Cholesky-Based Reduced-Rank Square-Root Kalman Filtering
J. Chandrasekar, I. S. Kim, D. S. Bernstein, and A. Ridley
Abstract— We consider a reduced-rank square-root Kalman
filter based on the Cholesky decomposition of the state-error
covariance. We compare the performance of this filter with
the reduced-rank square-root filter based on the singular value
decomposition.
I. I NTRODUCTION
The problem of state estimation for large-scale systems has
gained increasing attention due to computationally intensive
applications such as weather forecasting [1], where state
estimation is commonly referred to as data assimilation.
For these problems, there is a need for algorithms that are
computationally tractable despite the enormous dimension
of the state. These problems also typically entail nonlinear
dynamics and model uncertainty [2], although these issues
are outside the scope of this paper.
One approach to obtaining more tractable algorithms is
to consider reduced-order Kalman filters. These reducedcomplexity filters provide state estimates that are suboptimal
relative to the classical Kalman filter [3–7]. Alternative
reduced-order variants of the classical Kalman filter have
been developed for computationally demanding applications
[8–11], where the classical Kalman filter gain and covariance
are modified so as to reduce the computational requirements.
A comparison of several techniques is given in [12].
A widely studied technique for reducing the computational
requirements of the Kalman filter for large scale systems is
the reduced-rank filter [13–16]. In this method, the errorcovariance matrix is factored to obtain a square root, whose
rank is then reduced through truncation. This factorizationand-truncation method has direct application to the problem
of generating a reduced ensemble for use in particle filter
methods [17, 18].
Reduced-rank filters are closely related to the classical
factorization techniques [19, 20], which provide numerical
stability and computational efficiency, as well as a starting
point for reduced-rank approximation.
The primary technique for truncating the error-covariance
matrix is the singular value decomposition (SVD) [13–18],
wherein the singular values provide guidance as to which
components of the error covariance are most relevant to the
accuracy of the state estimates. Approximation based on the
SVD is largely motivated by the fact that error-covariance
This research was supported by the National Science Foundation, under
grants CNS-0539053 and ATM-0325332 to the University of Michigan, Ann
Arbor, USA.
J. Chandrasekar, I. S. Kim, and D. S. Bernstein are with Department of
Aerospace Engineering, and A. Ridley is with Department of Atmospheric,
Oceanic and Space Sciences, University of Michigan, Ann Arbor, MI,
dsbaero@umich.edu
truncation is optimal with respect to approximation in unitarily invariant norms, such as the Frobenius norm. Despite
this theoretical grounding, there appear to be no theoretical
criteria to support the optimality of approximation based on
the SVD within the context of recursive state estimation. The
difficult is due to the fact that optimal approximation depends
on the dynamics and measurement maps in addition to the
components of the error covariance.
In the present paper we begin by observing that the
Kalman filter update depends on the combination of terms
Ck Pk , where Ck is the measurement map and Pk is the error
covariance. This observation suggests that approximation of
Ck Pk may be more suitable than approximation based on
Pk alone.
To develop this idea, we show that approximation of Ck Pk
leads directly to truncation based on the Cholesky decomposition. Unlike the SVD, however, the Cholesky decomposition does not possess a natural measure of magnitude that is
analogous to the singular values arising in the SVD. Nevertheless, filter reduction based on the Cholesky decomposition
provides state-estimation accuracy that is competitive with,
and in many cases superior to, that of the SVD. In particular,
we show that, in special cases, the accuracy of the Choleskydecomposition-based reduced-rank filter is equal to the accuracy of the full-rank filter, and we demonstrate examples
for which the Cholesky-decomposition-based reduced-rank
filter provides acceptable accuracy, whereas the SVD-based
reduced-rank filter provides arbitrarily poor accuracy.
A fortuitous advantage of using the Cholesky decomposition in place of the SVD is the fact that the Cholesky
decomposition is computationally less expensive than the
SVD, specifically, O(n3 /6) [21], and thus an asymptotic
computational advantage over SVD by a factor of 12. An
additional advantage is that the entire matrix need not be
factored; instead, by arranging the states so that those states
that contribute directly to the measurement correspond to
the initial columns of the lower triangular square root, then
only the leading submatrix of the error covariance must
be factored, yielding yet further savings over the SVD.
Once the factorization is performed, the algorithm effectively
retains only the initial “tall” columns of the full Cholesky
factorization and truncates the “short” columns.
II. T HE K ALMAN
FILTER
Consider the time-varying discrete-time system
xk+1 = Ak xk + Gk wk ,
(2.1)
yk = Ck xk + Hk vk ,
(2.2)
where xk ∈ Rnk , wk ∈ Rdw , yk ∈ Rpk , vk ∈ Rdv , and
Ak , Gk , Ck , and Hk are known real matrices of appropriate
sizes. We assume that wk and vk are zero-mean white
processes with unit covariances. Define Qk , Gk GT
k and
Rk , Hk HkT , and assume that Rk is positive definite for
all k > 0. Furthermore, we assume that wk and vk are
uncorrelated for all k > 0. The objective is to obtain an
estimate of the state xk using the measurements yk .
The Kalman filter provides the optimal minimum-variance
estimate of the state xk . The Kalman filter can be expressed
in two steps, namely, the data assimilation step, where the
measurements are used to update the states, and the forecast
step, which uses the model. These steps can be summarized
as follows:
Data Assimilation Step
Kk = Pkf CkT (Ck Pkf CkT + Rk )−1 ,
Pkda
xda
k
=
=
Pkf
xfk
− Pkf CkT (Ck Pkf CkT +
+ Kk (yk − Ck xfk ).
(2.3)
−1
Rk )
Ck Pkf ,
(2.4)
(2.5)
Forecast Step
xfk+1 = Ak xda
k ,
f
Pk+1
Pkf
=
Ak Pkda AT
k
n×n
The matrices
∈ R
and
error covariances, that is,
Pkda
(2.6)
+ Qk .
∈ R
n×n
(2.7)
are the state-
da T
Pkf = E[efk (efk )T ], Pkda = E[eda
k (ek ) ],
given by
P = Un Σn UnT ,
where Un is orthogonal. For q 6 n, let ΦSVD (P, q) ∈ Rn×q
denote the SVD-based rank-q approximation of the square
1/2
root Σq of P given by
ΦSVD (P, q) , Uq Σq1/2 .
The following standard result shows that SS , where S ,
ΦSVD (P, q), is the best rank-q approximation of P in the
Frobenius norm.
Lemma III.1. Let P ∈ Rn×n be positive semidefinite,
and let σ1 · · · > σn be the singular values of P . If S =
ΦSVD (P, q), then
min
rank(P̂ )=q
2
kP − P̂ k2F = kP − SS T k2F = σq+1
+ · · · + σn2 . (3.4)
The data assimilation and forecast steps of the SVD-based
rank-q square-root filter are given by the following steps:
Data Assimilation step
−1
f
f
,
(3.5)
CkT + Rk
Ks,k = P̂s,k
CkT Ck P̂s,k
−1
da
f
f
P̃s,k
= P̂kf − P̂kf CkT Ck P̂s,k
CkT + Rk
Ck P̂s,k
, (3.6)
f
f
xda
s,k = xs,k + Ks,k (yk − Ck xs,k ),
In the following sections, we consider reduced-rank
square-root filters that propagate approximations of a squareroot of the error covariance instead of the actual error
covariance.
III. SVD-BASED R EDUCED -R ANK S QUARE -ROOT
F ILTER
Note that the Kalman filter uses the error covariances
Pkda and Pkf , which are updated using (2.4) and (2.7). For
computational efficiency, we construct a suboptimal filter that
uses reduced-rank approximations of the error covariances
Pkda and Pkf . Specifically, we consider reduced-rank approximations P̂kda and P̂kf of the error covariances Pkda and Pkf
such that kPkda − P̂kda kF and kPkf − P̂kf kF are minimized,
where k · kF is the Frobenius norm. To achieve this approximation, we compute singular value decompositions of the
error covariances at each time step.
Let P ∈ Rn×n be positive semidefinite, let σ1 > · · · > σn
be the singular values of P , and let u1 , . . . , un ∈ Rn be
corresponding orthonormal eigenvectors. Next, define Uq ∈
Rn×q and Σq ∈ Rq×q by


σ1


..
U q , u 1 · · · u q , Σq , 
 . (3.1)
.
σq
With this notation, the singular value decomposition of P is
(3.7)
where
(2.8)
(2.9)
(3.3)
T
where
da
efk , xk − xfk , eda
k , xk − xk .
(3.2)
f
f
S̃s,k
, ΦSVD (P̃s,k
, q),
(3.8)
f
P̂s,k
(3.9)
,
f
f
S̃s,k
(S̃s,k
)T .
Forecast step
xfs,k+1 = Ak xda
s,k ,
f
P̃s,k+1
=
da T
Ak P̂s,k
Ak
(3.10)
+ Qk ,
(3.11)
where
da
da
S̃s,k
, ΦSVD (P̃s,k
, q),
(3.12)
da
P̂s,k
(3.13)
,
da
da T
S̃s,k
(S̃s,k
) ,
f
P̃s,0
and
is positive semidefinite.
Next, define the forecast and data assimilation error cof
da
variances Ps,k
and Ps,k
of the SVD-based rank-q square-root
filter by
f
Ps,k
, E (xk − xfs,k )(xk − xfs,k )T ,
(3.14)
da
da
da T
Ps,k , E (xk − xs,k )(xk − xs,k ) .
(3.15)
Using (2.1), (3.7) and (3.10), it can be shown that
da
f
T
Ps,k
= (I − Ks,k C)Ps,k
(I − Ks,k C)T + Ks,k Rk Ks,k
, (3.16)
f
da T
Ps,k+1
= Ak Ps,k
Ak + Qk .
(3.17)
T
T
f
f
f
da
da
da
Note that S̃s,k
S̃s,k
6 P̃s,k
and S̃s,k
S̃s,k
6 P̃s,k
.
f
f
Hence, even if P̃s,0
= Ps,0
, it does not necesarily follow that
f
f
da
da
P̃s,k = Ps,k and P̃s,k = Ps,k
for all k > 0. Therefore, since
f
Ks,k does not use the true error covariance Ps,k
, the SVDbased rank-q square-root filter is generally not equivalent
to the Kalman filter. However, under certain conditions,
the SVD-based rank-q square-root filter is equivalent to the
Kalman filter. Specifically, we have the following result.
f
f
Proposition III.1. Assume that P̃s,k
= Ps,k
and
f
da
da
f
f
rank(Ps,k ) 6 q. Then, P̃s,k = Ps,k and P̃s,k+1 = Ps,k+1 . If,
f
f
f
in addition, Ps,k
= Pkf , then Ks,k = Kk and Ps,k+1
= Pk+1
.
Proof. Since rank(P̃kf ) 6 q, it follows from Lemma III.1
that
T
f
f
f
f
S̃s,k
= P̃s,k
.
(3.18)
P̂s,k
= S̃s,k
Hence, it follows from (3.5) that
f
f
CkT + Rk
Ks,k = Ps,k
CkT Ck Ps,k
while substituting (3.18) into (3.6) yields
−1
,
(3.19)
da
f
f
f
f
P̃s,k
= P̃s,k
− P̃s,k
CkT (Ck P̃s,k
CkT + Rk )−1 Ck P̃s,k
. (3.20)
f
f
Next, substituting (3.19) into (3.16) and using P̃s,k
= Ps,k
in the resulting expression yields
da
da
P̃s,k
= Ps,k
.
(3.21)
f
Since rank(P̃s,k
)
da
rank(P̃s,k ) 6 q, and
da
P̂s,k
6 q, it follows from (3.20) that
therefore Lemma III.1 implies that
T
da
da
da
= S̃s,k
S̃s,k
= P̃s,k
.
(3.22)
Hence, it follows from (3.11), (3.17), and (3.21) that
f
f
P̃s,k+1
= Ps,k+1
.
(3.23)
Finally, it follows from (2.3) and (3.19) that
Ks,k = Kk .
(3.24)
Therefore, (2.4) and (3.16) imply that
da
Ps,k
=
Pkda .
(3.25)
Hence, it follows from (2.7), (3.17) and (3.25) that
f
f
Ps,k+1
= Pk+1
.
f
f
Corollary III.1. Assume that xfs,0 = xf0 , P̃s,0
= Ps,0
, and
rank(P0f ) 6 q. Furthermore, assume that, for all k > 0,
rank(Ak ) + rank(Qk ) 6 q. Then, for all k > 0, Ks,k = Kk
and xfs,k = xfk .
xfs,0
Since P ∈ Rn×n is positive semidefinite, the Cholesky
decomposition yields a lower triangular Cholesky factor L ∈
Rn×n of P that satisfies
LLT = P.
Partition L as
L=
IV. C HOLESKY-D ECOMPOSITION -BASED
R EDUCED -R ANK S QUARE -ROOT F ILTER
The Kalman filter gain Kk depends on a particular subspace of the range of the error covariance. Specifically,
Kk depends only on Ck Pkf and not on the entire error
covariance. We thus consider a filter that uses reduced-rank
approximations P̂kda and P̂kf of the error covariances Pkda
and Pkf such that kCk (Pkda − P̂kda )kF and kCk (Pkf − P̂kf )kF
are minimized. To achieve this minimization, we compute a
Cholesky decomposition of both error covariances at each
time step.
L1
···
Ln
n
,
where, for i = 1, . . . , n, Li ∈ R has real entries
T
Li = 01×(i−1) Li,1 · · · Li,n−i+1
.
(4.2)
(4.3)
Truncating the last n − q columns of L yields the reducedrank Cholesky factor
ΦCHOL (P, q) , L1 · · · Lq ∈ Rn×q .
(4.4)
Lemma IV.1. Let P ∈ Rn×n be positive definite, define
S , ΦCHOL (P, q) and P̂ , SS T , and partition P and P̂
as
P1
P12
P̂1
P̂12
, (4.5)
P =
, P̂ =
(P12 )T P2
(P̂12 ) P̂2
where P1 , P̂1 ∈ Rq×q and P2 , P̂2 ∈ Rr×r . Then, P̂1 = P1
and P̂12 = P12 .
Proof. Let L be the Cholesky factor of P . It follows from
(4.3) that Li LT
∈ Rn has the structure
i
0i−1
0(i−1)×(n−i+1)
Li LT
=
,
(4.6)
i
0(n−i+1)×(i−1)
Xi
where Xi ∈ R(n−i+1)×(n−i+1) . Therefore,
n
X
0q×q 0q×r
T
Li Li =
,
0r×q
Yr
(4.7)
i=q+1
where Yr ∈ Rr×r . Since
P =
n
X
Li LT
i ,
(4.8)
i=1
it follows from (4.4) that
P = P̂ +
xf0 ,
Proof. Since
=
(2.8) and (3.14) imply that
f
f
Ps,0
= P0f . It follows from (3.16) that if rank(Ps,k
) 6
da
q, then rank(Ps,k ) 6 q, and hence (3.17) implies that
f
rank(Ps,k+1
) 6 q. Therefore, using Proposition III.1 and
induction, it follows that Ks,k = Kk for all k > 0. Therefore,
(2.5), (2.6), (3.7) and (3.10) imply that xfs,k = xfk for all
k > 0.
(4.1)
n
X
Li LT
i .
(4.9)
i=q+1
Substituting (4.7) into (4.9) yields P̂1 = P1 and P̂12 = P12 .
Lemma IV.1 implies that, if S = ΦCHOL (P, q), then the
first q rows and columns of SS T and P are equal.
The data assimilation and forecast steps of the Choleskybased rank-q square-root filter are given by the following
steps:
Data Assimilation step
“
”−1
f
f
Kc,k = P̂c,k
CkT Ck P̂c,k
CkT + Rk
,
“
”−1
da
f
f
f
f
P̃c,k
= P̂c,k
− P̂c,k
CkT Ck P̂c,k
CkT + Rk
Ck P̂c,k
,
xda
c,k
=
xfc,k
+ Kc,k (yk −
Ck xfc,k ),
(4.10)
(4.11)
(4.12)
where
f
f
S̃c,k
, ΦSVD (P̃c,k
, q),
(4.13)
f
P̂c,k
(4.14)
,
f
f
S̃c,k
(S̃s,k
)T .
Forecast step
xfc,k+1 = Ak xda
c,k ,
(4.15)
f
da T
P̃c,k+1
= Ak P̂c,k
Ak + Qk ,
(4.16)
where
da
da
S̃c,k
, ΦSVD (P̃c,k
, q),
(4.17)
da
P̂c,k
(4.18)
,
da
da T
S̃c,k
(S̃c,k
) ,
f
and P̃c,0
is positive semidefinite.
Next, define the forecast and data assimilation error cof
da
variances Pc,k
and Pc,k
, respectively, of the Cholesky-based
rank-q square-root filter by
h
i
f
, E (xk − xfc,k )(xk − xfc,k )T ,
(4.19)
Pc,k
h
i
da
da T
Ps,k
, E (xk − xda
,
(4.20)
c,k )(xk − xc,k )
f
da
that is, Pc,k
and Pc,k
are the error covariances when the
Cholesky-based rank-q square-root filter is used. Using (2.1),
(4.12) and (4.15), it can be shown that
da
Pc,k
f
Pc,k
=
=
f
(I − Kc,k C)Pc,k
(I
da T
Ak Pc,k
Ak + Qk .
T
T
Kc,k Rk Kc,k
,
− Kc,k C) +
(4.21)
(4.22)
T
T
da
f
da
f
S̃c,k
6 P̃kda .
S̃c,k
6 P̃kf and S̃c,k
Note that S̃c,k
f
f
Hence, even if P̃c,0
= Pc,0
, the Cholesky-based rank-q
square-root filter is generally not equivalent to the Kalman
filter. However, in certain cases, the Cholesky-based rank-q
square root filter is equivalent to the Kalman filter.
Proposition IV.1. Let Ak and Ck have the structure
A1,k
0
Ak =
, Ck = Ip 0 ,
(4.23)
A21,k A2,k
f
where A1,k ∈ Rp×p and A2,k ∈ Rr×r , partition Pkf and P̃c,k
as
Pkf =
»
f
P1,k
f
P21,k
f
)T
(P21,k
f
P2,k
f
f
P1,k
, P̃c,1,k
–
f
, P̃c,k
=
»
f
P̃c,1,k
f
P̃c,21,k
f
(P̃c,21,k
)T
f
P̃c,2,k
–
, (4.24)
f
f
where
∈ R
and P2,k
, P̃c,2,k
∈ Rr×r , and
f
f
f
f
assume that q = p, P̃c,1,k = P1,k , and P̃c,21,k = P21,k
. Then,
f
f
f
f
Kc,k = Kk , P̃c,1,k+1 = P1,k+1 , and P̃c,21,k+1 = P21,k+1 .
Proof. Let
p×p
have entries
da
da T
P1,k (P21,k
)
=
,
da
da
P21,k
P2,k
(4.25)
da
da
where P1,k
∈ Rp×p is positive semidefinite and P2,k
∈
r×r
R . It follows from (2.4) that
da
f
f
f
f
P1,k
= P1,k
− P1,k
(P1,k
+ Rk )−1 P1,k
,
(4.26)
da
P21,k
(4.27)
=
f
P21,k
−
f
f
P21,k
(P1,k
−1
+ Rk )
(Q21,k )T
Q2,k
.
(4.31)
f
da
f
Define P̂c,k
and P̂c,k
by (4.14) and (4.18), and let P̂c,k
da
and P̂c,k have entries
f
P̂c,k
=
–
» da
–
f
f
da
P̂c,1,k
(P̂c,21,k
)T
P̂c,1,k (P̂c,21,k
)T
da
, P̂c,k
= da
, (4.32)
f
f
da
P̂c,21,k P̂c,2,k
P̂c,21,k P̂c,2,k
»
da
f
where P̂1,k
, P̂1,k
∈ Rp×p are positive semidefinite and
da
f
r×r
P̂2,k , P̂2,k ∈ R . Substituting (4.32) into (4.10) yields
"
#
f
P̂1,k
f
Kc,k =
(P̂1,k
+ Rk )−1 .
(4.33)
f
P̂21,k
f
f
Since Sc,k
= ΦCHOL (P̃c,k
, q) and we assume that q =
f
f
f
f
p, P̃c,1,k = P1,k , and P̃c,21,k
= P21,k
, it follows from
Lemma IV.1 that
f
f
f
f
= Pc,21,k
.
, P̂c,21,k
P̂c,1,k
= Pc,1,k
(4.34)
Hence, Kc,k = Kk .
Next, substituting (4.14) into (4.11) yields
da
f
f
f
f
P̃c,k
= P̂c,k
− P̂c,k
CkT (Ck P̂c,k
CkT + Rk )−1 Ck P̂c,k
. (4.35)
da
Let P̃c,k
have entries
"
da
P̃c,k
=
da
P̃c,1,k
da
P̃c,21,k
da
(P̃c,21,k
)T
da
P̃c,2,k
#
,
(4.36)
da
da
where P̃c,1,k
∈ Rp×p is positive semidefinite and P̃c,2,k
∈
r×r
R . Therefore, it follows from (4.23) and (4.32) that
da
f
f
f
f
P̃c,1,k
= P̂c,1,k
− P̂c,1,k
(P̂c,1,k
+ Rk )−1 P̂c,1,k
,
(4.37)
da
P̃c,21,k
(4.38)
=
f
P̂c,21,k
−
f
f
P̂c,21,k
(P̂c,1,k
−1
+ Rk )
f
P̂c,1,k
.
Substituting (4.34) into (4.37) and using (4.25) and (4.26)
yields
da
da
da
da
P̃c,1,k
= P1,k
, P̃c,21,k
= P21,k
.
Moreover, since
Lemma IV.1 that
da
S̃c,k
=
da
ΦCHOL (P̃c,k
, q),
(4.39)
it follows from
da
da
da
da
P̂c,1,k
= P̃c,1,k
, P̂c,21,k
= P̃c,21,k
.
(4.40)
It follows from (4.16) and (4.23) that
Pkda
Pkda
where Qk has entries
Q1,k
Qk =
Q21,k
f
P1,k
.
Substituting (4.23) into (2.3) yields
f P1,k
f
Kk =
(P1,k
+ Rk )−1 .
f
P21,k
(4.28)
Furthermore, (2.7) and (4.23) imply that
f
da T
P1,k+1
= A1,k P1,k
A1,k + Q1,k ,
(4.29)
f
da
da T
P21,k+1
= A2,k P21,k
AT
1,k + A21,k P1,k A1,k + Q21,k ,
(4.30)
da T
f
P̃c,1,k+1
= A1,k P̂1,k
A1,k + Q1,k ,
(4.41)
f
da
da T
P̃c,21,k+1
= A2,k P̂21,k
AT
1,k + A21,k P̂1,k A1,k + Q21,k .
(4.42)
Therefore, (4.29), (4.39), and (4.40) imply that
f
f
f
f
P̃c,1,k+1
= P1,k+1
, P̃c,21,k+1
= P21,k+1
.
Corollary IV.1. Assume that Ak and Ck have the strucf
f
f
f
ture in (4.23). Let q = p, P̃c,1,0
= P1,0
, P̃c,21,0
= P21,0
,
f
f
and xc,0 = x0 . Then for all k > 0, Kc,k = Kk , and hence
xfc,k = xfk .
Proof. Using induction and Proposition IV.1 yields
Kc,k = Kk for all k > 0. Hence, it follows from (2.5),
(2.6), (4.12), and (4.15) that xfc,k = xfk for all k > 0.
for i = 1, . . . , r − 1. Using (4.11) in (5.12) yields
V. L INEAR T IME -I NVARIANT S YSTEMS
In this section, we consider a linear time-invariant system
and hence assume that, for all k > 0, Ak = A, Ck = C,
Gk = G, Hk = H, Qk = Q, and Rk = R. Furthermore,
we assume that p < n and (A, C) is observable so that the
observability matrix O ∈ Rpn×n defined by


C
 CA 


O,
(5.1)

..


.
CAn−1
has full column rank. Next, without loss of generality we
consider a basis such that
In
O=
.
(5.2)
0(p−1)n×n
Therefore, (5.1) and (5.2) imply that, for i ∈ {1, . . . , n} such
that ip 6 n,
CAi−1 = 0p×p(i−1) Ip 0p×(n−pi) .
(5.3)
The following result shows that the Cholesky decomposition based rank-q square-root filter is equivalent to the
optimal Kalman filter for a specific number of time steps.
Proposition V.1. Let r > 0 be an integer such that 0 <
q = pr < n. Then, for all k = k0 , . . . , k0 + r − 1, Kc,k =
Kk .
f
P̂c,k+1
(Ai−1 )T C T
(5.13)
h
i
f
f
T
f
T
−1
f
i T T
= A P̂c,k − P̂c,k C (C P̂c,k C + R) C P̂c,k (A ) C
+ Q(Ai−1 )T C T ,
Therefore, (5.6) implies that, for all i = 1, . . . , r − 1,
P̂ f c,k+1 (Ai−1 )T C T
(5.14)
f
i T T
f
T
= ψ P̂c,k (A ) C , P̂c,k C , A, Q, C, R, i .
Hence, it follows from (5.5)-(5.7) that for i = 1, . . . , r − 1,
f
P̂k+i
CT
f
Since P̃c,k
= Pkf0 , it follows from Lemma IV.1 that for
0
i = 1, . . . , r,
f
CAi−1 P̂c,k
= CAi−1 Pkf0 .
0
Hence, it follows from (5.7) and (5.15) that for
1, . . . , r − 1,
P̂kf0 +i C T = Pkf0 +i C T .
Hence, for i = 1, . . . , r − 1,
“
”
f
Pk+1
(Ai−1 )T C T = ψ Pkf (Ai )T C T , Pkf C T , A, Q, C, R, i , (5.5)
where
ψ(X, Y, A, Q, C, R, i)
(5.6)
= AX − AY (CY + R)
i−1 T
CX + Q(A
T
) C .
Therefore,
P f k+i C T
=
i =
(5.17)
Kc,k0 +i = Kk0 +i .
f
However, in general Pc,k
6= Pkf for k = k0 , . . . , k0 + r − 1,
Kc,k .
VI. E XAMPLES
= APkf AT − APkf C T (CPkf C T + R)−1 CPkf AT + Q. (5.4)
−1
(5.16)
Finally, (2.3) and (4.10) imply that for i = 0, . . . , r − 1
Proof. Substituting (2.4) into (2.7) yields
f
Pk+1
(5.15)
f
f
f
= ϕ(P̂c,k
(Ai )T C T , P̂c,k
(Ai−1 )T C T , . . . , P̂c,k
C T , A, Q, C, i).
(5.7)
We compare performance of the SVD-based and
Cholesky-based reduced-rank square-root Kalman filters using a compartmental model [22] and 10-DOF mass-springdamper system.
A schematic diagram of the compartmental model is
shown in Fig 1. The number n of cells is 20 with one state
per cell. All ηii and all ηij (i 6= j) are set to 0.1. We assume
that the state of the 9th cell is measured, and disturbances
enter all cells so that the disturbance covariance Q has full
rank.
ϕ(Pkf (Ai )T C T , Pkf (Ai−1 )T C T , . . . , Pkf C T , A, Q, C, i).
f
da
and P̂c,k
by
Define P̂c,k
f
f
f
da
da
da T
P̂c,k
, S̃c,k
(S̃c,k
)T , P̂c,k
, S̃c,k
(S̃c,k
) .
(5.8)
It follows from Lemma IV.1 and (5.3) that for all k > 0 and
i = 1, . . . , r,
da
da
f
f
. (5.9)
= CAi−1 P̃c,k
, CAi−1 P̂c,k
= CAi−1 P̃c,k
CAi−1 P̂c,k
Note that
f
da T
P̃c,k+1
= AP̂c,k
A + Q.
(5.10)
Multiplying (5.10) by (Ai−1 )T C T yields
f
da
P̃c,k+1
(Ai−1 )T C T = AP̂c,k
(Ai )T C T + Q(Ai−1 )T C T . (5.11)
Substituting (4.40) into (5.11) yields
f
da
P̂c,k+1
(Ai−1 )T C T = AP̃c,k
(Ai )T C T + Q(Ai−1 )T C T , (5.12)
Fig. 1.
Compartmental model involving interconnected subsystems.
We simulate three cases of disturbance covariance and
compare costs J , E(eT
k ek ) in Figure 3 where (a) shows
the cost when Q = I, ((b) shows costs when Q is diagonal
with entries
(
Qi,i =
4.0, if i = 9,
1.0, else,
(6.1)
and (c) shows costs when Q is diagonal with entries
(
0.25, if i = 9,
Qi,i =
(6.2)
1.0, else,
We reduce the rank of the square-root covariance to 2 in
all three cases. In (a) and (b) of Figure 3, the Choleskybased reduced-rank square-root Kalman filter exhibits almost
the same performance as the optimal filter whereas the
SVD-based reduced-rank square-root Kalman filter shows
degraded performance in (a). Meanwhile, in (c), the SVDbased has a large transient and large steady-state offset
from the optimal, whereas the Cholesky-based reduced-rank
square-root Kalman filter behaves close to the full-order
filter.
The mass-spring-damper model is shown in Figure 2. The
total number of masses is 10 with two states (displacement
and velocity) per mass. For i = 1, . . . , 10, mi = 1 kg,
and kj = 1 N/m, cj = 0.01 Ns/m, j = 1, . . . , 11. We
assume that the displacement of the 5th mass is measured and
disturbances enter all states so that disturbance covariance Q
has full rank.
Fig. 2.
Mass-spring-damper system.
We simulate three cases of disturbance covariance and
compare costs J in (a),(b) and (c) of Figure 4. (a) shows
costs when Q = I, (b) shows costs when Q is given by
(6.1), and (c) shows costs when Q is given by (6.2). We
reduce the rank of square-root covariance to 2 in all three
cases. In (a) and (b), the costs for the Cholesky-based and
SVD-based reduced-rank square-root Kalman filters are close
to each other but larger than the optimal. Meanwhile, in (c),
the SVD-based shows unstable behavior, while the Choleskybased filter remains stable and nearly optimal.
VII. C ONCLUSIONS
We developed a Cholesky decomposition method to obtain
reduced-rank square-root Kalman filters. We showed that
the SVD-based reduced-rank square-root Kalman filter is
equivalent to the optimal filter when the reduced-rank is
equal to or greater than the rank of error-covariance, while
the Cholesky-based is equivalent to the optimal when the
system is lower triangular block-diagonal according to the
observation matrix C which has the form [Ip×p 0] and p
is equal to the reduced-rank q. Furthermore, the Choleskybased rank-q square root filter is equivalent to the optimal
Kalman filter for a specific number of time steps. In general
cases, the Cholesky-based does not always perform better
that the SVD-based and vice versa. Finally, using simulation
examples, we showed that the Cholesky-based exhibits more
stable performance than the SVD-based filter, which can
become unstable when the strong disturbances enter the
system states that are not measured.
6
10
Optimal KF
RRSqrKF−Chol
RRSqrKF−SVD
R EFERENCES
5
10
4
J
10
3
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2
10
1
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50
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time index
300
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400
(a)
6
10
Optimal KF
RRSqrKF−Chol
RRSqrKF−SVD
5
10
4
J
10
3
10
2
10
1
10
50
100
150
200
250
time index
300
350
400
(b)
6
10
Optimal KF
RRSqrKF−Chol
RRSqrKF−SVD
5
10
4
10
J
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3
10
2
10
1
10
50
100
150
200
250
time index
300
350
400
(c)
Fig. 3. Time evolutions of the cost J , E(eT
k ek ) for the compartmental
system. For this example, R = 10−4 , P0 = 102 I, the rank of reduced
rank filters is fixed at 2 and energy measurement is taken at compartment 9.
Disturbances enter compartments 1,2,. . .,20. In (a), the cost J is when
disturbance covariance is Q = I20×20 , (b) is for the case when Q(9,9)
is changed to 4.0 and (c) is for the case when Q(9,9) is changed to
0.25.
6
10
Optimal KF
RRSqrKF−Chol
RRSqrKF−SVD
5
J
10
4
10
3
10
100
200
300
400 500 600
time index
700
800
900 1000
(a)
6
10
Optimal KF
RRSqrKF−Chol
RRSqrKF−SVD
5
J
10
4
10
3
10
100
200
300
400 500 600
time index
700
800
900 1000
700
800
900 1000
(b)
20
10
Optimal KF
RRSqrKF−Chol
RRSqrKF−SVD
15
J
10
10
10
5
10
0
10
100
200
300
400 500 600
time index
(c)
Fig. 4. Time evolutions of the cost J = E(eT
k ek ) of the mass-springdamper system. Here R = 10−4 , P0 = 1002 I, the rank of reduced rank
filters is fixed at 2, and a displacement measurement of m5 is available.
Velocity and force disturbances enter mass 1,2,. . .,10. (a) shows the cost J
when disturbance covariance is Q = I20×20 while (b) is for the case when
Q(9,9) is changed to 4.0 and (c) is for the case when Q(9,9) is changed to
0.25.