Active Disturbance Rejection for Boundary Control Systems

Technical Summary: Active Disturbance Rejection for Boundary Control Systems

Problem Statement
This paper addresses the stabilization of infinite-dimensional linear systems and partial differential equations (PDEs) subject to unknown input disturbances and unmodeled nonlinearities. Specifically, the authors consider abstract boundary control systems of the form:
$$\dot{x}(t) = Ax(t), \quad x(0) = x_0$$
$$Bx(t) = u(t) + d(t) + \phi_s(x(t)) + \phi_o(y(t))$$
$$y(t) = Cx(t)$$
where $x$ is the state, $u$ is the control input, and $y$ is the output. The system is corrupted by an unknown disturbance signal $d(t)$ and unmodeled nonlinearities $\phi_s$ (state-dependent) and $\phi_o$ (output-dependent). The goal is to design a controller that estimates and rejects the "total disturbance" $d_{tot} = d + \phi_s(x) + \phi_o(y)$, driving the state $x$ to zero as $t \to \infty$ while ensuring external well-posedness.

Methodology
The authors employ the Active Disturbance Rejection Control (ADRC) approach within the framework of abstract boundary control systems. The core methodology involves:

1. Total Disturbance Estimation: The controller utilizes an "inverted" system to estimate the total disturbance. By reinterpreting the boundary conditions that define the input as measurements and the output boundary traces as new boundary conditions, the authors construct an estimator. If the inverted system possesses well-defined and stable dynamics, the estimation error converges to zero.
2. Controller Structure: The proposed controller consists of three main components:
   • A Separator: A pre-stabilizing block that removes the control input $u$ from the original system dynamics before inversion.
   • An Inverter: A dynamic system that inverts the input-output relationship to estimate the unknown part of the input ($d_{tot}$).
   • A Luenberger-type Observer: A linear observer for the original system that utilizes the disturbance estimate $\hat{d}$ to approximate the unknown input and asymptotically cancel its effect.
3. Mathematical Framework: The analysis relies on the theory of well-posed boundary nodes and linear systems in Hilbert spaces. The authors define classical and generalized solutions for the closed-loop system, which includes the plant and the controller (itself a system with an internal loop). They utilize transfer functions, semigroup theory, and resolvent estimates to establish stability and well-posedness.

Key Contributions and Results
The paper presents two main theorems establishing the theoretical foundation for ADRC in abstract boundary control systems:

• Theorem 3.2 (Standard Well-Posedness): Under specific assumptions regarding the well-posedness of the inverted system (specifically, that the inverted boundary node has a well-posed output map), the authors prove:

  • The existence and uniqueness of classical and generalized solutions for the nonlinear closed-loop system.
  • Exponential convergence of the disturbance estimation error ($\hat{d} - d_{tot}$) to zero.
  • Exponential stability of the closed-loop state and output, provided the underlying semigroups are exponentially stable.
  • External incremental well-posedness of the closed-loop system.

• Theorem 3.6 (Relaxed Conditions for Parabolic Systems): Recognizing that the inverses of certain PDE systems (notably heat equations) are ill-posed, the authors provide a modified result. They replace the strict well-posedness requirement of the inverted system with a condition involving the uniform boundedness of a specific product of transfer functions. This allows the ADRC design to be applied to parabolic PDEs where the inverse dynamics are not well-posed in the standard sense, leveraging the smoothing effect of the parabolic models.

Applications to PDEs
The authors demonstrate the applicability of their abstract results by designing controllers for:

1. One-Dimensional Wave Equation: A system with collocated boundary inputs and outputs. The controller successfully rejects boundary disturbances and compensates for unmodeled nonlinearities, achieving exponential energy decay.
2. One-Dimensional Heat Equation: A system where the inverse is characteristically ill-posed. The authors apply Theorem 3.6 to show that the controller stabilizes the system and rejects disturbances despite the ill-posedness of the inverse, relying on the parabolic smoothing properties.

Significance and Claims
The paper claims to advance the state of the art in ADRC for PDEs in several specific ways:

• Exponential Convergence: Unlike many existing references that only show asymptotic convergence, this work establishes exponentially fast convergence of both the disturbance estimation and the controlled state.
• Handling Nonlinearities: The controller is capable of handling boundary nonlinearities ($\phi_o \neq 0$), whereas earlier results often assumed linear outputs or ignored output-dependent nonlinearities.
• External Well-Posedness: The authors establish external incremental well-posedness for the nonlinear closed-loop system, a property not always guaranteed in previous designs.
• Abstract Framework: The methodology is developed for abstract boundary control systems, providing a unified framework that avoids the complexity of finding representations as regular linear systems or system nodes.
• Weaker Conditions: The controller ensures the existence of closed-loop solutions under weaker conditions than required for the open-loop system, specifically without requiring global Lipschitz properties for $\phi_s$, bounds on the Lipschitz constant of $\phi_o$, or passivity/monotonicity assumptions.

The authors note that while infinite-dimensional systems theory has been used in ADRC for specific PDE models, this methodology has not previously been employed in the context of abstract infinite-dimensional systems. The results strengthen existing conclusions by providing a rigorous structural analysis and a set of conditions suitable for efficient controller tuning.