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We consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations (PDEs). These equations serve as a model for such physical phenomena as traffic flows, chemical reactors, and heat exchangers. We design controllers using a backstepping method, which has been recently developed for parabolic PDEs. With the integral transformation and boundary feedback...
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...
This tutorial paper presents backstepping boundary control designs for fluid systems. It starts with an introduction of an output-feedback controller for the problem of vortex shedding around a cylindrical bluff body in the flow. It is then followed by a presentation of stabilization of the thermal convection loop. Finally, the paper concludes with a control design for the turbulent channel flow.
After briefly overviewing the application problems that call for boundary control method and the existing techniques for control of PDEs, the paper introduces the key concepts for spatially continuous backstepping control design for PDE systems. After that, a general procedure for parabolic PDEs of a "spatially causal" class is presented, followed by a discussion of the main design equations...
In a companion ACC '06 paper we present a boundary control law, designed using the backstepping method, that stabilizes the 3D Navier-Stokes system linearized around the much studied Poiseuille flow profile. This control law employs three controllers that actuate solely along one boundary. The controller that acts in the wall normal direction prepares the system for the backstepping method, while...
We present an extension from 2D to 3D of a boundary control law which stabilizes the parabolic profile of an infinite channel flow. In this 3D case, we include an additional controller in the spanwise direction which, similar to the streamwise and wall normal velocity controllers, actuates solely along one boundary. The controller that acts in the wall normal direction transforms the system into a...
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