in general we want our lives to be
linear as shown on this graph this might
be in terms of success income or
happiness but in reality life is not
linear it is full of up and downs and
sometimes it gets even more complicated
if you're an engineer you will often
need to deal with nonlinear systems to
help you we're going to discuss
nonlinear state estimators previously we
use a simplified linear car model to
discuss state estimation through Kalman
filters however if the system is modeled
such that it takes into account
nonlinearities due to rod friction then
the state transition function becomes
nonlinear hear the noise enters the
system linearly but there may be systems
where the noise is not additive in a
general system either the state
transition function or the measurement
function or both may be nonlinear for
all these cases we need to use a
nonlinear state estimator instead of a
common filter as common filters are only
defined for linear systems here's an
example that shows the problem with
using a common filter for state
estimation of a nonlinear system the
common filter assumes a Gaussian
distribution if the state transition
function is linear then after undergoing
the linear transformation the
distribution maintains its Gaussian
property although it's not shown here
the same is true for the measurement
function G of X however if f of X is
nonlinear then the resulting state
distribution may not be Gaussian and
therefore the common filter algorithm
may not converge in this case you can
implement an extended Kalman filter
which linearizes the nonlinear function
around
of the current state estimates at each
time step the linearization is performed
locally and the resulting Jacobian
matrices are then used in the prediction
and update states of the Kalman filter
algorithm when the system is nonlinear
and can be well approximated by
linearization then extended Kalman
filter is a good option for state
estimation however it has a following
drawbacks it may be difficult to
calculate the Jacobins analytically due
to complicated derivatives there might
be a high computational cost to
calculating them numerically you cannot
apply an extended Kalman filter to
systems with a discontinuous model since
the system is not differentiable and the
jacobins wouldn't exist and
linearization doesn't provide a good
approximation for highly nonlinear
systems in the last case linearization
becomes invalid since the nonlinear
function cannot be approximated well
enough by a linear function and doesn't
describe system dynamics to address the
issues with extended Kalman filters you
can instead use another estimation
technique called the unscented Kalman
filter did he know that the creator of
the filter came up with this name after
noticing it deodorant on his co-workers
desk now back to the filter instead of
approximating a nonlinear function as an
extended Kalman filter does unscented
Kalman filters approximate the
probability distribution what we mean by
that is the following this is a
probability distribution an unscented
Kalman filter selects a minimal set of
sample points such that their mean and
covariance is the same as this
distribution these are referred as Sigma
points and are symmetrically distributed
around me
each Sigma point is then propagated
through the nonlinear system model the
mean and covariance of the non linearly
transform points are calculated and an
empirical Gaussian distribution is
computed which is then used to calculate
the new state estimate note that in the
linear camera filter algorithm the error
covariance P is calculated using the
state transition function in the
prediction step and then it is updated
using the measurement however in the
unscented Kalman filter we don't
calculate it in the same way because we
get it in prickley instead another
nonlinear state estimator based on a
very similar principle is the particle
filter it also uses sampled points
referred as particles the significant
difference from an unscented Kalman
filter is that a particle filter
approximates any arbitrary distribution
so it's not limited to a Gaussian
assumption and to represent an arbitrary
distribution that is not known
explicitly the number of particles that
a particle filter needs is much larger
than you'd need for an unscented Kalman
filter for comparison here are the
properties of the filters that we'll
discuss so far a common filter only
works on linear systems for state
estimation of nonlinear systems you can
use an e KF UKF or PF note that for an e
KF to precisely estimate States it needs
good linearization of the nonlinear
system model otherwise the filter
provides poor estimation a particle
filter is the only one that works for
any arbitrary distribution and we see
that the computational cost grows as we
move down the column particle filter is
protection only the most expensive
filter since it requires a large number
of particles to approximate an arbitrary
distribution in this video we discuss
the basic concepts behind different
nonlinear state estimators now if you
need to deal with any nonlinearities
such as the rod friction in the car
example you don't have to estimate
states of interest of your nonlinear
system for more information on e KS UK
snps explore the resources in the
descriptions of this video