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On the strict-feedback form of hyperbolic distributed-parameter systems

Nicole Gehring

Abstract

The paper is concerned with the strict-feedback form of hyperbolic distributed-parameter systems. Such a system structure is well known to be the basis for the recursive backstepping control design for nonlinear ODEs and is also reflected in the Volterra integral transformation used in the backstepping-based stabilization of parabolic PDEs. Although such integral transformations also proved very helpful in deriving state feedback controllers for hyperbolic PDEs, they are not necessarily related to a strict-feedback form. Therefore, the paper looks at structural properties of hyperbolic systems in the context of controllability. By combining and extending existing backstepping results, exactly controllable heterodirectional hyperbolic PDEs as well as PDE-ODE systems are mapped into strict-feedback form. While stabilization is not the objective in this paper, the obtained system structure is the basis for a recursive backstepping design and provides new insights into coupling structures of distributed-parameter systems that allow for a simple control design. In that sense, the paper aims to take backstepping for PDEs back to its ODE origin.

On the strict-feedback form of hyperbolic distributed-parameter systems

Abstract

The paper is concerned with the strict-feedback form of hyperbolic distributed-parameter systems. Such a system structure is well known to be the basis for the recursive backstepping control design for nonlinear ODEs and is also reflected in the Volterra integral transformation used in the backstepping-based stabilization of parabolic PDEs. Although such integral transformations also proved very helpful in deriving state feedback controllers for hyperbolic PDEs, they are not necessarily related to a strict-feedback form. Therefore, the paper looks at structural properties of hyperbolic systems in the context of controllability. By combining and extending existing backstepping results, exactly controllable heterodirectional hyperbolic PDEs as well as PDE-ODE systems are mapped into strict-feedback form. While stabilization is not the objective in this paper, the obtained system structure is the basis for a recursive backstepping design and provides new insights into coupling structures of distributed-parameter systems that allow for a simple control design. In that sense, the paper aims to take backstepping for PDEs back to its ODE origin.
Paper Structure (10 sections, 2 theorems, 30 equations, 5 figures)

This paper contains 10 sections, 2 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Transformation into strict-feedback form
  4. Volterra integral transformation
  5. Fredholm-type integral transformation
  6. Strict-feedback form
  7. Strict-feedback form for PDE-ODE systems
  8. Problem statement
  9. Transformation into strict-feedback form
  10. Concluding remarks

Key Result

Lemma 1

The kernel equations eq:trafo1_kerneleq1--eq:trafo1_kerneleq2 admit a piecewise continuous solution $K(z,\zeta)$ on $\mathcal{T}$.

Figures (5)

  • Figure 1: Coupling structure of the heterodirectional hyperbolic system \ref{['eq:sys']}. The arrows in blue () highlight actions related to controllability.
  • Figure 2: Coupling structure of \ref{['eq:sys_trafo1']} for $\Delta n=0$. Red arrows () indicate a violation of the strict-feedback form due to $A_0^+(z)\bar{x}^+(0,t)$.
  • Figure 3: Strict-feedback form \ref{['eq:sys_trafo2']} of the heterodirectional hyperbolic system \ref{['eq:sys']}, highlighted by the arrows in blue (), for the case $\Delta n=0$.
  • Figure 4: Coupling structure of the PDE-ODE system \ref{['eq:po:sys']}. The arrows in blue () highlight actions related to controllability.
  • Figure 5: Strict-feedback form \ref{['eq:po:sys-trafo']} of the PDE-ODE system \ref{['eq:po:sys']}, highlighted by the arrows in blue (), for the case $\Delta n=0$.

Theorems & Definitions (6)

  • Remark 1
  • Lemma 1: Volterra kernel
  • Remark 2
  • Definition 1: strict-feedback form
  • Remark 3
  • Lemma 2: controllability of ODE