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Smoothing for multipoint boundary value models
Systems & Control Letters 17 (1991) 445-452 North-Holland
445
Smoothing for multipoint boundary value models * Howard L. Weinert Department of Electrical and Computer Engineering, The Johns Hopkins UniL~ersity,Baltimore, MD 21218, USA Received 26 April 1991 Revised 6 August 1991
Abstract." We show how to solve the linear least-squares smoothing problem for multipoint boundary value models. The complementary model is derived and is used to determine the Hamiltonian equations for the smoothed state estimate and its error covariance. Stable algorithms are obtained using an invariant imbedding/multiple shooting procedure.
Keywords: Boundary value models; multipoint boundary conditions; smoothing; invariant imbedding; shooting.
I. Introduction In this p a p e r we study the fixed-interval smoothing problem for linear state models with multipoint boundary conditions. One context in which multipoint boundary conditions arise is that of flexible structures, like bridges and moorings, which consist of components coupled end-to-end (see, for example, [4]). We will derive efficient, stable algorithms for the smoothed state estimate and its error covariance. A number of authors have studied structural properties and the smoothing problem for the case of two-point boundary conditions (see [2,3,5,7-12,14]). As we shall show, the multipoint case has certain characteristics which do not appear in the two-point case. The multipoint boundary value model of interest here is given by 2(x) =Az(x)
+Bu(x),
N-I E Viz(xi) =r,
x ~ [ x o, XN_I] ,
X0
(1) (2)
i=0
where x is typically a spatial variable and u is a zero-mean, unit-intensity white noise with m components, r is a zero-mean random n-vector with Err' = Q z is the n-component state vector, and A, B, V0, V1. . . . . VN- 1 are constant matrices of appropriate size. We assume that u and r are uncorrelated. We have continuous measurements given by y(x)
= Cz(x)
+ v(x),
x ~
[x0, xN_,]
(3)
where C is a p × n matrix and u is a zero-mean, unit-intensity white noise which is uncorrelated with u or r. Given the m e a s u r e m e n t s y, our objective is to find the smoothed estimate (in the linear least-squares sense) of the state z, and its associated error covariance. As a simple example of a multipoint boundary value model, consider a static Euler-Bernoulli beam with flexural rigidity/3 and a continuously distributed load with density u. In state space form, the beam * This work was supported by the Office of Naval Research under Grant N00014-89-J-1642. 0167-6911/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
H.L. Weinert / Smoothing for multipoint boundary value models
446
0]
model is given by (1) with
0 0 0
A=
1 0 0
0 ,
B=
1/~
where the first component of the state vector is the deflection of the beam at point x. Now suppose there are boundary conditions consisting of pins at x 0 and x 2 and a clamp at x 1. Then these can be represented by (2) with
Vo =
[i °° i] v [000 °1 l°°° i] 0 0 0
0 0 0
0 ,
1 0 0
0 1 0
0 0 0
0 0 ' 0
~ =
0 0 I
0 0 0
0 0 0
and with Q quantifying the looseness, or uncertainty, of the pins and clamp.
2. The complementary model The Hamiltonian equations for the smoothed estimate are most readily derived using the method of complementary models [1,2,6,13,15]. If Y is the closed linear span of the components of y, then a complementary model is, in the case at hand, a multipoint boundary value model which generates complementary variables, denoted Yc and 0, whose components span the orthogonal complement of Y in the Hilbert space spanned by u, v and r. In order to derive a complementary model, we must first solve for z in terms of r and u. Integrating (1) gives
z( x) = q'( x, Xo)Z( Xo) + fxq)(x ~)Bu(~) d~
(4)
X 0
where q~ is the state transition matrix. Substituting into (2) yields N-I
N-I
r = E Vjcl)(x~, Xo)Z(Xo) + E Vjfxi"I'( x j, ~)Bu(~) dE. j=o j=o
(5)
The boundary value problem will be well-posed (that is, u and r will give rise to a unique z) if and only if the matrix EN=~l~q~(xj, x 0) is nonsingular. By multiplying (2) on the left by the inverse of this matrix, we can always ensure that N-1
Z ~.ffg(xj, Xo)=I.
(6)
j=0
In what follows it will be assumed that (6) holds. Solving (5) for z(x o) and substituting into (4), we get
z(x) =,/,(x, xo)r + fXN-'G(x, a)Bu(#) d( X 0
(7)
H.L. Weinert / Smoothing for multipoint boundary value models
447
where the multipoint Green's function G ( x , ~) is given by
• (x, xo)
VkeP(xk,~),
x>~,
U-~
G(x'~)=|-~(x,
E V,~(x~,~), x<~,
[
Xj <~
(8)
k=j+l
for j = 0, 1. . . . . N - 2. Now the complementary variables are given by
B'fx~ 'G'(¢, x)C'v(~) d~:,
Yc(X) = u ( x ) -
DO = r - QfXN_,~p,(~, x o ) C ' v ( ~ )
(9)
d~,
(10)
X 0
where D is any full-rank matrix satisfying
D D ' = Q. The proof is very similar to that in [13] (see also [1,2,6,15]) and will be omitted. Defining a complementary state by
Zc(X) =
fxlN-'G'(#, x)C'v(e)
d~:
(11)
we see that
y c ( x ) = u ( x ) - B'Zc(X ) .
(12)
Note that in contrast to the two-point case in which it is continuous, the complementary state here has jumps at the points x = x,, x 2 .... , XN_ 2. Just as with (9), we would like to express (10) in terms of z c. Defining 6 j = Z c ( X j - ) --Zc(Xj-~-),
j = 1, 2 . . . . . U - 2 ,
(13)
and using (8) and (11), we find that Zc( XN_I) = --Vl~_; f x u ,¢p,(~, x o ) C ' v ( ~ )
d~,
(14)
X o
6 j = -V/L~N-'qO'(~, x o ) C ' v ( ~ ) d~:, !
XN-I
Zc(X0) = V~L,,
(15)
!
dp (~, x o ) C ' v ( ~ ) d~.
(16)
Using the above along with (6), we can rewrite (10) as
DO=r-Q
Zc(Xo)-
E cb'(xj, xo)6 j - ~ ' ( x u _ l , x o ) z c ( x u _ , )
.
(17)
j=l
Next, we want to express z c as the state vector of a multipoint boundary value model like (1)-(2). Differentiating (11) gives Zc(X) = --AtZc(X ) -- C r y ( x ) ,
x E [ x 0 , XN_I] kk{Xj}j=l.U-2
(18)
H.L. Weinert / Smoothing for multipoint boundary, value models
448
The boundary conditions that go with (18) must yield a unique z c which matches (11). Since (18) is valid in each of N - 1 subintervals, we need a total of ( N - 1)n independent boundary conditions for a unique solution. Therefore the boundary conditions for the complementary state must have the form N
2
Kozc(Xl,) + ~ Kj6., +KN_,z~(x~_,)
= 0
(19)
j-I
for some suitable set of matrices {K~} of size (N - 1)n × n. In fact, if K 0 = [V 1
K,=[I
V2
0
K N 1=[0
VN_I]' '
-.-
""
O]'-Ko@'(xl, xo) ,
...
0
(20)
l ] ' - K 0 @ ' ( x N ,,xo),
it can be verified (see the appendix) that the boundary value problem (18)-(19) is well-posed and that its solution matches (11). Eqns. (12), (17)-(20) constitute a complementary model for (1)-(3).
3. The Hamiltonian equations The value of a complementary model now becomes clear because by cross-substituting between the original model and the complementary model, we can eliminate u, r, and t,, and then project onto the space spanned by the observations to obtain the Hamiltonian equations
~(x) =A£(x) +BB'£c(x ),
x ~ [ x o, x x 1],
~,.(x) = -A'2c(X ) +C'C£(x) - C ' y ( x )
'
(21)
x ~ [ x o, XN_,]\{Xj} u-z j=l'
(22)
with boundary conditions
where the 'hat' denotes linear least-squares estimate. Note that since the complementary variables yc and 0 are uncorrelated with y, their estimates are zero. By using invariant imbedding, we can transform (21)-(23) so that the resulting differential equations are partially decoupled and asymptotically stable. Let N(x) be the solution to the following Riccati equation on [x 0, Xu_l]:
N(x)= -N(x)A-A'N(x)+N(x)BB'N(x)-C'C,
N(XN_I)=O.
(24)
Then the coordinate change p(x) = N(x)£(x) + £~(x) transforms the above Hamiltonian equations into
~( x) = ( A - BB'N( x))2( x) + BB'p( x),
(25)
[x0, xN-,],
t
N-~
f~(x) = - ( A - B B ' N ( x ) ) p(x) - C'y(x),
(26)
with boundary conditions
Q@'(xj, Xo) ][2( xJ) ]= 0 -KoN(xo)
go
P(Xo) + i=l
0
where we have defined
{p( xj.-) -p(x ~P'=
p( XN 1 ) ,
+),
j=l,2
..... N-2,
j=N-1.
Kj
]t
~P'
(27)
H.L. Weinert / Smoothingfor muhipoint boundary calue models
449
If (A, B) is stabilizable and (A, C) is detectable, then (25) and (26) will be asymptotically stable in the directions of increasing and decreasing x, respectively. Note that the Riccati equation is identical to one that arises in smoothing for initial value models [1,15]. Other transformations are possible, for example using two Riccati equations, but the one used here is the most efficient in terms of the amount of computation required to find the smoothed estimate and its error covariance [14].
4. The smoothed state estimate
The transformed Hamiltonian equations can now be solved using a multiple shooting procedure• First solve the following initial value systems:
tAo(X) = - ( A - B B ' N ( x ) ) ' p o ( x ) - C ' y ( x ) ,
Po(XN_1) =O,
~o( X ) = ( A - BB'N( x ) ) 2o( X ) + BB'po( X ),
(28)
20(Xo) = 0.
(29)
The solution of (29) can be interpreted as the smoothed estimate of z(x) when V0 = I, Vj = 0 for j > 0, and Q =-0; or in other words, when the original model is an initial value model with zero initial condition [15]. Now it can be checked by differentiation that for xk_ ~ ~
2 ( x ) =So(X )
N-1
+ qt(x, Xo)~(Xo) + Y'. tit(x, xj)M(xi)3o + ~_, M(x)qt'(xi, x)6pj j=l
(30)
j=k
and for xk_ ~ < x
p(x)
=po(X) + •
qt'(x],
x)~p,
(31)
j=k
where qt is the state transition matrix of
A -BB'N(x)
and
I(I(x)=(A-BB'N(x))M(x)+M(x)(A-BB'N(x))'+BB',
M(xo)=O.
We can determine the unknown quantities in (30) by obtaining 2(Xk), k = 1, 2 . . . . , N - 1, and from (30) and (31), respectively, and substituting into (27)• This results in the following system:
[
/ Zo(X1)
=F'[20(X:N_,)
(33)
where
I q'(Xl,x0)-~(xl, x0) ], -K o
L=
p(x o)
[ po(Xo)
e(X0)
[ I + F1F2]
(32)
0
'
-A
-L
'
q.(x._~,x0)- ~(xN-t,Xo) M( xl)qt'( Xu_l, x,) ]
M(Xl)
M( x1)att'( x 2, Xl)
~( x2, Xl)M( Xl)
M(x2)
M ( X 2 ) a I t ' ( X N _ 1, X2)
ltt( XN- 1, X l ) M ( x l )
lI-f( XN_ 1, x z ) M ( x z )
M(XN-1)
H.L. Weinert / Smoothing for multipoint boundary ualue models
450
The well-posedness of (1)-(2) together with the uniqueness of linear least-squares estimates guarantee the nonsingularity of I + F 1F2. The components of the vector on the right side of (33) are obtained from (28)-(29). The solution of (33) is used in (30) to produce the smoothed estimate.
5. The error covariance
The smoothing error :~(x) is defined by
~(~) =z(~) - ~ ( ~ ) . Equations (1)-(3) and (21)-(23) imply that the smoothing error is also characterized by a multipoint boundary value problem as follows:
~(x) = A Y ( x ) +BB'(-2~.(x)) + B u ( x ) ,
x ~ [x 0, x N l], N-2
-~c(~) : -A'(-2c(X))+C'C~(x) +C'~'(x), x~[Xo, xN_,]\{xAj= , , ' ~ ~(xj)+
j =0
0
Ko
(-2c(X0))+ E
j=1
-gi)
Kj
QO'( xN l, A change of coordinates with
~(x) =U(x)~.(x) -2c(X),
tS~=
O(x1-)-~(xj +), ~:(XN-1),
/=1,2 ..... N-2,
j=U-
1,
transforms the above equations into
z(x)=(A-BB'N(x))~(x)+BB'~(x)+Bu(x),
X~[X,,,XN_,],
((X) = -- ( A - B B ' U ( x ) ) ' ~ ( x ) + N ( x ) B u ( x ) + C ' L ' ( x ) , Vo + QN(xo)
-Q
x E [ X o , Xu i ] \ {
(34)
Xj}j=IN-2,
xo)
(35) (36)
- K o N ( xo) As before, if we let
(o(X)= - ( A - B B ' N ( x ) ) ' ~ o ( X )
+ N ( x ) B u ( x ) + C ' U(x),
Zo( X ) = ( A - BB'N( x ) ) £o( x ) + BB'~o( x ) + Bu( x ),
~o(XN_,)=O,
ZT0(X0)=0,
(37)
(38)
then f o r x k l~
N-I
z T ( x ) = Y o ( X ) + ~ ( x , Xo)Y(Xo)+ Y'~ ~ ( x , xi)M(xj)a¢ + Y'~ M ( x ) q " ' ( x j , x)6¢, j=l
where
xo] Ilox oxo / ';' L a N_,
L~o(Xu-,)
(39)
j=k
(40)
H.L. Weinert / Smoothing for multipoint boundary value models
451
From [15] we know that
E[~o(Xo)~(Xo) ] =N(xo), E[5o(X)~6(Xo)] - 0 [ qt(x, s)M(s), x > s, E[5°(x)ia(s)] = ~M(x)gt'(s, x), x<~s.
(41) (42)
Therefore
P(x) =E[i(x)U(x)] = M ( x ) +R(x)(I+F1F2)
0
o, o]] F, (I+F, FE)'-tR'(x)
- R( x)( I + F, Fz)-'[ K°S'o( X) ] - [s( x)K o O]( I + F1F2)'-' R'( x ) where f o r x k_l~
"'"
(43)
M(x)alt'(Xu-I,X)]
and
R(x)=[qt(X, Xo) S ( x ) ] .
Appendix From (18),
zc(xo) = But Zc(X1 - ) =
Xo)Zc(X,-) + fxi'¢,'(~, Xo)C'v(~) d~.
~'(xl,
61 +Zc(X l + ) and from (18),
Zc(X , +1 = q~'(x 2, XllZc(X 2 - ) + f~2dP'(~, xllC'v(~ ) d~. Therefore
zc(xo) = q~'(xl, Xo)6 , + ~ ' ( x 2 , Xo)Zc(X2- ) + fx:dp'(~, xo)C'v(~ ) d~. Xo Continuing in the same fashion, we get N-2 Zc(X0) =
E j=l
t~C(Xj' XO)~jd-CI) ' ( X u - l , XO)Zc(XN-1)-'l-
fXN-'cP'(~, xo)C'v(~ ) d~. Xo
Substituting into (19) gives N-2
j=l
[Kj + Koq~'( xj, x0)]8 j + [ KN_ , + KO@'( XN_I, Xo)lZ~( XN_,) XN-I
t
+Kof£ ° • (~, xo)C'v(~) d~=0. With the {K i} given by (20), Zc(XN_ 1) and the {ai} can be obtained uniquely from the above equation, and
452
H.L. Weinert / Smoothing for multipoint boundar).' calue models
in fact a r e g i v e n by ( 1 4 ) - ( 1 5 ) . I n t e g r a t i n g (18) w i t h t h e v a l u e s g i v e n in ( 1 4 ) - ( 1 5 ) will p r o d u c e T h e r e f o r e t h e b o u n d a r y v a l u e p r o b l e m ( 1 8 ) - ( 1 9 ) is w e l l - p o s e d a n d its s o l u t i o n is g i v e n b y (11).
(11).
References [1] R. Ackner and T. Kailath, Complementary models and smoothing, IEEE Trans. Automat. Control 34 (1989) 963-969. [2] M.B. Adams, A.S. Willsky, and B.C. Levy, Linear estimation of boundary value stochastic processes, IEEE Trans. Automat. Control 29 (1984) 803-821. [3] A. Bagchi and H. Westdijk, Smoothing and likelihood ratio for Gaussian boundary value processes, IEEE Trans. Automat. Control 34 (1989) 954-962. [4] G. Chen, M.C. Delfour, A.M. Krall, and G. Payre, Modeling, stabilization, and control of serially connected beams, SIAMJ. Control. Optim. 25 (1987) 526-546. [5] C.E. Economakos and H.L. Weinert, The 2D Poisson equation: Computing smoothed estimates with discrete data, Proc. 25th Conf. Inform. Sci. Syst., Baltimore, MD (1991). [6] T. Kailath and M. Wax, A note on the complementary model of Weinert and Desai, IEEE Trans. Automat. Control 29 (1984) 551-552. [7] A.J. Krener, Reciprocal processes and the stochastic realization problem for acausal systems, in: C.l. Byrnes and A. Lindquist, Eds., Modelling, Identification, and Robust Control (Elsevier, Amsterdam, 1986) 197-209. [8] B.C. Levy, M.B. Adams, and A.S. Willsky, Solution and linear estimation of 2-D nearest-neighbor models, Proc. IEEE 78 (1990) 627-641. [9] B.C. Levy, R. Frezza, and A.J. Krener, Modeling and estimation of discrete-time Gaussian reciprocal processes, 1EEE Trans. Automat. Control 35 (1990) 1013-1023. [10] R. Nikoukhah, M.B. Adams, A.S. Willsky and B.C. Levy, Estimation for boundary value descriptor systems, Circuits Systems Signal Process. 8 (1989) 25-48. [11] L.R. Riddle and H.L. Weinert, Fast algorithms for the reconstruction of images from hyperbolic systems, Proc. 25th Conf Decision and Control (1986) 173-178. [12] L.R. Riddle and H.L. Weinert, Recursive linear smoothing for dissipative hyperbolic systems, Mech. Systems Signal Process. 2 (1988) 77-96. [13] H.L. Weinert, The complementary model in continuous-discrete smoothing, in: E. Parzen, Ed., Time Series Analysis of Irregularly Obserced Data (Springer-Verlag, Berlin-New York, 1984) 353-363. [14] H.L. Weinert, A note on efficient smoothing for boundary value models, Internat. J. Control 53 (1991) 503-507. [15] H.L. Weinert and U.B. Desai, On complementary models and fixed-interval smoothing, IEEE Trans. Automat. Control 26 (1981) 863-867.
445
Smoothing for multipoint boundary value models * Howard L. Weinert Department of Electrical and Computer Engineering, The Johns Hopkins UniL~ersity,Baltimore, MD 21218, USA Received 26 April 1991 Revised 6 August 1991
Abstract." We show how to solve the linear least-squares smoothing problem for multipoint boundary value models. The complementary model is derived and is used to determine the Hamiltonian equations for the smoothed state estimate and its error covariance. Stable algorithms are obtained using an invariant imbedding/multiple shooting procedure.
Keywords: Boundary value models; multipoint boundary conditions; smoothing; invariant imbedding; shooting.
I. Introduction In this p a p e r we study the fixed-interval smoothing problem for linear state models with multipoint boundary conditions. One context in which multipoint boundary conditions arise is that of flexible structures, like bridges and moorings, which consist of components coupled end-to-end (see, for example, [4]). We will derive efficient, stable algorithms for the smoothed state estimate and its error covariance. A number of authors have studied structural properties and the smoothing problem for the case of two-point boundary conditions (see [2,3,5,7-12,14]). As we shall show, the multipoint case has certain characteristics which do not appear in the two-point case. The multipoint boundary value model of interest here is given by 2(x) =Az(x)
+Bu(x),
N-I E Viz(xi) =r,
x ~ [ x o, XN_I] ,
X0
(1) (2)
i=0
where x is typically a spatial variable and u is a zero-mean, unit-intensity white noise with m components, r is a zero-mean random n-vector with Err' = Q z is the n-component state vector, and A, B, V0, V1. . . . . VN- 1 are constant matrices of appropriate size. We assume that u and r are uncorrelated. We have continuous measurements given by y(x)
= Cz(x)
+ v(x),
x ~
[x0, xN_,]
(3)
where C is a p × n matrix and u is a zero-mean, unit-intensity white noise which is uncorrelated with u or r. Given the m e a s u r e m e n t s y, our objective is to find the smoothed estimate (in the linear least-squares sense) of the state z, and its associated error covariance. As a simple example of a multipoint boundary value model, consider a static Euler-Bernoulli beam with flexural rigidity/3 and a continuously distributed load with density u. In state space form, the beam * This work was supported by the Office of Naval Research under Grant N00014-89-J-1642. 0167-6911/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
H.L. Weinert / Smoothing for multipoint boundary value models
446
0]
model is given by (1) with
0 0 0
A=
1 0 0
0 ,
B=
1/~
where the first component of the state vector is the deflection of the beam at point x. Now suppose there are boundary conditions consisting of pins at x 0 and x 2 and a clamp at x 1. Then these can be represented by (2) with
Vo =
[i °° i] v [000 °1 l°°° i] 0 0 0
0 0 0
0 ,
1 0 0
0 1 0
0 0 0
0 0 ' 0
~ =
0 0 I
0 0 0
0 0 0
and with Q quantifying the looseness, or uncertainty, of the pins and clamp.
2. The complementary model The Hamiltonian equations for the smoothed estimate are most readily derived using the method of complementary models [1,2,6,13,15]. If Y is the closed linear span of the components of y, then a complementary model is, in the case at hand, a multipoint boundary value model which generates complementary variables, denoted Yc and 0, whose components span the orthogonal complement of Y in the Hilbert space spanned by u, v and r. In order to derive a complementary model, we must first solve for z in terms of r and u. Integrating (1) gives
z( x) = q'( x, Xo)Z( Xo) + fxq)(x ~)Bu(~) d~
(4)
X 0
where q~ is the state transition matrix. Substituting into (2) yields N-I
N-I
r = E Vjcl)(x~, Xo)Z(Xo) + E Vjfxi"I'( x j, ~)Bu(~) dE. j=o j=o
(5)
The boundary value problem will be well-posed (that is, u and r will give rise to a unique z) if and only if the matrix EN=~l~q~(xj, x 0) is nonsingular. By multiplying (2) on the left by the inverse of this matrix, we can always ensure that N-1
Z ~.ffg(xj, Xo)=I.
(6)
j=0
In what follows it will be assumed that (6) holds. Solving (5) for z(x o) and substituting into (4), we get
z(x) =,/,(x, xo)r + fXN-'G(x, a)Bu(#) d( X 0
(7)
H.L. Weinert / Smoothing for multipoint boundary value models
447
where the multipoint Green's function G ( x , ~) is given by
• (x, xo)
VkeP(xk,~),
x>~,
U-~
G(x'~)=|-~(x,
E V,~(x~,~), x<~,
[
Xj <~
(8)
k=j+l
for j = 0, 1. . . . . N - 2. Now the complementary variables are given by
B'fx~ 'G'(¢, x)C'v(~) d~:,
Yc(X) = u ( x ) -
DO = r - QfXN_,~p,(~, x o ) C ' v ( ~ )
(9)
d~,
(10)
X 0
where D is any full-rank matrix satisfying
D D ' = Q. The proof is very similar to that in [13] (see also [1,2,6,15]) and will be omitted. Defining a complementary state by
Zc(X) =
fxlN-'G'(#, x)C'v(e)
d~:
(11)
we see that
y c ( x ) = u ( x ) - B'Zc(X ) .
(12)
Note that in contrast to the two-point case in which it is continuous, the complementary state here has jumps at the points x = x,, x 2 .... , XN_ 2. Just as with (9), we would like to express (10) in terms of z c. Defining 6 j = Z c ( X j - ) --Zc(Xj-~-),
j = 1, 2 . . . . . U - 2 ,
(13)
and using (8) and (11), we find that Zc( XN_I) = --Vl~_; f x u ,¢p,(~, x o ) C ' v ( ~ )
d~,
(14)
X o
6 j = -V/L~N-'qO'(~, x o ) C ' v ( ~ ) d~:, !
XN-I
Zc(X0) = V~L,,
(15)
!
dp (~, x o ) C ' v ( ~ ) d~.
(16)
Using the above along with (6), we can rewrite (10) as
DO=r-Q
Zc(Xo)-
E cb'(xj, xo)6 j - ~ ' ( x u _ l , x o ) z c ( x u _ , )
.
(17)
j=l
Next, we want to express z c as the state vector of a multipoint boundary value model like (1)-(2). Differentiating (11) gives Zc(X) = --AtZc(X ) -- C r y ( x ) ,
x E [ x 0 , XN_I] kk{Xj}j=l.U-2
(18)
H.L. Weinert / Smoothing for multipoint boundary, value models
448
The boundary conditions that go with (18) must yield a unique z c which matches (11). Since (18) is valid in each of N - 1 subintervals, we need a total of ( N - 1)n independent boundary conditions for a unique solution. Therefore the boundary conditions for the complementary state must have the form N
2
Kozc(Xl,) + ~ Kj6., +KN_,z~(x~_,)
= 0
(19)
j-I
for some suitable set of matrices {K~} of size (N - 1)n × n. In fact, if K 0 = [V 1
K,=[I
V2
0
K N 1=[0
VN_I]' '
-.-
""
O]'-Ko@'(xl, xo) ,
...
0
(20)
l ] ' - K 0 @ ' ( x N ,,xo),
it can be verified (see the appendix) that the boundary value problem (18)-(19) is well-posed and that its solution matches (11). Eqns. (12), (17)-(20) constitute a complementary model for (1)-(3).
3. The Hamiltonian equations The value of a complementary model now becomes clear because by cross-substituting between the original model and the complementary model, we can eliminate u, r, and t,, and then project onto the space spanned by the observations to obtain the Hamiltonian equations
~(x) =A£(x) +BB'£c(x ),
x ~ [ x o, x x 1],
~,.(x) = -A'2c(X ) +C'C£(x) - C ' y ( x )
'
(21)
x ~ [ x o, XN_,]\{Xj} u-z j=l'
(22)
with boundary conditions
where the 'hat' denotes linear least-squares estimate. Note that since the complementary variables yc and 0 are uncorrelated with y, their estimates are zero. By using invariant imbedding, we can transform (21)-(23) so that the resulting differential equations are partially decoupled and asymptotically stable. Let N(x) be the solution to the following Riccati equation on [x 0, Xu_l]:
N(x)= -N(x)A-A'N(x)+N(x)BB'N(x)-C'C,
N(XN_I)=O.
(24)
Then the coordinate change p(x) = N(x)£(x) + £~(x) transforms the above Hamiltonian equations into
~( x) = ( A - BB'N( x))2( x) + BB'p( x),
(25)
[x0, xN-,],
t
N-~
f~(x) = - ( A - B B ' N ( x ) ) p(x) - C'y(x),
(26)
with boundary conditions
Q@'(xj, Xo) ][2( xJ) ]= 0 -KoN(xo)
go
P(Xo) + i=l
0
where we have defined
{p( xj.-) -p(x ~P'=
p( XN 1 ) ,
+),
j=l,2
..... N-2,
j=N-1.
Kj
]t
~P'
(27)
H.L. Weinert / Smoothingfor muhipoint boundary calue models
449
If (A, B) is stabilizable and (A, C) is detectable, then (25) and (26) will be asymptotically stable in the directions of increasing and decreasing x, respectively. Note that the Riccati equation is identical to one that arises in smoothing for initial value models [1,15]. Other transformations are possible, for example using two Riccati equations, but the one used here is the most efficient in terms of the amount of computation required to find the smoothed estimate and its error covariance [14].
4. The smoothed state estimate
The transformed Hamiltonian equations can now be solved using a multiple shooting procedure• First solve the following initial value systems:
tAo(X) = - ( A - B B ' N ( x ) ) ' p o ( x ) - C ' y ( x ) ,
Po(XN_1) =O,
~o( X ) = ( A - BB'N( x ) ) 2o( X ) + BB'po( X ),
(28)
20(Xo) = 0.
(29)
The solution of (29) can be interpreted as the smoothed estimate of z(x) when V0 = I, Vj = 0 for j > 0, and Q =-0; or in other words, when the original model is an initial value model with zero initial condition [15]. Now it can be checked by differentiation that for xk_ ~ ~
2 ( x ) =So(X )
N-1
+ qt(x, Xo)~(Xo) + Y'. tit(x, xj)M(xi)3o + ~_, M(x)qt'(xi, x)6pj j=l
(30)
j=k
and for xk_ ~ < x
p(x)
=po(X) + •
qt'(x],
x)~p,
(31)
j=k
where qt is the state transition matrix of
A -BB'N(x)
and
I(I(x)=(A-BB'N(x))M(x)+M(x)(A-BB'N(x))'+BB',
M(xo)=O.
We can determine the unknown quantities in (30) by obtaining 2(Xk), k = 1, 2 . . . . , N - 1, and from (30) and (31), respectively, and substituting into (27)• This results in the following system:
[
/ Zo(X1)
=F'[20(X:N_,)
(33)
where
I q'(Xl,x0)-~(xl, x0) ], -K o
L=
p(x o)
[ po(Xo)
e(X0)
[ I + F1F2]
(32)
0
'
-A
-L
'
q.(x._~,x0)- ~(xN-t,Xo) M( xl)qt'( Xu_l, x,) ]
M(Xl)
M( x1)att'( x 2, Xl)
~( x2, Xl)M( Xl)
M(x2)
M ( X 2 ) a I t ' ( X N _ 1, X2)
ltt( XN- 1, X l ) M ( x l )
lI-f( XN_ 1, x z ) M ( x z )
M(XN-1)
H.L. Weinert / Smoothing for multipoint boundary ualue models
450
The well-posedness of (1)-(2) together with the uniqueness of linear least-squares estimates guarantee the nonsingularity of I + F 1F2. The components of the vector on the right side of (33) are obtained from (28)-(29). The solution of (33) is used in (30) to produce the smoothed estimate.
5. The error covariance
The smoothing error :~(x) is defined by
~(~) =z(~) - ~ ( ~ ) . Equations (1)-(3) and (21)-(23) imply that the smoothing error is also characterized by a multipoint boundary value problem as follows:
~(x) = A Y ( x ) +BB'(-2~.(x)) + B u ( x ) ,
x ~ [x 0, x N l], N-2
-~c(~) : -A'(-2c(X))+C'C~(x) +C'~'(x), x~[Xo, xN_,]\{xAj= , , ' ~ ~(xj)+
j =0
0
Ko
(-2c(X0))+ E
j=1
-gi)
Kj
QO'( xN l, A change of coordinates with
~(x) =U(x)~.(x) -2c(X),
tS~=
O(x1-)-~(xj +), ~:(XN-1),
/=1,2 ..... N-2,
j=U-
1,
transforms the above equations into
z(x)=(A-BB'N(x))~(x)+BB'~(x)+Bu(x),
X~[X,,,XN_,],
((X) = -- ( A - B B ' U ( x ) ) ' ~ ( x ) + N ( x ) B u ( x ) + C ' L ' ( x ) , Vo + QN(xo)
-Q
x E [ X o , Xu i ] \ {
(34)
Xj}j=IN-2,
xo)
(35) (36)
- K o N ( xo) As before, if we let
(o(X)= - ( A - B B ' N ( x ) ) ' ~ o ( X )
+ N ( x ) B u ( x ) + C ' U(x),
Zo( X ) = ( A - BB'N( x ) ) £o( x ) + BB'~o( x ) + Bu( x ),
~o(XN_,)=O,
ZT0(X0)=0,
(37)
(38)
then f o r x k l~
N-I
z T ( x ) = Y o ( X ) + ~ ( x , Xo)Y(Xo)+ Y'~ ~ ( x , xi)M(xj)a¢ + Y'~ M ( x ) q " ' ( x j , x)6¢, j=l
where
xo] Ilox oxo / ';' L a N_,
L~o(Xu-,)
(39)
j=k
(40)
H.L. Weinert / Smoothing for multipoint boundary value models
451
From [15] we know that
E[~o(Xo)~(Xo) ] =N(xo), E[5o(X)~6(Xo)] - 0 [ qt(x, s)M(s), x > s, E[5°(x)ia(s)] = ~M(x)gt'(s, x), x<~s.
(41) (42)
Therefore
P(x) =E[i(x)U(x)] = M ( x ) +R(x)(I+F1F2)
0
o, o]] F, (I+F, FE)'-tR'(x)
- R( x)( I + F, Fz)-'[ K°S'o( X) ] - [s( x)K o O]( I + F1F2)'-' R'( x ) where f o r x k_l~
"'"
(43)
M(x)alt'(Xu-I,X)]
and
R(x)=[qt(X, Xo) S ( x ) ] .
Appendix From (18),
zc(xo) = But Zc(X1 - ) =
Xo)Zc(X,-) + fxi'¢,'(~, Xo)C'v(~) d~.
~'(xl,
61 +Zc(X l + ) and from (18),
Zc(X , +1 = q~'(x 2, XllZc(X 2 - ) + f~2dP'(~, xllC'v(~ ) d~. Therefore
zc(xo) = q~'(xl, Xo)6 , + ~ ' ( x 2 , Xo)Zc(X2- ) + fx:dp'(~, xo)C'v(~ ) d~. Xo Continuing in the same fashion, we get N-2 Zc(X0) =
E j=l
t~C(Xj' XO)~jd-CI) ' ( X u - l , XO)Zc(XN-1)-'l-
fXN-'cP'(~, xo)C'v(~ ) d~. Xo
Substituting into (19) gives N-2
j=l
[Kj + Koq~'( xj, x0)]8 j + [ KN_ , + KO@'( XN_I, Xo)lZ~( XN_,) XN-I
t
+Kof£ ° • (~, xo)C'v(~) d~=0. With the {K i} given by (20), Zc(XN_ 1) and the {ai} can be obtained uniquely from the above equation, and
452
H.L. Weinert / Smoothing for multipoint boundar).' calue models
in fact a r e g i v e n by ( 1 4 ) - ( 1 5 ) . I n t e g r a t i n g (18) w i t h t h e v a l u e s g i v e n in ( 1 4 ) - ( 1 5 ) will p r o d u c e T h e r e f o r e t h e b o u n d a r y v a l u e p r o b l e m ( 1 8 ) - ( 1 9 ) is w e l l - p o s e d a n d its s o l u t i o n is g i v e n b y (11).
(11).
References [1] R. Ackner and T. Kailath, Complementary models and smoothing, IEEE Trans. Automat. Control 34 (1989) 963-969. [2] M.B. Adams, A.S. Willsky, and B.C. Levy, Linear estimation of boundary value stochastic processes, IEEE Trans. Automat. Control 29 (1984) 803-821. [3] A. Bagchi and H. Westdijk, Smoothing and likelihood ratio for Gaussian boundary value processes, IEEE Trans. Automat. Control 34 (1989) 954-962. [4] G. Chen, M.C. Delfour, A.M. Krall, and G. Payre, Modeling, stabilization, and control of serially connected beams, SIAMJ. Control. Optim. 25 (1987) 526-546. [5] C.E. Economakos and H.L. Weinert, The 2D Poisson equation: Computing smoothed estimates with discrete data, Proc. 25th Conf. Inform. Sci. Syst., Baltimore, MD (1991). [6] T. Kailath and M. Wax, A note on the complementary model of Weinert and Desai, IEEE Trans. Automat. Control 29 (1984) 551-552. [7] A.J. Krener, Reciprocal processes and the stochastic realization problem for acausal systems, in: C.l. Byrnes and A. Lindquist, Eds., Modelling, Identification, and Robust Control (Elsevier, Amsterdam, 1986) 197-209. [8] B.C. Levy, M.B. Adams, and A.S. Willsky, Solution and linear estimation of 2-D nearest-neighbor models, Proc. IEEE 78 (1990) 627-641. [9] B.C. Levy, R. Frezza, and A.J. Krener, Modeling and estimation of discrete-time Gaussian reciprocal processes, 1EEE Trans. Automat. Control 35 (1990) 1013-1023. [10] R. Nikoukhah, M.B. Adams, A.S. Willsky and B.C. Levy, Estimation for boundary value descriptor systems, Circuits Systems Signal Process. 8 (1989) 25-48. [11] L.R. Riddle and H.L. Weinert, Fast algorithms for the reconstruction of images from hyperbolic systems, Proc. 25th Conf Decision and Control (1986) 173-178. [12] L.R. Riddle and H.L. Weinert, Recursive linear smoothing for dissipative hyperbolic systems, Mech. Systems Signal Process. 2 (1988) 77-96. [13] H.L. Weinert, The complementary model in continuous-discrete smoothing, in: E. Parzen, Ed., Time Series Analysis of Irregularly Obserced Data (Springer-Verlag, Berlin-New York, 1984) 353-363. [14] H.L. Weinert, A note on efficient smoothing for boundary value models, Internat. J. Control 53 (1991) 503-507. [15] H.L. Weinert and U.B. Desai, On complementary models and fixed-interval smoothing, IEEE Trans. Automat. Control 26 (1981) 863-867.