Adaptive Control of Linear Hyperbolic PDEs
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Coupled first-order linear hyperbolic partial differential equations (PDEs) can be used to model flow and transport phenomena. Relevant physical systems that can be modeled as hyperbolic PDEs are, to mention a few, transmission lines, timedelays, traffic flow and multiphase flow. Linear hyperbolic PDEs have therefore been subject to extensive research, and a number of techniques and feedback laws have been proposed in the literature. In the last one and a half decade, the use of infinite-dimensional backstepping has proven itself to be a powerful tool in the design of controllers and observers for linear PDEs. Starting in the early 2000s with non-adaptive stabilization of the heat equation, the backstepping method quickly found its application in adaptive control problems for parabolic PDEs. Several results on adaptive control of more general parabolic PDEs using the backstepping method followed in the later years, and even a book was published on the topic. Backstepping was applied for controller design of linear hyperbolic PDEs for the first time in 2008, and it was extended to more and more complicated systems of coupled linear hyperbolic PDEs in milestone papers published in 2011, 2013 and 2016, respectively. However, the first use of backstepping for adaptive stabilization of a linear hyperbolic PDE was as late as in 2014, and the amount of material on adaptive control of linear hyperbolic PDEs was therefore very limited at the time the work underlying this thesis started. The overall goal of this thesis is to extend the comprehensive work on adaptive control of parabolic PDEs to hyperbolic PDEs, and derive adaptive observers and controllers for systems of coupled linear hyperbolic PDEs with uncertain parameters at both the boundaries and in the interior domain. Additionally, some selected non-adaptive control and observer problems are considered. These are within optimality, output tracking and disturbance rejection. This thesis is a collection of papers that is categorized into six topic-based chapters. Chapter 2 contains all non-adaptive control and observer results of the thesis. Chapter 3 contains results on estimation of parameters, leaving out any control results. Adaptive state feedback controllers are considered in Chapter 4 for systems with uncertain in-domain parameters. Adaptive output-feedback controllers are then in Chapter 5 presented for systems with uncertain boundary conditions. Adaptive output-feedback controllers are derived in Chapter 6 for systems with uncertain in-domain parameters, and the closely related problem of model reference adaptive control is jointly considered. Finally, Chapter 7 contains a paper that discusses how to establish convergence in adaptive systems, and offers a new convergence result for this matter.