Ungarala, Sridhar

English

EngagedScholarship @ Cleveland State University

Cleveland State University
EngagedScholarship@CSU
Chemical & Biomedical Engineering Faculty
Publications Chemical & Biomedical Engineering Department
4-2009
Comments on "Robust and Reliable Estimation Via
Unscented Recursive Nonlinear Dynamic Data
Reconciliation"
Sridhar Ungarala
Cleveland State University, s.ungarala@csuohio.edu
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NOTICE: this is the author’s version of a work that was accepted for publication in Journal of
Process Control. Changes resulting from the publishing process, such as peer review, editing,
corrections, structural formatting, and other quality control mechanisms may not be reflected in this
document. Changes may have been made to this work since it was submitted for publication. A
definitive version was subsequently published in Journal of Process Control, 19, 4, (April 2009) DOI
10.1016/j.jprocont.2008.08.001
Repository Citation
Ungarala, Sridhar, "Comments on "Robust and Reliable Estimation Via Unscented Recursive Nonlinear Dynamic Data Reconciliation"" (2009).
Chemical & Biomedical Engineering Faculty Publications. 48.
https://engagedscholarship.csuohio.edu/encbe_facpub/48
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Original Citation
Ungarala, S. (2009). Comments on “Robust and reliable estimation via unscented recursive nonlinear dynamic data reconciliation”.
Journal of Process Control, 19(4), 717-718. doi:10.1016/j.jprocont.2008.08.001
Comments on "Robust and reliable estimation via unscented recursive 
nonlinear dynamic data reconciliation" 
In their paperVachhani, Narasimhan and Rengaswamy Uournal 
of Process Control 16 (2006) 1075- 10861. proposed sta te estima-
tion wi th constraints by solving least squares optimization with re­
spect to arbitrary sigma points around the state. In this note it is 
shown that the unscented recursive nonlinear dynamic data recon­
cilia t ion (URNOOR) approach fai ls to reduce to Kalman filter for 
unconstrained linear Gaussian systems. 
In the description of the procedure of unscented Kalman filter 
(UKF ). Vachhani et a!. missed a key step. After the sigma points 
are propagated and the mean and covariance are calculated for 
the predicted state (Eqs. ( 13 ) and (14) on p. 1078), new sigma 
points must be calculated around the predicted mean using the 
predicted covariance matrix. If this step is not performed the 
covariance matrix of the system noise Q does not affect the filter 
gain. 
In the URNOOR formulation, the authors did not justify the ba­
sis for posing the least squares problem for solving the "optimal" 
sigma points (p. 1079). State estimation as optimization can be 
based on minimizing errors or maximizing the probability density 
with respect to the state. This least squares obje<:tive function has 
no basis to be applicable to arbitrarily chosen sigma points. More­
over, the equali ty and inequality constraints of the states are not in 
general applicable to the sigma points. especially if the constraints 
are multivariate relationships. Since the proposed URNDDR ap­
proach is based on this flawed optimization setup it lacks te<:hnical 
rigor. In the foll owing discussion, it is shown that URNOOR results 
are incorrect. 
Consider state estimation of a scalar, linear, unconstrained. 
Gaussian process Xk<-I - A,l(k+ W k, Y"" I - CXk< 1+Vk<-I' where Wk .... 
N(O,Q) a~d .V"..I .... N(~, R ). Give~ the estimate i~:l and variance ~kjk' 
the predictions are: x~"' l lk = AX~:k and Pk. 11k- A Pkik +Q. By solVing 
the unconstrained optimization problem 
min (Yhl - CXk+II*+1)2 + (Xk+llt+1 - i k+11t )2 
"". ,~. , R Phl[l: 
Kalman fi lter (KF) corrector equations are obtained 
. CP~"' I Ii:Yhl + RXh llk 
Xk+II-\'+ 1 "" C' P R 
h i k + 
P _ Ph11kR  
k+II*+1 - C' P R· k+llk + 
The objective function may bejustified using maximum likelihood, 
Bayesian inference, least squares or orthogonal projection among 
others. It can be shown that UKF reduces to KF for this case as it 
is expected. Define sigma points and weights at time k 
K 
WO = l + K' 
1.k:U ,2 = Xt:k ± J(1 +K)Pk.k; W 1.2 = 2(1 ~ K) 
The predictions are Zh l1*J = A Z.tlkJ, which lead to the predicted 
state and variance , 
Xh l lk = L W lh +l lk., "" AXk:t. ,., 
Pk+l .k = L
, 
W I(h +l lk" - Xk+!ld +Q = A2pk;k +Q. 
i~O 
The sigma points are recalculated around the predicted state 
1.k+l lk.O = i t "" k , 
1.k+I[I:. 1.2 = Xk+l[I: ± J( l + K )Pk+I[I:' 
In their paper Vachhani et at. did not include this step. which 
makes Kalman gain incorrectly independent of Q, The original sig­
ma points may be used if the state vector is augmented with noise 
terms. The measurement function is evaluated at sigma points as 
h +l[l:,1 = CXh!lk.J to compute the predicted measurement and the 
variance , 
Yk+! = L ,., W;h+!,k.1 = CXk+l [1:, 
2
P....k+1 "" L
, 
Wi(h"' lk.1 - Yk+tl + R = C2Pk+llk + R. 
;~ 
The covariance between predicted state and measurement is , 
P"".k+l = L ,., W r(1.k+l[k.J - Xk+l)(Yk .. l lkJ - Yk+l) = CPt +1•k. 
The UKF fi lter gain is K"", - PXVJc- l{P ....Jc-I . The state estimate and 
variance are 
which is the Kalman filter. 
Now consider the URNOOR approach for the same problem. 
After the sigma points are propagated we have h +l lk.O = Xk+ l lk 
and Zk+ l ;k.l,2 = X~+ l lk ± AJ (l + K)Pk:t . Note that the varia nce of 
these points is A2Pkl", which is not Pk' 1Ik. The following least 
squares optimization problem is posed with respect to each 
sigma point although it is not apparent why they should be subject 
to it 
min (Yk+ 1 - Cb+1Ik+Li + (;(.+l lk+1.I - Xk+l lk,;l 
h. ,Ik+1J R Pk+ 1[k 
Since the objective function is of the same form, the solutions for 
"optimalH sigma points are in the form of Kalman fi lter corrector 
equation for the mean 
CPk+ 1ikYk+l + RZk+1[k.i 
C2Pk+ l lk + R 
Hence, the "optimal H sigma points are rewritten as 
1k+l lk+1.0 = Xk+l lk+h 
: _ RA J ( l + K)Pk;k 
Xt+1Ik+1.1.2 = Xk+1[h l ±.-2 R·pL- k+l [k + 
Computing the mean and covaria nce of the sigma points gives 
L
, 
W;1k+1[k+ U = Xk+1[k+l ' 
i_O ,
L W; (Xo+11k+11 - Xk+IIk+1)2 
i~O 
Although the mean matches the Kalman fi lter estimate. the vari­
ance is clearly something else. It appears like the variance obtained 
by URNDDR approach will be lower than the Kalman fi lter esti­
mate. Hence, URNDOR does not reduce to the Kalman filter for this 
special case, which invalidates the optimization approach to com­
pute the sigma poi nts. In view of the growing awareness about UKF 
among process systems engineering community the authors are 
urged to address this issue, 


Comments on  Robust and Reliable Estimation Via Unscented Recursive Nonlinear Dynamic Data Reconciliation

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