The red lines correspond to “ground truth” in our example.

The green points in Figure 9b show the noisy measurements of velocity at different time steps, assuming the noise is modeled by a Gaussian with variance 8. The blue lines show the a posteriori estimates of the velocity and position. It can be seen that the a posteriori estimates track the ground truth quite well even when the ideal system model (the gray lines) is inaccurate and the measurements are noisy. The cyan bars in the right figure show the variance of the velocity at different time steps. Although the initial variance is quite large, application of Kalman filtering is able to reduce it rapidly in few time steps.

Discussion. We have shown that Kalman filtering for state estimation in linear systems can be derived from two elementary ideas: optimal linear estimators for fusing uncorrelated estimates and best linear unbiased estimators for correlated variables. This is a different approach to the subject than the standard presentations in the literature. One standard approach is to use Bayesian inference. The other approach is to assume that the a posteriori state estimator is a linear combination of the form , and then find the values of At and Bt that produce an unbiased estimator with minimum MSE. We believe that the advantage of the presentation given here is that it exposes the concepts and assumptions that underlie Kalman filtering.

Most presentations in the literature also begin by assuming that the noise terms wt in the state evolution equation and vt in the measurement are Gaussian. Although some presentations1, 10 use properties of Gaussians to derive the results in Figure 3, these results do not depend on distributions being Gaussians. Gaussians however enter the picture in a deeper way if one considers nonlinear estimators. It can be shown that if the noise terms are not Gaussian, there may be nonlinear estimators whose MSE is lower than that of the linear estimator presented in Figure 6d. However, if the noise is Gaussian, this linear estimator is as good as any unbiased nonlinear estimator (that is, the linear estimator is a minimum variance unbiased estimator

Kalman gain is not a dimensionless value here. If Ht = I, the computations in Figure 6d reduce to those of Figure 6c as expected.

Equation 39 shows that the a posteriori state estimate is a linear combination of the a priori state estimate and the measurement (zt). The optimality of this linear unbiased estimator is shown in the Appendix. It was shown earlier that incremental fusion of scalar estimates is optimal. The dataflow of Figures 6(c,d) computes the a posteriori state estimate at time t by incrementally fusing measurements from the previous time steps, and this incremental fusion can be shown to be optimal using a similar argument.

Example: falling body. To demonstrate the effectiveness of the Kalman filter, we consider an example in which an object falls from the origin at time t=0 with an initial speed of 0 m/s and an expected constant acceleration of 9. 8 m/s2 due to gravity. Note that acceleration in reality may not be constant due to factors such as wind, and air friction.

The state vector of the object con-
tains two components, one for the
distance from the origin s(t) and one
for the velocity v(t). We assume that
only the velocity state can be mea-
sured at each time step. If time is dis-
cretized in steps of 0.25 seconds, the
difference equation for the dynamics
of the system is easily shown to be
the following:
where we assume and
The gray lines in Figure 9 show
the evolution of velocity and distance
with time according to this model.
Because of uncertainty in modeling
the system dynamics, the actual evo-
lution of the velocity and position
will be different in practice. The red
lines in Figure 9 show one trajectory
for this evolution, corresponding to
a Gaussian noise term with covari-
ance in Equation 32 (because
this noise term is random, there are
many trajectories for the evolution,
and we are just showing one of them).

We have shown
that Kalman
filtering for state
estimation in linear
systems can be
derived from two
elementary ideas:
optimal linear
estimators for
fusing uncorrelated
estimates and best
linear unbiased
estimators
for correlated
variables.