Abstract
We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with an in-domain nonlinearity is considered first. For this system a nonlinear feedback law, based on gain scheduling, is derived explicitly, and a proof of local exponential stability, with an estimate of the region of attraction, is presented for the closed-loop system. Control designs (without proofs) are then presented for a string PDE and a shear beam PDE, both with Kelvin–Voigt (KV) damping and free-end nonlinearities of a potentially destabilizing kind. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization based design.
Issue Section:
Research Papers
Keywords:
gain scheduling,
PDE backstepping,
boundary control,
nonlinear control,
stabilization,
motion planning,
hyperbolic PDEs,
wave equation,
string,
beam,
asymptotic stability,
beams (structures),
closed loop systems,
control nonlinearities,
control system synthesis,
damping,
feedback,
nonlinear differential equations,
partial differential equations,
scheduling,
structural engineering
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.Copyright © 2011
by American Society of Mechanical Engineers
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