Application of Post Kalman Filtering Techniques to Tracking Problems
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Application of
Post Kalman Filtering Techniques
to Tracking Problems
A Thesis
Submitted in Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy
at the
Jadavpur University
By
M. SRINIVASAN
Department of Electrical Engineering
Jadavpur University
Kolkata, India
2007Synopsis
SYNOPSIS
Application of Post Kalman Filtering Techniques to Tracking
Problems
Submitted By
M. SRINIVASAN
Department of Electrical Engineering, Jadavpur University,
for the Ph. D (Engg.), 2007
1 Background
State and parameter estimation are two very important and technically interesting areas of
control systems. Tracking of moving objects primarily requires state estimation and occasionally
parameter estimation. Since the 1960's decade, the Kalman filter (KF) and its variants like
Extended Kalman Filter (EKF) had been the mainstay of tracking.
The present work deals with Filtering and Estimation for non-linear signal models. In several
tracking problem, the EKF has shown the tendency to diverge. The past ten years saw a new wave
of filtering techniques, which we call 'post Kalman filtering' techniques, which promise improved
performance. These improved algorithms rely on the availability of computational power within
reasonable constraints of cost, volume, space and power requirement.
The post Kalman filtering techniques are capable of accommodating nonlinear systems and
non-Gaussian noise and are therefore considered/claimed to be powerful. Some of the post
Kalman filtering techniques had been proposed to exploit the increased computing power of
computers towards better and more reliable estimation.
Though it is fairly easy to demonstrate the superiority of any novel filtering technique using
carefully chosen mathematical problems, real-life tracking operates in a non-ideal situation, where
the simplifying assumptions used for modelling a problem often are not valid. Unlike simple
filters like "alpha beta gamma" filters, least square filters, KF and EKF, where a vast amount of
application experience exists, such a body of experience is not available for post Kalman filters.
Primary motivation for the present work was to obtain a deeper insight into the applicability,
shortcomings and relative strengths of these post Kalman filtering techniques, by applying them
to (i) known tough (non-tracking) problems, (ii) selected (and admittedly simple but highly
nonlinear) tracking problems.Synopsis
The second motivation had been to use the above insights towards obtaining improved
estimation and filtering methods for tracking problems related to Tactical Ballistic Missile (TBM)
tracking. For reasons other than purely technical, the application area of this work had to be more
focussed towards defence application of tracking.
1.1 Tracking, Interception and Guidance
Traditionally, tracking had been employed for military applications. A common example of
military application is a Radar Tracker, which orients the Radar dish antenna (in modern "Phased
Array" Radars, the antenna may be actually stationary but the Radar beam is oriented) towards an
aircraft. Obviously, the same Radar Tracking system can be employed in civil applications such
as in Air Traffic Control in an airport. Other tracking example in aerospace is tracking of a
satellite in orbit.
One common military application of the tracking system is intercepting an enemy aircraft, or
a large missile say a TBM. The interception is generally done by another guided missile. In order
to successfully guide the interceptor towards the threat and 'home' on to it, the position and
velocity (and sometimes acceleration) of the threat object must be made available from the
tracking system.
The algorithm for generating appropriate guidance command from the tracking information is
generally called the "guidance law". The tracking can be carried out from ground or a mother
ship/aircraft or from the interceptor (using a seeker or a homing head, mentioned earlier) itself or
a combination of both. The related issues are discussed somewhat more elaborately in Chapter 3.
The main outputs from an onboard tracking system (seeker) are the bearings and/or sight line
rate and optionally range and range rate. Such outputs from the trackers are generally imperfect
and noisy, due to many factors discussed in Chapter 3. Tracking sensor noise can have several
consequences like poor guidance performance leading to missing the target, losing track of the
target long before the final engagement (track loss).
The tracking signal, therefore, has to be filtered from such noise. If the disturbances are high
frequency in nature (out of band), compared to the desired signal, a low pass filter could have
been adequate. However, substantial components of such 'noise' and disturbances could be "in
band", which makes traditional low-pass filters inadequate.
The best option therefore is to employ state estimators (many of which also go by the name
filter, for example Kalman Filter).
2 Problem Statement and Scope
The present work envisages to extend existing nonlinear filtering techniques, develop new
nonlinear filtering techniques and to evaluate and characterize such filters for their suitability for
onboard target tracking applications. The characterized filters (modified and novel) are essentially
9.Synopsis
of the post Kalman type, defined earlier. Gaussian noise excitation has been used throughout,
though the nonlinearities in the system would eventually produce non-Gaussian noise.
Two major and important areas have deliberately been kept outside the scope of the present
work. The first of these is the scenario where the target is highly manoeuvring. A separate breed
of filtering [Bar-Shalom2001] known as multiple-model based techniques are required in such
situation (see chapter 3). While techniques developed in the present work may be adopted in a
multiple model scenario, no attempt has been made to explore such possibility.
The second to be left outside the scope of this thesis are extensions of the Monte Carlo
simulation based techniques, such as Particle Filters [Ristic2004, Arulampalam2002]. One reason
is that such filters are very computation intensive and may not be suitable for onboard application.
The second reason is that co-workers in the same laboratory are working on Particle Filters and
their results were made available to the present worker for cross validation. Another significant
set of approach, which was not covered in the present work is the family of filtering techniques
using mixed Gaussian [Ito2000] approach.
3 The Approach and Methodology
As first step, selected filters have been tested for robustness in nonlinear filtering situations.
In the process, the behaviour of these filters have been characterized. The filters which survived
the robustness characterization studies were tested against two classes of target tracking scenarios,
namely the naval target tracking with slow moving targets and tracking of Tactical Ballistic
Missiles in the post re-entry phase. Both exo-atmospheric and endo-atmospheric engagements
have been tried.
Performance of the filters have been based on Monte Carlo runs. Care has been exercised to
ensure that the Monte Carlo population size is adequate. Towards the above goal, simplified but
nonlinear models have been evolved, which are simple enough for transparency but contain the
essential elements of tracking problem.
From the performance of EKF and UKF the pattern of track losses were first characterized
and a systematic criterion was evolved to detect (and count) track loss cases. The risk sensitive
filters were chosen as a candidate because such filters had the promise of enhanced robustness. As
the Extended Risk Sensitive Filters (ERSF) did not show substantial improvement, the CDRSF
was evolved. However, the newer filtering algorithms proposed have been evaluated like the other
filters.
The experience in evaluating the filters against each other and against Cramer Rao Bound,
and the reasonable result obtained by Sadhu et al [Sadhu2004] led to the development of a
tentative approach for the preliminary design of tracking systems, which potentially can also be
used in the co-design of Homing Guidance Systems.Synopsis
4 Main Contribution of the Thesis
The contribution of this work may be summarized as follows:
• Defining the probability of track-loss and advocating its use as a performance measure
for tracking filters. Other performance measures have also been suggested and utilised
e.g. the so-named Q-sensitivity and the initial error covariance sensitivity.
• Selecting a set of test problems for evaluation of filters. Difficult nonlinear filtering
problems have been chosen from the literature. Further, variants of the parallel
approach tracking scenario have been evolved as test problems to mimic the ballistic
object tracking problem.
• Characterization of known filters like EKF, ERSF and variants of UKF with the help of
the test problems
• The extra degree of freedom available in the scaled UKF was utilised as a design trade-
off mechanism and its efficacy has been demonstrated with the BOT test case.
• The unsealed square root UKF algorithm has been demonstrated with the ballistic target
tracking problem.
• Extending the capability of known filtering algorithms, leading to the (i) Adaptive Grid
Filter (ii) Nonlinearity Compensated EKF and (iii) the Constrained EKF.
• The variants of risk sensitive filtering algorithms has been extended and applied to
tracking problems.
• A novel-filtering algorithm, called Central Difference Risk Sensitive Filters (CDRSF)
has been developed and tested.
• Closed form recursive relations for obtaining the CRLB of nonlinear systems with
singular Q matrix have been evolved.
• Proposing how the CRLB may be employed for preliminary design of Tracking systems
and homing guidance systems.
5 Organisation and Content of the Thesis
The thesis is broadly partitioned in 4 parts. The introductory part (Part-A) contains an
Introduction, literature survey (chapter 2), and discussions on the tracking problem (chapter 3).
The second and third chapters of the dissertation perform a literature survey on the methods and
the application areas and provide an appreciation of a significant fraction of the literature cited.
The following table gives a break up of the date of publication and number of literature, out of a
total of 245.Synopsis
Number of Papers/reports/books cited
Year of BEFORE 1990-1995 1996-1999 2000-2003 2004-2007
Publication
1990
40 26 43 74 62
Publications before 1990 are generally textbooks and also publications related to the areas of
tracking and guidance. The second part (Part-B) concerns characterization of Post Kalman Filter
techniques and also introduces the new and modified filter algorithms. This part starts with
Chapter 4, which presents the test problems used for characterizing the filters. The same chapter
also includes discussions of how the characterizing parameters like track loss etc are defined.
The filtering techniques closely related to the Extended Kalman filters are discussed in
chapter 5. The techniques include Iterated EKF, Nonlinearity compensated EKF (NCEKF) and
Constrained EKF (CEKF). These techniques are already described in literature and descriptions
are brief, putting more emphasis on characterization by simulation studies. Alternative algorithms
of NCEKF and CEKF are also provided.
Characterization of linear regression Kalman filters (LRKF) is described in Chapter 6.
Chapter 6 characterize the UKF and CDF. Besides, variants of UKF including the scaled UKF
and square root UKF (SRUKF) are explored. One of the main contributions of this thesis is the
Adaptive Grid Technique. The approach and algorithm for this is described in fair amount of
detail in Chapter 7.
The Risk sensitive filtering concepts and Extended Risk sensitive Filters are described and
evaluated in Chapter 8. Another contribution of this thesis is the Central-Difference Risk
Sensitive filter (CDRSF). The algorithm of CDRSF and its characterization are described in
Chapter 9. The CDRSF algorithms for posterior and prior estimation are presented and evaluated.
The third part concerns TBM tracking applications. The techniques already developed have
been applied to TBM tracking problems. The exo-atmospheric tracking problem is discussed in
Chapter 10, whereas the endo-atmospheric tracking problem, including estimation of Ballistic
coefficient is discussed in Chapter 11. The CRB- based method for evolving the preliminary
design of a tracking system and the guidance system is presented in Chapter 12. This is in the
form of conjecture as no numerical examples were provided.
The last part (Part-D) contains only one chapter (Chapter 13), covering discussions,
conclusions and scope for further work. The concluding part is followed by a list of literature
cited. The appendices provide algorithms and flowcharts, which have been used in this work but
not embedded within the chapters.
5Synopsis
6 Main Results
6.1 Test Cases Evolved and Used
Two types of test cases have been chosen: (a) difficult nonlinear filtering problems
chosen/adapted from the literature, (b) simplified tracking scenarios which contained some
essential ingredients of target tracking problems in aerospace and surface motions.
For the type-a problems as above, it has been found that these problems were able to
discriminate between competing filtering methods. This is not a coincidence, because (i) the
worker had been pro-active in selecting problems where EKF performs poorly, also, (ii) some of
the signal models with low discriminating power have been excluded from the thesis
The family of parallel approach tracking scenarios form the bulk of the type-b problems. It
was shown in chapter-4 that BOT problem is a special case of this family. While surveying the
literature on TBM tracking, it was found that performance of the same filter differed with
different choice of the coordinate system. The algebra and trigonometry of the axis transformation
seemed to overshadow the essence of filtering. It was therefore decided to use the family of
parallel approach tracking scenarios, wherever possible. A specific instance of this type of
problem is the bearing-only tracking (BOT).
As several previous workers have studied the BOT problem, it provided opportunity to
validate the results obtained by the present worker and to also compare the performance of more
number of filters (which were not developed by the present worker). Also included in the set of
test problems is the determination of unknown Ballistic Coefficient problem. A test problem from
the literature has been taken.
6.2 Performance Measures Evolved
For characterization of filters (previously known, extended by the present worker and the new
filters developed), three new metrics (performance measures) for tracking filters have been
advocated, in addition to the usual metrics of Root Mean Squared Error and Mean error.
Three new metrics are (a) probability of track-loss (b) Q-sensitivity and (c) the initial error
covariance sensitivity. These metrics have been defined and their applications have been
illustrated. Arguments for suitability of these metrics have been provided. It has been shown that
in a number of cases these metrics are able to discriminate between competing filtering methods.
6.3 EKF, EKF Variants and Augmentation
As EKF had hitherto been the filtering method of choice, the performance of EKF had been
analysed for almost all cases as a basis for comparison. It has been demonstrated and reconfirmed
in the present work that EKF fails miserably in several tracking scenarios, including the BOT.
6Synopsis
Several earlier workers obtained performance improvement using the iterated EKF method.
This was tried on a few test problems. However, there was little or no performance improvement,
specially in the track loss situations. Considering the additional computational loads, this filtering
method was not considered further.
A fairly novel idea of improving the accuracy (and possibly robustness) of EKF was
suggested by [Fujimotol997], who called this as Nonlinearity Compensated EKF (NLCEKF).
This method was examined and put into a more general framework by this worker. While the
NLCEKF method, worked for the test case suggested by its originator [Fujimotol997], it failed to
provide any improvement in other test cases tried by the present worker. Though this method is
not recommended for onboard tracking in its present form, further work in this direction is
recommended.
Another fairly novel idea of improving the accuracy (and possibly robustness) of EKF was
suggested by [Nordsjo2004], who suggested a form of constrained minimisation and named the
technique as the Constrained EKF (CEKF). The CEKF method was examined in detail and put
into a more general framework in this dissertation. The method, though worked for the test case
suggested by the originator, failed to provide any improvement in other test cases tried by the
present worker.
In fact, even for the signal model provided by the originator, if the initial conditions are
changed, no benefit was obtainable from this method. Due to the observed fragility of this
method, it was not considered a candidate as a potential tracking filter.
6.4 Linear Regression Filters
Linear Regression type of filters (LRKF), particularly UKF and the Central Difference Filter
(CDF) were applied and analysed in this work with a fair amount of detail. With different types of
test cases, it has been shown that these filters generally perform equal to or better than EKF.
Additionally, a trade-off mechanism was evolved for the UKF based method.
The extra degree of freedom available in the scaled UKF was utilised as a design trade-off
mechanism and its efficacy has been demonstrated with the BOT test case. UKF based methods
have been tried in almost all the test cases and only some of the significant results have been
presented in relevant chapters.
The CDF method differs from the UKF in small details but in a significant way. In many test
cases where UKF was found applicable, the CDF results were found to closely match that for
UKF. Such results were not presented here. However, the CDF method lacked the extra degree of
freedom of the scaled UKF. The CDF method has been extended for the Risk Sensitive Filtering.
The LRKF family of filters had been characterised in this chapter using the BOT and other
problems.
7Synopsis
6.5 Adaptive Grid Methods
The Grid Based Method of filtering has been modified by the present worker leading to the
Adaptive Grid Filter (AGF). The Adaptive Grid Filter retains much of the inherent simplicity of
the Grid Based Filter but improves the numerical efficiency by one or more orders of magnitude.
The rationale behind this approach has been explained and the algorithm has been reported.
The ordinary grid based method has been all but discarded by workers due to its poor
numerical efficiency, particularly after the Particle Filter based methods were made available.
There were some earlier attempts to improve the efficiency of this type of filter. However, in the
adaptive grid method, developed in the present dissertation, the adaptation is far more extensive,
as the following features have been utilised: (i) use of EKF as "proposal" for guessing the
centroid and the span of the posterior distribution, (ii) using non-uniform spacing of grids to suit
the proposal distribution, (iii) orienting the grids along the principal axes of the 1-sigma ellipsoid.
Each of these contributed to the improvement of efficiency.
However, even after this improvement, this technique is still computation intensive and may
not be preferred for on-board implementation. Two facts may however justify the efforts put for
developing the AGF. Firstly, during the present research work, the AGF provided a benchmark
and validation tool for other non-linear filtering methods. This provided additional confidence in
the developed and modified filtering methods. Secondly, a co-worker [Bhaumik2005] has
subsequently applied this technique to demonstrate the Adaptive Grid Risk Sensitive Filter
(AGRSF). Currently, AGRSF is one of the general methods for solving risk sensitive nonlinear
filtering problems.
6.6 Risk Sensitive Filters and CDRSF
The present work is possibly the first in applying risk-sensitive filters to tracking problems,
including the Bearing-only tracking and TBM tracking problems. During the course of this
research it was realised that the chosen metrics like (a) probability of track-loss (b) Q-sensitivity
and (c) the initial error covariance sensitivity are basically robustness measures in a wider sense
of the term. It was therefore natural to verify whether the Risk Sensitive Filter, which was claimed
by the originators to be robust, performs better with respect to these measures. A few examples of
better robustness for linear signal models were available in the literature.
When the present work was undertaken, algorithm for risk sensitive filtering for general
nonlinear signal models were not available. An approximate form, the Extended Risk Sensitive
filter (ERSF) was therefore used first. The test cases considered are (i) the BOT problem and (ii)
Track While Scan (TWS) problem. The results of case studies indicate that the performance of
ERSF is somehow better than the conventional EKF. One more interesting result was that the
robustness improved for risk-seeking filters rather than risk-averse filters however, the improved
performances were worse than that obtainable by UKF.
8Synopsis
The performance of ERSF for highly nonlinear problem was not at all encouraging as
would be evident from the results provided here. It was therefore necessary to verify
whether the above result was due to the gross approximation inherent in E R S F or it was
due to the Risk sensitive approach itself. It was therefore necessary to verify whether the
above result was due to the gross approximation inherent in ERSF or it was due to the Risk
sensitive approach itself. Accordingly a better method called the Central Difference Risk
Sensitive Filter (CDRSF) was developed. This has since been published [Sadhu2007]. The
superiority of CDRSF over ERSF has been demonstrated in the present work for non-tracking but
highly nonlinear estimation problems.
In the case of tracking problems (BOT), however, even CDRSF did not yield significant
results showing improvement in robustness. The range of the positive risk sensitivity parameter
where the filter will converge often turned out to be rather small in several test cases. The number
of track-loss cases for CDRSF was significantly less compared to the ERSF but almost equal to
the (risk neutral) CDF. This ruled out any significant improvement by introducing risk sensitivity.
(Only in the case of estimating unknown ballistic coefficient, some noticeable improvement was
obtained as shown in a later chapter 11).
On the whole, though new insights have been obtained with the CDRSF, Risk sensitive filtering
algorithm can not be recommended for onboard tracking filter without reservation.
6.7 E x o - a t m o s p h e r i c T r a c k i n g of T B M
Estimation for exo-atmospheric ballistic target tracking by applying nonlinear filtering
techniques and Cramer Rao bound using four different combinations of measurements are
described in Chapter 10. The process model is linear and obtained from the simplified assumption
that the final part of homing trajectory is nearly parallel as explained in chapter 4.
The CRLB studies were used to find out how well the different filters perform with respect to
the CR bound. Also studied in this chapter is the effect of sampling time in tracking performance.
As expected, tracking performance improves with smaller sampling time. This chapter also refers
to results of other chapters and that of earlier studies by co-workers which show that the nature of
the performance of good filters provide the same trend as CRLB performance. These important
findings are utilized in chapter 12.
The EKF, UKF, ERSF and PMF are applied to the exo-atmospheric ballistic tracking problem
with assumption that only SLA is available as measurement. With large MC runs, RMSE and
track loss analysis are studied. The simulation results show that the error performance of UKF and
PMF is superior to other filters and it is close to CRLB. PMF has the advantage that it requires
less computational load. EKF and ERSF (positive mu) have shown unacceptable number of
failures. ERSF was found to be very sensitive to the risk factor and cannot therefore be
recommended with confidence.
9Synopsis
By introducing SLA and range as measurements, the performance of the filters have shown
substantial improvement. Both EKF and UKF exhibited performance close to CRLB. Both the
position and velocity errors quickly converged. When SLA and SLR measurement combination is
used, there was some performance improvement over SLA-only case. Performance of both EKF
and UKF were comparable. When all the three measurements viz. SLA, SLR and range were
used, tracking performance improved but were very close to that for SLA and range case.
6.8 E n d o - a t m o s p h e r i c T r a c k i n g of T B M
This chapter concerns the endo-atmospheric ballistic target tracking. The significant aspect of
endo-atmospheric tracking problems is atmospheric drag, which is expressed in terms of the
ballistic coefficient (BC), square of the velocity and an altitude dependent density term. These
make the system equation nonlinear. In this chapter, several estimation algorithms were studied
in three different endo-atmospheric tracking scenarios. For each case, several measurement
options are tried out.
The vertical falling body with stationary observer and unknown BC is a well-known and
standard problem [Ristic2003, Ito2000, Zhang2003]. The estimation results obtained by EKF,
UKF and PF are already reported in [Ristic2003]. We have applied other numerically efficient
filters including SR-UKF and CDF for the given problem. In this dissertation, the robustness
against uncertainty present in the process noise covariance is explored by applying CDRSF and
ERSF.
A novel aspect of this dissertation is simplification of ballistic tracking problem by the
parallel approach model where both target and platform move parallel but in opposite directions.
The parallel approach endo-atmospheric scenario is studied with two cases; one is known BC
(N2-3) and another is unknown BC (N3-3). The well-known suboptimal tracking algorithms
including EKF and UKF are applied for problem (N2-3) and its tracking performance with
bearings-only (SLA) measurement is presented. The endo-atmospheric tracking problem (N3-3) is
also studied by applying EKF and UKF.
As the main motivation of this chapter is to estimate unknown BC, the ballistic target tracking
with unknown BC is analyzed by considering different measurements. The bearings (SLA only),
bearings rate (SLR only) and, SLR and range measurement are used. The RMSE performance of
applied filters (EKF and UKF) with each measurement case is studied.
The parallel approach, slantly falling body case is studied with known BC as case study-I. .
With the initialization condition chosen here, if the initial error covariance of the filter is different
from the truth, the UKF was observed to fail though in lesser numbers compared to EKF. This
emphasizes the importance of proper initialization of filter.
The estimation of unknown ballistic coefficient was studied first with a simplistic case for a
ballistic vertically falling body, with range as the only measurement. The filters used were
variants of sigma point filters (LRKF)and the EKF. It was found from the nominal (error
10Synopsis
covanance matrices) condition results, that the EKF tends to diverge, with large bias and rms
errors. The performance of other LRKFs was comparable and the CDF was the fastest.
With perturbed Q (the process noise covariance assumed by the filter being different from the
true process noise ), the effect of risk sensitive filtering was studied next. Under nominal Q, the
risk sensitive and risk neutral filters provided the same result (not shown in detail), however, in
the perturbed case, the risk sensitive filter produced somewhat better rms error performance. The
effect was more visible for the BC estimation.
The slant falling target tracking with unknown BC is. denoted as case study-Ill, where, the
EKF and UKF are applied with combinations of bearing (SLA), sight line rate (SLR) and range
measurement. It was found that the performance of estimators with SLA measurement only was
superior to the cases with SLR only measurement. With range measurement included, the rms
error for position improved dramatically after the first few iterations. This rapid error correction
in position, resulted in somewhat increased rms error in velocity in cases with higher platform
motion noise, which however settled down subsequently.
The platform motion noise was seen to have a profound effect. With reasonable values of
platform motion noise covariance (about 100 m squared) the position error settled down to about
10m. This would be acceptable for an interception duty. With high platform motion noise, this
could become 60m. It is therefore, important to assign realistic value for this noise. For the
particular values of parameters studied, performance improvement with UKF was noticeable in a
number of failure cases.
6.9 C R L B a n d T r a d e Off M e c h a n i s m for P r e l i m i n a r y Design
Availability of the Crammer Rao Lower Bound (CRLB), the theoretical lower bound of filter
error covariance for nonlinear signal models has been found to be a very useful tool. As a part of
this research, closed form recursive relations for obtaining the CRLB of nonlinear systems with
singular Q matrix have been evolved. Note that though the results were known for null Q matrix
case, the same was not available for singular non-zero Q matrix for a general nonlinear case.
With the help of test cases, it has been shown that the CRB captures the essence of filtering
problems without getting into the specifics of the filtering algorithms. Unlike linear filtering
problems, the error covariance predicted by the simpler filters like EKF is often unreliable.
Separate and elaborate MC runs are necessary to estimate the estimation error covariance. The
CRB also demonstrated to be capable of quantifying the influence of sampling time, engagement
kinematics, measurement noise and similar parameters, thereby helping to make choices of
sensors and signal processing computer.
With the above results, it is conjectured that the CRLB may be used as a tool not only to
benchmark other filters but as a design tool for designing the guidance configuration at least in the
preliminary phase.
11Synopsis
It has been shown further that the use of CRLB allows certain amount of parallelism in designing
interceptor guidance system. However, for using such a preliminary design method some
knowledge about the preliminary kinematics, a good quantification of process and measurement
noise covariances (reliable parameterization) and inputs from an experienced filter designer in the
form of pessimism factors are required.
7 Conclusion of the Thesis
i. A number of Post Kalman Filters for nonlinear signal models have been studied and
characterized with a set of test cases (suggested by the present worker), and performance
measures, evolved as part of the present work.
ii. It was found that the test cases evolved and used in this dissertation have adequate
discriminating ability, i.e. could bring out deficiencies of filtering techniques.
iii. Three new metrics, namely (a) probability of track-loss (b) Q-sensitivity and (c) the
initial error covariance sensitivity have been defined and their applications have been
illustrated. Arguments for suitability of these metrics have been provided. It has been
found that in a number of cases these metrics are able to discriminate between competing
filtering methods.
iv. Linear Regression Kalman Filters (LRKF) have been evaluated with the help of the test
cases.lt has been found that Unscented Kalman filters generally perform equal to or
better than EKF.
v. For the UKF based method, it was found that trade off mechanism can be easily
incorporated by utilising the extra degrees of freedom that exists in UKF.
vi. The Adaptive Grid Filter has been developed and it retains much of the inherent
simplicity of the Grid Based Filter and improves the numerical efficiency by one or more
orders of magnitude. Even after this improvement, this technique is still computation
intensive and may not be suitable for on-board implementation of tracking filters.
vii. For the Nonlinearity Compensated EKF, no significant benefits were observed and this
algorithm is not recommended for tracking application.
viii.No significant benefits were observed also for the Constrained EKF (CEKF). Moreover,
this method failed to converge in the test problem tried. This algorithm is not
recommended for tracking application.
ix. Extended risk sensitive filters (ERSF) was applied to the test problems and it is concluded
that (i) for severe nonlinear problems, often ERSF loses track, (ii) the nature of risk
sensitive filters could not be adequately explored by ERSF. This type of filter is therefore
not recommended for tracking application.
12Synopsis
x. A novel-filtering algorithm, called Central Difference Risk Sensitive Filter (CDRSF) has
been developed and tested as a part of the present work. This allowed further exploration
of Risk Sensitive filtering. For Ballistic coefficient estimation, improved robustness (with
respect to process noise sensitivity) was observed. Further investigation in this direction
is recommended.
xi. Closed form relations for obtaining the CRLB of nonlinear systems with singular Q
matrix have been obtained and informal proof has been provided.
xii. It has been found that CRLB has the potential to be employed for preliminary design of
Tracking systems and homing guidance systems.
xiii.lt has been found that with the present state of knowledge, nonlinear filtering algorithms
suitable for onboard implementation which will reliably track even non-manoeuvring
targets in the presence of all possible initialisation errors, process noise and measurement
patterns (measurements and noise component) may be difficult to obtain. Further research
should be directed in this area.
Koikata-700 012
13You can also read
