Browsing by Subject "Nonlinear estimation"

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    Improving the consistency of nonlinear estimators: analysis, algorithms, and applications
    (2013-02) Huang, Guoquan
    Autonomous robots are emerging as candidates for performing increasingly complex tasks, such as surveillance and environment monitoring, search and rescue, and planetary exploration. Nonlinear estimation (i.e., estimating the state of a nonlinear system from noisy measurements) arises in all these applications. For instance, robot localization - which is considered as one of the fundamental problems in robotics - seeks to determine the robot's pose (position and orientation) using measurements from onboard sensors (e.g., an odometer and a camera). Another closely-related and important example is target tracking, where the objective is to estimate the target's state using remote sensor observations. Even though many different algorithms, such as the extended Kalman filter (EKF) and the batch maximum a posteriori (MAP) estimator, have been developed for solving these problems, substantial empirical evidence shows that most existing nonlinear estimators tend to become inconsistent (i.e., the state estimates are biased and the error covariance estimates are smaller than the true ones). Moreover, a significant limitation is that the causes of inconsistency have not been sufficiently studied in the literature; if an estimator is inconsistent, the accuracy of its estimates is unknown, which makes the estimator unreliable. The objective of this dissertation is to investigate the main causes of inconsistency of nonlinear estimation and develop new algorithms for improving consistency. As one of the main research thrusts, we study in depth the inconsistency problem in robot localization, including simultaneous localization and mapping (SLAM) and multi-robot cooperative localization (CL). In particular, we show for the first time ever that one fundamental cause of inconsistency is the mismatch between the observability properties of the underlying nonlinear system and the linearized system used by the estimator. By performing observability analysis, we prove that the linearized error-state system used by standard filtering/smoothing algorithms - the EKF, the unscented Kalman filter (UKF), and the sliding-window filter (SWF) - has an observable subspace of higher dimension than that of the underlying nonlinear system. This implies that these estimators gain spurious information (more specifically, about the global orientation) from the measurements, which unjustifiably reduces the uncertainty of the state estimates and causes inconsistency. Based on this key insight, for unobservable nonlinear systems, we propose a novel methodology for designing consistent linearized estimators. Specifically, we develop a family of Observability-Constrained (OC)-estimators - including the OC-EKF, the OC-UKF, and the OC-SWF - whose Jacobians are computed in a way to ensure that the estimator's linearized system model has an observable subspace of the same dimension as that of the underlying nonlinear system. Furthermore, we investigate the inconsistency of estimators for observable nonlinear systems, such as target tracking using distance or bearing measurements, whose cost functions are non-convex and often have multiple local minima. In such cases, we discover that the inconsistency of a standard linearized estimator, such as the EKF, is primarily due to the fact that the estimator is able to find and track only one local minimum. To address this issue, we convert the estimator's nonlinear cost function into polynomial form and employ algebraic geometry techniques to analytically compute all its local minima. These local minima are used as initial estimates by a bank of MAP estimators to efficiently track the most probable hypotheses for the entire state trajectory. Moreover, we adapt this idea to particle filters (PFs) and develop an Analytically-Guided-Sampling (AGS)-PF. Specifically, the AGS-PF employs an analytically-determined Gaussian mixture as proposal distribution which not only takes into account the most recent measurement but also matches all the modes of the posterior (optimal proposal) distribution. As a result, the AGS-PF samples the most probable regions of the state space and hence significantly reduces the number of particles required. As precise long-term localization and tracking are essential for a variety of robotic applications, by introducing a solid theoretical framework for improving the consistency of nonlinear estimators, this work offers significant benefits for robots employed in these tasks. Moreover, the proposed solutions constitute novel paradigms for engineers to follow when designing consistent estimators for other nonlinear systems, and hence have the potential to benefit applications beyond robotics.
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    Observer Design for Non-Monotonic Nonlinear Systems and Interesting Contemporary Applications
    (2022-08) Movahedi, Hamidreza
    This dissertation analyzes observer design for non-monotonic nonlinear systems and develops globally stable observer design techniques for such systems. Non-monotonic nonlinear systems are frequently encountered in many practical applications, including vehicle tracking, magnetic position estimation, robotics, state of charge (SoC) estimation in Li-ion batteries, and infectious disease spread dynamics. Very few papers in literature have recognized the challenge that existing nonlinear observer design methods do not work for non-monotonic systems. This dissertation demonstrates that current LMI-based observer design methods do not have feasible solutions for many non-monotonic systems. This motivates the need for new observer design techniques. Such techniques are used in three major applications in this thesis: state of charge estimation in Li-ion batteries, magnetic position estimations, and infectious disease spread dynamics estimation.First, a class of systems in which the process dynamics and output equations contain nonlinear functions of only scalar arguments are considered. A Lyapunov approach is utilized to develop an LMI-based observer design method for this class of nonlinear systems. Then, the failure of LMI-based methods to provide constant observer gains for non-monotonic systems is rigorously analyzed, and it is demonstrated that, no matter how small the Lipschitz constant or the Jacobian bounds of the involved nonlinear functions, these methods cannot provide a stabilizing constant observer gain if all the functions of the system are non-monotonic. Based on this theoretical result, a technique to extend the design method to include switched gain observers is presented and its global asymptotic stability is rigorously proven. The developed observer design methodology is utilized to estimate the SoC in a lithium-ion battery, using measurements of terminal voltage and bulk force. The challenge in this application is that the bulk force applied to the casing of the battery, as a result of Lithium-ion intercalation and deintercalation, is a non-monotonic function of the SoC. Hence, a switched gain observer is devised and applied. Using detailed simulations of possible mismatches in the battery model, the robustness of the observer is compared with that of the extended Kalman filter, and the observer is showed to be less susceptible to these model errors. Experimental results corroborate this finding. Position estimation in electro-hydraulic actuators using non-contacting magnetic sensors is another subject that is considered. Magnetic measurements in this application are all non-monotonic, hence based on the theoretical findings of this dissertation it is clear that more than one magnetic sensor is needed for this estimation problem. Subsequently, the minimum singular values of the observability matrix are utilized as a metric for minimizing the number of sensors and optimizing sensor locations. Extensive experimental results are provided to demonstrate the optimality of the sensor locations and the accuracy of the switched gain observer designed for this application. Next, the hysteresis in Li-ion batteries is analyzed. A nonlinear double capacitor model is used for this problem that contains a measurement equation with two nonlinear functions, one of them being significant hysteresis in voltage of the battery as a function of the SoC. Previously, researchers in this field used a differential equation to model the hysteresis. In this dissertation, it is shown that this popular method loses observability and a modified Preisach method is suggested as an alternative. Then a nonlinear Lipschitz observer is designed for this application, and it is shown that the observer provides accurate SoC estimates based on experimental data in the presence of hysteresis. Another research application explored in this dissertation is centered around infectious disease spread dynamics and the real-time estimation problem of variables characterizing disease spread. The COVID-19 epidemic is studied for this purpose, and a new nonlinear dynamic model is developed to enhance the traditional SEIR epidemic model to include additional variables. Subsequently, a cascaded observer is developed to estimate the real-time values of the infection rate, and the basic reproduction number of COVID-19 spread in Minnesota. Finally, the use of the nonlinear observer design techniques for handling sensor noise and disturbance rejection is considered. A H∞-based globally stable nonlinear observer design technique was explored to provide design flexibility equivalent to the popular locally stable extended Kalman filter which is based on linearization of plant dynamics. Furthermore, comparing with the performance of the extended and unscented Kalman filters in the presence of non-Gaussian pulse disturbances and through detailed simulations, it is shown that the H∞ nonlinear observer performs better and can provide a guaranteed upper bound on the estimation error.