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We consider the problem of boundary stabilization of a one-dimensional wave equation with an internal spatially varying anti-damping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant...
Smith Predictor-like designs for compensation of arbitrarily long input delays are commonly available only for finite-dimensional systems. Only very few examples exist where such compensation has been achieved for PDE systems, including our recent result for a parabolic (reaction-diffusion) PDE. In this paper we address a more challenging wave PDE problem, where the difficulty is amplified by allowing...
Much of the boundary control of wave equations in 1D is based on a single principle-passivity-under the assumption that control is applied through Neumann actuation on one boundary and the other boundary satisfies a homogeneous Dirichlet boundary condition.We have recently expanded the scope of tractable problems by allowing destabilizing anti-stiffness (a Robin type condition) on the uncontrolled...
Past papers on adaptive control of unstable PDEs with unmatched parametric uncertainties have considered only parabolic PDEs and first-order hyperbolic PDEs. In this note we introduce several tools for approaching adaptive control problems of second-order-in-time PDEs. We present these tools through a benchmark example of an unstable wave equation with an unmatched (non-collocated) anti-damping term,...
We present the controller and observer designs for hyperbolic PDE systems. The main ideas are introduced on the example of a wave equation - a model of an undamped, vibrating string. Both a full state feedback boundary controller and a boundary sensor based observer are introduced. The string results are followed by a presentation of controller and observer designs for shear beam and Timoshenko beam...
We consider the problem of stabilization of a one- dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated "boundary damper" feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more...
In this paper we present the first extension of the backstepping methods that we have developed so far for control of parabolic PDEs (thermal, fluid, and chemical reaction dynamics) to second-order PDE systems (often referred loosely as hyperbolic) which model flexible structures and acoustics. We introduce controller and observer designs capable of adding damping to a model of beam dynamics using...
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