On the Accuracy of State Estimators for Constant and Time-Varying Parameter Estimation Yousaf Rahman * , Jiapeng Zhong † , Alexey Morozov ‡ , and Dennis S. Bernstein § University of Michigan, 1320 Beal Ave., Ann Arbor, MI 48109 Nonlinear estimation techniques are often used to estimate constant and time-varying parameters. The purpose of this paper is to use illustrative examples to compare the accu- racy of several estimation techniques (the extended Kalman filter, the unscented Kalman filter, and the ensemble adjustment Kalman filter) along with retrospective cost model refinement. Both constant and time-varying examples are considered. Each algorithm is tuned to illustrate its capabilities for the given examples. I. Introduction In many modeling and control applications, the structure of the model is known, but the parameters may be uncertain. Within the context of system identification, models of this type are called white box models. In contrast, models whose structure is either partially or fully unknown are called grey-box and black-box models, respectively. Parameter-estimation is related to, but distinct from, state estimation, where states evolve due to external inputs and their interaction with other states. In contrast, an unknown parameter may either be constant or time-varying in a pre-specified manner that is independent of initial conditions and outputs. Although a constant or time-varying parameter is not technically a state, it can be modeled as a state by assigning it fictitious dynamics and stochastic forcing. In continuous time, these dynamics are ˙ x = w, whereas, in discrete time, these dynamics are x k+1 = x k + w k , where w is the external forcing. For a system with linear dynamics, the resulting state estimation problem is nonlinear due to the multiplication between “real” and “fictitious” states. State-estimation techniques are widely used for parameter estimation [1–3]. Among the earliest works is the classic paper [4], which analyzes the accuracy of the extended Kalman filter within the context of linear dynamics. Convergence analysis of the extended Kalman filter is provided in [5]. Beyond the extended Kalman filter, nonlinear estimation techniques have been developed based on a wide variety of techniques, including stochastic ensembles [6–8], deterministic ensembles [9,10], Gaussian mixtures [11], density estimators [12], Fokker-Planck solutions [13], moving horizon techniques [14], and adaptive estimators [15,16]. Each of these techniques can potentially be applied to parameter estimation. * Graduate Student, Aerospace Engineering Department, University of Michigan, Ann Arbor † Graduate Student, Control Science and Engineering Department, Harbin Institute of Technology, Harbin, China ‡ Graduate Student, Aerospace Engineering Department, University of Michigan, Ann Arbor § Professor, Aerospace Engineering Department, University of Michigan, Ann Arbor 1 of 15 American Institute of Aeronautics and Astronautics Downloaded by UNIVERSITY OF MICHIGAN on April 2, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2013-5192 AIAA Guidance, Navigation, and Control (GNC) Conference August 19-22, 2013, Boston, MA AIAA 2013-5192 Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Guidance, Navigation, and Control and Co-located Conferences