**Nonlinear State and Parameter Estimation Using Iterated Sigma Point Kalman Filter: Comparative Studies Nonlinear State and Parameter Estimation Using Iterated Sigma Point Kalman Filter: Comparative Studies**

Marwa Chaabane, Imen Baklouti, Majdi Mansouri, Nouha Jaoua, Hazem Nounou, Mohamed Nounou, Ahmed Ben Hamida and Marie‐France Destain Nouha Jaoua, Hazem Nounou, Mohamed Nounou, Ahmed Ben Hamida and Marie‐France Destain Additional information is available at the end of the chapter

Marwa Chaabane, Imen Baklouti, Majdi Mansouri,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/63728

#### **Abstract**

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116 Nonlinear Systems - Design, Analysis, Estimation and Control

In this chapter, iterated sigma‐point Kalman filter (ISPKF) methods are used for nonlinear state variable and model parameter estimation. Different conventional state estimation methods, namely the unscented Kalman filter (UKF), the central difference Kalman filter (CDKF), the square‐root unscented Kalman filter (SRUKF), the square‐ root central difference Kalman filter (SRCDKF), the iterated unscented Kalman filter (IUKF), the iterated central difference Kalman filter (ICDKF), the iterated square‐root unscented Kalman filter (ISRUKF) and the iterated square‐root central difference Kalman filter (ISRCDKF) are evaluated through a simulation example with two comparative studies in terms of state accuracies, estimation errors and convergence. The state variables are estimated in the first comparative study, from noisy measure‐ ments with the several estimation methods. Then, in the next comparative study, both of states and parameters are estimated, and are compared by calculating the estimation root mean square error (RMSE) with the noise‐free data. The impacts of the practical challenges (measurement noise and number of estimated states/ parameters) on the performances of the estimation techniques are investigated. The results of both comparative studies reveal that the ISRCDKF method provides better estimation accuracy than the IUKF, ICDKF and ISRUKF. Also the previous methods provide better accuracy than the UKF, CDKF, SRUKF and SRCDKF techniques. The ISRCDKF method provides accuracy over the other different estimation techniques; by iterating maximum a posteriori estimate around the updated state, it re‐linearizes the measurement equation instead of depending on the predicted state. The results also represent that estimating more parameters impacts the estimation accuracy as well as the convergence of the estimated parameters and states. The ISRCDKF provides improved state accuracies than the other techniques even with abrupt changes in estimated states.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Keywords:** Kalman filter, sigma point, state estimation, parameter estimation, nonlin‐ ear system

## **1. Introduction**

Dynamic state‐space models [1–3] are useful for describing data in many different areas, such as engineering [4–8], biological data [9, 10], chemical data [11, 12], and environmental data [8, 13–15]. Estimation of the state and model parameters based on measurements from the observation process is an essential task when analyzing data by state‐space models. Bayesian estimation filtering represents a solution of considerable importance for this type of problem definition as demonstrated by many existing algorithms based on the Bayesian filtering [16– 25]. The Kalman filter(KF) [26–29] has been extensively utilized in several science applications, such as control, machine learning and neuroscience. The KF provides an optimum solution [28], when the model describing the system is supposed to be Gaussian and linear. However, the KF is limited when the model is considered to be nonlinear and present non‐Gaussian modeling assumptions. In order to relax these assumptions, the extended Kalman filter (EKF) [26, 27, 30–32], the unscented Kalman filter (UKF) [33–36], the central difference Kalman filter (CDKF) [37, 38], the square‐root unscented Kalman filter (SRUKF) [39, 40], the square‐root central difference Kalman filter (SRCDKF) [41], the iterated unscented Kalman filter (IUKF) [42, 43], the iterated central difference Kalman filter (ICDKF) [44, 45], the iterated square‐root unscented Kalman filter (ISRUKF) [46] and the iterated square‐root central difference Kalman filter (ISRCDKF) [47] have been developed. The EKF [26] linearizes the model describing the system to approximate the covariance matrix of the state vector. However, the EKF is not always performing especially for highly nonlinear or complex models. On behalf of linearizing the model, a class of filters called the sigma‐point Kalman filters (SPKFs) [48] uses a statistical linearization technique which linearizes a nonlinear function of a random variable via a linear regression. This regression is done between *n* points drawn from the prior distribution of the random variable, and the nonlinear functional evaluations of those points. The sigma‐point family of filters has been proposed to address the issues of the EKF by making use of a deterministic sampling approach. In this approach, the state distribution is approximated and represented by a set of chosen weighted sample points which capture the true mean and covariance of the state vector. These points are propagated through the true nonlinear system and capture the posterior mean and the covariance matrix of the state vector accurately to the third order (Taylor series expansion) for any nonlinearity. As part of the SPKF family, the UKF [26, 27, 33] has been developed. It uses the unscented transformation, in which a set of samples (sigma points) are propagated and selected by the nonlinear model, providing more accurate approximations of the covariance matrix and mean of the state vector. However, the UKF technique has the limit of the number of sigma‐points which are not so large and cannot represent complicated distributions. Another filter in the SPKF family is the central difference Kalman filter (CDKF) [37, 38]. It uses the Stirling polynomial interpolation formula. This filter has the benefit over the UKF in using only one parameter when generating the sigma‐point. To add some benefits of numerical stability, the SRUKF and the SRCDKF [41] have been developed. The advantage of these filters is that they ensured positive semidefiniteness of the state covariances. The iterated sigma‐point Kalman filter (ISPKF) methods employ an iterative procedure within a single measurement update step by resampling the sigma‐point till a termination criterion, based on the minimization of the maximum likelihood estimate, is satisfied.

The objectives of this chapter are threefold: (i) To estimate nonlinear state variables and model parameters using SPKF methods and extensions through a simulation example. (ii) To investigate the effects of practical challenges (such as measurement noise and number of estimated states/parameters) on the performances of the techniques. To study the effect of measurement noise on the estimation performances, several measurement noise levels will be considered. Then, the estimation performances of the techniques will be evaluated for different noise levels. Also, to study the effect of the number of estimated states/parameters on the estimation performances of all the techniques, the estimation performance will be studied for different numbers of estimated states and parameters. (iii) To apply the techniques to estimate the state variables as well as the model parameters of second‐order LTI system. The perform‐ ances of the estimation techniques will be compared to each other by computing the execution times as well as the estimation root mean square error (RMSE) with respect to the noise‐free data.
