Adaptive observer design for parabolic partial differential equations

• Main Author: Ascencio, Pedro
• Other Authors: Astolfi, Alessandro ; Parisini, Thomas
• Published: Imperial College London 2017
• Subjects: 515
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.721598

This thesis addresses the observer design problem, for a class of linear one-dimensional parabolic Partial Differential Equations, considering the simultaneous estimation of states and parameters from boundary measurements. The design is based on the Backstepping methodology for Partial Differential Equations and extends its central idea, the Volterra transformation, to compensate for the parameters uncertainties. The design steps seek to reject time-varying parameter uncertainties setting forth a type of differential boundary value problems (Kernel-PDE/ODEs) to accomplish its objective, the solution of which is computed at every fixed sampling time and constitutes the observer gains for states and parameters. The design does not include any pre-transformation to some canonical form and/or a finite-dimensional formulation, and performs a direct parameter estimation from the original model. The observer design problem considers two cases of parameter uncertainty, at the boundary: control gain coefficient, and in-domain: diffusivity and reactivity parameters, respectively. For a Luenberger-type observer structure, the problems associated to one and two points of measurement at the boundary are studied through the application of an intuitive modification of the Volterra-type and Fredholm-type transformations. The resulting Kernel-PDE/ODEs are addressed by means of a novel methodology based on polynomial optimization and Sum-of-Squares decomposition. This approach allows recasting these coupled differential equations as convex optimization problems readily implementable resorting to semidefinite programming, with no restrictions to the spectral characteristics of some integral operators or system's coefficients. Additionally, for polynomials Kernels, uniqueness and invertibility of the Fredholm-type transformation are proved in the space of real analytic and continuous functions. The direct and inverse Kernels are approximated as the optimal polynomial solution of a Sum-of-Squares and Moment problem with theoretically arbitrary precision. Numerical simulations illustrate the effectiveness and potentialities of the methodology proposed to manage a variety of problems with different structures and objectives.