Once you have mastered the multi-dimensional (or multivariate) Kalman Filter, you are ready to tackle the Non-linear Kalman Filters.

The Kalman Filter solves the estimation problem for linear systems. However, most real-life systems are non-linear.

For non-linear systems handling, the Linear Approximation techniques shall be applied. This part describes two common modifications of the Kalman Filter that perform Linear Approximation:

- Extended Kalman Filter (EKF)
- Unscented Kalman Filter (UKF) or Sigma-point Kalman Filter (SPKF)

The EKF performs analytic linearization of the model at each point in time. EKF is the most common non-linear Kalman Filter.

While the standard Linear Kalman Filter (LKF) is an optimal filter since we minimize the estimate uncertainty (see "Kalman Gain Equation Derivation" section), all Kalman Filter modifications for non-linear systems are sub-optimal since we use approximated models.

This chapter describes the EKF and the UKF methods. We discuss the advantages and disadvantages of each method. Furthermore, each method is exemplified by numerical examples.

Before diving into the problem solution, we must understand the problem itself. What are the non-linear systems, and why does the standard Linear Kalman Filter fail with non-linear systems?

We distinguish between two types of non-linearities:

- State-to-measurement non-linear relation
- Non-linear system dynamics

We will treat each type of non-linearity separately and then combine them.

Assume an air balloon that can move only upwards or downwards with constant acceleration dynamics. We are interested in estimating the balloon altitude.

The balloon system dynamic model is linear.

We can describe the balloon system dynamics as follows: