kalman filtering & smoothing to estimate real-valued states & integer constants
DESCRIPTION
Kalman Filtering & Smoothing to Estimate Real-Valued States & Integer Constants. Mark L. Psiaki, Sibley School of Mechanical & Aerospace Engineering Cornell University. Goal. Improve estimation algorithms for systems that have integer measurement ambiguities - PowerPoint PPT PresentationTRANSCRIPT
AIAA GNC, 11 Aug. 2009
Mark L. Psiaki, Sibley School of Mechanical & Aerospace EngineeringCornell University
Kalman Filtering & Smoothing to Estimate Real-Valued States & Integer Constants
GNC/Aug. ‘09 2 of 21
Goal Improve estimation algorithms for systems that have
integer measurement ambiguities CDGPS with double-differenced integer ambiguities Systems using carrier-phase measurements of TDMA
signals
Use SRIF/LAMBDA-type formulation to deal with mixed real/integer problem
Develop optimal & suboptimal Kalman filter & smoother algorithms Optimal: keep all ambiguities & treat as integers Suboptimal: retain integers in a finite time window
Strategies
GNC/Aug. ‘09 3 of 21
Outline of TalkI. Related researchII. Problem definitionIII. Mixed real/integer Kalman filter
Optimal, retains all past integers Suboptimal, retains finite window of past integers
IV. Mixed real/integer fixed-interval smoother Optimal, retains all integers of fixed interval Suboptimal, retains finite window of past & future integers
relative to each time point
V. Truth-model simulation & resultsVI. Conclusions
GNC/Aug. ‘09 4 of 21
Related Research: Batch estimation w/integer ambiguities
The LAMBDA method, Teunissen (1995) & follow-ons Other methods, e.g., Chen & Lachapelle (1995) SRIF LAMBDA-like method, Psiaki & Mohiuddin (2007)
Kalman filtering w/integer ambiguities Standard Covariance EKF, Kroes et al. (2005) SRIF-based EKF, Mohiuddin & Psiaki (2008) Sub-optimal dropping of each integer ambiguity immediately
after its last occurrence in a measurement Smoothing w/integer ambiguities
Nothing
GNC/Aug. ‘09 5 of 21
Dynamics ModelReal-state dynamics:
Partitioning of integer states by affected measurement sample times (past, past & present, past, present & future):
kkkkkk ΓΦ wxx 1
kk
k nnn 1
Growth of integer state with sample number
kk
kk
ΠΔ
Δ
n
nn
n
1 Or dynamic re-partitioning
kk
kkk Π~ n
nnn
11
GNC/Aug. ‘09 6 of 21
Measurement Model
111
11~
ykkk
nkkwkkxkk HHH nnwxy
… using integer vector partitions
… using full integer vector
1111 ykknkkwkkxkk HHH nwxy
111 000000~
knknk ΠI
IHH
GNC/Aug. ‘09 7 of 21
Example Sensitivities of Different Measurement Types to Different Integers
0 1000 2000 3000 4000 5000 6000 7000 8000 90001
2
3
4
5
6
7
8
9
10
11x 10
-3
Time (sec)
Am
bigu
ity S
ensi
tiviti
es, h
tilde
(m)
GNC/Aug. ‘09 8 of 21
Kalman Filtering/Smoothing Problem find: x0, …, xk+1, w0, …, wk, & nk+1 = [n0; …; nk] to minimize:
subject to: xj+1 = jxj + jwj + j for j = 0, 1, 2, ..., k nk+1 is an integer-valued vector
][][ 000T
00021
xxxxxx ˆRˆRJ zxzx
k
jjwwjjwwj RR
0
T21 ][][ ww
k
jj
jnjjwjjxj HHH
0
T1
012
1 ][{ yn
nwx
}][ 10
1
j
jnjjwjjxj HHH y
n
nwx
GNC/Aug. ‘09 9 of 21
Stage-k a posterior info:
Combined information eqs. w/dynamics substitution for xk:
New stage-(k+1) a posterior info after QR factorization:
Optimal SRIF Kalman Filterxkxkkxnkkxxk RR ˆˆˆ znx
nknkknnkR ˆˆ zn
111
1
1
1
111
11
ˆˆ
ˆˆˆ
0
]0,ˆ[00]0,ˆ[ˆˆ
00
yknkxkwk
kkxkk
nkkkxxkxk
k
kk
nkkxkkkxkwk
nnk
xnkkxxkkkxxk
wwk
ΦH
ΦR
HΦHΓΦHHRRΦRΓΦR
R
yz
z
nxw
1
1
1
1
1
1
1
11
11
11
ˆˆˆ
ˆˆˆ
000
ˆ00
ˆˆ0
ˆˆˆ
rk
nk
xkwk
rk
nk
xkwk
k
kk
nnk
xnkxxk
wnkwxkwwk
RRRRRR
zzzz
nxw
GNC/Aug. ‘09 10 of 21
Measurement Update via Integer Linear Least-Squares Solution Solve integer linear least-squares problem to
determine integer a posteriori estimate
Back-substitute to compute real-valued states:
]ˆˆ[]ˆˆ[ 111T
11121
nkknnknkknnk RR znzn1
ˆkn
]ˆˆˆ[ˆˆ 1111
11
kxnkxkxxkk RR nzx
)(min 1kJ n
GNC/Aug. ‘09 11 of 21
Suboptimal KF Retention of Exact Integers within a Window of Samples
0 1000 2000 3000 4000 5000 6000 7000 8000 90001
2
3
4
5
6
7
8
9
10
11x 10-3
Time (sec)
Am
bigu
ity S
ensi
tiviti
es, h
tilde
(m)
Measurements used in tk = 3000 secsub-optimal filter
tk = 3000 sec +0/- i*deltat windowfor considering exact integers
kk
i,k
k Δ
Δ
nn
n
m )1(max
GNC/Aug. ‘09 12 of 21
Stage-k a posterior info:
Combined information eqs. w/dynamics substitution for xk & mk
New stage-(k+1) a posterior info after QR factorization:
Suboptimal SRIF Kalman Filter xkxkkxmkkxxk RR ˆˆˆ zmx
mkmkkmmkR ˆˆ zm
111
1
1
1
111
11
ˆˆ
ˆˆˆ
0
0
ˆ00ˆˆˆˆˆ
000
ykmkxkwk
kkxkk
mkkkxxkxk
k
kkik
mkkxkkkxkwk
mmbkmmak
xmbkkxxkkkxxkxmak
wwk
ΦH
ΦRΔ
HΦHΓΦHHRRRΦRΓΦRR
R
yz
z
mxwn
1
1
1
1
1
1
1
1
1
11
11
11
ˆˆˆˆ
ˆˆˆˆ
0000
ˆ000
ˆˆ00
ˆˆˆ0
ˆˆˆˆ
rk
mk
xkwkΔnk
rk
mk
xkwkΔnk
k
kkik
mmk
xmkxxk
wmkwxkwwk
ΔnmkΔnxkΔnwkΔnΔnkΔ
RRRRRRRRRR
zzzzz
mxwn
GNC/Aug. ‘09 13 of 21
Terminal sample K initialization:
1-sample backwards recursion starts w/filtered wk & smoothed xk+1 info. eqs. & uses dynamics to get
QR factorize to isolate smoothed xk info. eq.
Optimal RTS Smoother in SRIF Form
,ˆ K*K nn ,RR xxK
*xxK KxnKxK
*xK ˆRˆ nzz K
*K xx i.e.,
*xk
wk
k*xxk
*xk
*kwnkkwxkwk
kk
k*xxkk
*xxk
kwxkkwxkwwk ˆ
R
RRˆ
RR
RRR
111
111
11
11
z
nzxw
*xk
*wk
*xk
*wk
kk
*xxk
*wxk
*wwk
R
RR
zz
xw
0 )]0 [
known(with
1*k
*k I nn
GNC/Aug. ‘09 14 of 21
Suboptimal RTS Smoother Retention of Exact Integers within a Window of Samples
0 1000 2000 3000 4000 5000 6000 7000 8000 90001
2
3
4
5
6
7
8
9
10
11 x 10-3
Time (sec)
Am
bigu
ity S
ensi
tiviti
es, h
tilde
(m)
Additional measurements used intk = 3000 sec sub-optimal smoother
Measurements used in tk = 3000 secsub-optimal filter & smoother
tk = 3000 sec +/- i*deltat rangefor considering exact integers
)11( ik,Kmin
kk
k
n
nm
l
GNC/Aug. ‘09 15 of 21
Terminal sample K initialization:
1-sample backwards recursion starts w/filtered wk & nk-i & smoothed xk+1 & lk+1 info. eqs. & uses dynamics & integer permutation/partitions to get
Suboptimal RTS Smoother (1 of 2)
,RR xxK*xxK
,ˆ xK*xK zz
,RR xmK*xlK ,RR mmK
*llK
mK*lK zz
*lk
*xk
wkΔnk
*lk
k*xxk
*xk
kwxkwk
kΔnxkΔnk
kkkik
*llak
*llbk
*xlakk
*xxkk
*xxk
*xlbk
wlkkwxkkwxkwwk
ΔnlkkΔnxkkΔnxkΔnwkˆˆ
R
RˆRˆ
RR
RRΓRR
RRΓRRRRΓRR
1
1
1
11
1
1
11
11
11
00
00
z
z
zz
lxwn
GNC/Aug. ‘09 16 of 21
Suboptimal RTS Smoother (2 of 2) New stage-k smoothed xk & lk square-root information
equations after QR factorization
is the integer vector that minimizes
The real part of the state is determined by back substitution:
*lk
*xk
*wk
*nk
*lk
*xk
*wk
*nk
kkkik
*llk
*xlk
*xxk
*wlk
*wxk
*wwk
*nlk
*nxk
*nwk
*nkn
RRRRRRRRRR
zzzz
lxwn
00000
0
*kl
][][)( T *lkk
*llk
*lkk
*llkk RRJ zlzll
)()( 1 *k
*xlk
*xk
*xxk
*k RR lzx
GNC/Aug. ‘09 17 of 21
Example 1-Dimensional CDGPS-Type Problem with 3rd-Order Dynamics Dynamics:
Measurements:
kkk
k
kkkkk
k ΔtΔtΔt
qΔtΔtΔtΔt
wxx
01/331/3/500)34/(4/5000)52/(
10010
0.5122
1
k
kk
kk
s
s
kΔtΔt
ΔtΔt
kkr
kxy
2
2
10.1255.01
0.1255.01
)/1(000000)/1( {1
1
3205
1921
1283
1920531
3205
1921
1283
1920531
2 }11
000000
yks
s
s,k
s,k
kkk
kkr
k
kkr
k
n
n
h~
h~
qΔtΔt w
GNC/Aug. ‘09 18 of 21
x1 Errors for Three Kalman Filters
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
-0.1
-0.05
0
0.05
0.1
Time (sec)
x 1 filte
r erro
r (m
)
OptimalSuboptimal, i = 40Suboptimal, no integers
GNC/Aug. ‘09 19 of 21
x1 Errors for Three Smoothers
0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Time (sec)
x 1 sm
ooth
er e
rror (
m)
OptimalSuboptimal, i = 40Suboptimal, no integers
GNC/Aug. ‘09 20 of 21
Integer-Part Computational Cost of Optimal & Suboptimal Algorithms
0 1000 2000 3000 4000 5000 6000 7000 8000 90000
0.5
1
1.5
2
2.5
3
3.5
4 x 104
Time (sec)
Cum
ulat
ive
L tot IL
LS E
xecu
tion
Cos
t Met
ric
Optimal KFSuboptimal KF, i = 40Suboptimal KF & Smoother, i = 40
GNC/Aug. ‘09 21 of 21
Summary & Conclusions Developed optimal & suboptimal Kalman filters & fixed-
interval smoothers for mixed real/integer estimation problems Constant integer ambiguities enter only measurements Optimal algorithms consider all integers in data batch Suboptimal algorithms drop integers that affect measurements
only in remote past or future Tested using data from truth-model simulation
Optimal & suboptimal filter achieve modest accuracy gains vs. all-reals approximate filter
Filter accuracy gains may be greater for different problem Optimal & suboptimal smoother significantly more accurate
than all-reals smoother Suboptimal smoother nearly as accurate as optimal smoother Suboptimal algorithms reduce required processing by at least
65% through reductions in dimensions of measurement update integer linear least-squares problems