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filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space (as opposed to a discrete state space as in the hidden Markov model). The Kalman filter is a estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows, the notation represents the estimate of at time n given observations up to, and including at time m ≤ n

. The state of the filter is represented by two variables: , the state estimate at time k given observations up to and

including at time k;

, the a posteriori error covariance matrix (a measure of the

estimated of the state estimate).

The formula for the updated estimate and covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the section.

Invariants

If the model is accurate, and the values for and accurately reflect the distribution of the initial state values, then the following invariants are preserved: (all estimates have a mean error of zero)

is the expected value of , and covariance matrices accurately reflect the where

covariance of estimates

Example application, technical[edit]

Consider a truck on frictionless, straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocity is. We show here how we

derive the model from which we create our Kalman filter. Since are constant, their time indices are dropped.

The position and velocity of the truck are described by the linear state space

where is the velocity, that is, the derivative of position with respect to time.

We assume that between the (k 1) and k timestep uncontrolled forces cause a constant acceleration of ak that is normally distributed, with mean 0 and standard deviationσa.

From Newton's laws of motion we conclude that

(note that there is no term since we have no known control inputs. Instead, we applies that effect to the state

assume that ak is the effect of an unknown input and

vector) where

and

so that

where

and

At each time step, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also normally distributed, with mean 0 and standard deviation σz.

where

and

We know the initial starting state of the truck with perfect precision, so we initialize

and to tell the filter that we know the exact position and velocity, we give it a zero

covariance matrix:

If the initial position and velocity are not known perfectly the covariance matrix should be

initialized with a suitably large number, say L, on its diagonal.

The filter will then prefer the information from the first measurements over the information already in the model.

Deriving the a posteriori estimate covariance matrix

Starting with our invariant on the error covariance Pk | k as above

substitute in the definition of

and substitute

and

and by collecting the error vectors we get

Since the measurement error vk is uncorrelated with the other terms, this becomes

by the properties of vector covariance this becomes

which, using our invariant on Pk | k 1

and the definition of Rk becomes

This formula (sometimes known as the "Joseph form" of the covariance update equation) is valid for any value of Kk. It turns out that if Kk is the optimal Kalman gain, this can be simplified further as shown below.

Kalman gain derivation

The Kalman filter is a minimum mean-square error estimator. The error in the a

posteriori state estimation is

We seek to minimize the expected value of the square of the magnitude of this

vector, . This is equivalent to minimizing the trace of the a

. By expanding out the terms in the equation

posterioriestimate covariance matrix

above and collecting, we get:

The trace is minimized when its matrix derivative with respect to the gain matrix is zero.

Using the gradient matrix rules and the symmetry of the matrices involved we find that

Solving this for Kk yields the Kalman gain:

This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used.

Simplification of the a posteriori error covariance formula

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