# 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.

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.

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