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# 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.