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Explicit and parameter-adaptive boundary control laws for parabolic partial differential equations

  • Author(s): Smyshlyaev, Andrey S.
  • et al.
Abstract

The dissertation introduces a new constructive approach to the problem of boundary stabilization of linear parabolic partial differential equations (PDEs). The approach is based on an infinite-dimensional extension of the backstepping method. Using a Volterra integral transformation, the unstable plant is converted to an exponentially stable target system and the control gain is found as a solution of a certain well-posed hyperbolic PDE. In addition to stabilizing controllers, we also introduce inverse optimal controllers, which provide low control effort and robustness margins. For many physically motivated cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. For the case when the measurements are available only at the boundary, we develop exponentially convergent observers that are dual to the state feedback controllers. These observers are then combined with the controllers to obtain the solution to the output feedback problem with actuator and sensor located on the same or the opposite boundaries. For the plants with constant unknown parameters we design the certainty equivalence adaptive controllers with two types of identifiers: passivity-based identifiers and swapping identifiers. For the plants with unknown spatially-varying parameters we design the adaptive scheme based on the Lyapunov method. The control gain is computed through an approximate solution of a linear PDE or through a limited number of recursive integrations. We show the robustness of the proposed scheme with respect to an error in the online gain computation. Finally, we consider a problem of output feedback stabilization of PDEs with unknown, spatially varying reaction, advection, and diffusion parameters. Both sensing and actuation are performed at the boundary. We construct a transformation of the original system into the PDE analog of "observer canonical form," with unknown parameters multiplying the measured output. Input and output filters are implemented so that a dynamic parametrization of the problem is converted into a static parametrization where a gradient estimation algorithm is used. We also solve the problem of the adaptive output tracking of a desired reference trajectory prescribed at the boundary

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