Multiple Model Estimation for Linear Stochastic
Hybrid Systems with Non-Homogeneous Transition
RCHIVES
Probabilities
MASSAC HUSETTS INSTITUTE
OF TECHNOLOGY
by
Michael William Kasperski
0 T114 2015
B.S., The University of Texas at Austin (2013)
LI 3RARIES
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Masters of Science in Aeronautics and Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
Author ..
I.. epartnment eronautics and Astronautics
August 20, 2015
Certified by........
Signature redacted..
Hamsa Balakrishnan
Associate Professor of Aeronautics and Astronautics
Thesis Supervisor
Accepted by............
Signature redacted
Paulo C. Lozano
I
Associate Professor of Aeronautics and Astronautics
Chair, Graduate Program Committee
2
Multiple Model Estimation for Linear Stochastic Hybrid
Systems with Non-Homogeneous T'ransition Probabilities
by
Michael William Kasperski
Submitted to the Department of Aeronautics and Astronautics
on August 20, 2015, in partial fulfillment of the
requirements for the degree of
Masters of Science in Aeronautics and Astronautics
Abstract
This thesis investigates the field of stochastic hybrid estimation. A broad introduction to the framework surrounding estimation, filtering, and multiple model based
systems is presented. More specifically, the often made assumption of a constant
time-invariant mode transition probability matrix is relaxed. Recent work done in
the area of non-Markov jump stochastic hybrid systems is explored, including semiMarkov systems, non-homogeneous transition probability matrices, and continuousstate-dependent mode transitions. Algorithms needed to develop linear multiple
model based filters with non-homogeneous transition probabilities are detailed. Finally, a case study for the practical implementation of an extended Kalman filter in
the application of attitude heading and reference systems is conducted.
Thesis Supervisor: Hamsa Balakrishnan
Title: Associate Professor of Aeronautics and Astronautics
3
4
Acknowledgments
I would like to extend my heartfelt thanks to Jason McKnight, for all your help and
support. Without your constant encouragement, this thesis would not have existed.
I would like to thank my advisor Hamsa Balakrishnan for your understanding,
patience, and help throughout my time at MIT.
Finally, thanks Mom and Dad, for pushing me forward, and for always being there
when I needed someone to talk to, even if I didn't know it.
5
6
Contents
1
1.1
Estim ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2
F iltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
M odel Uncertainty
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
Multiple Model Estimation . . . . . . . . . . . . . . . . . . . .
16
1.3.1
2
. . . . . . . . . . . . . . . . . .
21
. . . . . . . . .
22
2.2
Semi-Markov Jump Systems . . . . . . . . . . . . . . . . . . . . . . .
23
2.3
Non-Homogeneous Markov Transitions
. . . . . . . . . . . . . . . . .
24
2.4
Linear Hybrid Systems with Non-Homogeneous Transition Probabilities 25
Markov Jump Systems and the IMM
2.1.1
The IMM Estimator: A Common Application
29
Filtering and Multiple Model Estimation
3.1
The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
. . . . . . . . . . . .
32
Multiple Model Estimation . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.1
General Formulation
. . . . . . . . . . . . . . . . . . . . . . .
33
3.2.2
The Interacting Multiple Model Estimator
. . . . . . . . . . .
34
3.2.3
Continuous-State-Dependent Mode Transition Probabilities. .
36
3.1.1
3.2
4
19
Literature Review
2.1
3
13
Introduction
The Kalman Filter Recursion Equations
Case Study: Attitude Heading and Reference Systems
4.1
Variables and Coordinate Definitions . . . . . . . . . . . . . . . . . .
7
39
40
4.3
The Extended Kalman Filter.
44
4.4
Process Model . . . . . . . .
45
4.5
Measurement Model.....
47
4.6
Simulation Model . . . . . .
49
4.7
Results . . . . . . . . . . . .
51
4.8
Summary
. . . . . . . . . .
56
.
.
.
.
.
.
Sensor Characteristics
59
Discussion . . . . . . . ..
.
. . . . . . . . . . . . . . . . . . . . .
59
5.2
Future Work. . . . . . ..
.
. . . . . . . . . . . . . . . . . . . . .
60
.
.
5.1
.
Conclusion
.
5
42
4.2
8
List of Figures
[31
. . . .
1-1
State estimation, a concise representation, reproduced from
4-1
Body fixed reference frame (left), and inertial reference frame (right)
14
[2 3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4-2
Simulated roll, pitch, and yaw for (a) Model 1 and (b) Model 2 . . . .
50
4-3
Simulated side slip and climb angles used in Model 3
. . . . . . . . .
51
4-4
Estimation divergence for the (a) designed filter, under the same noise
conditions as the (b) original filter . . . . . . . . . . . . . . . . . . . .
4-5
52
Monte Carlo simulation for (a) filer with altitude measurements and
(b) without altitude measurements, for the same sensor noise parameters 53
4-6
Sample trajectory for one of the fifty iterations . . . . . . . . . . . . .
54
4-7
Mean of the roll and pitch error for the fifty iterations . . . . . . . . .
54
4-8
Mean innovations sequences for the fifty iterations . . . . . . . . . . .
55
9
10
List of Tables
4.1
Standard definitions of relevant variables . . . . . . . . . . . . . . . .
40
4.2
Noise in the simulated sensors . . . . . . . . . . . . . . . . . . . . . .
51
4.3
Error statistics for the Monte Carlo simulation . . . . . . . . . . . . .
53
4.4
Error statistics for miss-matched model . . . . . . . . . . . . . . . . .
56
11
12
Chapter 1
Introduction
In modern systems, many applications can be classified as hybrid, or systems with
both discrete and continuous dynamics
141.
There has been a wealth of studies on
how to perform estimation on a class of hybrid problems coined as jump Markov
systems. These systems behave according to a finite number of dynamic modes, with
one mode being active at any given time, and the transition between modes being
modeled as a Markov chain.
The investigation in this thesis focuses on relaxing
common assumptions about the structure of the Markov chain governing the mode
transitions, such as its often assumed time invariance and constant state-independent
nature.
1.1
Estimation
Estimation, in the simplest terms, is making a "best guess" of some aspect of a
system based on incomplete, indirect, intermittent, and uncertain observations. In
other words, to map a set of measurements of the systems behavior to an estimate
of the needed state variable. It is an incredibly deep field, with a variety of unique
applications. For instance, consider a standard GPS receiver, where there is no information about the current location of the object, but multiple "noisy" (uncertain)
measurements off different satellites. Estimation will fuse those different indirect observations into the "best" estimate of the receivers position. Applications can range
13
System
t
Dynamic
System
Prior
Information
Measurement
Error Source
Error
Source
ttState
System
State ,Measurement
System
Measurement,
State
Estimator
Estimate,
State
Uncertainties
Figure 1-1: State estimation, a concise representation, reproduced from [31
from tracking problems, to parameter or model determination, to stochastic control,
and to many more.
The main focus of this thesis is on the topic of state estimation.
illustrates the process of state estimation.
Figure 1-1
Here, the observer has access only to
the measurements. There can be uncertainty in both the measurements taken, and
the understanding of the dynamic system in question. The observer must use both
the uncertain understanding of the system, the uncertain measurements, and prior
knowledge to infer some aspect of interest in the system.
1.2
Filtering
In order to implement estimation in practice, recursive algorithms are often used.
That is, given the previous estimate of the state, use a new piece of information
(measurement) in order to compute a new estimate of the state. Of all the estimators,
perhaps the most well known and widely used is the Kalman filter. It provides a
recursive framework that computes the optimal state estimate in the sense of leastsquared error for linear Gaussian systems.
Often, state and measurement equations used to model physical systems are nonlinear. Although the Kalman filter will not provide optimal estimates in that case, an
approximate solution can be computed through linearization. The extended Kalman
filter does this using first order Taylor expansions within the state and measurement
equations. Several other methods for non-linear filtering exist depending on the application and system being considered, such as the unscented Kalman filter or particle
14
filter.
1.3
Model Uncertainty
Kalman filter design, both in the linear and nonlinear cases, relies on some design
model that represents the system being investigated. The uncertainties in these types
of filters often consists of additive white noises in the system and measurement equations. That is, the underlying system and measurement model are well understood,
including the transition matrices, input, gain, and noise properties. However, in many
practical applications, much like other parts of estimation, there exists some uncertainty in these parameters. The system may be too complex, have hard to determine
parameters, or have uncertain noise. In addition, a broader class of problems emerges
when the system being considered dramatically changes over time in a stochastic
sense.
In general, there are two approaches to handling model uncertainty [33, 3].
1. Robust estimation: Design a fixed filter that performs for all expected variations
in the model parameters
(a) Robust Kalman (72) filtering: Design the filter that gives small error for
all models that might be active in the system. That is, reach a compromise
between all expected behaviors of the system.
(b) W... filtering: Consider the noise process to be deterministic, and attempt
to minimize the energy of the estimation error signals relative to the noise
energy. W,4 filtering is especially efficient at handling external noise sources
with unknown statistics.
2. Adaptive state estimation: Design a filter that estimates both the state, and
the model parameters (i.e. the system matrices)
(a) Joint filtering of state and parameters: Add the state parameters to the
state vector and create a bi-linear system. Particle filter methods often
show good results with this type of estimation
15
(b) On-line noise tuning: Implement simple rules that modify noise levels
when a filter is misbehaving, often through monitoring innovations and
their covariance whiteness.
(c) Multiple Model (MM) estimation: Divide the system dynamics into a number of modes (discrete states), and within each mode classify the continuous
state dynamics as set of difference/differential equations. Estimate both
the continuous state and the discrete mode of the system.
1.3.1
Multiple Model Estimation
In the multiple model approach, the system in question behaves according to one of
a finite number of models. The uncertainty in which model, or "mode", is active is
a discrete uncertainty. The combination of the continuous uncertainties (i.e. noise)
and discrete uncertainties (mode) classify this type of problem as stochastic hybrid
estimation. In the dynamic sense, the actual system may switch between modes of
operation, but only one mode will be active at a given time. The multiple model filter
keeps track of both the continuous statistics (the state estimate and covariance) and
the discrete statistics (mode probabilities)
For example, when tracking an aircraft there is a large difference in system dynamics when it is undergoing uniform motion compared to when it is maneuvering. If the
filter used to track the aircraft is based on the more prevalent mode of flight, uniform
motion, then the portions of flight where the aircraft is maneuvering will cause the
filter to give less accurate estimates. As discussed above, one way to handle this is
through the use of robust Kalman filtering, where a compromise is reached between
the two modes, giving an acceptable level of root-mean squared (RMS) error when
tracking either mode of flight. However, another approach is to use a multiple model
filter, where there are two distinct filters, each providing estimates for one mode of
flight, which are combined by keeping track of mode probabilities. It has been shown
that this form of multiple model estimation gives much better performance in both
modes of flight [7, 211.
16
In the multiple model approach approach, the continuous state probability density functions (pdfs) are represented as a mixture of Gaussian hypotheses, each conditioned on a mode of the system. The transitions between modes is modeled by a
Markov chain.
A bank of Kalman filters, matched to the modes of the system, is
used to propagate the conditional Gaussian pdfs based on new measurements.
As
shown in [3, 1, 7], the optimal state estimator for hybrid systems relies on the state
estimates of each individual mode, and must include all possible mode histories. It
can be shown that this leads to an exponential growth in the number of hypotheses
being tracked. Even in the most basic case of linear Markov switching, the algorithm
requires exponentially growing memory.
In practice, to implement a filter it is necessary to have a recursive algorithm
where the complexity does not grow with time.
Thus, a sub-optimal estimator is
required. These sub-optimal estimators differ in that they use a Gaussian mixture as
an approximate information state to curb the growth of complexity. There are several
such estimators, including generalized pseudo-Bayesian algorithms (GPB1, GPB2),
and the interacting multiple model (IMM) algorithm [1, 31. The differences between
the algorithms lies in the timing and method of hypotheses reduction.
1. History pruning: A na~ive approach, where the hypotheses with low probability
are discarded and the remaining ones have the probabilities renormalized.
2. History merging: Use fixed depth merging where equivalent Markov chain paths
are merged to a single Gaussian pdf.
For example, the GPB algorithms will
merge histories after the measurement update occurs.
Much of the framework surrounding multiple model estimation assumes that the
mode transitions are governed by a Markov process where the transition probabilities
between different modes are constant, well defined, and depend only on the previous
mode. However, in practical applications this is not always the case. For example, an
aircraft navigating through sectors in the national air space (NAS), it is much more
likely to start a maneuver near a waypoint, and mode transitions will be a function
of position. As mentioned, this thesis is primarily concerned with the case where the
17
transition probabilities are allowed to vary.
18
Chapter 2
Literature Review
Multiple model estimation has been the focus of research in many different disciplines.
There exists a wide array of systems that can be modeled as a finite set of dynamic
modes with well understood transition properties, or as jump Markov systems (JMS).
A system that exhibits dynamics that abruptly change within a finite set can be fit
into the multiple model framework. These abrupt changes in system behavior can be
due to a variety of factors, such as external environmental changes or internal system
component failures. Hybrid systems are defined by two main components [25]:
1. State: The configuration of an object at a given time, that changes based on a
stochastic difference/differential equation.
2. Mode (or Model): One of a finite number of possible modes (each governed by a
different set of difference/differential equations). The mode switches according
to a set of transition probabilities, and is thus governed by a discrete stochastic
process.
In this sense, a multiple model estimator is able to provide more accurate estimates
by using primarily the state estimator matched to the current system dynamics, and
switch estimators when the mode switches. This approach provides an advantage
when it is not possible to classify the dynamic behavior of a system with a single
model.
19
Many different fields in science and engineering have utilized these types of filters
in theory and practice. Much attention has been payed to target tracking, or trying to
estimate the state of a maneuvering object through external measurements. For example, the process of tracking an aircraft through ground based radar measurements
[2, 21, 31, 32], terrain-based ground target tracking 116], road constrained vehicle
tracking [19], or multi-target grid based particle filter tracking [28].
The process
of tracking has also been extended to intent based trajectory prediction for aircraft
[381. Another field which has received extensive focus is failure detection in dynamic
systems. Here, failures in components or subsystems can be represented as abrupt
changes in the parameters of the system. This can range from simple detection in
dynamic systems [34, 37]; targeted detection of sensor/actuator failure in complex
systems like F16 jets [26]; detection of interference, jamming, and spoofing in GPS
systems [36].
Similarly, repaired components can also cause this abrupt change in
system parameters [5]. In the same vein, the area of diagnostics and fault diagnosis
has been investigated, which goes beyond identifying a failure has occurred and seeks
to answer why and how
113, 15, 20]. Multiple model based filters have also seen ex-
tensive use in signal processing, with specific focus on detection of Bernoulli-Gaussian
processes [12]. Also these filters have been used in sensor management problems [141.
Another interesting extension of the multiple model framework is to the approximate
modeling of nonlinear systems by a set of linearized models which encompass the systems dynamic range
[27, 35]. Multiple model estimation has also seen use in medical
applications, where it has been used to track tumors [301 and estimate cardiac phases
in echographic images [291.
Many complex dynamic applications exist that can be put in the framework of hybrid systems with a continuous state and set of discrete modes. In order to effectively
handle estimation for these types of systems, multiple model algorithms have seen
extensive development in the past decades. These algorithms approach the problem
of multiple model estimation in a systematic and practical way. In general these algorithms are characterized by a bank of filters, each tuned to the dynamics of one of the
finite modes of a system (mode-matched filter) and an underlying rule that merges
20
the individual filters into a single estimate [25]. The bank of filters can be comprised
of simple Kalman filters (KF), extended KFs (EKF), and other similar algorithms.
Work has also been done to utilize more complex methods such as particle filtering
[8, 9, 20].
2.1
Markov Jump Systems and the IMM
One of the most basic forms of multiple model estimation is for Jump Markov linear
systems. Consider a system of difference equations, with state
Xk
and measurements
Zk:
+ w(k, Mk)
(2.1)
H(Mk)xk + v(k, Mk)
(2.2)
Xk = A(Mk)k-1
Zk
where Mk
- {1, 2,... ,r} is a finite Markov chain representing the systems mode at
sampling period k, has known transition probabilities:
7rij = P{Mk = jlMk_
1
=
i}
(2.3)
and the noise statistics of w and v are mutually independent white Gaussian processes
uncorrelated with the initial estimate.
In this context, one algorithm that has emerged as the dominant method of estimating linear jump Markov systems is the interacting multiple model (IMM) algorithm.
It has been shown to be a superior compromise between computational
complexity and estimate accuracy [7, 21]. Blom and Bar-Shalom demonstrate that
for this class of system the exact optimal estimator is characterized by exponentially
increasing complexity due to the need to condition on a growing number of mode
histories [7]. In practice, such an estimator can not be implemented. They highlight
that prior to their paper there were several approaches to implementing a sub-optimal
estimator, which differ in how the prior mode histories are managed. As outlined in
section 1.3.1 there are two approaches to managing the mode histories. First, the
21
histories with low probability can be "pruned" or discarded, which is done in the
detection estimation (DE) algorithm. Second, the histories can be merged as in the
generalized pseudo Bayes (GPB1/GPB2) algorithms.
Blom and Bar-Shalom saw room for improvement in the area of hypotheses (history) reduction. Their algorithm, similar to GPB takes advantage of merging, but
uses a different timing. They coined this novel filter as the interacting multiple model
(IMM) algorithm. In their paper they demonstrated that the IMM algorithm, which
has linear computational complexity (as a function of the number of modes/filters),
performed about the same as algorithms with quadratic complexity (GPB2) [7].
2.1.1
The IMM Estimator: A Common Application
One of the first applications of the IMM algorithm after it was proposed in [71 was for
air traffic control (ATC) tracking [211. The ATC problem demonstrates several key
ideas behind multiple model estimation. In standard civilian aviation, most aircraft
will exhibit two main modes of operation. They will either be undergoing uniform
motion (UM), i.e.
non-accelerating strait line motion, or undergoing a maneuver
(usually coordinated turns or climbing/descent).
The goal for ATC is to provide
accurate estimates of the speed, location, flight mode, and altitude of aircraft occupying an increasingly dense national air space (NAS). The measurements of the
system (aircraft) available to ATC are given by radar reports, which are limited in
both accuracy and reliability 1211. Given that the measurements are an uncertain
sequence of position readings, the goal is to infer not only a more accurate position
estimate, but also velocity, flight mode, etc.. At the time of the paper, the current
method of estimating an aircrafts state was to use a single Kalman filter. Thus, the
filter was basically an average of the different dynamic modes (using noise manipulation). In order to capture the maneuvering sections of the trajectory, there was a
large compromise in terms of error (RMS) during the UM portions of flight. Li and
Bar-Shalom demonstrated that the IMM filter provided much better estimated for
both modes of flight, along with the ability to detect which mode the aircraft was
currently in [211.
22
2.2
Semi-Markov Jump Systems
One of the first assumptions to receive further scrutiny was the Markov nature of
Approximate Bayesian filters for semi-Markov jump linear
the mode transitions.
systems were investigated in [6, 11, 24]. Even before Blom and Bar-Shalom published
the IMM results, Mathews and Tugnait looked at hybrid stochastic estimation for
linear discrete-time systems with fixed-lag [24].
That is, given some time t, use
measurements up to time t + N, where N is defined as the lag. The changing system
parameters, i.e. multiple modes, are modeled as a semi-Markov chain with known
transition statistics.
In this type of semi-Markov chain, the transition probability
matrix is the same as a standard IMM filter, (i.e.
assumed constant and known
to the filter designer), but time between transitions are assumed to be an arbitrary
random variable. The idea behind processing with fixed lag was based on the fact that
in well-defined systems with known parameters, fixed-lag smoothing leads to better
estimates.
Thus, the idea was to apply the same methodology to systems in the
hybrid class. One application they mention is the case of off-line tracking analysis of
maneuvering targets, where time-delay has little effect. Mathews and Tugnait utilize
a detection estimation algorithm (DEA), which is a sub-optimal method that uses
history "pruning" to get around the inherent history growth.
They show that, as
expected, the addition of a fixed-lag delay in processing observations can lead to an
improvement in both detection and estimation for hybrid systems.
Later, Campo et. al. further investigated the issue of fixed-lag based estimators
for hybrid systems [111].
They looked at the case where the modes switch after a
random sojourn time, and in some cases the switching probabilities are a function of
the sojourn time. This leads to a "sojourn-time-dependent-Markov" (STDM) chain
which is used in the underlying mode switching model.
This semi-Markov model
is used in combination with the hypothesis merging techniques in the, at the time,
recently developed IMM algorithm
[7].
They demonstrated through simulation, that
this STDM-IMM algorithm gave a stable estimation filter for both two and three
model cases. Also, they showed that the could rapidly detect the true system model.
23
Blom furthered this development of semi-Markov based estimators [6].
In his
paper, he developed the full Bayesian filter equations, both in the optimal (noncomputationally effecient) sense, and the sub-optimal sense. He further reduced the
computational burden of the STDM-IMM filter developed by Campo et. al., reducing
the number of computations involved in hypotheses merging. The key development in
his paper was the modeling of semi-Markov mode switching as state-dependent mode
switching, and deriving the exact Bayesian equations for that case. He also proposed
two additional semi-Markov IMM algorithms (SM-IMM) which proved to be much
more cost effective than the previous STDM-IMM and SM-DEA algorithms.
2.3
Non-Homogeneous Markov Transitions
A natural further extension to the research is to look at non-homogeneous or uncertain Markov transitions. That is, further relax the assumption that the transition
probabilities are fixed and known to the designer. There has been a fair amount of
research done in this area. In practical applications, non-Markov jumps are the rule
rather than the exception. For instance, an aircraft taxiing at an airport is much
more likely to make a maneuver in certain positions, such as intersections. Similarly,
aircraft in flight are much more likely to start a turn near fixed waypoints, or begin
maneuvers at fixed positions or velocities.
One proposed method was to consider the case of estimating the state of a hybrid system with state-dependent mode transition probabilities. Koutsoukos et. al.
[20] and Blom and Bloem [8, 9] looked at this case through the application of particle filtering methods.
Blom and Bloem noted that there exists a large challenge
implementing nonlinear filtering when there are non-Markov jumps, and that this
challenge can be countered by applying particle filter methods [8]. Their paper extends the exact Bayesian filter characterization to this much larger class of problems,
and develops a novel particle filter for stochastic hybrid systems. This IMM particle filter (IMMPF) was tested through simulation and demonstrated to have better
performance for situations with frequent mode switching. When the mode switches
24
were infrequent it performed about as well as a standard IMM, but is still much
more useful for deviations from the standard Markov jump linear system (i.e. it can
handle'nonlinearities and non-Markov mode switching). However, even though these
methods can be applied to a large class of systems (nonlinear, non-Gaussian noise,
non-Markov transitions, etc.), many problems can emerge when trying to implement
particle filters. Computational complexity and stability concerns emerge, especially
when the dimension of the state space is large. Blom and Bloem showed that in order
to estimate the state in a hybrid system when the mode transitions are functions of
the state vector, a multivariate integration of the continuous state pdfs and mode
transition probabilities must be computed
19].
Some work has been done attempting
to approximate this integration or simplify it in some way.
Zhang looked at the case where the transition probability matrix is piecewise
homogeneous [39].
That is, it is time-varying, but does so within a finite set of
possible matrices. He considers variations of two types: arbitrary and stochastic. He
notes that this is a special case of non-homogeneity within a larger class of these
types of transition probability matrices. The filtering problem is developed within
the '.. framework rather than IMM. Long and Yang furthered this line of thinking
by looking at fault detection based on a delta operator approach [221. Like Zhang,
they looked at the case where the transition probability matrix varies within a finite
set, but further defined the set as a finite polytope set. Their approach was developed
to investigate systems that have mode-dependent delays, non-linearities, and a class
of non-homogeneous Markov switching.
2.4
Linear Hybrid Systems with Non-Homogeneous
Transition Probabilities
The work done by Koutsoukos et. al., Blom and Bloem highlights one of the challenges in implementing a filter for stochastic hybrid systems with non-homogeneous
transition probabilities. That is, when the mode transition probabilities are a func25
tion of the continuous state, a multivariate integral must be evaluated in order to
compute the recursion of the transition probabilities [8, 9, 20]. Their formulation
relied on particle filtering, which despite covering a large class of estimation problems, is sometimes too complex. Seah and Hwang investigated the specific case of
linear hybrid systems, and made approximations or simplifications to evaluate the
multivariate integral [31, 32].
In their first paper, Seah and Hwang look at the case of tracking aircraft [31]. They
use the IMM algorithm to counter the exponentially growing mode histories, and a
Monte Carlo (MC) integration method to solve the multivariate integral that comes
from the state-dependent mode transition probabilities. They show that, for some
applications, the computational cost associated with the MC integration is small. As
for the mode transition probabilities, they condition them based on the state-variables
associated with the flight plan and pilot's intent modeling. Specifically, they base the
transition probabilities on waypoints, navaids, or fixes in the pre-described flight plan,
and make the assumption that the pilot will adhere to that plan. They key novel
concept they present is the use of MC integration along with approximations similar
to those found in the IMM (merging) in order to efficiently evaluate the multivariate
integral.
They provide a standard ATC example involving coordinated turn and
uniform motion modes of flight. Their algorithm exhibited lower peak and average
estimation error and more accurate mode prediction when compared to a standard
naive IMM estimator with a constant transition probability matrix.
Seah and Hwang furthered their work on continuous-state-dependent mode transitions by looking at the simplification of continuous state "guard conditions" [32].
They point out that in many cases, a mode transition will occur when the continuous state reaches a certain subset. They define this as a "guard condition" of the
continuous state space. They reference the specific example of ATC, where the majority of flight profiles follow fixed patters. For example, when an aircraft is taking off
from the runway, it will perform an initial acceleration until reaching a critical speed
(guard condition), then climb at a constant velocity until reaching a certain altitude
(guard condition) before resuming a gradual accelerating climb. To solve the problem
26
of multivariate integration, the method Seah and Hwang propose in this paper is a
form of analytical integration where the computational cost depends on the dimension of the guard condition, and is independent of the continuous state space. They
also consider uncertainty within the guard conditions, which more realistically models
practical systems, and extend their work to cover stochastic guard conditions. They
put the conditional mode probabilities in terms of Gaussian pdfs (they also do a version with cdfs). The algorithm developed is coined as the state-dependent-transition
IMM (SDT-IMM). Two examples using a standard ATC example are presented.
Zhao and Lie approach the problem of linear non-homogeneous jump Markov
systems in a slightly different fashion [401. They look at the case where the transition
probability matrix is time variant and takes values from a finite set randomly at each
time step, and also reference Blom and Bloems particle filter work [9]. They regard
the matrix as a continuous stochastic variable, allowing for an optimal recursion to
find the posterior matrix pdfs. Each transition probability matrix in the set is used
to run a different bank of KFs in a second mixing step.
This proves to be more
efficient that the PF methods developed by Blom and Bloem, but has the trade-off
of only working when the transition probability matrix exhibits this structure. They
provide simulation benchmarks comparing their algorithm to a standard IMM filter,
where the underlying simulation is structured around the defined class of transition
probability matrices.
As expected, their filter is more accurate, while being more
computationally demanding that the standard IMM.
27
28
Chapter 3
Filtering and Multiple Model
Estimation
As discussed, there is a wealth of literature and research done on the topic of filtering
and multiple model based estimation. Several key concepts emerge when studying
this class of problems. Fist, some form of filter is needed to recursively provide model
matched estimates. Second, a Bayesian framework is applied to the stochastic nature
of the discrete mode estimation. Lastly, sub-optimal techniques are needed to reduce
both the complexity of the multiple model estimator, and the multivariate integration
that arises in formulating the mode probability posterior when the mode probabilities
are a function of the state.
3.1
The Kalman Filter
As mentioned, the Kalman filter (KF) is a widely used optimal estimator given the
problem fits within certain constraints and assumptions. Most filter derivations are
done in the discrete time case, and assume that the noise statistics on the system
and measurements are zero mean, uncorrelated, white, and Gaussian. Of course, it is
possible to relax and modify most of those assumptions. For instance, the filter can
easily be put into the continuous time framework. Also, the zero-mean assumption
can be taken away by simple static shifting, and whiteness can be dealt with by pre29
conditioning the filter. In most cases, practical problems can be framed in a context
that will make the Kalman filter the optimal linear estimator.
For the purposes of this thesis the Kalman filter will be kept in its most basic
form. That is, a linear estimator with well behaved noise statistics. This produces
a simple recursive algorithm that provides estimates for a stochastic system, where
the estimate takes the form of a Gaussian random variable with a given mean and
covariance.
First, consider the linear discrete-time system with state and measurement difference equations given by:
Xk+1 =AkXk + Wk
Zk = HkXk +Vk
k = 0,1, 2, ...
(3.1)
k = 1,2,...
(3.2)
where Xk is the nx-dimensional state vector and zk is the nz-dimensional measurement
vector. If necessary, an input vector (eg. control, forcing, etc.) can be included in
the state equation without much problem.
The noise vectors wk and Vk are both a sequence of zero-mean white Gaussian
noise, with covariance Wk and Rk and mutually independent:
E[WkWk'] = Wk
(3.3)
E[vkvk] = Rk
(3.4)
E[wkvkJ
(3.5)
=
0
In this context, the system matrices and noise covariances are all assumed known
and can be time varying (Ak, Hk, Wk, and Rk). In addition the initial state is a
modeled as a Gaussian random variable and is independent of both the process and
measurement noise.
The two values being computed by the algorithm are the state estimate and covariance. Keeping in mind that due to the stochastic nature of the problem the state
30
is a random variable, then the estimate is defined as:
k~k
(3.6)
E[xkIZk]
where:
(3.7)
Zk A {zIz2 ,...,zk}
is the sequence of measurements up to and including time k. Specifically, the estimate
of the state at time k is the mean of the random state conditioned on the past
measurements Zk.
The other value of interest, the covariance, is essentially the uncertainty associated
with the estimate of the state. If the state estimation error is defined as:
Xkjk
Xk
-
(3.8)
XkIk
then the covariance associated with the estimate is defined:
Qkjk A E[(xk - -k k)(Xk - 4k )T|Zk]
(3.9)
Thus, the KF seeks to propagate forward the statistics of the estimate given
stochastic knowledge of the system, and new information in the form of stochastic
measurements of the system. Put another way, at time index k there is an estimate
'kIk
with uncertainty Qkjk based on measurements Zk. Then, at time index k + 1
filter receives a new measurement
Zk+1.
Through a simple set of matrix equations, the
filter is able to produce the new optimal estimate
.k+11k+1
and uncertainty Qk+l|k+1,
and can do so recursively for future time steps.
The underlying principles behind the Kalman filter can be applied to situations
that do not meet the many assumptions made in deriving the optimal linear filter.
Tools like the extended Kalman filter or unscented Kalman filter make extending the
recursive nature of the Kalman filter to nonlinear systems possible.
31
3.1.1
The Kalman Filter Recursion Equations
For a detailed derivation, refer to [3]. First, starting with the state estimate
:kk
and
covariance QkIk at the current time step k propagate the statistics forward using the
state equation:
ik+l|k = A:kk
(3-10)
Qk+11k = AQkIkA T + W
(3.11)
then, calculate the optimal gain, measurement innovation, and innovation covariance:
Vk+1 = Zk+1 - H-k +11k
(3.12)
Sk+1 = HQk+l1 kH T + R
(3.13)
Lk+1 = Qk+llkH TS-l
(3.14)
finally, update the state and covariance using the new information contained in the
measurement:
k+1|k+1 = &k+1|k
Qk+llk+1 =
(3.15)
+ Lk+lvk+1
(I - Lk+1H)Qk+llk
This represents one cycle of the Kalman filter recursion.
(3.16)
Note, that the system
matrices and noise covariances could still depend on time k but was left time-invariant
for notational simplicity.
3.2
Multiple Model Estimation
There are several ways to derive estimators in a multiple model sense. As mentioned
in Chapter 2, it is possible to derive linear sub-optimal estimators
17,
24], or opti-
mal estimators based on particle filtering methods [8, 9, 20]. In general, there exists
trade-offs between filter complexity and computational load, and often, sub-optimal
32
estimators perform well enough in terms of accuracy, while incurring much less computational burden [1].
General Formulation
3.2.1
Consider the linear system given by:
Xk+1 =
Zk
where
Xk
A(Mk)xk + w(k, Mk)
= H(Mk)xk + v(k, Mk)
is the nx-dimensional state vector and
vector; Mk E {1, 2, . .
, r}
Zk
k = 0, 1, 2, ...
(3.17)
k = 1, 2, ...
(3.18)
is the n,-dimensional measurement
is one of r discrete modes active at time k.
As before, the noise statistics on w and v are Gaussian, zero-mean, white, and
uncorrelated with each other.
w(k, Mk = j) ~ j(O, Qj)
(3.19)
v(k, Mk = j) ~ .(0,
(3.20)
Rj)
The transitions between modes is governed by a Markov chain. In the standard
derivation, probabilities governing the transitions are assumed to be constant:
7rij = P{Mk+1 = jIMk = i}
(3.21)
where the mode history, or sequence of modes is:
Mk, = [M1 , M2 , ..., Mk]
(3.22)
,Mk E {1,2, ... , }(3.23)
M1, M2, ....
l = 1,2,...rk
(3.24)
and is exponentially growing in time as a function of rk.
Consider that the mode history given above is a mutually exclusive, collectively
33
exhaustive set of events. Employing the total probability theorem gives the pdf of the
state conditioned on the measurements, or the state estimate, produces a Gaussian
mixture with exponentially increasing complexity[3]:
p[xk|Zk] = E p[xk|M k'1, Zk]P{Mk'I|Zk}
(3.25)
1=1
where the second term is the probability of a given mode history, and can be calculated
by Bayes rule [31:
(3.26)
P{Mk'l|Zk} = P{Mk'I|Zk-1, zk}
= p[zk|jMk, Zk-1]P{'MlkIZk-1}
p[zkI Zk-1]
=
=
0p[zk| Mk'I, Zk-1]P{AMk-l's, Mk|Zk-1}
(3.27)
(3.28)
C
1
p[zk lMkl, zk-1]P{MkImk-1's, Zk-1}P{Mk-1'sjZk-1}
(3.29)
C
where if, as assumed above, the Markov chain is only dependent on the previous mode
becomes:
P{Mk'l|Zk} - -p[zk|Mk', Zk-1]P{Mk|Mk_1}P{Mk-1's|Zk-1}
C
(3.30)
This, combined with Equation 3.25, show that the optimal estimator requires
conditioning on the entire mode history, even in Jump Markov systems [3]. This issue
is easily handled by using some form of sub-optimal estimator, which as discussed in
Chapter 2 is often the interacting multiple model (IMM) algorithm.
3.2.2
The Interacting Multiple Model Estimator
One of the key pieces to performing multiple model estimation in a practical sense
is some form of sub-optimal estimator. In the recent past, the interacting multiple
model estimator has emerged as one of the best options for state estimation of hybrid
systems [1]. At its core, the IMM estimator is simply a bank of filters (often Kalman
34
filters), each matched to a separate dynamical mode of the system. Through the
clever use of a Gaussian mixture approximation, the algorithm is able to perform
with O(r) complexity, as opposed to the O(rk) complexity of the optimal estimator.
The IMM algorithm can be broken down succinctly in three distinct steps [11:
1. Mixing: Each filter in the bank calculates a state estimate and covariance separately.
At the start of the recursion, these mode-conditioned estimates are
"mixed" in a Gaussian sense using the discrete mode probabilities.
2. Filtering: Each mixed estimate is passed to the corresponding filter to produce
an appropriate mode and measurement conditioned estimate.
3. Mode Probability Update: Based on the output of each filter and the transition
probabilities the discrete mode probabilities are updated
Consider the system outlined in Equations 3.17-3.20, where the superscript j E
{1, 2, ...
, r}
denotes the discrete mode. Let the discrete mode probability at time k
be:
= P{Mk = J}
(3.31)
S = 1 k = 0, 1, . .
(3.32)
14
r
j=1
The following represents one recursion of the IMM filter from time index k -I to k,
in the case where the mode transition probabilities are constant as in Equation 3.21.
At the start of the recursion, the previous filter estimate for each mode
.
_-|l_1 and
covariance Qilk-1 along with the discrete mode probabilities Pi_ 1 are available.
First, calculate the predicted mode probability using the Markov transition:
Il-i_
=i
Z}
1
= P{M
(3.33)
then the IMM mixing probability is given by:
pili A P{M_ 1 IM ,Zk-} = Iit
35
i
(3.34)
and each filter in the bank can then be initialized as:
ZQ-1k-1
-l|k-1 k ilk -1k-
kllk-1 T
k1k-1
(3.35)
-11k-11.'
klk-
k
The initial values in Equations 3.35-3.36 are passed to the bank of j = 1, 2, ...
,r
filters. For instance, the Kalman filter outlined in Section 3.1.1 can be used if the
system is linear. In any case, the filters each produce their own estimate, covariance,
innovations, and innovation covariance:
kik, Qilk, , v,
and Si.
To update the mode probabilities, a likelihood function based on the innovations
is formed as:
Ai =
(vA;0, S') =27rS Iexp(-1/2vj' S'v')
(3.37)
and the mode update becomes:
^3
=
k
A3
-1
iklk-lAi
(3.38)
Although the recursion does not explicitly calculate, or in fact need, the combined
state estimate and covariance, it is now possible to compute it as a simple weighted
Gaussian:
Xk4k
klkllk
=Z
(3.39)
ii
3.2.3
Continuous-State-Dependent Mode Transition Probabilities
In addition to the complication of increasing histories, consider the case where the
mode transition probabilities are not constant as in Equation 3.21. Blom and Bloem
investigated the exact Bayesian filter in the case where the mode transitions depended
36
on the continuous state [8]. Consider the more general system of stochastic difference
equations:
Xk = a(Xk-1, Mk, Wk)
(3.41)
Zk = h(xk, Mk, Vk)
(3.42)
Mk = c(Mk1, Xk_1, uk)
(3.43)
where (Xk, Mk) is now the hybrid system state, and a,h, and c represent general
measurable mappings.
Filtering,, in general, has the goal of finding the optimal estimate of a random
variable. Therefore, in this problem, the goal is to find the density of the joint state
(Xk, Mk)
conditioned on the measurement sequence Zk: p[Xk, MkIZkI.
From Equation 3.43 the current mode Mk is now defined as a function of the
previous mode Mk_1, the previous state Xk_1, and the noise sequences {uk}, {Wk}.
Therefore, the process {Mk} no longer represents a Markov chain. However, it can
be shown the the joint process {Xk, Mk} satisfies the Markov property [8]. In their
paper, Blom and Bloem derive the exact representation of the conditional density
of the joint state. First, the state-dependent mode transition probabilities can be
defined as:
Irij(x) = p[Mk = iIMk_1 = i,
Xk1]
H(x) = {rij(x)},=1,...,r
(3.44)
(3.45)
then, the joint distribution can be found as [8]:
p[xk, MkZk ] = p[zklxk, Mk]
Sj
p[klxk-1, Mkl Z[H(x')p[xkk-1, Mk_1|Zk-']dx/ct
j=1
(3.46)
Here, Blom and Bloem apply particle filtering methods in conjunction with IMM
like approximations to form an IMM based particle filter (IMM-PF) [8].
37
However, in practice it is sometimes more desirable to have a simpler filter for
computational and implementation concerns. As mentioned in Section 2.4 Seah and
Hwang investigated the special case of linear filtering with state-dependent transition
probabilities [31, 32].
Consider the system of linear stochastic equations given in 3.17-3.20, along with
the state-dependent mode transition probabilities defined in 3.44-3.45.
Seah and
Hwang show that in this case, there are two challenges with implementing a filter
[32]. First, the continuous state can be calculated by:
Z p[xkIMk,1
p[xkIZk] =
, Zk]P{Mk'I|Zk}
(3.47)
1=1
which as discussed in Section 3.2.1 is plagued by the issue of exponentially increasing
complexity, and can be approximated effectively with the IMM algorithm. Second,
the update involved in calculating the conditional state transition probabilities, given
by 1311:
p[Mk = jIMk-1', Zk-1] =
j
7rij(X)P[Xk-1 = xIMk-1', Zk-l]dx
(3.48)
which as outlined in Section 2.4 presents problems in obtaining an analytical solution.
Seah and Hwang tackled this problem in several ways. First, they utilized a brute
force MC integration method [31]. Second, they present two approximations for the
conditional mode transition probabilities; a Gaussian pdf:
7rij (x) = aij + bijA/(Lijx; pi, Eij)
(3.49)
= aij + bij Dq(Lij x; pij, Eig)
(3.50)
or Gaussian cdf:
7rij(x)
which proves useful in deriving a specific case of state driven mode transitions. They
cite examples of both positional based waypoints in tracking (pdf), and inequality
"fcut-off" based limits in manufacturing (cdf) [321.
38
Chapter 4
Case Study: Noise Bounds for Pitch
and Roll Estimation using Attitude
Heading and Reference Systems
The operation of a low cost attitude and heading reference system (AHRS) aboard a
fixed wing aircraft was derived and simulated. Parts of this section were presented in
a term paper for 16.322: Stochastic Estimation and Control 118]. An AHRS is used
to estimate the roll, pitch, and yaw of an object relative to a fixed reference frame.
This estimate is formed by filtering noisy measurements taken with accelerometers,
gyroscopes, and other sensing devices. These systems are used in a variety of applications, ranging from large scale navigation and control problems such as aircraft, to
small scale applications such as camera stabilization and small unmanned vehicles.
Typically, the components used in an AHRS are very expensive. However, advancements in micro electro-mechanical system (MEMS) technology has provided
access to smaller, more affordable sensors. These low cost micro sensors are often be
characterized by low accuracy and noisy outputs, so it is necessary to provide some
correction, and understand the limitations of the estimated state. Also, using only
inertial measurements from gyroscopes will lead to increasing error over time, as the
noisy measurements are integrated to form the state estimate.
To combat this estimate "drift," measurements from other sources must be added
39
to the MEMS measurements of body rates and accelerations. For an aircraft, these
could include: airspeed, altitude, GPS, magnetometers, etc. Given data available
from multiple sensors, each with their own noise characteristics, some technique is
needed to combine the information into an estimate of the state. In this case, due
to structure of the nonlinear dynamics, The fusion of measurements from multiple
sources, each with distinct noise characteristics, is done through the use of an Extended Kalman filter.
Here, the estimate under consideration is the roll and pitch of an aircraft. Four
measurement devices are used: a gyroscope, an accelerometer, an airspeed sensor, and
an altimeter. The goal is to evaluate the effect that different levels of sensor noise
has on the estimate of the state. Ultimately, it would be useful to provide a bound
on the gyroscope noise variance and drift that keeps the roll and pitch angle errors
under 5'. This would be useful for informing the selection of instruments used in the
inertial measurement unit (IMU) of the AHRS, specifically when considering the costs
of different instruments. The following demonstrated some of the practical problems
involved in implimenting a filter, especially when considering non-linear dynamics.
Several useful noise characterizations are presented, along with common quantifiable
outputs and measurements of a filters usefulness.
Variables and Coordinate Definitions
4.1
Throughout the chapter, the variable definitions in Table 4.1 will be used.
u
v
w
4
0
0
Var
a
3
Table 4.1: Standard definitions of relevant variables
p Body rate about the x-axis
Airspeed in the body x-axis
q Body rate about the y-axis
Airspeed in the body y-axis
r Body rate about the z-axis
Airspeed in the body z-axis
fx Accelerometer reading in the x-axis
Roll angle
fy Accelerometer reading in the y-axis
Pitch angle
fz Accelerometer reading in the z-axis
Heading
h Altitude
Airspeed
dt Resample time
Angle of attack
y Climb angle
Side slip angle
40
C:
North
center of gravity
Xu
ZN
South
9Earth
Figure 4-1: Body fixed reference frame (left), and inertial reference frame (right) [231
Several reference frames are useful when describing and sensing the motion of an
aircraft.
This is because it is often easier to express different motions in different
frames. For instance, it is typically easier to understand the attitude, or orientation,
of an aircraft relative to some fixed reference.
Typically, for aircraft applications
coordinate systems are expressed in two categories: "body"-frame coordinates and
"internal"-frame coordinates. The body frame is fixed to the aircrafts center of gravity,
and moves with the aircraft as it translates and rotates. The inertial frame is defined
relative to a fixed point, and does not change with time.
Figure 4-1 shows the body fixed reference frame
YB
on the left, which is attached
to the aircrafts center of gravity. The x-axis pointing out the nose and the z-axis points
down, both along the aircrafts planes of symmetry. The y-axis forms a right-handed
orthogonal system. The accelerometer and gyroscope both take measurements in this
frame.
Also, Figure 4-1 shows the inertial North-East-Down (NED) reference frame
YN
on the right. If a flat-Earth assumption is made, the XN and YN axes form the local
horizon, with the x-axis pointing North, and the y-axis pointing East. The z-axis
points down, towards the center of the Earth. Additionally, assuming the Earth is
perfectly spherical, the gravity vector will point along the z-axis in the NED reference
frame, g9
=
[0
0
g]
.
The roll, pitch, and yaw are defined relative to this frame
in that they represent a rotation between the NED frame and body frame.
41
In fact, it is a simple process to move between the two reference frames using a
standard coordinate transformation. To transform from the NED frame to the body
frame a sequence of rotations is performed:
CI
1
0
Cq 0 cos# sin# 0 1 0
0
-sin#
0
-sin 0 cos 0 0
cos#
(4.1)
= Cx(#)Cy(O)Cz(P)
cos0
0
-sinO
cos@
sing'
0
0
cos0
0
0
1
(4.2)
sin0
CB = (
)
(4.3)
were it can be seen that the pitch and roll are directly related to the orientation of
one frame with respect to the other.
4.2
Sensor Characteristics
For this problem, four sources of measurements were considered: a gyroscope, accelerometer, airspeed sensor, and altimeter.
A MEMS gyroscope is used to take measurements of the angular velocity about
a set of body-fixed axes. Typically, a set of three sensors will be used on mutually
orthogonal axes, providing measurements of the rates about each axis. There are
many different classes of gyroscopes, covering a wide spectrum of accuracy and pricing. In order to design a low cost AHRS package, it is desirable to use a less expensive
gyroscope, which will often be characterized by larger error. Typically this error can
come from a variety of sources, such as bias, scale factor error, misalignment, and
random noise. For simplicity, and because of simulation concerns, only the random
noise and bias terms are considered in the analysis. The measurement taken by the
gyroscope (pm) is modeled as the sum of the true state (p), the time varying bias in
42
the gyroscope (bp) and a stochastic white process (Ep):
PM
p + bp +Ep
qm
=
r
q+bq+Eq
(4.4)
r + br + Er]
where the bias drift is modeled as a random walk process:
(4.5)
bi = Ebi
To acquire the pitch and roll of an aircraft, the body rates must be integrated.
Due to the error in measuring these rates, the error in state estimation will tend to
grow over time. This leads to the requirement of additional sensor information to
properly correct the state estimates.
An accelerometer is a device that measures acceleration. It typically consists of
three distinct sensors on orthogonal axes in order to measure acceleration in three
dimensions. The output of the accelerometer is the true vehicle acceleration minus
the gravitational acceleration:
fm = a - g
(4.6)
The measurement obtained from the accelerometer is treated as the true value
with additive white noise:
f .
fX + Ef1
fy[
fy + E
fz +
[z"
(4.7)
EfzJ
Lastly, there are two additional sensor readings typically available on an aircraft.
First, true airspeed is measured using a pitot-static probe or similar device. Again,
this is measurement is modeled as the true value plus additive white noise. Similarly,
the altitude can be measured through pressure, and is also modeled as the true value
plus additive white noise:
Va
= Va + Eva
43
(4.8)
hm = h+Eh
4.3
(4.9)
The Extended Kalman Filter
As detailed in Section 3.1, the Kalman filter is a tool used to estimate the state of
a linear process given the presence of uncertainty in the process and measurements.
The observations of a system are often incomplete, indirect, intermittent, and inexact,
and the KF has ways of handling this [101. The filter forms estimates by propagating statistics through time and minimizing the variance of the error. The extended
Kalman filter (EKF) is an extension of the KF to non-linear systems.
The filter used for estimating the pitch and roll follows the EKF with a couple
assumptions made in order to reduce computational time. A general continuous-time
state space model with discrete measurements for nonlinear systems is:
J(t) = f (x(t), u(t), t) + G(x(t), t)w(t)
(4.10)
z(tk) = h(x(tk), u(tk), tk) + v(tk)
(4.11)
where w(t) and V(tk) are mutually uncorrelated white noise sequences with covariance's W(t) and R(tk) respectively.
First, the state and covariance is propagated according to the process equation.
To do the one-step prediction, Euler's formula is used to perform the integration of
the nonlinear state. This reduces the computational load significantly:
Xk+11k = XkIk
+ f(x(tk), u(tk))dt
(4.12)
Then, the system is then linearized:
F(tk) =
Of
af
(4.13)
and discretized with the following approximations, again to reduce computational
44
load:
I + F(tk)dt + 0.5(F(tk)dt)2
(4.14)
(tk+1, T)G(T)Q(r)GT (T) 1 T (tk+l, T)dT
(4.15)
= eF(tk)dt
Wdk
=
k
'k
W
+ G(tk+1)W(tk+1)G T (tk+1))
~0.5dt('IkG(tk)W(tk)GT (tkT
(4.16)
The one step covariance prediction follows:
(.17
Pk+1k = 4kPkIk k + Wdk
Second, the predicted values are updated with the measurements of the system.
First, the measurement equation is linearized:
Hk+1 =
ax-h
(4.18)
,k+1jkU(tk+1)
The Kalman gain:
Lk+1 = Pk+1kH T+1[Hk+1Pk+1IkHkT+l
+ Rk-
1
(4.19)
Finally, the updated state and covariance are given by:
k+1k+1 =
k+1k
+ Lk+1(zk+1
Pk+1lk+1 =
4.4
(I
-
-
h(Xk+llk, U(tk+1)))
Lk+lHk+1)Pk+llk
(4.20)
(4.21)
Process Model
The states of interest to this estimation problem are the pitch and roll of an aircraft.
As outlined, measurements for the body angular rates are available from the gyroscope
readings. Through kinematics, the body angular rates can be related to the pitch and
45
roll rates by:
S
1
sin 0tan 0
cos 0tan 0
0
cosq$
-sinG
1
q
(4.22)
There are two main issues of note. First, it is safe to ignore the yaw angle and yaw rate
since the dynamics do not depend on either. Second, the state equations include the
values of body rates, which are directly being measured. However, the measurements
of body rates depend both on white noise and a time varying bias term. In general,
this bias will not be observable, and it would be necessary to estimate it. Therefore,
the bias terms are added to the state vector. In addition, the dynamics of the altitude
is easy to model which makes it a convenient state variable:
6 b,
x(t)
(4.23)
b, br h]
(4.24)
u(t) = [Pm qm rm Va]
- 6q)sin ptan&+(rm
(pm - bp - cp)+(qm -bq
- br - 6r)cosotan9
6
(qm - bq - 6q)cos - (rm- br -Er)sin#
bp
Ebp
bq
Ebq
br
Ebr
h
usinG -w sin cos0 -w cos
sin0
(4.25)
where the body velocities
U
V
]
T
can be related to the current state variables
and measurements as detailed in Section 4.5.
46
The general state space model in Equation 4.10 becomes:
(Pm - bp) + (q, - bq) sin 0 tan 0 + (r, - br) cos 0 tan 9
(qm
-
bq) cos
0
- (r, - b,) sin#
0
f(x(t), u(t),t) =
(4.26)
0
0
Va sin 9 cos 0 - Va cos
-1
G(x(t), t) =
w(t) =
-sinocos9
-cosotan9
#
cos 9 sin 0
0
0
0
0
0
-cos#
sin9
0
0
0
0
0
1
0 0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0 0
,W =diag({o2 0,
Cbq
(4.27)
(4.28)
r2 ' r70
2, , 921)
Ebr
Ebp
Ebq
4.5
Measurement Model
The body frame accelerations are measured through accelerometers. The body frame
accelerations can be related to the body velocity rates of change, body velocity, angular rates, roll, pitch, and gravity through:
fx
qw - rv
[
=
fy
fz
B
i
t>
+
ru -pv
pv - qu
47
1
-sin
-
cos0 sin#
cos 9 cos
4
g
(4.29)
Measurements for the airspeed are also available, and the airspeed can be related
to the body velocities through:
[
cos a sin 3
v
(4.30)
sin
Va
sin acos
.
w
For the purpose of modeling the filter, it is assumed that the side slip angle is small,
0
0, the pitch angle is approximately the angle of attack 0 ~ a, and the time rate
of change of the airspeed is small V ~ 0. With these assumpti ons, the body velocities
and body velocity rates are given by:
cos0
u
Va
v
0
sin 0
w'v
[l
-0cos0
0
,j=Va
Low sin 0
I
(4.31)
As previously outlined in Equation 4.24, the airspeed is treated as an input. The
measurement vector is given by:
T
Z(tk) = [f,,
fm
fzm
(4.32)
hm
Equations 4.29-4.31 can be put in the general form described in Equation 4.11.
h(x(t), u(t), t) =
Va((r,
Va((qm - bq) cos
-
-
(r,
- br) sin #) sin 0
br) cos 0 - (p,
# - (r,
-
+ Va(q - bq) sin 0 + g sin 0
bp) sinG) - g cos 0 sin
#
-Va((q m - bq) cos #
- br) sin 0) cos 0 - Va(qm - bq) cos 0 - g cos 0 cos q
h
(4.33)
48
Ea.
v(k) =Ea
, R = diag([,
r
, a or2])
(4.34)
Eaz
LEh
4.6
Simulation Model
In order to test an estimation scheme, it is necessary to compare the estimate to
some form of "truth". One of the characteristics of aircraft estimation is that it is no
suitable way to directly measure the true state accurately. Often, when developing
new sensors or estimation algorithms, the estimates are compared to more accurate
sensor outputs.
Another approach is to obtain a representation of the true state
through simulation.
Three distinct simulation cases were considered.
In each, the trajectory of an
aircraft was generated over 290s. In the first two, the simulation matched the model
exactly. That is, steady flight with no side slip and no climb angle was considered
(Va = 0,
y
=
0, and a = 0). In this case, the pitch and roll angles were chosen, and
the yaw angle and rate were calculated through:
sin#
=
(4.35)
Integration was done numerically using the trapezoidal rule. The roll and pitch rates
we found through numerical differentiation using simple finite difference approximation to reduce computational load.
The first model used piecewise linear functions for the pitch and roll, having one
section where the aircraft rolled at a fixed angle, and one section where it pitched at
a fixed angle. The second model used a general spline through 24 reference points for
both the pitch and roll. Model 1 is shown in Figure 4-2 (a), and Model 2 in Figure 4-2
(b).
Through use of the kinematic and dynamic equations introduced earlier, the angles
49
(b)
(a)
20
00
50
----
100
w 20
- - - --- - - - --------- ---200
250
300
. - 1 0 ..----- ...--
---...
-----...
--.
-..--.....
50
100
150
100
50
-------
200
0
250
0
50
200
150
300
100
200
250
300
200
250
300
-.....
- - --.
-200
0
Time [s]
150
Time [9]
--
tM 0
100
250
- ----------
.20
300
it20
50
200
20 ........................................................
Tim. [a]
0
10
rime Is)
---20 -------~1o-------
S0
0
150
Tim. [s]
-
-
3
50
100
150
Time [s]
200
250
300
Figure 4-2: Simulated roll, pitch, and yaw for (a) Model 1 and (b) Model 2
were used to compute "truth" for the generated quantities:
&V
fy
--
.-
v.
q
-
W
r
]
,
7i
(4.36)
fz
The third, and final, trajectory relaxed two of the main modeling assumptions by
allowing a non-zero side slip and climb angle. The pitch, roll, and yaw trajectories
were kept the same as Model 2. However, since a non-zero side slip and climb angle
were added through a similar spline procedure, this causes the aircraft is no longer
undergo steady coordinated flight and changes the body velocities in Equation 4.30:
cos a sin
Va
[wj
sin3
sin a cos #J
(cos 9 cos y + sin
9
cos -y) sin
sin /3
= Va
(4.37)
(sin 9 cos -y - cos 9 sin -y) cos
#
rul
This, along with its time derivative, can be used in Equation 4.29 to calculate the
"true" body accelerations.
50
(a)
S
51 -----------
.
................
K
..
--------
0
-
-
0
.
0
----------- ----------- I--------
------------
--------------
----------
-2
Ca
_'6
50
100
150
Time [s]
200
250
50
300
50
100
150
200
250
300
Time [91
Figure 4-3: Simulated side slip and climb angles used in Model 3
4.7
Results
The ultimate goal is to find the maximum permissible noise values on the measurements instruments that keeps the error under 50. As a starting point, some realistic
values for sensor noise and initial conditions were taken from
[23].
It was assumed
that the accelerometer has the same noise statistics for each axis, and that each axes
was uncorrelated. A similar assumption was made for the gyroscope. The filter noise
statistics were then "tuned" using the simulation Model 2 until either the pitch or roll
errors were close to 50. Three different versions of the filter were tested. In the first
case, no bias or altitude measurements was included. The second filter added bias to
the rate measurement. The final case included both bias and measurements of the
altitude.
9a
-gy
Ub
Table 4.2: Noise in the simulated sensors
Starting Point Tuned Values
0.5 m/s 2
0.2 M/s 2
Accelerometer noise
0.3 o /2
0.05 o /s2
Gyroscope Noise
211 o /hr
211 o /hr
Bias rate variance
ov.
Airspeed noise
0.6 m/s
3 m/s
Uh
Altitude noise
N/A
20 m
0 0 0
(4.38)
51r 57r 57r 27r 27r 27r
180' 180' 180' 180' 180' 180
(4.39)
oo[ 0 0
Two main design choices were changed during the construction of the filter. First,
51
(a)
TuS tate
0- - Estimated State
so
0
100
IO
ime [a]
(b)
200
250
Estimated State'
lo-
050
300
100
150
'ime [s)
200
250
300
Figure 4-4: Estimation divergence for the (a) designed filter, under the same noise
conditions as the (b) original filter
the filter was originally designed without the inclusion of gyroscope bias. Performance suffered in terms of error. In addition, this did not really model the physical
problem well. Although the addition of bias as a random walk process to the state
variables increased the accuracy in terms of error, it also produced the phenomenon
of trajectory divergence in the roll state. An example of this is shown in Figure 4-4,
where (b) shows the trajectory for the original filter without bias (a) shows an untuned filter with bias included. This stresses the importance of properly testing any
non-linear filter, especially in a Monte-Carlo sense. The observed divergent trajectory
occurred 1/10 times for different sets of measurements. Interestingly, Figure 4-4 (b)
also shows that not having bias in the measurements and process state produced lag
in the estimate.
The second main design choice was made to counter the issue of roll error divergence after it was observed in the estimate. To accomplish this, another type of
measurement was added to the system in the form of altitude. A comparison of the
filter with and without altitude measurements included, using the final tuned noise
values in both cases, is shown in Figure 4-5. Figure 4-5 (b) shows that given the
same noise parameters, a lack of altitude measurements was most likely the cause for
filter divergence. This makes sense, since the rate of change of the altitude is a function of both pitch and roll under the assumptions made, and so indirectly provides
52
(a)
10
5-
50
0
100
150
200
250
300
200
250
300
Time [s]
(b)
-
40
-
Z20
-20-
0
50
100
150
Time [s]
Figure 4-5: Monte Carlo simulation for (a) filer with altitude measurements and (b)
without altitude measurements, for the same sensor noise parameters
more information for the estimator to use. Interestingly, the divergence amplitude
can be reduced, or all but eliminated, by adding more measurement noise to the
accelerometers and gyroscopes, but at the cost of increasingly large raw estimation
error.
Several relevant metrics for the final tuned filter are summarized in the following
figures and table. The filter was run for fifty iterations of measurements generated
from the same "truth".
Table 4.3: Error statistics for the Monte Carlo simulation
2a
Maximum
State Error Mean
#
0
Roll angle
Pitch angle
0.1560
0.0080
2.5150
0.0977'
4.1860
0.1710
It was decided that the maximum estimation error should be bounded.
Inter-
estingly, In the choice of noise parameters there is some freedom. For instance, the
gyroscope error could be increase, while the accelerometer error could be decreased,
and still result in a 50 bound on the error. For the actual selection of low cost sensors,
a cost/benefit analysis would have to be done on real values for the noise.
Finally, the third model detailed in Section 4.6, that is, the one with side slip and
climb angle added to the trajectory was passed through the "tuned" filter. This is
53
(a)
20
-
10
-
-
True State
Estimated State
a
S-10
so
100
150
Time [s]
300
250
200
(b)
15
-
10
-
True State
Estimated State
5
cc
0
C
-5
)
-I
_0
50
100
150
Time [s]
200
300
250
Figure 4-6: Sample trajectory for one of the fifty iterations
(a)
61
4
I
_
0
-2
50
100
150
Time [s]
(b)
300
250
200
0.2r-
w
0.1
0
I~
Vi
-0.1
Y
-
Z
50
100
150
Time [s]
200
250
Figure 4-7: Mean of the roll and pitch error for the fifty iterations
54
300
(a)
0
0
-0
0
150
150
Time [s]
(b)
200
200
250
250
50L
100
160
Time [s]
(c)
200
250
3 00
50
100
150
Time [s]
200
250
3 00
F00020
0
0
-10
3
300
100
50
3
-
U,0.5
0
C -0. 60
(d)
-
150
"
1 -500
50
'
0
__
IAPO
100
150
150
Time [9]
200
20
250
250
Figure 4-8: Mean innovations sequences for the fifty iterations
55
3
3 00
a case where the filter model does not exactly match the real trajectory. The error
plots look similar to the figures presented for the base case, but with slightly more
noise. The statistics of the error are given in Table 4.
Table 4.4: Error statistics for miss-matched model
Maximum
2State Error Mean
6.6670
0.1560 5.8860
<$ Roll angle
0.6000
0 Pitch angle 0.0080 0.215'
In this case, there is more than 50 of error in the 95% interval of the roll angle.
All things considered, this side slip and climb angles considered in Model 3 were
fairly small, but they still drove the error in estimation higher. This suggests that
perhaps a more robust model would better suit the problem for general trajectories.
Or maybe not, if the aircraft in question is expected to fly on a coordinated trajectory.
A less tightly "tuned" filter could also be considered, to reduce error in this type of
uncoordinated flight, at the expense of more accurate sensors.
4.8
Summary
The roll and pitch of an aircraft was estimated using an EKF. Four measurement
devices are used: a gyroscope, an accelerometer, an airspeed sensor, and an altimeter.
The goal is to evaluate the effect that different levels of sensor noise has on the estimate
of the state, and demonstrate the practical implementation of a non-linear dynamic
filter Ultimately, a bound on the gyroscope noise variance and drift was given that
keeps the roll and pitch angle errors under 5'. This process should be useful for
informing the selection of instruments used in the inertial measurement unit (IMU)
of the AHRS, specifically when considering the costs of different instruments.
Several issues arose during the implementation of the filter. First, it was interesting that a lack of sufficient measurements could lead to divergence in the non-linear
filter. This was most likely due to the integration error not being properly corrected
in the filter, either through lack of measurements, or improper noise selection (i.e.
measurement trust vs. prediction trust).
56
Second, several assumptions were made
on the model during the filter design process. It was shown, that when even when
the true process deviates slightly from the assumptions, the filter still functions, although not quite as well. This is fairly standard for miss-modeling a problem, but
was interesting nonetheless.
Moving forward, it would be interesting to consider a slightly more complicated
model that captures the effects of climb angle and side slip, maybe in terms of a
multiple model like filter. Also, it might be interesting to add in more sources of
measurements, such as GPS or magnetometers to see if the estimation error can be
reduced.
57
58
Chapter 5
Conclusion
5.1
Discussion
Throughout the thesis, the concept of estimation with regards to stochastic hybrid
systems was thoroughly investigated.
A broad outline of the topics of estimation
and multiple model based systems was provided, along with several realizations of
practical filters and algorithms. A literature review of the concepts underlying the
development of stochastic hybrid estimation from early studies and filters, to more
advanced concepts was provided. Specifically, the often made assumption of a time invariant transition probability matrix, which classifies a system as jump Markovian was
looked at. Recent work relaxing this assumption was explored, from the concept of
semi-Markov jump systems, to more advanced Bayesian realizations and particle filter
implementations of continuous-state-dependent mode transition probabilities. More
practical linear filters using Gaussian assumptions and continuous-state-dependent
mode transitions were also outlined, along with the concept of a finite set of transition probabilities used to model time-invariance.
A brief outline of the derivations and equations needed to implement both simple
Kalman filters and more advanced extended Kalman filters was provided. In addition,
a general framework for multiple model estimation was provided, along with the equations needed to implement an interacting multiple model algorithm in practice. Also,
some of the challenges involved in the realization of a continuous-state-dependent
59
mode transition probability matrix were discussed, including the multivariate integral that arises from the derivation. Two approximations to the mode transitions in
the form of Gaussian pdfs and cdfs were provided.
Finally, a case study on the implementation of an extended Kalman filter algorithm
for the application of attitude heading and reference systems was provided.
This
highlighted several issues that can appear when using non-linear filters, including
divergence of the estimate, and the usefulness of additional sources of information in
the form of measurements. It was also shown that if the true system process deviates
from the model used in the filter, additional problems can manifest.
5.2
Future Work
Many avenues exist for extending the concepts presented in this thesis. One interesting area is to explore additional modeling problems, and implement time varying
transition probabilities in other systems and applications. Also, several other concepts often explored in the framework of filtering such as smoothing and prediction
deserve further attention.
In addition, the underlying nature of the non-constant mode transition probability matrix deserves further consideration. Here, only two concepts of sub-optimal
implementations were presented; continuous-state-dependent mode transition probabilities, and time invariance in the context of a finite set of possible mode transition
probabilities.
60
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