Introduction to Kalman filtering

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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
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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

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Schedule
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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

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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

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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
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Car-following example

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

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Car-following example

• Now, let the state xk be defined by
• Then we can write the discretized car following
  model in the following form

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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

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Combining model Kalman filter

• Express the accuracy of the input / model by
  adding random noise w

Model
kk-1
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Combining model Kalman filter2

• Introduction of the Kalman filter

Model
Filter
kk-1
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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

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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

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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?

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Minimum variance estimator

• Minimum variance estimator

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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

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Discrete Kalman filter

• We consider linear stochastic discrete time
  systems
• Model and measurement noise are assumed Gaussian
  and mutually independent the following holds

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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?

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Examples of state-space formulation

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

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Discrete Kalman filter

• Initial conditions for the state x0
• Note that the time step between two times k and
  k1 need not be constant!

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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

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Discrete Kalman filter

• Important property if
• is Gaussian, then all of the above estimators
  are identical

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Example

• Reconsider our model
• with
• Then obviously we have

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Example

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

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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

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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

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The Kalman Filter

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

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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!

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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

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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

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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

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Application example

• Time k 1
• Time k 2

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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

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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

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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

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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

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Example 2

• What can you say about the filter behavior when
  there is no information about the initial
  conditions?

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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

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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)

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Proof of the Kalman theorem

• For the covariance matrix we have

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Proof of the Kalman theorem

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

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Stationary filter

• Consider the time-independent discrete time
  model
• Recall that
• and thus
• Let P?
• then we have

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Bicycle example

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

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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

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Filter divergence

• Different ways to analyze filter divergence
• Simple approach is based on analyzing the
  residuals
• Theoretically, we have
• and

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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

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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)

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Extended Kalman filter

• Consider the following non-linear system
• Assume that we can somehow determine a reference
  trajectory
• Then
• where

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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.

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The Extended Kalman Filter

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

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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