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An Introduction to the Extended Kalman Filter

Kalman Filters are a form of predictor-corrector used extensively in control systems engineering

for estimating the unobservable states of a process. The estimated states may then be used as

part of a strategy (i.e. control law) for controlling one or more of the unobservable states.

This tutorial discusses the Extended Kalman Filter which was developed for non-linear discrete-

time processes (or at least processes that may be modeled with sufficient accuracy as a non-

linear discrete-time process). Other tutorials discuss other types of Kalman filters: the original

Kalman Filter (for linear processes); the Kalman-Bucy Filter(for continuous-time systems); and

the Unscented Kalman Filter (which is an extension of the Extended Kalman Filter).

Figure 1: Non-linear discrete-time process with input and measurement noise.

Consider the non-linear discrete-time process shown in Figure 1which may be written in the

standard state-space form,

Here

kdenotes a discrete point in time (with k-1 being the immediate past time point).

uk is a vector of inputs

xk is a vector of the actual states, some of which may be unobservable.

yk is a vector of the actual process outputs.

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is a vector of the measured process outputs.

wkand vkare process and output noise respectively. They are assumed to be zero mean

Gaussian with covariance Qkand Rk respectively.

f(.) and h(.) are generic non-linear functions relating the past state, current input, and

current time to the next state and current output respectively.

The purpose of the Extended Kalman Filter is to estimate the unobservable states and the actual

process outputs given the input, measured output and assumptions on the process and output

noise. This is shown inFigure 2 where the estimated states are , and are the estimated

measured outputs.

Figure 2: Input-output of the Extended-Kalman Filter.

As with the original Kalman Filter, the Extended Kalman Filter uses a 2 step predictor-corrector

algorithm. The first step involves projecting both the most recent state estimate and an estimate

of the error covariance (from the previous time period) forwards in time to compute a predicted (or

a-priori) estimate of the states at the current time. The second step involves correcting the

predicted state estimate calculated in the first step by incorporating the most recent process

measurement to generate an updated (or a-posteriori)state estimate.

However, due to the non-linear nature of the process being estimated the covariance prediction

and update equations cannot use fand hdirectly. Rather they use the Jacobian of fand h. The

Jacobians are defined as

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For the Extended-Kalman Filter, mathematically, the predictor step is given by

And the corrector step is given by,

In the above equations Pk is an estimate of the covariance of the measurement error and Kk is

called the Kalman gain. After both the prediction and correction steps have been performed then

is the current estimate of the states and can be calculated directly from it. Both and Pk

are stored and used in the predictor step of the next time period.

If the process is linear then the above equations collapse to the equations of the original (i.e.

linear) Kalman Filter. However, unlike theKalman Filter, the Extended-Kalman Filter is not

optimal in any sense. And further, if the process model is inaccurate then due to the use of the

Jacobians -- which essentially represent a linearization of the model -- the Extended-Kalman

Filter will likely diverge leading to very poor estimates.

However, in practise, and when used carefully, the Extended-Kalman Filter can lead to very

reliable state estimation. This is particularly the case when the process being estimated can be

accurately linearized at each point along the trajectory of the states.

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A simple example demonstrating how to implement an Extended Kalman Filter in Simulink can be

found here.

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