September, 1984
LIDS-P-1403
LINEAR SMOOTHING FOR DESCRIPTOR SYSTEMS
Milton B. Adams*
Bernard C. Levy** Alan S. Willsky**
The Charles Stark Draper Laboratory
555 Technology Square
Cambridge, MA. 02139
The linear smoother for an n-th order two-point
boundary value descriptor system (TPBVDS) [1] is formulated by using the general linear estimator solution
developed by the method of complementary modelshin [2].
The smoother is shown to take the form of a 2n
order
TPBVDS. By employing the solutions to generalized
Riccati equations 13] it is shown that the smoother
th
dynamics can be decomposed into two n
order descripdnmc
ca
bdeoosdittonostochastic
tor form equations. The implementation of the smoother
solution is also addressed.
1.
Introduction
The class of descriptor systems was introduced by
Luenberger [1] to describe the dynamics of certain
linear systems for which the standard state-space representation is not applicable. The properties of
processes of this type have been studied by a number of
investigators
and
and
(e.g. the , optimal
investigators (e.g. 141, [51 and [6]) and the optimal
control problem has been examined in [7]. In this paper
we discuss the fixed-interval smoothing problem for an
nth order TPBVDS. The solution is formulated as a 2nth
order TPBVDS by an application of the general linear
estimator solution developed in [21 via the method of
complementary models. The implementation of the smoother solution is also addressed. A general solution for
a "regular"* TPBVDS is established along with a condition for well-posedness. This general solution is
developed as a linear combination of (i) the solution
to stable forward and backward recursions corresponding to processes with standard state space dynamics and
(ii) the two-point boundary value. Following the methodology in [8] for the decomposition of the smoother for
two-point boundary value processes satisfying statespace form dynamics, the dynamics of the 2ntn order
TPBVDS are also decomposed into two nth order descriptor form representations. This decomposition can be
viewed as the generalization to descriptor systems of
the Mayne-Fraser smoothing formula [131 for standard
linear systems. An application of the general solution
to these two lower order descriptor form systems yields
the smoothed estimate.
Some aspects of the smoother and its implementation remain unresolved. First, the forward/backward
general solution is only applicable for a regular
TPBVDS.
General conditions for regularity of the
Sesmoother solution have not yet been established.
h
cond, the decomposition of the 2n
order TPBVDS smoother into two nth order descriptor forms requires the
Massachusetts Institute of Technology
77 Massachusetts Ave.
Cambridae, MA. 02139
existence of solution into two nth order descriptor
forms requires the existence of solutions to "generalized" Riccati equations [31, and general conditions
for the existence of such solutions have not been
established.
2.1
The Model
In this
section the model for the discrete linear
TPBVDS is introduced,and aforward/backward
form of its general solution is developed.
Let u, be
form of its general solution is develooed. Let u, be
an mxl white sequence on [O0,K-ll with constant covariance matrix Q. Let E, A and B be nxn, nxn and nxm
matrices respectively with the pair (EA} comprising
is not reouired to be
a regular Dencil [91. Thus, E is not required to be
K ]
partitions as [VO:V
,and let v be an nxi random vecpartitions as [
and
et
be an nx random vector uncorrelated with u and with covariance matrix
v
Then the discrete TPBVDS satisfies the difference equation:
Exl
=Axk
B
(la)
with
two-point boundary condition
0
K
(lb)
v = V XO + V xK .
The realization of the smoother developed in [21
by the method of complementary models requires anoperator representation of the process dynamics and boundary
condition.
The appropriate operator representation for
(1) is developed as follows. Let the set of points
between 0 and K-1 be represented by Q = [0,K-1], and
let the boundary of this region be defined as Q2= {0,K}.
With D representing the unit delay, the dynamics of the
nxl vector process in (la) are given by the first order
difference operator:
n(D U)
(2a)
2
2
defined notationally as
L
L = (D-1E-A)
and operationally as
k = Exk+l
Viewing the matrix V as the operator
V: R2n
Rn
the dynamics and boundary condition in (la) and
can be expressed as
L = Bu
vx
c
= v
2c)
(2d)
(lb)
(3a)
(3b)
This work was supported by the National Science Foundation under Grant ECS-83-12921.
C.S. Draper Laboratory, 555 Technology Square,
Cambridge, MA. 02139.
.
Department of Electrical Engineering and Computer Science, and the Laboratory for Information and Decision
Systems. M.I.T., Cambridge, MA. 02139.
*Conditions
for regularity are given in Section 2.
~~~·3Cl~~~~illCII~~~~~tr··~~~*^·;.I
~~
;:~~~
=
2.2
(3c)
A General Solution for the TPBVDS
A general solution for the TPBVDS in (1) is form.ulated here as a linear combination of two stable recursions, one forward and one backward. This form of
the solution both lends insight into the properties of
where
o
descriptor systems defined by (I) (e.g. weil-posedness)
and will provide a means for implementing the smoother
that is developed later in Section 3.
that later
is developed
in Section 3.
Given that {E,A} comprises a regular pencil there
exist nonsingular nxn matrices T and F such that
and
(4a)
and AJ
Af
01
FAT
Ffb
(k)FOlv v
[Xf b
where
k)
f
Txregular
and x
that
K
-
'K
.
from (9a) into
the solution
Vb0b 0
x
]
(10a)
L
0:
9b(kL,K
(lOb)
Applying the inverse of the transformation in (5), the
original process xk is recovered by
-1
T
k
(
,
In this way, we have have constructed a stable, forward/backward two-filter recursive implementation of the
general solution for the TPBVDS defined by (la) and
(lb) with the only constraint being that {E,Ai form a
pencil.
Xfk h(
Tx(5)
Lbk
K
K) + V]
-(k,
fb (kf
~x
f'k-i
O
Vb
Finally substituting for x
(8a) and (8b), it can be swn
to (6) is given by
(4b)
where As is nlxnl,A is no x n2 n
n
+ n2, and all
eigenvalues of Af and A. lie within the unit circle.
These properties of Af and Ab can be obtained as a modification of the decomposition of a regular matrix pencil as described in (9]. The decomposition in [9]
splits the pencil into forward dynamics corresponding
to a pencil of the type zI-A; and into backward dynamics
corresponding to z-iI-Ab where Ab is nilpotent. The
modification (4) simoly shifts the unstable forward
modes of Af into the backward dynamics Ab . Employing
T in (4) to define the equivalence transformation
K
+ Vf(K,
The well-oosedness condition for (la) and (lb) is
kthat
and multiplying the dynamics in
(1) becomes decoupled as
x fk+l = A xf
- B
ff k+l 4f
= Afxf,k
k
(la) on the left by F,
-uk
(6a)
(6a)
F-b in (9b) be invertible. That is, for x to be
uniquely defined on [O0,K] by the input uk and boundary
value v,Ffb must be invertible.
This invertibility
criterion can be used to show that if E is singular,
(lb) must be a two-point (or multi-point) boundary condition (i.e. (lb) cannot be an initial condition) as
follows.
k
A l - B(6b)First note the E singular implies that Ab is singular and consequently that Ob is singular. For an
initial value problem, VO
I and Vi * 0, and therefore, Vf = 0 and Vb = 0. Thus, for an initial value
(
problem Ffb becomes
(6c)
fb = [Vf
0b
b( 0,K)]
A D Xb,k+l -BbUk
%b,k
where
F Bf1
= FB
- I
Bb
F
.
Note that A
is forward stable and A
is backward sta=
ble. Given the transformation in (5T, the boundary
condition in (lb) takes the form
0V.
v
VI
f
bf
fo +
:
0
LXxb,oK
[fV
'
Vb
[V
V
f
(7a)
0°
b(OK
Clearly, a necessary condition for F
lar is that 9. (O,K) be nonsingular.
to be nonsinqu-
However, this does
not hold whenDE is singular.
where
2.3
0. 0
[V .V
0T-1
-V T 1and
£ K
vV<] E V T
(7b)
Employing the forward/backward representation of
the dynamics of the TPBVDS in (6), a general solution
,k as ths soluDefine
to (1) is derived as follows.
tion to (6a) with zero initial condition'
and xbre
fosution
hk
toas
the solution to (6b) with a zero final condition.' Note
that both can be computed by numerically stable recursions.
With f the transition matrix for A and
the
transition matrix for A, one can write
0
xf
k
= )f(k,O)xf
+x
k f
f,4O
(8a
(Ba),k
Green's Identity
In addition to the operator representation of the
TPBVDS in (3) the smoother solution we develop later
also requires the formal adjoint difference operator L
which is obtained through Green's identity for L on Q
Green's identity for discrete processes is ob[21.
tained from summation by parts of the inner product
K-1
=
2 <0.-l
2
Yk
k
=[
K-i
K
Yk(EXk+l
A-xk)
(12)
and
Xbk = )b(kOK)xK
Xok
(8b)
Substituting for xf K and xb
(7a) and solving fof
K
xf 0 and
[f
- ------
- -----
-
`N"~~.i
F1
vKO
V
-
~ ~
i·
'`(
O
from (8a) and (8b) into
xb, gives
- V0
(9a)
`~·r
Summation by parts can be interpreted as the counterpart of integration by parts as follows:
jIntergration by Parts
T
udv - uv
!
-
T v
vdu
0
iSummation by Parts
*__ _ _ _ _ _ _
K-1
Uk+L(Vk+l
}
-
-u v ) -
((u v
form for the estimator in equation (3.17) of
-)
v(
k
__
The term on the right hand side of the identity for
sunmmation by parts in (13) has been obtained simply by
shifting the index of summation on the left hand side
To
and adding (ukv-uOv) to account for the shift.
01
QB
,
_ k+l
E k
- l-C'R C
A
k
.
kCR
(20a)
K-1
0(Ex+l
.
(EXk+l-'kk0
Yk
k
x'Iwith
(Er'yk-AIYk)
xk k
k-I'AY k
Z
boundary condition
(W'T61W + V'Tv
W·Ilb
T
vb
xEy_ + x
_1
K
K-lL
and and r
=o{-1
X' =
b
=
0
1
L
(20b)
1-0
L K-1
As mentioned earlier, it will be convenient to
write the smoother in terms of the shifted process A
defined in (17b).
In this way the apparent four-point
boundary condition in (20b) becomes a two-point bouncondition. Furthermore, if we were to sDeciaiize
to the case of causal processes (E=I, VO=t, V--'O), the
smoother takes the traditional form of the discrete
fixed-interval smoother (see e.g. [101).
Thus, in
terms of X and Ab (20a) and (20b) become
f1~dary
(t15b
(
-
L
fK-1
°0 +
,i
(15a)
= DE' - A' ,
E.
V)
(14)
,_'yK
.
Defining
L
In
The formal adjoint
ply
..given by matrix transpositions.
difference operator L , the matrix*F, and xb and y
(temporarily y will be used in place of ) have all
been determined in the derivation of Green's identity.
The resulting smoother dynamics are given by
_.
_
the form of (13), we perform the
put (12) into
same type of index shifting to write
[2].
, C. W and V are all sim-
this case, the adjoints of
kuk+l -k
(13
k-l
(19b)
The minimum variance estimator of x given the observations y and y can be written by substituting
from the operator description of the TPBVDS in (3) and
Green's identity in (15) and (16) into the operator
)
! k=O
I K K 0 0°
i __________
v _= Wx + r
b
-b
W b
_ __
cd)
E
Green's identity can be written directly from (14) as
-Lx,'>
<x
-L Yx>
>E
1
+
2
2
-
C'R
0E
C
Ak+
k4-l
Yk
~k
(i.e. X = Dy)
.
(17a)
XklC
k
A'
Yk
(2 la)
We will see later that the estimator can be expressed in a simpler form if written in terms of a
shifted version of y, which we denote by A:
W
' bYb
LWKJ
=
°
0.'0
+ W0'RblW0
LVK-lV
IvK',-l vO
+W w
wkh
b1
-E'
0
0
°
0
I-IA
In terms of the shifted process A, yb is given by
b
b
(17b)
]
AK
+ LV
v -[v
K'
V
Given Green's identity as expressed in (15) and (16),
in the next section we establish an internal difference
realization of the smoother for the discrete TPVBDS.
E
E =
3.
3.1
The Smoother and Two-Filter Imolementation
o
E
1
-1 K
b W
'
1K
W
(21b)o
K
'-BQ B'
--_
-A'
(22a)
The Smoother as a TPBVDS
The observations for the fixed-interval smoothing
problem are comprised of an observation of the process
x at each point in the interval [O,K-1] as well as a
Let C be a
boundary observation as described below.
pxn matrix, and let W be a full rank qx2n matrix with
fin, with the rows of W linearly independent of those
of V in (2.3) and with qxn partitions:=
O3:CK
W= [W W]
(18)
Let rk be a pxl white noise process over [O,K-lI whose
covariance matrix R is nonsingular and constant on
[0,K-11.
Let r, be a qxl random vector with nonsinoular covariance iimatrixb. In addition, u, v, r and
rb are assumed to be mutually orthogonal. The observations are defined by a process Yk
k
uk = C-k + rk on
,K-l]
and a qxl boundary observation
(19a)
A
0
-E'
A =
C
(22b)
1
and
(22c)
CR
the smoother dyanmics in (21a) can be written in descriptor form as
kll
E
I
k+l
*
In [21
.
k
'
(23)
Y
is denoted by E.
A change in notation is
required since E is used here in defining the dynamics of th TPBVDS in (la).
3.2
Impelmentation of the Smoother
k
Assuming* that the pair {E,A} is regular, one
could immediately employ available numerical techniques
(111 to find matrices F and T analogous to those in
(4a) and (4b) to obtain a solution for [(,Al as described in (8) through (11).
Rather than taking this
numerical route directly, we will take an intermediate
step by which (23) is further simplified revealing
some additional underlying structure of the smoother
dynamics.
As will become apparent, this intermediate
step has been motivated by the two-filter fixed-interval smoother solutions for non-descriptor form systems
(i.e. systems for which E - I) [8].
In particular,
we will find 2nx2n matrix sequences Fk and T
which de-1
k
compose E and A as
ia
[-1
copsr.n
fk
'I
FkETk+
-
0
and FAT
-
Abk T22
I
fk
Tk
F
-
k
QB'
k
k
+F
A'
(28g)
k
A +F
C
C'R
(28h)
*
(29)
Eb,k
[e·b
(29) along with the eight relations in (28)
yields
[
(25)
S
(28f)
Lec
Employimg
Eb,kl
f,k
k+l
F
Tk
2b
,kTkII= k(
A
E'
1
decomposition and defining
f,k+l
(28e)
12
k
E f,kk
Ee f
A
A[I - (C'R
fk
f
f,k
F
T
A
E
21
T 12
12
the 2n
order smoother dynamics in (23) become decoupled into two nth order descriptor forms
E
k
F
f,k
,
(24b)
-
[f
k+l
A. k T
ok
21
A
d
22
(24a)
A kc+l
Given this
k
21
Ab,k.
-
k
21
Motivated by the decomposition of the smoother dynamics
for state-space form two-point boundary value systems
in
[81 and (121, choose
0
[_
-
b,k
k
bbk
(30a)
f,k
C
k)
+
C'R -C]
-A(C'R-1C + efk
(30b)
ef
fk
fk
-)
C'R
(30c)
E'
(30d)
+
+(26a)
B
f,k
f,k Yk26a)
and
Bbk = C'R
bo,k Xbk+l
Ab,k
b,k + Bb,k Yk
(26b)
Ab k
(30e)
-
)
A'tI + (BQB' + bk -
BQB
-
(30f)
where
-F
]*
…
F
(26c)
k
It can be shown that in order to achieve the decomoosition in (26a) and (26b), the partitions of F and T
k
k
denoted by
Kwhere
k
F11
k
k
Fk
k
kc
k
F12
11
k
k
and
Tk =
Tk
22
21
E
E
b,k
22
E
12
k
(28a)
k
28b)
-
F1 1
BQ'k
BQB'
An outstanding problem is
+ F1 2
k
A'
f,k+l
E'
A(C'RC
+
)
f k
f,k
A'
+ BQB'
(31a)
22
,'9bk
E - A'(BQB'
+ 8b,k+l )
A + C'R
C .
(31b)
Equations (30a) through (30f) along with equations
(26a) and (26b) completely define the decoupled dynamics for the transformed processes xf,k and Xb, k .
An expression for the boundary condition for the
tranformed processes can be developed as follows.
First rewrite the boundary condition (21b) for the
untransformed smoother processes in a more compact
form:
22
E +l
f,k
~~b,k
-1
(27)
F11
T
~
(3
E
kc+
~
Riccati equations:
must satisfy
Ef,k T11
-.
k
8f and 8b satisfy the following generalized
Ic
T12
k
-1
'-
(28c)
the determination of the
conditions for which (E,A} is regular.
0,KI
4
][
[
0
K
EI
With the inverse of equivalence transformation Tk given
by
It
-1
fz]
k-1
' O
' f,k
f,k
k
3
T
- - --- - - - - - (33a)
E'
X
e k
-1
yields a representation for the estimation error dynamics and boundarv condition.
Emplovinq equations (3.20)
and (3.23) in [21, the smoothing error for the TPBVDS
can be shown to, satisfy descriptor form dynamics:
E[Eu]
where
+[xkl
ik+J
3fk
Zk
(
'k
+ E'9b,k E
kEB
f,k
(33b)
EV + V)B
,k
f,k
the boundary condition (32)
(33c)
can be written in
the transformed processes xf and xi. as
kJ
(36a)
with boundary condition
v
-IV
v
WIff
La
lr
O
[
+ E'b,0
E) Z0
f
-I)E
f i
f- -0
-- I
0
(IE'SO
f=
V0,KZo
VO K'ff,00
V:V SK
1,K ZK
EO
V
[~~~~v0,K
x
xK
-E',
o~~~~0
0
.
-
K
K
E6
ffO
f
-1
0K fK
fK K K
- - - - - - - bk-1
K
+I- Ef,,
-b.
0
XO
terms of
W'!T
~
yb+
(V
rk
-C'R
.
(36b)
where the error is defined as
x - x,
and A are
defined in (22), and the boundary condition (36b) is
written with the same notation as that used to express
the smoother boundary condition in (32).
Lbi
The covariance of the smoothing error can be computed by an anolication of the difference equations
formulated in the Appendix.
In order to apply those
equations, the error dynamics (36a) must first be
transformed to the decoupled forrard/backward form in
(6a) and (6b) with the boundary condition transformed
(34)
There is an alternative to directly transforming
Given the expression for the inverse transformation in
(33a), the estimate x can be recovered from the forward and backward processes by inverting (25) to give
(36a) into forward backward fornm. That is, given the
similarity between the error dynamics (36a) and the
smoother dynamics (23,
the transformations Tk and
in (29) and (30) can be used to first decouple the k
=j
That is,
Xb
fk + I
the estimate
E'ef*lfZk
k
is
(5
k
linear' combination of the
error dynamics into two nth order descriptor forms
f]k
transformed forward and backward processes.
Of course to recover
solve
(26)
xk
e
Tk
via (35) we must first
eb
T
(37)
kk
with boundary condition (34) to get xf and
xb. However, the general solution presented in Section
2 is only applicable for the case that E
E. , Af, Ab
Bf and B, are constant.
To satisfy this ondition, we
can evaluate (30a) through (30f) with the steady state
and then solve for the error covariance in
these dynamically decoupled processes.
solutions to the generalized Riccati equations for
and 9
in (31a) and (31b).
eb o
c
e
s
e
o
a
o
t
s
r
The objective has been to transform the smoother
The smoother and smoothing
tf
error equations for a
TPBVDS have been derived.
A stable method for obtaining a numerical solution for a regular mPSVDS as a
linear combination of forward and backward recursions
dynamics into two lower order descriptor forms. The
Potential benefits are twofold.
The first is that the
computation required to compute the two nth order solu-
and the two-point boundary value has been developed
an
th
w
-node
cnio.
alon(6
with a well-osedness condition.
th
tion is less than that required for a single 2 nth order
system.
The second and most important is that the
descriptor form dynamics of the lower order systems may
be more easily analyzed in terms of whether or not they
The smoother for an n
order system is shown to
be a 2nth order TPBVDS.
With respect to the smoother,
the following remain as areas for further investigation: (1) well-posedness or smoothability conditions
represent a regular pencil.
For instance, if we assume
that a steady state solution 93 exists for (31a) then
[14] need to be established, (2) conditions under
which the 2nth order TPBVDS smoother is regular as
the forward dynamics (26a) become
defined in Section 2, (3) conditoins for existence of
solutions to the generalized Riccati equations (31a)
and (31b), and (4) how might the smoother decomposition simplify the determination of smoother regularity.
E9,
x
A
(C'R
C +
-1
f
- A(C'R .* + 3)
CR Cf
xfk
f~k~l
f)
f
-L,
-l
I CR
k
~u
A question which is unanswered thus far is: given that
,E,A} forms a regular pencil, under what conditions do
1
IE,A[I-(C'R C + )f)-lC'R-kCl} form a regular pencil?
4.
The Smoothing Error
In addition to providing a general representation
'of the estimator dynamics and boundary condition, the
method of complementary models as employed in [21 also
terms of
Since the 1-D representation of the discretized
dynamics of 2-D systems can be written in the form of
a multi-point boundary value descriptor system (MPBVDS)
' [121, an extension to multi-coint problems of the
smoother solution developed here would be auite useful..
A general solution for ^1PBVDS similar to the forward/
backward form for TPBVDS in Section 2 has been developed in
121.
However, the extension of the smoother
solution to problems of this type is still
under
investigation.
5~r^-C;~Thesmoot error
ing
euf
o
r
a to
References
(11
and
D.G. Luenberger,"Dynamic Equations in Descriotor
Form," IEEE Trans. Auto. Contr., Vol. AC-22, pp.
312-321, June 1977.
[(21 M.B. Adams, A.S. Whisky and B.C. Levy. "Linear
21 M.B. Adams, A.S. Willsky and B.C. Levy, "Linear
Estimation of Boundary Value Stochastic Processes,
Part I: The Role and Construction of Complementarz Models", to appear in IEEE Trans. Auto. Contr.
0 nk)
fb
A.J. Laub, 'Schur Techniques in Invariant Imbedding Methods for Solving Two-Point Boundary Value
Problems," Proceeding 21St CDC, Orlando, Florida.
Dec. 1982.
[4]
D.G. Luenberger, "Time-invariant Descriptor System,
Automatica, Vol. 14, pp. 473-480, 1978.
(5)
G.C. Verghese, B.C. Levy and T. Kailath. "A Generalized State-space for Singular Systems, IEEE Trans.
Auto. Contr., Vol. AC-26, pp. 811-831, Aug. 1981.
(61
Lewis, F.L., "Inversion of Descriptor Systems,"
Proceedinqs 22nd CDC, San Antonio, Texas, Dec, 1983.
[7]
D. Cobb, "Descriptor Variable Systems and Optimal
State Regulation," IEEE Trans. Auto. Contr., Vol.
AC-28, pp. 601-611, May 1983.
(81
M.B. Adams, A.S. Willsky and B.C. Levy, "Linear
Estimation of Boundary Value Stochastic Processes.
Part II: 1-D Smoothing Problems," to appear in IEEE
Trans. Auto. Contr.
[91
F.R. Gantmacher, The Theory of Matrices, New York:
Chelsea, 1964.
(101 A.E. Bryson and M. Frazier, "Smoothing for Linear
and Nonlinear Dynamic Systems," TDR 63-119, Aero.
Sys Div., Wright-Patterson AFB, pp. 353-364, 1964.
0
fn b
k
Difference equations for each of these three covariances
0
can be derived by substituting expressions for x and
represented as weighted sums of
on
into each of
represented as weighted sums of u
the expectations in (A.l). In this manner, it can be
shown that
(1)
(3]
E0
Pf(n,k)
9f(n,k)Pf0(k,k)
where
Pf(k+l,k+l)
0
P (0,0)
I
(A.2a)
(k+l,k)P (k,k)$(k+l,k)
+ BF kB
_ kkfk'
(A.2b)
0
(
0
(2) P (n,k)
A
where
k
b
0
P 0 (k,k)1'(k,n)
b
b
(A.3a)
k
b
b
b
0
+ BP
(KK)
+'Bb~k~kB~,;
Pb(K,K) =
0
b,k kB ,k
b
and (3) for n > k
0
0
0
PLn,k) - G
n$'(k,n) - ~f(n,k)fk
fb
fbn b
f
fbk
where
0
0n
fb,k+l - f(k+l,k) fkl
(k+l,k)
+ B fkkB,k;QkB
k
and for n < k,
Pjn,k) =
A3b
(A.3b)
(A.4a)
0 .
0
(A.4b)
.
(A.4c)
Given these three covariances, the covariance of xk:
=E tExkxk]
=
Pk
E[xkx] f,k
- Tk{
can be written as
.x
]
lITT,(A.5a)
'[x
k(A.5a)
(k)'
+-(k)'
k
1+
Sk)
+
P = Tk
Pk
_kt
P
[11] P. Van Dooren, "The Computation of Kronecker's
Canonical Form of a Singular Pencil," Linear Alae-(k,k)
bra and its
ApDlications, Vol. 23, pp. 103-140,
1979.
fi
b(k)F-11 -lj't,
bfb
v fb
lr
Wk) + E-(k)
k
}
-1'
(A.b)
,k
0
b
where
[12] M.B. Adams, "Linear Estimation of Boundary Value
Stochastic Processes," Sc.D. Thesis, Dept. of Aero.
and Astro., MIT, Cambridge, MA., Jan. 1983.
[13] J. Wall et al., "On the Fixed Interval Smoothing
Problem," Stochast'ics, Vol. 5, pp. 1-41, 1981.
[141 A. Gelb, Aooiied Optimal Estimation, M.I.T. Press,
Cambridge, .A., 1974.
The Covariance of a TPBVDS
Appendix:
-l
K O
0
(k) = (k)FlffvKPO(K,k) + Pb(K,k)l
0
V
)
0
+ VbP b(k0) + P (0,k)]}
b
f
b
and
(k)
$(k)
(k)
= %lb(k)
Pf(K,K)
P (,O)
F -lVK 0
_
Fib [Vf:Vb]…iP '(K,O);
P 0(
)
fb
A set of difference equations from which the covariance of a TPBVDS can be computed are presented in
this appendix, Since the smoother error is given as a
TPBVDS in (36), these equations are especially useful in
evaluating the performance of the TPBVDS smoother derived in this paper.
The starting point for the development here is the
general solution in (10a) for the TPBVDS defined by (la)
and (lb).
Recall that the boundary value v is assumed
to be orthogonal to the inout uk throughout
- (0O,K-11.
Thus, v is orthogonal to x k and x
in (lOa) for all
-k
f
0 k
k in . so that the covariaiine of xk In (11) can be
-written as a linear combination of .v, the covariance of
v, and the following three covariances:
(1)
P (n,k)
f
S Etx
(2)
Pb(n,k) - E:x
f,n
x
'
f,k
nxb k'
I
,
(A.la)
(A.lb)
(A.5c)
(A.hc)
b
V
IV
jVb
(A.5d)
Thus, the covariance of the process xk can be computed
for any point k in i given the solution of the three
matrix difference equations (A.2), (A.3) and (A.4).