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We present a novel methodology for designing output-feedback backsteppingboundary controllers for an unstable 1-D diffusion-reaction partialdifferential equation with spatially-varying reaction. Using "folding"transforms the parabolic PDE into a 2X2 coupled parabolic PDE system withcoupling via folding boundary conditions. The folding approach is novel in thesense that the design of bilateral controllers can be generalized to centeringaround arbitrary points, which admit additional design parameters for both thestate-feedback controller and the state observer. The design can be selectivelybiased to achieve different performance indicies (e.g. energy, boundedness,etc). A first backstepping transformation is designed to map the unstablesystem into a strict-feedback intermediate target system. A second backsteppingtransformation is designed to stabilize the intermediate target system. Theinvertibility of the two transformations guarantees that the derivedstate-feedback controllers exponentially stabilize the trivial solution of theparabolic PDE system in the L^2 norm sense. A complementary state observer islikewise designed for the dual problem, where two collocated measurements areconsidered at an arbitrary point in the interior of the domain. The observergenerates state estimates which converge to the true state exponentially fastin the L^2 sense. Finally, the output feedback control law is formulated bycomposing the state-feedback controller with the state estimates from theobserver, and the resulting dynamic feedback is shown to stabilize the trivialsolution of the interconnected system in the L^2 norm sense. Some analysis onhow the selection of these points affect the responses of the controller andobserver are discussed, with simulations illustrating various choices offolding points and their effect on the stabilization.

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