Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

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.

Bio-Inspired Integrated Sensing and Control Flapping Flight for Micro Aerial Vehicles

“…A gain-scheduling based approach for nonlinear PDEs has been presented in Ref. [88], while Krstic and Smyshlyaev [43] derived an adaptive controller for parabolic PDEs.…”

“…There are several controller designs in the literature for stabilizing and controlling beam bending (see, for example, [32,88,101,27,40]). We design a perturbation observer-based controller to facilitate a motion planning-based tracking controller for bending.…”