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An Introduction to Kalman Filtering

- Serge P. Hoogendoorn Hans van Lint
- Transport Planning Department
- Delft University of Technology

Rudolf E. Kalman

Scope of course

- Introduction to Kalman filters
- Application of Kalman filters to training ANN
- Hands-on experience by exercises applied toyour

own problems or a problem we provide - Book review format
- Each week one chapter will be discussed byone of

the course participants - Take care both the book / lecture notes have a

very loose notational convention! - Important information resource
- http//en.wikipedia.org/wiki/Kalman_filtering

Schedule

Kalman filter motivation

- Modeling approaches
- Mathematical-physical models
- Black-box / time-series
- Main advantage of time-series modeling is ease

with which on-line estimation is achieved - Main disadvantage is the fact that many known

relations describing the systems behavior are not

used - Kalman filtering nice way to combine advantages

of mathematical physical models and time-series

models

Lord Kalman

- First publication on Kalman filteringin 1960 by

R.E. Kalman - Combine mathematical models with measurement

information in a way that is very efficient and

elegant - First important application navigation of

space-ships - Data from radar, gyroscopes, visual observations
- Knowledge regarding dynamic behavior of the

space-ship - Many other applications since then

Mathematical Physical model

- xk state (vector) that completely describes the

situation in the system at time k - Simulating systems behavior yields new state

xk1

Model

kk-1

Car-following example

- Simple car-following model
- The speed V(t) of leader is given (and exogenous)
- Discretized form

Car-following example

- Now, let the state xk be defined by
- Then we can write the discretized car following

model in the following form

State-space modeling

- Useful to determine mathematical properties of

system (stability, controllability,

observability) - Used in optimal control
- Exercise 1
- Consider a dynamical process from your own

research field and formulate it as a (discrete /

discretized) state-space description

Combining model Kalman filter

- Express the accuracy of the input / model by

adding random noise w

Model

kk-1

Combining model Kalman filter2

- Introduction of the Kalman filter

Model

Filter

kk-1

Use of the Kalman filter

- Main objectives of Kalman filter
- Measurement information is used to find and

eliminate modeling errors, errors in the input

and errors in the parameters - Model information is used to eliminate outliers

in the measurements - For the concept of filtering, the following are

important - Filtering aim to reconstruct the state at time

k, using all available information until that

time k - Predicting based on the measurements until time

k we forecast the system for times lgtk - Smoothing reconstruct the state for times l,

given measurements at times kgtl

Combining model Kalman filter3

- Kalman filter is used to based on available

measurements to correct the model predictions

in the most efficient way - This allows to correct the input quantities, the

model parameters and the prediction output - Example for unknown quantity x we have the

following model and have measurements z - Choose as estimator

Estimator variance

- We find
- Note for k0 (no weight to measurement)
- For k 1 (no weight to model)
- How to use information from both model and

measurements in an optimal way?

Minimum variance estimator

- Minimum variance estimator

Kalman filter

- Basics of the general Kalman filter are the same
- Multi-dimensional generalization based on

discrete-time dynamic system and measurement

equation - Minimum variance estimator
- Filter will be derived in ensuing of this

introduction

Discrete Kalman filter

- We consider linear stochastic discrete time

systems - Model and measurement noise are assumed Gaussian

and mutually independent the following holds

Examples of state-space formulation

- Consider AR(1) model
- Now consider AR(2) model
- AR(2) model is not in the correct form
- How to rewrite AR(2) model?

Examples of state-space formulation

- Consider AR(1) model
- Now consider AR(2) model
- Define new statethen

Discrete Kalman filter

- Initial conditions for the state x0
- Note that the time step between two times k and

k1 need not be constant!

Discrete Kalman filter

- Aim of the DKF (discrete Kalman Filter) is find

an estimator for the state that optimally used

the model and measurement information - Maximum a-posteriori estimator
- Conditional mean estimator
- Minimum variance estimator

Discrete Kalman filter

- Important property if
- is Gaussian, then all of the above estimators

are identical

Example

- Reconsider our model
- with
- Then obviously we have

Example

- The conditional p.d.f. for x given z equals
- Some tedious computation leaves us with

Example

- The maximum a-posteriori estimate in case of a

Gaussian estimate is equal to the mean - Note that this is equal to the minimum variance

estimator we derived earlier - Note that we did not make any prior assumptions

regarding the linear structure of the estimator - For general discrete linear processes with

Gaussian noise - Optimal filter is linear
- Kalman filter is optimal in the sense of minimum

variance

The Kalman Filter

- Multi-dimensional generalization of scalar

results - Let the minimum variance estimator of xk given

measurements z1,,zl be denoted by - And let Pkl denote the covariance matrix of this

matrix - For the initial conditions, we have

The Kalman Filter

- Time-propagation (prediction)
- Measurement adaptation (correction)
- Kalman gain

The Kalman Filter

- Note the clear predictor corrector structure
- The covariance Pkk-1 and Pkk and filter-gain

are not dependent on the measurements zk and can

thus be computed off-line (drawbacks?) - The filter produces besides the optimal estimate,

the covariance matrix Pkk which is an important

estimate for the accuracy of the estimate - The matrix also shows the effect of changing the

measurement settings on the accuracy of the

estimates!

Application example

- Consider a bicycle equipped with GPS driving with

10 km/h (appox. constant speed) , due to wind,

grades, etc. - Some variation in the speeds are present
- Per hour, we want to determine speed and location

of bicycle - We have the following initial conditions

Application example

- Assuming disturbances on the speed yields

decision to apply noise on the speed dynamics - For the position we have
- State dynamics with xk dk sk

Application example

- Assume that we have data on the vehicle positions
- with a measurement error R 2 km2
- If we assume that we have the observations z1 9

km, z2 19.5 km, z3 29 km, which can be used

to estimate the location and the speed - What are the results of application of a Kalman

filter? - Exercise code the Kalman filter in Matlab

Application example

- Time k 1
- Time k 2

Application example

- Time k 3
- Note that the speed can only be estimated

correctly when two positions at two different

time-steps are known - This becomes apparent from the covariances P

Example 2

- Consider a model where only the measurements are

subject to noise - Let P0 be the variance of the initial estimate
- Determine the covariance matrices Pkk-1 and Pkk

Example 2

- Consider a model where only the measurements are

subject to noise - Let P0 be the variance of the initial estimate
- Determine the covariance matrices Pkk-1 and Pkk
- We can prove that

Example 2

- Note that in this case
- For the filter gain, we have
- Since
- we see that the influence of the measurements

will be zero since Kk 0

Example 2

- What can you say about the filter behavior when

there is no information about the initial

conditions?

Example 2

- What can you say about the filter behavior when

there is no information about the initial

conditions? - In this case, we have
- Which in turn implies that
- This yields

Proof of the Kalman theorem

- Let us consider the mean value of the state xk1
- This describes the conditional mean estimator

(equal to the minimum variance estimator, and the

maximum a posteriori estimator)

Proof of the Kalman theorem

- For the covariance matrix we have

Proof of the Kalman theorem

- Important lemma (see notes)
- Remainder of proof by complete induction

Stationary filter

- Consider the time-independent discrete time

model - Recall that
- and thus
- Let P?
- then we have

Bicycle example

- Recall GPS equipped bicycle
- For k very large, we get

Filter divergence

- What to do if the model is not a good description

of reality? - In this case, the estimations of the filter can

be much worse than one would suspect based on the

covariance matrix (note that the covariance

matrix is not dependent on the actual

measurements) - If the model is incorrect then both the

covariance matrix and the filter gain are

incorrect the filter thinks the prior estimates

are accurate and will not weight current

measurements adequately (recall earlier example) - This phenomenon is referred to as

filter-divergence

Filter divergence

- Different ways to analyze filter divergence
- Simple approach is based on analyzing the

residuals - Theoretically, we have
- and

Filter divergence

- Since we know the theoretical statistics of the

residuals, we can compare these to the measured

statistics of the residuals - If these are alike, then the filter will function

properly - If not, we have a case of filter divergence

Oppressing filter divergence

- Two straightforward approaches (for details, see

literature) - Increase the model noise Qk, possibly each time

step (adaptive filtering) - Increase the weights of recent measurements

(reduction of the measurement covariance Rk)

Extended Kalman filter

- Consider the following non-linear system
- Assume that we can somehow determine a reference

trajectory - Then
- where

Extended Kalman filter

- For the measurement equation, we have
- We can then apply the standard Kalman filter to

the linearized model - How to choose the reference trajectory?
- Idea of the extended Kalman filter is to

re-linearize the model around the most recent

state estimate, i.e.

The Extended Kalman Filter

- Time-propagation (prediction)
- Measurement adaptation (correction)
- Kalman gain

Next week

- Hans will discuss application to Kalman filters

to adaptive parameter identification (in

particular for ANN) - Exercise with state-space modeling (exercise 1),

implementing a simple filter in Matlab (exercise

2) and setting up an extended Kalman filter

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