Folding Bilateral Backstepping Output-Feedback Control Design For an Unstable Parabolic PDE

We present a novel methodology for designing output-feedback backstepping boundary controllers for an unstable 1-D diffusion-reaction partial differential equation with spatially-varying reaction. Using "folding" transforms the parabolic PDE into a 2X2 coupled parabolic PDE system with coupling via folding boundary conditions. The folding approach is novel in the sense that the design of bilateral controllers can be generalized to centering around arbitrary points, which admit additional design parameters for both the state-feedback controller and the state observer. The design can be selectively biased to achieve different performance indicies (e.g. energy, boundedness, etc). A first backstepping transformation is designed to map the unstable system into a strict-feedback intermediate target system. A second backstepping transformation is designed to stabilize the intermediate target system. The invertibility of the two transformations guarantees that the derived state-feedback controllers exponentially stabilize the trivial solution of the parabolic PDE system in the L^2 norm sense. A complementary state observer is likewise designed for the dual problem, where two collocated measurements are considered at an arbitrary point in the interior of the domain. The observer generates state estimates which converge to the true state exponentially fast in the L^2 sense. Finally, the output feedback control law is formulated by composing the state-feedback controller with the state estimates from the observer, and the resulting dynamic feedback is shown to stabilize the trivial solution of the interconnected system in the L^2 norm sense. Some analysis on how the selection of these points affect the responses of the controller and observer are discussed, with simulations illustrating various choices of folding points and their effect on the stabilization.