American Institute of Aeronautics and Astronautics 092407 1 A Perturbation Method for Estimation of Dynamic Systems Manoranjan Majji 1 , John L. Junkins 2 and James D. Turner 3 Texas A&M University, College Station, TX, 77843-3141. I. Introduction erturbation methods have constituted an important tool for analysis of Nonlinear systems. Solution to several classical problems in history was provided by these powerful methods and they have become an inseparable component of the modern university physics and engineering curriculum. Books by Nayfeh ([1], [2]), Bellman[3] among others, document in detail several techniques involved in the analysis of Nonlinear Dynamical systems using perturbation methods along with applications. Estimation theory deals with the problem of estimating the state of a dynamical system from sensor measurements, usually corrupted by noise. The famous paper by Kalman [4] which solved the estimation problem for a linear system with measurements corrupted with Gaussian noise, was quickly applied to nonlinear problems via a local linearization of the dynamic system and measurement equations arriving at what is today known as the Extended Kalman Filter (EKF) . Detailed derivations of these algorithms are presented in the text books, ([5], [6], [7]). Recently, due to advancement of computational capabilities, more stringent estimator design requirements are being placed on the engineers. This led to a path for engineers to investigate possible generalizations of extended filtering strategies to accommodate the assumptions taken by the classical Kalmanesque framework. Work of Julier et al., [8] motivated by Athans [9], developed the unscented Kalman filter that takes in to account the second order terms in the Extended Kalman filter can be considered a testimony of the resurgence of interest in high order methods. Several other methods quickly followed, including the so called particle methods[10]. Although particle filters are an important choice approach, in most physical systems, the continuity of solutions and associated properties of the models, makes analytic methods of estimation remain extremely attractive. Many problems with accurate models are candidates for use with analytic methods. In most engineering applications, the estimator design is fraught with the interaction of three most important factors (challenges). First is the nonlinearity of the dynamic system in question and the measurement model. Dynamic system nonlinearity, forces non – Gaussian error propagation through the dynamical system and causes deviation of the case from Extended Kalman framework. Measurement system nonlinearity plays an important part in the decision process to enable the determination of the “best” state estimate. The second factor influencing the importance of high order terms in the state estimation is the sparsity of measurements. Long propagation phases enable the nonlinearities in the dynamic system to “matter” and a linearized covariance propagation might underpredict otherwise. The third factor is the initial condition uncertainty and process noise. While large uncertainty in initial conditions amplifies the effect of high order terms, the increase in process noise statistics tends to wash - out the model errors and, if a appropriately large estimate of the second moments of the noise is used, to inflate the covariance to retain a near - Gaussian nature. The study of these three important issues in a pristine environment is facilitated by retaining high order terms in the Extended Kalman Filter and producing an analytic filtering framework out of the same. This paper presents an alternative path to the development of such an analytic filtering architecture. One of the most important advantages of the architecture developed herein is that it facilitates the development of a systematic tool to aid in the determination of appropriate filter order and process/measurement noise levels, depending on the interaction of the above issues on a case by case basis. A perturbation expansion approach is utilized in describing the departure motion dynamics from a nominal trajectory of the Nonlinear dynamical system of interest. To develop equations governing the statistics of the unforced departure motion, we follow a perturbation technique often used in solution of complex physics problems (Chapter 7 of [11] and [12] present distant versions of this method). The method is a close cousin of the theory of regular perturbations([1],[2] and [3]), with the exception that the parameter introduced is artificial and used for ordering purposes. The unforced departure motion dynamics is written as a series in the artificial ordering parameter λ as ( ) (0) (1) 2 (2) ( ) ( ) 0 M n n k k i i i i i i k x z z z z z δ λ λ λ λ =∞ = = + + + + + = ∑ (1) Now, differentiating this unforced departure motion dynamics, we have that 12 1 1 * , ... 1 ˆ ˆ ˆ (): ((, , ), ) ((, , )) ... ! p p p M k k k k k i i i ikk k k k p x t f xtt x x t f xtt x f x x p λ δ δ δ δ − = = + − = ∑ (2) Substituting the assumed form from equation (1) for the departure motion dynamics, we arrive at the following differential equations governing ( ) () k i z t as ( ) 1 1 1 1 1 2 1 2 1 1 1 2 1 2 1 2 1 2 3 1 2 3 (0) * (0) , (1) * (1) * (0) (0) , , (2) * (2) * (0) (1) (1) (0) * (0) (0) (0) , , , 1 2! 1 1 ... 2! 3! i ij j i ij j ijj j j i ij j ijj j j j j ijjj j j j z f z z f z f z z z f z f z z z z f z z z = = + = + + + (3) 1 Graduate Research Assistant, Aerospace Engineering Department, 3141 TAMU, Student Member, AIAA. Email: majji@tamu.edu 2 Regents Professor, Distinguished Professor of Aerospace Engineering, Holder of the Royce E. Wisenbaker Chair in Engineering, Aerospace Engineering Department, 3141 TAMU, Fellow, AIAA. 3 Research Professor, Director of Operations, Consortium for Autonomous Space Systems (CASS), Aerospace Engineering Department, 3141, TAMU, Associate Fellow, AIAA. P