fraser solution and ours differ
significantly in one aspect. That is, for our
smoother the contribution of the forward filter
to the smoothed estimate at some point t depends
not only on past observations, as does the MayneFraser solution, but also on future observations
through the term
through the term Oc'P,(T)qf0(T) in (14). A
similar statement applies tor the backward
filter contribution.
5.
Following (8), an explicit expression for the
Ffb
Yb
ecPs(V]
E[
_[
{
b,
CONCLUSIONS
An internal differential realization of the
fixed-interval smoother for an nth order noncausal two-point boundary value stochastic process (TPBVP) has been shown to have the same 2nth
order Hamiltonian dynamics as the fixed-interval
smoother for causal processes. The simplicity of
this two-filter form is achieved by employing an
information form for the diagonalizing transformation with carefully chosen boundary conditions
for the differential equations governing its
elements. The significant difference between
our two-filter implementation and that for causal
processes is that in the noncausal case the
smoothed estimate at a given point in the interval is a noncausal function of each of the forward and backward processes (see (40) and (41c)).
In closing we note that differential realizations for estimators of both discrete and continuous parameter multidimensional stochasticems,"
processes can be formulated as well by the method
of complementary models. The solutions to these
multidimensional estimators and problems associated with their implementation are addressed in
[8].
[10]
Bryson, A.E. and Frazier, M., "Smoothing
for Linear and Nonlinear Dynamic Systems,"
TDR 63-119, Aero. Sys. Div., WrightPatterson AFB, pp. 353-364, 1964.
[11]
Fraser, D.C., "A New Technique for Optimal
Smoothing of Data," Ph.D. Dissertation,
Dept. of Aero. and Astro., MIT, 1967.
[12]
Mayne, D.Q., "A Solution of the Smoothing
Problem for Linear Dynamical Systems,"
Automatica, V. 4, pp. 73-92, 1966.
Automatica, V. 4, pp. 73-92, 1966.
Acknowledgements:
The work of M.B. Adams was
supported by a Draper Fellowship, and the work
of A.S. Willsky and B. C. Levy was supported by
the National Science Foundation under Grant
ECS-8012668.
REFERENCES
[1]
Krener, A.J., "Acausal Linear Systems,"
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Kailath, T.
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Adams, M.B., Willsky, A.S. and Levy, B.C.,
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Adams, M.B., Linear Estimation of Boundary
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