The dissertation introduces advancements in the theory and the applications of state estimation for systems described by partial differential equations (PDEs) with boundary measurements. State estimation is of great importance to engineering applications since typically full-state knowledge is required for control and decision making but only a limited number of states are directly available from measurements.

The estimation problems considered in the dissertation are solved via boundary observers, derived from the backstepping method for PDEs,

which is a constructive method for boundary control and estimation. The dissertation is confined to PDEs of parabolic type, including: reaction-diffusion equations, radial reaction-diffusion equations, systems of coupled reaction-diffusion equations, and systems including ODEs coupled with diffusion equations. Some contributions include: a robustness study for boundary observers with respect to uncertainty in parameters and measurement disturbances in the sense of input-to-state stability, the design of observers for coupled diffusion-reaction equations with an uncoupled target system, the design of observers for the expected value of a randomly switching diffusion-reaction PDE, and the design of observers for coupled ODEs with a radial diffusion equations.

The dissertation is motivated by the state estimation problem from electrochemical models of lithium-ion batteries. Lithium-ion technology is widely use in portable electronics, electrified vehicles, aerospace, medical devices, among many other industries. The safe and optimal operation of a battery relies on the battery management system which requires accurate and reliable estimates of internal states of the battery. Electrochemical models are first principle models that provide a detailed description for the time evolution of the battery states. Among electrochemical models, the single particle model (SPM) is a simple electrochemical model suitable for the design of control and estimation algorithms. The dissertation introduces state estimation methodologies from the SPM adapted for lithium-ion batteries with electrodes of multiple active material, electrodes with non-spherical particle geometries, temperature dependent parameters, state dependent parameters and materials with phase transitions. The results in the dissertation are not restricted to battery applications, in fact, the last chapter studies the state estimation problem for a wellbore reservoir model used in managed pressure drilling applications.