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Research Papers

Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

[+] Author and Article Information
Antranik A. Siranosian1

Miroslav Krstic, Andrey Smyshlyaev

Department of Mechanical and Aerospace Engineering,  University of California, San Diego, La Jolla, CA 92093

Matt Bement

 Los Alamos National Laboratory, Los Alamos, NM 87545

1

Corresponding author.

J. Dyn. Sys., Meas., Control 133(5), 051007 (Aug 01, 2011) (12 pages) doi:10.1115/1.4004065 History: Received December 12, 2008; Revised February 26, 2011; Published August 01, 2011; Online August 01, 2011

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.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Comparison of the nonlinearity f(u(0,t)) used in the string simulations, and its linear approximation f′(0)u(0,t).

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Figure 4

Comparison of the nonlinearity f(u(0,t)) used for the beam simulations, and its linear approximation f′(0)u(0,t), for F=1.

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Figure 2

String simulation results for the closed-loop system for boundary control using linearization based controller (dashed) and gain scheduling based nonlinear controller (solid), for initial tip displacements u0(0)={0.1,0.3,0.347,0.363}. The plots compare (a) the energy E(t)=||ut(t)||2+||ux(t)||2, (b) the tip displacement u(0,t), and (c) the boundary control effort ux(1,t).

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Figure 3

String simulation results comparing the tip displacement u(0,t) and reference trajectory ur(0,t) when boundary control is applied with linearization based control and gain scheduling based nonlinear control.

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Figure 5

Beam simulation results showing the closed-loop system with linearization based control (dashed) and gain scheduling based nonlinear control (solid). The beam is initialized with u0(x)=310(1-x)2 and α0(x)=-35(1-x) and zero velocities, and the nonlinearity strength is varied as F={0.3,0.53,2}. The plots compare the (a) energy E(t), (b) tip displacement u(0,t), and (c) boundary control effort ux(1,t).

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Figure 6

Beam simulation results comparing the tip displacement and reference trajectory when boundary control is applied with linearization based control and gain scheduling based nonlinear control.

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