Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves

Wim Michiels, Federico Bribiesca-Argomedo, Jean Auriol

Abstract

The paper addresses the boundary control of a class of hyperbolic PDEs, based on an equivalent representation in terms of an integral-difference equation. The situation is considered where direct compensation of reflection terms induces a fragile closed-loop system, in the sense of lack of strong stability. This is theoretically resolved by adding a low-pass filter to the control law, but the choice of its cut-off frequency is crucial in balancing robustness at high frequencies and performance at low frequencies. First, the maximum stability interval in parameter $T$ is determined, with $T$ the inverse of the filter's cutoff frequency. Next, model mismatch on the PDE parameters is considered and a sufficient stability condition is derived in terms of allowable mismatch and cut-off frequency, satisfied in a region in the combined parameter space with a conic shape around $T=0$. Finally, this qualitative behavior is confirmed by exact stability charts for a special case where all model mismatch is contained into one parameter. It is highlighted that the set of stability crossing curves exhibits a fractal-like structure, which is explained using a limit system with discrete delays.

On hyperbolic PDEs, filtered feedback control laws, and fractal-like stability crossing curves

Abstract

The paper addresses the boundary control of a class of hyperbolic PDEs, based on an equivalent representation in terms of an integral-difference equation. The situation is considered where direct compensation of reflection terms induces a fragile closed-loop system, in the sense of lack of strong stability. This is theoretically resolved by adding a low-pass filter to the control law, but the choice of its cut-off frequency is crucial in balancing robustness at high frequencies and performance at low frequencies. First, the maximum stability interval in parameter is determined, with the inverse of the filter's cutoff frequency. Next, model mismatch on the PDE parameters is considered and a sufficient stability condition is derived in terms of allowable mismatch and cut-off frequency, satisfied in a region in the combined parameter space with a conic shape around . Finally, this qualitative behavior is confirmed by exact stability charts for a special case where all model mismatch is contained into one parameter. It is highlighted that the set of stability crossing curves exhibits a fractal-like structure, which is explained using a limit system with discrete delays.
Paper Structure (26 sections, 11 theorems, 121 equations, 5 figures, 1 algorithm)

This paper contains 26 sections, 11 theorems, 121 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System under consideration
  3. Time-delay representation
  4. Stabilizing controller and robustness aspects
  5. Objective and contributions
  6. Strong stability condition of the open-loop system
  7. Filtered control law and closed-loop system
  8. Preservation of stability by the filter's inclusion
  9. Characterization of maximum $T$ to preserve stability
  10. Robustness against parametric perturbations
  11. Description of the model mismatch
  12. Qualitative description of the stability region
  13. Stability charts for a special case
  14. Behavior for small $T$ and $\varepsilon$
  15. Case $H_{11}\in(\frac{1}{2}\ ,1)$
  16. ...and 11 more sections

Key Result

Proposition 1

Exponential stability of sys-final in the sense of Definition def_exp_stab_IDE is equivalent to exponential stability of eq:hyperbolic_couple in the sense of Definition Def_stability.

Figures (5)

  • Figure 1: Sets $\mathcal{S}_0$ and $\mathcal{S}_{-1}$ for $H_{11}=0.6$. The tick black dots represents $2$ points on $\cup_{k\in\mathbb{Z}} \mathcal{S}_k$, whose off-spring of critical points, described by (\ref{['offspring']}), are indicated with crosses. The two dashed tangents to $\mathcal{S}_0$ and $\mathcal{S}_{-1}$, respectively, bound the largest cone that is contained in the stability region.
  • Figure 2: Stability crossing curves of (\ref{['plantpde']}) and (\ref{['controlpde']}) in the $(T,\varepsilon)$-parameter space when neglecting the two distributed delay terms. The lower pane is obtained by zooming in on the upper pane. The scaling property (\ref{['offspring']}) induces infinitely many stability crossing curves in the neighborhood of $(T,\varepsilon)=(0,0)$, and a fractal nature of the plot. Only finitely many curves are shown, while the superimposed triangles indicate the regions with a high concentration.
  • Figure 3: Angles $\alpha_+$ and $\alpha_-$ as a function of parameter $H_{11}$.
  • Figure 4: Stability region of (\ref{['plantpde']}) and (\ref{['controlpde']}) in the $(T,\varepsilon)$-parameter space. Only the stability crossing curves in the region adjacent to central stability region are shown. For $\varepsilon=0$, the critical values of $T$, computed by Algorithm \ref{['algsweep']}, are indicated by circle markers. The lower pane is obtained by zooming in on the upper pane. The superimposed triangles indicate the regions with a high concentration of stability crossing curves.
  • Figure 5: Schematic representation of the domain $\bar{D}_{ij}$.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Theorem 1
  • Proposition 4
  • Definition 4
  • ...and 7 more