I J E E C E International Journal of Electrical, Electronics ISSN No. (Online) : 2277-2626 and Computer Engineering 2(2): 7-12(2013) Special Edition for Best Papers of Michael Faraday IET India Summit-2013, MFIIS-13 Joint Estimation of Parameters and States of Nonlinear Systems using Adaptive Divided Difference Filter Aritro Dey * , Smita Sadhu * and Tapan Kumar Ghoshal * *Department of Electrical Engineering, Jadavpur University, Kolkata, (WB) India (Received 15 October, 2013 Accepted 01 December, 2013) ABSTRACT: An adaptive Divided Difference filter for joint estimation of parameters and states of a nonlinear system has been proposed in this work. The adaptive filter is proposed for improved estimation specifically in the situation when knowledge about the process noise statistics is unavailable. The innovation sequence has been employed for adaptation of the unknown process noise covariance. The evolved Adaptive Divided Difference filter has been evaluated with a benchmark nonlinear problem of ballistic object tracking. With the help of simulation, it has been demonstrated that even though the process noise covariance is unknown, the performance of proposed filter is superior compared to a non adaptive Divided Difference filter. Index Terms: Adaptive filtering, Divided Difference Filter, Innovation, Parameter estimation, State estimation, Ballistic object tracking I. INTRODUCTION This paper addresses the problem of joint estimation of parameter and states of a nonlinear system where accurate knowledge of the process noise statistics is unavailable. Process noise represents the modeling inaccuracy of system dynamics [1] and its covariance is often unknown. Inappropriate assumption of process noise can degrade the estimation accuracy of the filter and sometimes may cause divergence even for linear signal models [2] − [4 ]. In case of joint estimation of parameter and state, successful estimation of unknown parameter depends on discerning choice of noise covariance on the basis of trial and error method. The adaptive nonlinear filter has been proposed in this work which can replace this elaborate experimentation by online adaptation of the unknown noise covariance. Nonlinear estimation problem has been focused in this work. For parameter estimation problems, even for the linear systems the usual procedure is to model the unknown parameter as a state and to augment with the state vector resulting in the process dynamics to be nonlinear. Among the existing filters for nonlinear state estimation Extended Kalman Filter (EKF) [1, 5] often provides satisfactory estimation performance for mildly nonlinear problems. Adaptive forms of EKF [6] are fairly easy to implement as the structure of adaptive Kalman filter may be directly employed. However, the shortcomings of EKF are well known [5]-[7] and it may even face divergence problem specifically when the system dynamics has strong nonlinearities. To overcome the limitation of EKF, a large numbers of filters have been reported in literatures which can take care of the nonlinearities avoiding derivative calculation [7] − [9]. A subset of derivative free nonlinear filters is recognized as sigma point filters. One of the sigma point filter termed as Central Difference filter (CDF) based on interpolation method was first proposed in [9]. The authors of [10-11] later modified the method of [9] employing Stirling’s interpolation formula and named as Divided Difference Filter (DDF). In [12] performance accuracy of DDF is compared with Unscented Kalman Filter (UKF), Square Root UKF and observed that the performance of DDF is equivalent with the well established sigma point filters. However, formulation of adaptive DDF is not widely reported in literature apart from a few exceptions such as [13], [14], [15]. The focus of [13] is on robustness rather than adaptation. The adaptive DDF proposed in [14] used first order DDF framework and in [15] a robust adaptive divided difference filter has been proposed which emphasizes on robust estimation in presence of noise distributions with thicker tails. In these work the Q adaptation method may suffer from non positive definiteness of adapted Q as per the formula of adaptation. In this paper Q adaptation rule has been characterized differently inspired by [Mohamed 1999] so that positive definiteness of Q is guaranteed. There are a number of approaches for filter adaptation. An earlier work of [18] reported a number of methods of adaptation for linear estimation problem in Kalman filter frame work. In [6] the adaptive EKF is applied for a navigation problem which follows online tuning of noise covariance based on ML estimation. Similar methods have also been recognized from the recent trend of adaptive filters. Several methods of adaptive filtering proposed in [3] include weighted Kalman filter, scaling of noise covariance, innovation based adjustment of noise covariance and combination of last two methods. The adaptive filter proposed in [4] has been termed as Innovation based adaptive estimation (IAE) which is contrast with Multiple Model Adaptive Estimation (MMAE) method. Innovation based method of adaptive filtering has been focused in the present work. Innovation and residual based adaptive Kalman filters have been demonstrated for linear plants in [2]−[4], where the unknown process and (or) measurement noise covariance are adjusted using innovation or residual sequence.