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Optimal Boundary Control of Reaction–Diffusion Partial Differential Equations via Weak Variations

[+] Author and Article Information
Scott J. Moura

UC President's Postdoctoral Fellow
Mechanical & Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92093
e-mail: smoura@ucsd.edu

Hosam K. Fathy

Assistant Professor
Mechanical & Nuclear Engineering,
Pennsylvania State University,
University Park, PA 16802
e-mail: hkf2@psu.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 19, 2012; final manuscript received November 21, 2012; published online February 21, 2013. Assoc. Editor: Rama K. Yedavalli.

J. Dyn. Sys., Meas., Control 135(3), 034501 (Feb 21, 2013) (8 pages) Paper No: DS-12-1022; doi: 10.1115/1.4023071 History: Received January 19, 2012; Revised November 21, 2012

This paper derives linear quadratic regulator (LQR) results for boundary-controlled parabolic partial differential equations (PDEs) via weak variations. Research on optimal control of PDEs has a rich 40-year history. This body of knowledge relies heavily on operator and semigroup theory. Our research distinguishes itself by deriving existing LQR results from a more accessible set of mathematics, namely weak-variational concepts. Ultimately, the LQR controller is computed from a Riccati PDE that must be derived for each PDE model under consideration. Nonetheless, a Riccati PDE is a significantly simpler object to solve than an operator Riccati equation, which characterizes most existing results. To this end, our research provides an elegant and accessible method for practicing engineers who study physical systems described by PDEs. Simulation examples, closed-loop stability analyses, comparisons to alternative control methods, and extensions to other models are also included.

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Figures

Grahic Jump Location
Fig. 1

A visualization of the weak variations concept for optimal state and control trajectories

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Fig. 2

Simulation example of state trajectories for the (a) open-loop and (b) closed-loop systems

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Fig. 3

Boundary control input for the LQR controller

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Fig. 4

Real part of eigenvalues for closed-loop spectrum of infinite-time LQR controlled system

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Fig. 5

Block diagram for open-loop transfer function in negative feedback form to study stability robustness to uncertainty at the plant input

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Fig. 6

Nyquist plot of G(s), defined in (46)

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Fig. 7

Feedback control gains for the infinite-time LQR (solid lines) and backstepping (dashed) controllers. Parameter q corresponds to the state penalty: Q(x,y) = q sin(πx) sin(πy). Parameter γ is the reaction coefficient in the backstepping target system (55).

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