Table of Contents:
Developing Models for Kalman Filters
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 Selftune 
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 CorrelationBased Model 
Least Squares methods complete a correlationbased 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 pseudorandom 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 runtime 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. 