Infinite-dimensional backstepping for a class of parabolic distributed parameter systems.

摘要

The dissertation addresses the problem of feedback boundary control for parabolic Partial Differential Equations (PDEs). The main idea in our approach is to construct a nonsingular coordinate transformation, a special application of backstepping, that allows to convert the original system into a new set of coordinates where one can design a control law that stabilizes the system.; We start with the example of an unstable heat equation. The stabilization is achieved by constructing a coordinate transformation that, if thought of as an integral transformation, has a feedback “kernel” that is a known continuous function. The limitation of the design with continuous kernel is that it cannot handle arbitrarily large level of the system instability.; We then present two designs for nonlinear parabolic PDEs in 1D. The first design is for stabilization of a solid propellant rocket burning instability modeled by a single 1D PDE, while the second one is for the boundary control of chemical tubular reactors modeled by a system of two coupled 1D PDEs. The two designs are conceptually similar (the former one has a modification that handles a nonlinear destabilizing boundary condition imposed at the uncontrolled boundary) and guarantee stability for any finite discretization in space of the original PDE models.; Using the experience from the design for the unstable heat equation, a modified approach with coordinate transformations that result in sufficiently regular kernels (not continuous but L_∞) is developed for a class of linear parabolic PDEs motivated by engineering applications. The proposed design procedure can handle systems with an arbitrary finite number of open-loop unstable eigenvalues.; We then extend the design for nonlinear ID parabolic PDE systems to 2D in a systematic fashion for a 2D thermal convection loop. We also explain how to obtain the coordinate transforms and how to apply control for systems in 3D.; Finally, we show on the simulation examples for the heat convection loop, solid propellant rockets, and chemical tubular reactors that parabolic PDEs can be successfully stabilized for a variety of different simulation settings with low order backstepping controllers that use only a small number of state measurements.