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Research Papers

Dead-Time Compensation for Wave/String PDEs

[+] Author and Article Information
Miroslav Krstic

Department of Mechanical and Aerospace Engineering, University of California, San Diego La Jolla, CA 92093-0411krstic@ucsd.edu

J. Dyn. Sys., Meas., Control 133(3), 031004 (Mar 23, 2011) (13 pages) doi:10.1115/1.4003638 History: Received December 06, 2008; Revised April 05, 2010; Published March 23, 2011; Online March 23, 2011

Smith predictorlike designs for compensation of arbitrarily long input delays are commonly available only for finite-dimensional systems. Only very few examples exist, where such compensation has been achieved for partial differential equation (PDE) systems, including our recent result for a parabolic (reaction-diffusion) PDE. In this paper, we address a more challenging wave PDE problem, where the difficulty is amplified by allowing all of this PDE’s eigenvalues to be a distance to the right of the imaginary axis. Antidamping (positive feedback) on the uncontrolled boundary induces this dramatic form of instability. We develop a design that compensates an arbitrarily long delay at the input of the boundary control system and achieves exponential stability in closed-loop. We derive explicit formulae for our controller’s gain kernel functions. They are related to the open-loop solutions of the antistable wave equation system over the time period of input delay (this simple relationship is the result of the design approach).

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 7

The transformations among the four system representations in Fig. 6. Only the transformation (v,u)↦(z,w) is of backstepping type, whereas the other two have the role of moving the damping and moving the transport-wave interconnection from a boundary into the domain, which is a form that facilitates the analysis.

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Figure 8

The outline of the proof of exponential stability of the cascade system (z,ψ). The Lyapunov functional for the autonomous z-system is constructed in Lemma 9. The Lyapunov functional for the ψ-system is constructed in Lemma 7. The input-to-state stability (ISS) of the ψ-system with respect to the z-system is shown in Lemma 8. The Lyapunov stability of the overall (z,ψ)-system is shown in Lemma 10.

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Figure 9

The norms used in the stability analysis of the closed-loop system. Their equivalence is established in the lemmas indicated in the figure.

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Figure 10

A pickup on an electric guitar. Based on Faraday’s law of induction, it converts the string velocity into voltage. Connected into a high-gain amplifier, this system results in (domainwide) antidamping, which manifests itself as a swell in volume, up to a saturation of the amplifier, which guitarists refer to, simply, as feedback.

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Figure 1

Antistable wave PDE system with input delay

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Figure 2

Control of a thermoacoustic instability in a Rijke tube (61) (a duct-type combustion chamber)

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Figure 3

A diagram of a string with control applied at boundary x=1 and an antidamping force acting at the boundary x=0

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Figure 4

The process of finding the solution to the PDEs 18,19,20,21,22 for the gain function ρ(x,y). First, the undamped wave equation with homogeneous boundary conditions, Eqs. 60,61,62 or Eqs. 63,64,65, is solved for ϖ using Lemmas 4 and 5. Second, Lemma 3 yields the solution ϑ to the wave equation with domainwide antidamping λ, Eqs. 47,48,49 or Eqs. 50,51,52. Third, Lemma 2 yields the solution ς to the wave equation with boundary antidamping q, Eqs. 38,39,40,41,42. Finally, using Eq. 43, the gain functions ρ,ρx,ρy, which are needed in the controller Eq. 37, are found in Proposition 6.

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Figure 5

The graph of the function λ(q)

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Figure 6

A sequence of system transformations employed in the analysis. The original plant is (v,u), whereas the backstepping transformation converts the closed-loop system into the autonomous, exponentially stable system (z,w). Due to the boundary interconnection and the boundary damping at x=0, the stability of the system (z,w) is hard to analyze. The damping is moved from the boundary to the domain using the transformation (z,w)↦(z,φ). Finally, the boundary interconnection, in which an unbounded operator arises, is converted into an easier-to-analyze in-domain interconnection using the transformation (z,φ)↦(z,ψ). The transformations between the four representations are indicated in Fig. 7. The stability analysis of the (z,ψ)-system is outlined in Fig. 8. Stability of the (z,ψ)-system is studied in Lemmas 7–10, using the Lyapunov function Ω(t). Then, stability of the (z,φ)-system is shown in Lemma 11 and Proposition 12.

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