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Rapid Boundary Stabilization of Two-Dimensional Elastic Plates with In-Domain Aeroelastic Instabilities

Xingzhi Huang, Ji Wang

TL;DR

This work tackles aeroelastic flutter of a two-dimensional elastic plate modeled by coupled wave PDEs with in-domain instabilities. It introduces a rapid boundary stabilization strategy that uses Fourier modal decomposition to reduce the 2D problem to a family of 1D modes, then applies a multi-layer backstepping design to obtain a boundary control with designer-tunable exponential decay. An anti-collocated boundary observer enables output-feedback implementation, with kernel and observer designs proven well-posed and exponentially stable. Numerical simulations with a 3-mode truncation demonstrate rapid suppression of flow-induced vibrations from an open-loop unstable baseline, validating the approach for active wing flutter suppression in high Mach regimes.

Abstract

Motivated by active wing flutter suppression in high-Mach-number flight, this paper presents a rapid boundary stabilization strategy for a two-dimensional PDE-modeled elastic plate with in-domain instabilities, where the exponential stability is achieved with a decay rate that can be arbitrarily assigned by the users. First, the aeroelastic system is modeled as two-dimensional coupled wave PDEs with internal anti-damping terms, derived by Piston theory and Hamilton's principle. Using Fourier series expansion, the 2-D problem is decomposed into a parameterized family of 1-D systems. For each mode, a full-state boundary feedback controller is designed via PDE backstepping transformation. To enable output-feedback implementation, a state observer is further designed to estimate the distributed states over the two-dimensional spatial domain. Through Lyapunov analysis, the exponential stability of the 2-D elastic plate PDE under the proposed boundary control is established with a designer-tunable decay rate. Numerical simulations verify the effectiveness of the control strategy in suppressing flow-induced vibrations.

Rapid Boundary Stabilization of Two-Dimensional Elastic Plates with In-Domain Aeroelastic Instabilities

TL;DR

This work tackles aeroelastic flutter of a two-dimensional elastic plate modeled by coupled wave PDEs with in-domain instabilities. It introduces a rapid boundary stabilization strategy that uses Fourier modal decomposition to reduce the 2D problem to a family of 1D modes, then applies a multi-layer backstepping design to obtain a boundary control with designer-tunable exponential decay. An anti-collocated boundary observer enables output-feedback implementation, with kernel and observer designs proven well-posed and exponentially stable. Numerical simulations with a 3-mode truncation demonstrate rapid suppression of flow-induced vibrations from an open-loop unstable baseline, validating the approach for active wing flutter suppression in high Mach regimes.

Abstract

Motivated by active wing flutter suppression in high-Mach-number flight, this paper presents a rapid boundary stabilization strategy for a two-dimensional PDE-modeled elastic plate with in-domain instabilities, where the exponential stability is achieved with a decay rate that can be arbitrarily assigned by the users. First, the aeroelastic system is modeled as two-dimensional coupled wave PDEs with internal anti-damping terms, derived by Piston theory and Hamilton's principle. Using Fourier series expansion, the 2-D problem is decomposed into a parameterized family of 1-D systems. For each mode, a full-state boundary feedback controller is designed via PDE backstepping transformation. To enable output-feedback implementation, a state observer is further designed to estimate the distributed states over the two-dimensional spatial domain. Through Lyapunov analysis, the exponential stability of the 2-D elastic plate PDE under the proposed boundary control is established with a designer-tunable decay rate. Numerical simulations verify the effectiveness of the control strategy in suppressing flow-induced vibrations.
Paper Structure (32 sections, 3 theorems, 126 equations, 9 figures, 1 table)

This paper contains 32 sections, 3 theorems, 126 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Boundary control of two-dimensional elastic plates
  4. Higher-dimensional backstepping control of PDEs
  5. Main Contribution
  6. Notation
  7. Modeling
  8. Modeling of aerodynamic forces
  9. Equations of motion and boundary conditions via Hamilton's Principle
  10. Controller Design
  11. Decompose the 2D problem into infinite many 1D problems through Fourier Series
  12. Transformation to coupled transport PIDEs
  13. Backstepping Transformation and Target System
  14. Stabilizing control law and main result
  15. Observer Design
  16. ...and 17 more sections

Key Result

Lemma 1

Consider system eq_1dwave--eq_1dw for each mode $n$, with initial conditions $w_{n,0}, \alpha_{n,0}, \beta_{n,0} \in H^{1}(0,1)$, $w_{n,0t}, \alpha_{n,0t}, \beta_{n,0t} \in L^2$, under the control law eq_U1n--eq_U3n, for $\delta_1, \delta_2, \delta_3$ satisfying then there exists a solution $w_n(t,\cdot)$, $\alpha_n(t,\cdot)$, $\beta_n(t,\cdot)$$\in H^1(0,1)$, $w_{n,t}(t,\cdot)$, $\alpha_{n,t}(t,

Figures (9)

  • Figure 1: Flow-induced vibration wing: from the physical model to the mathematical plant.
  • Figure 2: Results in the open loop.
  • Figure 3: $w(t,x,y)$ under the proposed observer-based output-feedback boundary controller.
  • Figure 4: $\alpha(t,x,y)$ under the proposed observer-based output-feedback boundary controller.
  • Figure 5: $\beta(t,x,y)$ under the proposed observer-based output-feedback boundary controller.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 1
  • Claim 1
  • Claim 2