(Our
navigation buttons are at the TOP of each screen.)
Event detection as it relates to failure detection in navigation
systems and similarly for radar target maneuver detection (both achieved
by further processing Kalman filter outputs).
An important uptodate expose about the status of the Interactive
Multiple Model (IMM) bankofKalmanFilters is provided down below as it
relates to failure detection or target tracking when the system models are
nonlinear.
Similarly, an uptodate expose about the status of Generalized
Likelihood Ratio (GLR) approaches to event detection is provided below as
well.
Key Benefits:
 We are knowledgeable about various historical approaches, their
assumptions, their derivation, and their historical evolution. 
 We have gone through the rigorous supporting mathematics, yet we summarize
the results in a clear straightforward manner, expressed as simply as possible. 
 We are familiar with the application constraints associated with utilizing
these algorithms and are aware of what application constraints are usually
actively in force. 
Capabilities:
 We have historically developed a realtime failure event
detection algorithm for an ESG introduced into submarine INS
navigation
 We looked into all aspects in references [1][8], [18].
 We continue to monitor the field looking for new
developments
 We review and comment, as appropriate, in references [9][17].
 We maintain eternal vigilance in this specialty area!
Historical Account of our experience therein:
 [1] Kerr, T. H., “Poseidon Improvement Studies: RealTime
Failure Detection in the SINS\ESGM,” TASC Report TR41820, Reading,
MA, June 1974 (Confidential) for Navy, SP2413 (Jerome “Jerry” Katz).

 [2] Kerr, T. H., “A Two Ellipsoid Overlap Test for
RealTime Failure Detection and Isolation by Confidence Regions,”
Proceedings of IEEE Conference on Decision and Control,
Phoenix, AZ, December 1974.

 [3] Kerr, T. H., “Failure Detection in the SINS\ESGM
System,” TASC Report TR52831, Reading, MA, July 1975
(Confidential) for Navy, SP2413 (Jerome Katz).

 [4] Kerr, T. H., “Improving ESGM Failure Detection in the
SINS\ESGM System (U),” TASC Report TR67831, Reading, MA, October
1976 (Confidential) for Navy, SP2413 (Jerome Katz).

 [5] Kerr, T. H., “Failure Detection Aids for Human
Operator Decisions in a Precision Inertial Navigation System
Complex,” Proceedings of Symposium on Applications of Decision
Theory to Problems of Diagnosis and Repair, Keith Womer (editor),
WrightPatterson AFB, OH: AFIT TR 7615, AFIT\EN, Oct. 1976, sponsored
by the local Dayton Chapter of the American Statistical Association,
Fairborn, Ohio, pp. 98127, June 1976.

 [6] Kerr, T. H., “RealTime Failure Detection: A Static Nonlinear
Optimization Problem that Yields a Two Ellipsoid Overlap Test,” Journal of
Optimization Theory and Applications, Vol. 22, No. 4, pp. 509535, August 1977.

 [7] Kerr, T. H., “Statistical Analysis of a Two Ellipsoid
Overlap Test for RealTime Failure Detection,” IEEE Transactions
on Automatic Control, Vol. AC25, No. 4, pp. 762773, August 1980.

 [8] Kerr, T. H., “False Alarm and Correct Detection
Probabilities Over a Time Interval for Restricted Classes of Failure Detection
Algorithms,” IEEE Transactions on Information Theory, Vol. IT28, No.
4, pp. 619631, July 1982.

 [9] Kerr, T. H., “Examining the Controversy Over the Acceptability
of SPRT and GLR Techniques and Other Loose Ends in Failure Detection,” Proceedings
of the American Control Conference, San Francisco, CA, pp. 966977, 2224 June 1983.

 [10] Carlson, N. A., Kerr, T. H., Sacks, J. E., “Integrated
Navigation Concept Study,” Intermetrics Report No. IRMA321, 15 June 1984,
for ITT (Nutley, NJ) for ICNIA (Wright Patterson AFB).

 [11] Kerr, T. H., “Decentralized Filtering and Redundancy
Management Failure Detection for MultiSensor Integrated Navigation Systems,” Proceedings
of the National Technical Meeting of the Institute of Navigation (ION), San
Diego, CA, 1517 January 1985.

[12] Kerr, T. H., “Decentralized Filtering and Redundancy
Management for Multisensor Navigation,” IEEE Trans. on Aerospace and
Electronic Systems, Vol. AES23, No. 1, pp. 83119, Jan. 1987 (minor
corrections appear on p. 412 of May and on p. 599 of July 1987 issues of same
journal). This won
1988 M. Barry Carlton Award and $2000 honorarium for
Outstanding Paper to
appear in IEEE Aerospace and Electronic Systems Transactions in
’87
[for more details, please see page 822 of Vol. 24, No. 6,
Nov. 1988 of the aforementioned journal].
[13] Kerr, T. H., “Comments on ‘A ChiSquare Test for Fault
Detection in Kalman Filters’,” IEEE Transactions on Automatic Control,
Vol. AC35, No. 11, pp. 12771278, November 1990.
[14] Kerr, T. H., “A Critique of Several Failure Detection
Approaches for Navigation Systems,” IEEE Transactions on Automatic Control,
Vol. AC34, No. 7, pp. 791792, July 1989.
[15] Kerr, T. H., “On Duality Between Failure Detection and
Radar\Optical Maneuver Detection,” IEEE Transactions on Aerospace and
Electronic Systems, Vol. AES25, No. 4, pp. 581583, July 1989.
[16] Kerr, T. H., “Comments on ‘An Algorithm for RealTime
Failure Detection in Kalman Filters’,” IEEE Trans. on Automatic Control,
Vol. 43, No. 5, pp. 682683, May 1998.
[17] Kerr, T. H., “Comments on ‘Determining if Two Solid
Ellipsoids Intersect’,” AIAA Journal of Guidance, Control,
and Dynamics, Vol. 28, No. 1, pp. 189190, JanuaryFebruary 2005.
[18] Kerr, T. H., “Integral Evaluation Enabling Performance
Tradeoffs for Two Confidence RegionBased Failure Detection,” AIAA Journal of Guidance, Control, and
Dynamics,
Vol. 29, No. 3, pp. 757762, MayJune 2006 (click here to download a 229KB
version of this paper). or Click here.
[19] Kerr,
T. H., “Extending Decentralized Kalman Filtering
(KF) to 2D for RealTime
Multisensor Image Fusion and\or Restoration,” Signal Processing, Sensor
Fusion, and Target Recognition V, Proceedings
of SPIE Conference, Vol. 2755, Orlando, FL, pp. 548564, 810 April 1996.
The
approach, developed above by Tom Kerr, is independently endorsed in: Brumback, B. D., Srinath, M. D., “A
ChiSquare Test for FaultDetection in Kalman Filters,” IEEE Trans. on
Automatic Control, Vol. 32, No. 6,
pp. 532554, June 1987.
The above diagram is encountered in Kerr’s
original development of the Two Confidence Region approach to Failure Detection
for a submarine navigation application.
Again, this diagram is encountered in Kerr’s
original development of the Two Confidence Region approach to Failure Detection
for a submarine navigation application.
Two straight shooting, “noholesbarred”
exposes follow below with the corresponding references that are cited here
appearing underneath these discussions:
Prof.
Yaakov BarShalom (University of Connecticut)
(Click
here for more on what UCONN has pursued for defense)
http://www.ieee.org/about/awards/bios/picard_recipients.html
Please
click here to download a more detailed account of the current status of GLR
and IMM within a 213KByte pdf file.
Icelandic: “Ef satt skal segja.”
Status of Interactive
Multiple Model (IMM) Parallel BankofKalman filters Approach for Nonlinear
Applications
Since we have a working perspective into many other aspects
of Kalman filtering including its generalizations to approximate nonlinear
estimation [Ref. 4]; and its related concernsincluding having found and exposed
and corrected historical fallacies [Ref. 5], even those relating to random or
stochastic processes [Ref. 26]; we use this forum to also point out
perceived weaknesses that have not been widely publicized hithertofore that we, as
specialist in this area, perceive to exist in several other alternative
approaches to failure detection. Such considerations arise in reducing mere
theory to a final practical implementation instead of continuing to dwell on
ideal starting points of an event detection approach without considering the
realities of the constraints that exist in real implementations (one such being
the standard use of reducedorder filters, where filter residuals are no longer
ideally white and unbiased [Ref. 27] thus foiling or corrupting the original
approach of [Ref. 38] which explicitly relies on whiteness of residuals as a
gauge of normal unfailed behavior, as is also relied upon by GLR [Ref. 16]).
Gaussian confidence regions still persist as ellipsoidal Gaussians when
reducedorder filters are inserted in the application and certain reducedorder
filters still avail exact covariances online in realtime [Ref. 37] so CR2 is
therefore robust with respect to this aspect for mechanizations using this or
other similar reducedorder filter formulations. Moreover, Gaussian confidence
regions arise even when the pdf’s are from the more general exponential family in
situations where the important conditional and marginal distributions are still
Gaussian [Ref. 28, Chaps. 14].
While, by now, it is routine to consider the generalization
of Kalman filter estimation techniques from mere linear systems (for which
Kalman filters are optimal estimators [Ref. 1], [Ref. 2], [Ref. 3]) to nonlinear
systems (for which Extended Kalman filters or Iterated Extended Kalman filters [Ref.
4] are frequently useful, tractable approximate estimators for nonlinear
filtering [Ref. 5, Sec. 12]), as also discussed in [Ref. 1], [Ref. 2, Vol. 2,
1982], [Ref. 3]. Similar ideas should successfully generalize each of the Kalman
filters arising in the bankofKalman filters that occur in Interactive Multiple
Model (IMM) mechanizations as IMM is generalized beyond the exclusively linear
case for which it was originally rigorously derived as a two level approximation
(even in the purely linear case), where the sojourn times and Markov chain
transition probabilities are a new contrivance within IMM, useful by providing
additional parameters for tuning to better match potential application
situations by keeping alternative models more actively viable than they had been
for the Magill bankofKalmanfilters (1965); however, the associated IMM
probability calculations are more suspect in an attempted generalization to the
nonlinear case. Specifically, in each of the following three references [Ref. 6,
before Eq. 4], [Ref. 7, after Eq. 2], [Ref. 8, after Eq. 6], “the critical
mixture is assumed to be a sum of Gaussians, then the prior pdf is a Gaussian
mixture and can be approximated (via moment matching) with a single Gaussian....” (While sums of Gaussian random variables or sums of
Gaussian random
processes are always Gaussian, that is not the issue or situation here where the
topic instead is whether the resulting pdf of the output is a weighted sum of
the Gaussian pdf’s called a “Gaussian mixture”, as
claimed. Moreover, it has been asserted that the resulting Gaussian
mixture consisting of a weighting of several Gaussian pdf’s
can be subsumed as one Gaussian.)
Our objections to this aspect is fourfold:

For nonlinear systems, the estimates outputted by an
EKF are not Gaussian in general (unlike the assumption);

There are already existing analytic results [Ref. 9]
which caution that a single subsuming Gaussian pdf is not always possible
(nor usually possible) even if the individual participating pdf’s were in
fact Gaussian when the means of the various contributing pdf’s are not in
close enough proximity, as gauged by the spread of the associated
covariances. This topic has been an issue since the historically well known
GaussianSums approach of [Ref. 10], [Ref. 11] which also used a
bankofKalmanfilters structure (which also did not match “expectations”, so to speak ). Indeed, nonlinear filtering situations
frequently exhibit multimodal output estimates as a fact of life, as
discussed in [Ref. 12];

The “moment matching” called for in [Ref. 6], [Ref.
7], [Ref. 8] is not explained there nor is there an opportunity to do so
within the algorithm for each timestep k, as needed;

It is not clarified what is to be matched in “moment
matching” by what and to what and by what gauge will it be determined that
it matches closely enough.
Nothing about these aspects has been explained in the three references cited
above, which is why we raise the issue by pointing it out here now. Please see
Ref. [40] below and our comments in red attached to it. Go
to Top
Please
click here for a more detailed account in a 112KByte pdf file.
Status of ResidualBased Failure Detection or
Maneuver Detection when Quiescent Condition of Whiteness and Unbiasedness is Jeopardized
by use of Reduced Order Kalman Filters in Actual Mechanization
We now seek to point out apparent weaknesses that have not been widely publicized or even acknowledged hithertofore that we, as specialist in this area, perceive to exist in several alternative approaches to
failure detection (being a special case of event detection). Such considerations arise in reducing mere theory to a final practical implementation instead of continuing to dwell on ideal starting points of the original formulation of an event detection approach without explicitly considering the realities of the constraints that exist in implementation within the actual applications. One such prevalent constraint being the standard use of reducedorder
suboptimal filters, where filter residuals are no longer ideally white and
unbiased (specifically, filter residuals are white and unbiased if and only if the system and sensor model used in the Kalman filter are identical to what exists for the actual system or in
it’s
truth model representation used in the simulation, otherwise the residuals are either nonwhite or biased or both) thus degrading or corrupting the original idealized aspects of many detection approaches which explicitly
rely on an assumption of “whiteness and unbiasness of
residuals” as a gauge of normal unfailed behavior.
The “whiteness of Kalman filter residuals” is also relied upon in
another approach to a failure detection formulation using the Generalized Likelihood
Ratio (GLR), where, again, reducedorder filter usage introduces bias and nonwhiteness of the associated filter residuals even in the nominally unfailed situation. Such effects introduce ambiguity into the algorithmic decision of whether to declare that “a failure has
occurred” or to declare that “no failure is
present” since now the situation is less of a dichotomy for the decision algorithm after “the water has been
muddied” by the use of a reducedorder filter, as historically required in most applications (where similar issues also arise for use of reducedorder observers in application environments where noise is relatively less significant). Are the Kalman filter residuals now nonwhite and biased because of a failure occurring or because of the standard use of a reducedorder filter in the particular application? Such obscuring effects are consequentially timevarying when the associated navigation (NAV) filter structures which provoke or aggravate them are similarly timevarying).
The Generalized Likelihood Ratio (GLR) approach to event detection, where
maximum likelihood estimates of unknown parameters are utilized within the ratio
of the H_{1} pdf to the H_{0} pdf in lieu of not knowing the
actual requisite parameters of the mixed hypothesis since they are in fact
unknown, is presented and developed by Davenport and Root [Ref. 29]. Root went
further [Ref. 30] to investigate applicability of GLR techniques in the radar
detection problem of resolving closely spaced targets in a background of either
known arbitrary correlated Gaussian noise or in Gaussian white noise. However,
Root [Ref. 30] obtained explicit criteria that could be applied to indicate
conditions under which one could expect to not resolve two known signals
(of unknown amplitudes and parameters) and additionally pointed out a difficulty
of using GLR for this purpose.
McAulay and Denlinger [Ref. 31] advocated use of GLR in conjunction with a
Kalman filter in decisiondirected adaptive control applications. Finally,
Stuller [Ref. 32] defined an Mary GLR test that ostensibly overcame Root’s
original objections [Ref. 30] to GLR for this type of application. ([Ref. 32]
also provides a limited history of GLR developments for radar, excepting no
mention of [Ref. 31], which possibly eluded him as being so recent. The use of
GLR for failure detection was pioneered by Willsky and Jones [Ref. 16] using an identical
GLR formulation as presented by McAulay and Denlinger [Ref. 31]. While both
Willsky and McAulay claim optimality of the GLR, they never explicitly specify a
criteria by which it may be judged optimal nor do they supply a proof or
reference where such a claim is demonstrated (specifically, [Ref. 31] references
the proof to be in a English translation of an identified German textbook but
diligent followup on our part here revealed no such substantiation located
there).
While use of GLR has potential in many detection situations, it is not
without its drawbacks that are frequently overlooked:
 Selin found that some of the unknown parameters (such as unknown relative
carrier phase) must also be estimated in order to maximize the a
posteriori probability in the estimation of two similar signals in white
Gaussian noise [Ref. 33]; 
 Selin further identified four standard caveats [Ref. 34, p. 106]
associated with use of a maximum likelihood estimate of the unknown
parameters in a likelihood ratio (as occurs within GLR); 
 The GLR is not a Uniformly Most Powerful (UMP) test [Ref. 35, p.
92]; 
 There are cases where use of GLR can give bad results [Ref. 35, p.
96]; 
 That use of a Maximum likelihood estimate (MLE) is not necessarily statistically
consistent in general is explicitly demonstrated in a counterexample in
[Ref. 36, p. 146]. 
GLR is again being advocated for use in radar applications but those that do
so appear to ignore the historical objections for use of GLR in these types of
applications as well as the explicit counterexamples in [Ref. 39, 968 ff, App.
A, pp. 973974] that, apparently, have never been refuted. The new version of
GLR (called “Ed Kelly’s GLR”) is of a different form than used by the others
mentioned above [Ref. 21] and is apparently a pseudoGLR but useful
nonetheless for radar tracking but evidently somewhat lacking in its present
form for failure detection since it ignores the underlying onset time of the
detection event of interest. Use of the Entropy Maximization (EM) algorithm may
placate Selin’s and Roots’ concerns above but is a large computational burden
that may defy a realtime implementation.
References:
 Jazwinski,
A. H., Stochastic Processes and Filtering Theory, Academic Press,
N.Y., 1970.
 Maybeck,
P. S., Stochastic Models, Estimation, and Control, Part I, Academic
Press, N.Y., 1979.
 Gelb,
A. (Ed.), Applied Optimal Estimation, MIT Press, Cambridge, MA, 1974.
 Kerr,
T. H., “Streamlining Measurement Iteration for EKF
Target Tracking,” IEEE Trans. on AES, Vol. 27, No.
2, pp. 408420, Mar. 1991 (minor correction in Nov. 1991).
 Kerr,
T. H., “Exact Methodology for Testing Linear System
Software Using Idempotent Matrices and Other ClosedForm Analytic Results,”
Proceedings of SPIE, Session 4473: Tracking Small Targets, San Diego,
CA, pp. 142168, 29 Jul.3 Aug. 2001.
 BarShalom,
Y., Chang, K. C. and Blom, H. A. P., “Tracking a
Maneuvering Target Using Input Estimation Versus the Interactive Multiple
Model,” IEEE Trans. on AES, Vol. 25, No. 2, pp.
296300, Mar. 1989.
 Mehra,
R., Rago, C., Seereeram, S., “Autonomous Failure
Detection, Identification, and Faulttolerant Estimation with Aerospace
Applications,” Proceedings of IEEE Aerospace
Conference, Part 2, Vol. 2, Aspen, CO, pp. 133138, 2128 Mar. 1998.
 Rago,
C., Prasanth, R., Mehra, R. K., Fortenbaugh, R., “Failure
Detection and Identification and FaultTolerant Control using IMMKF with
Applications to the EagleEye UAV,” Proceedings of the
37^{th} IEEE Conference on Decision and Control, Part 4, Vol. 4,
Tampa, FL, pp. 42084213, 1618 Sept. 1998.
 Konstantellos,
A. C., “Unimodality Conditions for Gaussian Sums,”
IEEE Trans. on Automatic Control, Vol. 25, No. 4, pp. 838839, Aug.
1980.
 Alspack,
D. L., and Sorenson, H. W., “Nonlinear Bayesian
Estimation Using Gaussian Sum Approximations,” IEEE
Trans. on Automatic Control, Vol. 16, No. 4, pp. 439448, Aug. 1972.
 Alspack,
D. L., “A Gaussian Sums Approach to the Multitarget
Identification Tracking Problem,” Automatica, Vol.
11, pp. 285296, 1975.
 Netto,
N. L., Andrade, L., Gimeno, L., and Mendes, M. J., “On
the Optimal and Suboptimal Nonlinear Filtering Problem for DiscreteTime
Systems,” IEEE Trans. on Automatic Control, Vol.
23, No. 6, pp. 10621067, Dec. 1978.
 Kerr, T. H., “Critique of Some Neural Network
Architectures and Claims for Control and Estimation,” IEEE
Transactions on Aerospace and Electronic Systems, Vol.34, No. 2, pp.
406419, Apr. 1998.
 Kerr, T. H., “Decentralized Filtering and Redundancy
Management for Multisensor Navigation,” IEEE
Transactions on Aerospace and Electronic Systems, Vol. 23, No. 1, pp. 83119, Jan.
1987 (corrections: May, p. 412; Jul., p. 599). This won 1988 M. Barry Carlton Award and $2000 honorarium for
Outstanding Paper to
appear in IEEE Aerospace and Electronic Systems Transactions in
’87
[for more details, please see page 822 of Vol. 24, No.
6, Nov. 1988 of the aforementioned journal].
 Kerr, T. H., “A Critique of Several Failure Detection
Approaches for Navigation Systems,” IEEE Transactions
on Automatic Control, Vol. 34, No. 7, pp. 791792, Jul. 1989.
 Willsky, A. S., Jones, H. L., “A Generalized
Likelihood Ratio Approach to Detection and Estimation of Jumps in Linear
Systems,” IEEE Transactions on Automatic Control, Vol. 21, No.
1, pp. 108112, Feb. 1976.
 Kerr, T. H., and Chin, L., “A Stable Decentralized
Filtering Implementation for JTIDS RelNav,” Proceedings
of IEEE Position, Location, and Navigation Symposium (PLANS),
Atlantic City, NJ, pp. 318329, 811 Dec. 1980.
 Kerr, T. H., “Stability Conditions for the RelNav
Community as a Decentralized EstimatorFinal Report,”
Intermetrics, Inc. Report No. IR480, Cambridge, MA, 10 Aug. 1980.
 Kerr, T.H., and Chin, L., “The Theory and Techniques
of DiscreteTime Decentralized Filters,” in Advances
in the Techniques and Technology in the Application of Nonlinear Filters and
Kalman Filters, edited by C. T. Leondes, NATO Advisory Group for
Aerospace Research and Development, AGARDograph No. 256, Noordhoff
International Publishing, Lieden, pp. 31 to 339, 1981.
 Kerr, T. H., “On Duality Between Failure Detection and
Radar/Optical Maneuver Detection,” IEEE Transactions
on Aerospace and Electronic Systems, Vol. 25, No. 4, pp. 581583, Jul.
1989.
 Kerr, T. H., “New Lamps for Old: a Shell Game for
Generalized Likelihood Use in Radar? Or this isn’t your
father’s GLR!,”
Proceedings of SPIE, Session 4473: Tracking Small Targets, San Diego,
CA, pp. 476483, 29 Jul.3 Aug. 2001.
 Kerr, T. H., “Status of CRLike Lower bounds for
Nonlinear Filtering,” IEEE Transactions on Aerospace
and Electronic Systems, Vol. 25, No. 5, pp. 590601, Sept. 1989 (Author’s reply: Vol. 26, No.
5, pp. 896898, Sept. 1990).
 Kerr, T. H., UEWR Design NotebookSection 2.3: Track Analysis, TeK
Associates, Lexington, MA, (for XonTech, Hartwell Rd, Lexington, MA),
XonTech Report No. D74410300, 29 Mar. 1999.
 Kerr, T. H., “Emulating Random Process Target
Statistics (using MSF),” IEEE Transactions on
Aerospace and Electronic Systems, Vol. 30, No. 2, pp. 556577, Apr.
1994.
 Kerr, T. H., “Fallacies in Computational Testing of
Matrix Positive Definiteness/Semidefiniteness,” IEEE
Transactions on Aerospace and Electronic Systems, Vol. 26, No. 2, pp.
415421, Mar. 1990.
 Kerr, T. H., “Vulnerability of Recent GPS Adaptive
Antenna Processing (and all STAP/SLC) to Statistically NonStationary Jammer
Threats,” Proceedings of SPIE, Session 4473:
Tracking Small Targets, San Diego, CA, pp. 6273, 29 Jul.3 Aug. 2001.
 Boozer, D. D., McDaniel, W. I., “On Innovation
Sequence Testing of the Kalman Filter,” IEEE
Transactions on Automatic Control, Vol. 17, No. 1, pp. 158160, Feb.
1972.
 Arnold, B. C., Castillo, E., Sarabia, J. M., Conditional Specification
of Statistical Models, SpringerVerlag, NY, 1999.
 Davenport, W. B., Root, W. L., Introduction to the Theory of Random
Signals in Noise, McGrawHill, NY, 1958.
 Root, W. L., “Radar Resolution of Closely Spaced
Targets,” IRE Transactions on Military Electronics,
Vol. MIL6, pp. 197204, Apr. 1962.
 McAuley, R. J., Denlinger, E., “A DecisionDirected
Adaptive Tracker,” IEEE Transactions on Aerospace and
Electronic Systems, Vol. 9, No. 2, pp. 229236, Mar. 1973.
 Stuller, J. A., “Generalized Likelihood Signal
Resolution,” IEEE Transactions on Information Theory,
Vol. 21, No. 3, May 1975, pp. 276282.
 Selin, I., “Estimation of the Relative Delay of Two
Similar Signals of Unknown Phases in Gaussian Noise,”
IEEE
Transactions on Information Theory, Vol. 10, pp. 189191, Jul. 1964.
 Selin, I., Detection Theory, Princeton University Press, Princeton,
NJ, 1965.
 Van Trees, H. L., Detection, Estimation, and Modulation Theory, Part
I: Detection, Estimation and Linear Modulation Theory, John Wiley and
Sons, Inc., NY, 1968.
 Patel, J. K., Kapadia, C. H., Owen, D. B., Handbook of Statistical
Distributions, Marcel Dekker, Inc. NY, 1976.
 Kerr, T. H., “Computational
Techniques for the Matrix Pseudoinverse in Minimum Variance ReducedOrder
Filtering and Control,” in Control and Dynamic
SystemsAdvances in Theory and Applications, Vol. XXVIII: Advances in
Algorithms and computational Techniques for Dynamic Control Systems,
Part 1 of 3, C. T. Leondes (Ed.), Academic Press, NY, pp. 57107, 1988.
 Mehra, R. K., Peschon, J., “An Innovations Approach to
Fault Detection and Diagnosis in Dynamic Systems,” Automatica,
Vol. 7, No. 5, pp. 637640, 1971.
 Kerr, T. H., “The Controversy Over Use of SPRT and GLR
Techniques and Other LooseEnds in Failure Detection,” Proceedings
of American Control Conference, Vol. 3, San Francisco, CA, pp. 966977, 2224 Jun.
1983.

BarShalom, Y., Challa, S., Blom, H. A. P., “IMM
Estimator Versus Optimal Estimator for Hybrid Systems,”
IEEE Trans. on Aerospace and Electronic Systems, Vol. 41, No. 3, pp.
986991, July 2005. [If
M(k) were in fact known beforehand, then {x(k), M(k)} would, in fact, be a
Markov process as these authors rely on in their proofs. However, just how
the system will switch is unknown to the user or engineer a priori or even
immediately after the evolution in time of the system of interest having
this potential structure. Notice that the authors need to assume that the
underlying multiple model systems have a linear structure before a Gaussian
sums mixture model can be invoked. When models are nonlinear, probability
density function
calculations are infinite dimensional in general (i.e., the actual calculations
for the associated probabilities, while tractable, cannot be accomplished in
realtime since the indicated integrations to be performed are
infinitedimensional in all but the simplest of cases). You can write expressions
that involve integral evaluations but you are SOL for realtime computations
of the required answers. This is essentially Bucy’s representation
theorem of 30 years ago.]

He, T., BenDavid, S., Tong, L., “Nonparametric Change Detection and Estimation in LargeScale Sensor
Networks,” IEEE Trans. on Signal Processing, Vol. 54, No. 4, pp. 12041217, April 2006.

Nyberg, M., Frisk, E., “Residual
Generation for Fault Diagnosis of Systems Described by Linear
DifferentialAlgebraic Equations,” IEEE Trans. on Automatic Control, Vol. 51, No. 12, pp.
19951999, December 2006.

Stoica, P., “Performance Evaluation of Some Methods
for OffLine Detection of Changes in Autoregessive Signals,” Selected
Papers on Performance Evaluation of Signal and Image Processing Systems,
Firooz Sadjadi (editor), SPIE Milestone Series, 5, No. 4, pp. 134136,
1999. [T. H. Kerr’s comment: An assertion within
this paper is that existing fault detection or change detection schemes
involve monitoring test statistics that consist of quadratic forms of
subcollections of states that are Gaussianly distributed with zero
means, and that as a consequence, the test statistic has a Chisquare
distribution. We object in that while this statement is almost true for
many different approaches, it is not true for CR2 nor for GLR. We at TeK
Associates have worked out the statistics in detail and published them
in the open peerreviewed literature for CR2 for the transient
timevarying situation that faced us. If we had the luxury of waiting
until the timeinvariant steadystate was reached (which was impossibly
long and never arrived for the submarine navigation application for
which CR2 was developed), then the distribution of CR2 is also asymptotically
chisquare with mdegreesoffreedom, where m is dimension of the number
of states being simultaneously monitored. For GLR, the GLR test
statistic itself is the maximum of several different quadratic forms of
Gaussianly distributed states so it is the maximum of several different
Chisquared entities which results in something that is not chisquared
even if the situation were so benign as to have an identical chisquared
distribution in common (which it does not). Again the maximization
operation alters the distribution of the output in a predictable way
(via A. Papoulis’ textbook) but its calculation is not tractable
and has yet to be done.]

Cheng, Q., Varshney, P.
K., Michels, J. H., Belcastro, C. M.,
“Fault Detection in Dynamic Systems via Decision Fusion,” IEEE Trans. on Aerospace and Electronic Systems, Vol.
44, No. 1, pp. 227242, Jan. 2008.

Tudoroiu, N.,
Khorasani,
K., “Satellite
Fault Diagnosis using a Bank of Interacting Kalman Filters,” IEEE Trans. on Aerospace and Electronic Systems, Vol.
43, No. 4, pp. 13341350, Oct. 2007.

Chan, Y. K., Edsinger, R. W., “A
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