# Nonlinear Estimation

## 2. Literature review on quadrature based nonlinear estimation

Rahul Radhakrishnan, Abhinoy Kumar Singh, and Shovan Bhaumik
### 2.1 Introduction

Non linear filtering problem consists of estimating the states of a stochastic system
using noisy measurements. This problem has significant applications in radar
tracking, navigational and guidance systems, sonar ranging and satellite orbit determination.
Most widely used filtering technique for non linear problem was Extended
Kalman Filter (EKF). However, when applied on highly non linear filtering
problems, solution of filter tends to diverge. To tackle this, many variants of Extended
Kalman Filter was introduced. Later various filtering techniques like Central
Difference Filter (CDF), Unscented Kalman Filter (UKF), Cubature Kalman
Filter (CKF), Cubature Quadrature Kalman Filter (CQKF), Gauss-Hermite Filter
(GHF), Sparse Grid Gauss-Hermite Filter (SGHF) and Multiple Square Root
Quadrature Kalman Filter (MSQKF) were introduced. All these filters were developed
under the Bayesian framework and gives suboptimal solution because of the
inability to compute the Bayesian recursion. Solution of integrals which becomes
intractable for non linear systems makes the solution of these filters suboptimal.

### 2.2 Unscented Kalman filter

Unscented Kalman Filter (UKF) is a filtering technique used for dynamic statespace
models, based on the intuition that it is easier to approximate a probability
distribution than to approximate an arbitrary nonlinear transformation. This filter
uses deterministic sampling approach, called unscented transformation, to represent
a random variable using a number of deterministically selected sample points
called sigma points

[Anoniou 2007].
These sample points capture the mean and covariance of
the random variable and it captures the posterior mean and covariance accurately
to the second order, when propagated through the nonlinear system. Since the
filter only propagates the mean and variance of the distribution, information loss
can occur when applied to non-Guassian distributions.

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