Observer Design for Non-Monotonic Nonlinear Systems and Interesting Contemporary Applications

Abstract

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.