American Institute of Aeronautics and Astronautics 1 Relaxation of Stability Requirements for Extended Kalman Filter Stability within GPS/INS Attitude Estimation Matthew Rhudy * , Yu Gu † , and Marcello Napolitano ‡ Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV, 26506 The Extended Kalman Filter (EKF) is a widely used estimator for nonlinear systems. The stability characteristics of the EKF have been examined, but have not been rigorously proven for realistic assumptions on the initial error and noise disturbances. The existing stability analysis methods were first implemented in order to obtain a baseline calculation of the requirements on the system’s initial error and noise disturbances. Since these requirements were determined to be too strict for realistic application, modifications were applied to the stability analysis in order to relax these assumptions and prove the stability with more realistic assumptions. Significant improvements in the initial error and noise bounds were achieved. Using a case study of low-cost attitude estimation, experimental flight data was utilized to reinforce and demonstrate the theoretically derived results. While the stability characteristics were examined in the context of a specific problem, the derived methods can be applied to any EKF application. I. Introduction tate estimators are a commonly used tool for state space systems in order to estimate states that cannot be directly measured, which is useful for applications such as full state feedback control 1 . For control applications that use the estimated states for feedback, it is important to ensure that the estimation algorithm is stable. The stability of a state estimator is defined in terms of the convergence of the state estimate to the true state, or, equivalently, the state estimate error converges to zero. A commonly used state estimator for linear systems is the linear Kalman filter 2 , which has been proven to have exponentially stable state estimation error by various authors 3-7 . However, state estimators for nonlinear systems introduce additional difficulties, and therefore the stability is not as clearly defined. Using the linear Kalman filter stability work as a basis, various work has been done on the topic of Extended Kalman Filter (EKF) 8 stability. Ljung analyzed, using techniques derived in 9 , the asymptotic behavior of the EKF for continuous-time parameter identification of a linear system 10 . Baras et al. presented a method to derive dynamic observers as asymptotic limits of recursive filters for both linear and nonlinear systems with no inputs and linear observations in continuous time 11 . Song and Grizzle provided a proof that the Kalman filter is a global observer for discrete-time linear time-varying systems, and expanded this result to show that the EKF is a quasi-local asymptotic observer 12 for discrete-time nonlinear systems with no inputs 13 . La Scala et al. expanded upon the work of Song and Grizzle 13 , giving sufficient conditions for stability of the discrete-time EKF for a nonlinear system without inputs and with linear observation equations. The frequency tracking problem was used as an example to demonstrate the bounds on the tracking error 14 . Boutayeb et al. presented a convergence analysis of the EKF for deterministic discrete-time nonlinear systems with inputs. This paper considered the deterministic case, i.e., no process or measurement noise, and therefore presented results in terms of an arbitrary measurement noise matrix, and introduced two additional matrices that are used to control the stability and convergence of the EKF 15 . In another work, Boutayeb and Aubry analyze the stability of a strong tracking Extended Kalman Observer (EKO) 16 . Xiong et al. presented a stability analysis of the Unscented Kalman Filter (UKF) 17 , which was later pointed out by Wu et al. 18 to apply to a more general set of filters, including the EKF. Some influential work on EKF stability was done in the late 1990’s by Konr ad Reif with various co-authors. First, Reif et al. proposed a modification to the continuous-time EKF that introduced an additive term of instability * Ph.D. Candidate, Mechanical and Aerospace Engineering (MAE) Department, PO Box 6106, matthew.rhudy@gmail.com, now a Visiting Assistant Professor at Lafayette College, Easton, PA 18042. † Assistant Professor, MAE Department, PO Box 6106, yu.gu@mail.wvu.edu, AIAA Senior Member. ‡ Professor, MAE Department, PO Box 6106, AIAA Senior Member. S Downloaded by Yu Gu on March 14, 2014 | http://arc.aiaa.org | DOI: 10.2514/6.2014-0446 AIAA Guidance, Navigation, and Control Conference 13-17 January 2014, National Harbor, Maryland AIAA 2014-0446 Copyright © 2014 by Matthew Rhudy, Yu Gu , Marcello Napolitano. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. AIAA SciTech