Advanced Kalman Filter Theory and Practice

This intensive course is designed for engineers and
scientists who need to deepen their understanding of the mathematical foundations
of Kalman filtering and learn new practical techniques to build and maintain
real world Kalman filters.

**Prerequisites**:An
undergraduate degree in engineering, natural science (physics, chemistry, etc.)
or mathematics. An advanced degree in
one of these fields will make it easier to grasp a few of the concepts presented, but
nothing in the course presupposes such advanced work.
Some practical acquaintance and/or experience with applications
of Kalman filters in real world situation will certainly enrich the learning
experience in this course but, again, are not required.

To increase
your understanding of Kalman filter theory and provide hands-on experience
in solving practical problems associated with the design and construction of
Kalman filters.
**Course Style**: This
is an intensive, highly interactive seminar.
The participants will not be mere passive spectators at a series of
lectures. Class participants will have
ample opportunity to exercise what they learn.
The course is divided up into segments with each segment lasting
approximately 2 hours. There are 4
segments each day. Each segment is 2
hours long and is structured as follows:

q 1-hour
interactive presentation of new material

q 30-minute
exercise session

q 15-minute
exercise review period

q 15-minute
break

**
The 30-minute exercise session usually consists of two parts: an individual
part (15 minutes) and a workgroup part (15 minutes). The student will have
the opportuntity to work both alone and as part of a workgroup. The
workgroups are arranged so that the level of skill and experience is
balanced across workgroups.**

** Sample
content from Day 1, Segment 1, including the exercise session at the end,
are provided here in a PDF file: Set Theory (PDF). (Note this
segment is only about the half the length of other segments, because it is
preceded by an introductory session on Day 1.)**

**Most of the segments also have Matlab m-files and/or Simulink
models associated with them. For example, the segment on linear systems
theory uses the following pair of files (among others): State Space Generation m-File and Generic State Space Model. **

**Course Content:**

** **

**
Stochastic dynamic systems (both linear and non-linear) play a central role
in Kalman filter theory. Thus the course will provide a
solid grounding in the mathematical concepts and techniques underlying such systems.
Also, a brief introduction to Hilbert Space
theory will be provided, since optimal estimation techniques such as the
Kalman filter can be briefly characterized a orthogonal projection in
Hilbert space. Many fields of modern science and engineering employ Hilbert
Space concepts and techniques; for example, in engineering they play
a key role in digital signal processing and in physics they are used in
quantum mechanics. Hilbert Space is not as scary as it
sounds. It is really only a generalization of 3-dimensional Euclidean space
to an infinite number of dimensions. The concepts of inner product (aka dot
product), orthogonality, norm (aka vector magnitude), and linear transformations are
already familiar from 3-space. In addition to these more mathematical
topics, useful practical techniques for identifying appropriate state models
based on empirical input-output data will be taught. Hands-on problem
solving using these techniques will help class participants acquire solid
engineering tools.**

** **

**Day 1:
Foundational Math I: **

q **Segment
1 – Basic Set Theory: sets, set operations, Cartesian products, functions,
n-ary operations, relations, equivalence relations. Practical applications:
probability event algebras.**

q **Segment
2 – Algebra: algebraic structures,
homomorphisms, groups, rings, fields, vector spaces, bases, inner products,
norms, matrices**

q **Segment
3 – Topological spaces, open / closed / compact sets, continuous functions; ****s-algebras,
measures, probability measures; Borel sets; integration**

q **Segment
4 – Infinite dimensional vector spaces; Banach space; Hilbert space; orthogonal
projection, least squares approximation, recursive formulations**

**Day 2:
Foundational Math II:**

q **Segment
1 – Probability theory, random variables, distributions, density functions,
expectation, moments, conditional expectation**

q **Segment
2 – Stochastic processes, second-order processes, Gaussian processes,
Wiener-Levy processes, white noise, random walk, random constants, random
ramps; Allan variance**

q **Segment
3 – Correlation, autocorrelation, cross-correlation, power spectral density.
Practical applicatons: system identification techniques (SI/SO -> MI/SO)**

q **Segment
4 – Mean square calculus: convergence,
derivatives, integration, stochastic differential equations**

**Day 3: Dynamic
Systems (deterministic):**

q **Segment
1 – Ordinary differential operators, differential equations, dynamic systems,
state space representation**

q **Segment
2 – Distributions: linear functionals, dual space, delta functional, Heaviside
operator**

q **Segment
3 – Transforms: Laplace, Fourier,
Hilbert, Mellin**

q **Segment
4 – Solutions to differential equations:
state transition matrix; transient / steady-state response; frequency
domain analysis: stability criteria**

**Day 4: Dynamic
Systems (stochastic) and Kalman Filtering**

q **Segment
1 – Stochastic differential equations, stochastic dynamic systems**

q **Segment
2 – Error and Uncertainty: The error
covariance matrix and covariance analysis; matrix Riccatti equation**

q **Segment
3 – Measurement model; observability criteria;**

q **Segment
4 – Continuous formulation of the filtering problem and derivation of
solutions: static and dynamic estimation problems; alternative derivations**

**Day 5:
Advanced Topics**

q **Segment
1 – System identification: state vector augmentation; model identification
and evaluation; innovations sequence monitoring. **

q **Segment
2 – Discrete formulation: difference
equations**

q **Segment
3 – Nonlinear problems: Linearization
techniques and the extended Kalman filter**

q **Segment
4 – Adaptive Kalman filtering**