טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentSigalov Daniel
SubjectState Estimation in Linear Systems with Random Parameters
DepartmentDepartment of Applied Mathematics
Supervisor Professor Yaakov Oshman


Abstract

    State estimation in dynamical systems is an area of active research that has been making significant progress since the celebrated work of Rudolf Kalman in the early 60s. Being optimal in the MMSE sense under the linear-Gaussian assumptions, the Kalman filter loses optimality when applied to the family of hybrid systems, where a discrete mode variable dictates the dynamical and measurement models active at each time. Given bounded computational resources, the MMSE estimator in a general hybrid system cannot be computed, and considerable effort has been dedicated to obtaining efficient suboptimal algorithms.


    Two such approaches may be considered: 1) seeking for linear MMSE algorithms, and 2) using adaptive methods that are based on reasonable, yet heuristic approximations of the MMSE estimator of the state. Each approach has advantages and disadvantages relative to the alternative. The theoretical rigor of LMMSE filters, and their robustness to noise distribution uncertainty, are counterbalanced by their typical relative inferior performance, stemming from the fact that they cannot "adapt to the data".


    The estimation problem is further complicated in scenarios entailing data of uncertain origin. These are common in tracking applications in the presence of spurious measurements. The optimal estimator of the target state requires exponentially growing resources, thus calling for the implementation of suboptimal techniques.


    In this work we address the problem of state estimation in systems with random coefficients and consider three research questions. First, the commonly considered modes are either deterministic, white, or Markov. Are there situations where none of these can model the problem appropriately and what algorithms can one use then? Second, the lack of LMMSE algorithms motivates the search for such filters for problems currently having only heuristic solutions. Finally, the existing variety of different algorithms for problems involving measurement uncertainty calls for an attempt to obtain a unified formulation and solution of various problems.


    In this research we aim at answering these questions. First, we describe a class of problems in which the modes cannot be modeled directly using the Markov assumption. Specifically, we consider cases where the natural mode variable may switch between its states a bounded number of times. The proposed solution rests on redefining the system mode using augmentation, which renders the augmented mode Markov, thus facilitating the use of standard estimation techniques.


    Second, we develop LMMSE methods for systems with random coefficients. We begin with a filtering problem where the mode affects the system's dynamics equation, and proceed with deriving the LMMSE estimator in jump linear systems augmented by two feedback terms. These are motivated by closed-loop control approaches, and by the fact that, typically, sensors use validation windows to discard unlikely returns. The approach is demonstrated in an example of tracking in clutter.


    Finally, we propose a unified framework for state estimation under data and model uncertainties. We show that a large variety of problems may be formulated using a single state-space dynamical system with random coefficients. Consequently, all may be solved using standard tools such as the IMM algorithm or related methods.