Chapter	Title

Section 1
Chapter 1	Why Model?
What is a model? What is a linear model? How is this related to Kalman Filters? For what purpose?
Chapter 2	Linearity
How do strict and conventional notions of linearity differ? How is this related to Kalman Filter models?
Chapter 3	State Models and Kalman Filters
Why are the linear models discussed so far all unsuitable for purposes of Kalman Filters — what is missing?
Chapter 4	State Transition Dynamic Models
Detailing the special form your model must have to be suited for classic Kalman Filtering — and why.
Chapter 5	Feedback in State Models
Some adjustments you will need in your model if you choose to — or must — employ feedback stabiliization as your system operates.
Chapter 6	Linear Least Squares Review
An essential tool: how to formulate and solve a Linear Least Squares problem to evaluate "best fit" model parameters.
Chapter 7	Practical Linear Least Squares Calculations
Introducing the Octave package, and showing how simple it is to perform Linear Least Squares calculations in practice.
Chapter 8	Least Squares Dynamic Fit
A lot of ideas come together in an attempt to build a dynamic mode using Least Squares methods.
Chapter 9	Adapting ARMA Models
Another approach: fabricate a suitable state transition model from an ARMA model obtained by regression analysis methods.
Chapter 10	Verifying the Model
How to numerically simulate and evaluate your proposed model using actual system I/O data.
Chapter 11	Updating and Sequential Least Squares
It is not necessary to collect huge data sets and grind them down all at once for Least Squares fitting — you can consolidate as you go.
Chapter 12	Scaling, Weighting, and Least Squares
An important feature... a serious hazard... how you present data to a least squares problem affects the solution you will get.
Chapter 13	Fading Memory in Least Squares Problems
Critical adjustments are required to allow Least Squares updates to systematically prefer new data over old.
Chapter 14	Introducing Recursive Least Squares (RLS)
Begin the search for efficient Least Squares solutions when frequent updates of parameter values are needed.
Chapter 15	Efficient RLS Computation
Inverse updates provide the missing piece for the RLS method.
Chapter 16	Adaptive "Recursive Least Squares" Applications
Avoiding disaster when using RLS methods for system models that change "adaptively" over time.
Chapter 17	Total Least Squares Methods
An alternative to Linear Least Squares when all system inputs and outputs are noisy.
Chapter 18	LMS: Let the Model Self-tune
Applying the LMS method to let a state transition model incrementally improve itself, based on test data — and patience.
Chapter 19	Restructuring State Models, Part 1
State transition models are not unique! Introducing transformations that can produce equivalent models.
Chapter 20	Restructuring State Models, Part 2
Introducing Householder transformations, for sparse and efficient state model systems.
Chapter 21	Restructuring State Models, Part 3
Introducing Eigenvector transformations, for producing state model systems with minimal state interaction.
Chapter 22	Model Order
Discussing the importance of representing the correct number of internal states in the model.
Chapter 23	Randomized Test Signals
How to effectively collect system test data for determining system order.
Chapter 24	Autocorrelation in Test Signals
Distinguishing artificial side effects of testing in correlation data.
Chapter 25	Measuring System Correlation
How to perform a correlation analysis on system I/O data — it's easy.
Chapter 26	Recognizing State Effects on Correlation
Patterns in correlation data that indicate the presence of internal states.
Chapter 27	Test Case: Identifying States in Correlation
A numerical example of counting internal states using correlation methods.
Chapter 28	Transition Matrix for Response Modes
Exploratory validation of correlation analysis results by constructing a model.
Chapter 29	Finishing Correlation-Based Model
Least Squares methods complete a correlation-based model to see if results are credible.
Chapter 30	LMS: Experiments in Tuning New States
LMS methods are used to splice an additional behavior onto an existing model.
Chapter 31	Reducing Overspecified Models
Reducing model order: removing redundant and undesirable elements from an existing model.
Chapter 32	Compacting the Reduced Model
Numerical cleanup after model order reduction, to obtain a compacted equivalent model.
Chapter 33	Adjusting Time Steps for Discrete Models
Transforming a state transition model to operate at a different time interval than the original model.
Chapter 34	State Observers: Theory
Introducing the other approach: trying to adjust the model's internal state variables rather than the model itself.
Chapter 35	State Observers: Design
Exploring how to set the observer parameters — its gains — to tune observer performance.
Chapter 36	State Observerse with a Weak Model
Benefits and hazards of observers: making good models work better, hiding the deficiency of a bad model.
Chapter 37	Reformulating the State Observer
A mathematical reformulation that combines the functions of state transition prediction and observer.
Chapter 38	Minimalist Observer: the Alpha Beta Filter
Observers with a model so weak that it barely qualifies as a model — yet, sometimes completely sufficient.
Chapter 39	Quantifying Variation
How variance is used in Kalman Filters to represent the properties of random noise.
Chapter 40	Initial Variance
How an interpretation of variance is employed to represent highly uncertain initial system conditions.
Chapter 41	Variance Propagation
How initial uncertainty and new random noise sources interact to affect progression of state uncertainty over time.
Chapter 42	Generating Correlated Random Vectors
How to produce pseudo-random noise with specified correlation properties, so that Kalman Filters can be simulated.
Chapter 43	Simulating the Noisy Observer
Experiments testing observer response to correlated noise.
Chapter 44	Observer Optimization
How to determine the "Kalman Gains" that achieve Wiener optimal tracking of the system state by the observer.
Chapter 45	The Steady State Kalman Filter
For fixed transition and noise models, how the complicated run-time variance updates can be eliminated.
Chapter 46	Consistent Covariances
An accurate noise model might not be possible — but it can at least be consistent with the actual system.
Chapter 47	The Dreaded Kalman Divergence
What happens when bad models produce seriously bad results, and why this doesn't need to happen to you.
Chapter 48	Considerations for Data Smoothing
How Kalman Filters can be tweaked to produce optimal estimates at past and future times, not just the next step.

Table of Contents:

Developing Models for Kalman Filters