On observer forms for hyperbolic PDEs with boundary dynamics

Abstract: A hyperbolic observer canonical form (HOCF) for linear hyperbolic PDEs with boundary dynamics is presented. The transformation to the HOCF is based on a general procedure that uses so-called observability coordinates as an intermediate step. These coordinates are defined from an input-output relation given by a neutral functional differential equation (FDE), which, in the autonomous case, reduces to an autonomous FDE for the output. The HOCF coordinates are directly linked to this FDE, while the state transformation between the original coordinates and the observability coordinates is obtained by restricting the observability map to the interval corresponding to the maximal time shift appearing in the FDE. The proposed approach is illustrated on a string-mass-spring example.

• The paper presents a systematic procedure for constructing observer canonical forms for hyperbolic PDE-ODE systems, enabling explicit state reconstruction from boundary measurements.
• It employs observability coordinates via input–output FDE analysis and Volterra-type integral transforms to capture distributed dynamics effectively.
• The framework facilitates robust observer and output feedback design, validated through examples like string–mass–spring systems.

Observer Forms for Hyperbolic PDEs with Boundary Dynamics: Theory and Construction

Introduction and Motivation

This paper presents an explicit and systematic framework for constructing observer canonical forms for a broad class of linear hyperbolic PDEs with boundary dynamics, including PDE–ODE cascade systems. Building on notions from finite-dimensional observer canonical forms, this work generalizes these concepts to distributed-parameter systems, leveraging connections to functional differential equations (FDEs) and input–output relations that encode the underlying system dynamics.

Traditional backstepping approaches and canonical form methodologies for hyperbolic systems have advanced controller and observer design, yet canonical observer forms explicitly tailored to include boundary ODE dynamics have remained less developed. By introducing observability coordinates derived from input–output relations in the FDE framework, the paper establishes a direct procedure for constructing the hyperbolic observer canonical form (HOCF), providing both theoretical and algorithmic contributions.

Hyperbolic Observer Canonical Form and Observability Coordinates

The core result is the formal construction of the HOCF for general linear single-input single-output (SISO) hyperbolic PDE-ODE systems. The key methodology proceeds as follows:

1. Identification of the Input–Output FDE: By integrating the hyperbolic transport PDE along characteristics and using the boundary coupling conditions, the system output trajectory is shown to satisfy a neutral FDE, typically of the form

$$\sum_{i=0}^{n-1}a_iy^{(i)}(t)+y^{(n)}(t+\hat\tau)+a_{n}^* y^{(n)}(t)=0,$$

where $a_i$ are scalar coefficients, $a_n^*$ a possibly unbounded operator, and $\hat\tau$ the maximal time shift.

1. Definition of Observability Coordinates: Defining $\bar y(\tau, t) = y(t+\tau)$ for $\tau \in [0, \hat\tau]$, the state is parameterized by the restriction of the output trajectory to a finite time interval, leveraging the bijectivity of an appropriately defined observability map. This yields a representation of the distributed state as a function of the output and its delayed versions.
2. Construction of the HOCF: The observer form combines an integrator chain representing the finite-dimensional boundary dynamics and a transport PDE with output injection. The distributed state in the HOCF, $\eta_{n+1}(\tau, t)$, evolves according to

$$\partial_t \eta_{n+1}(\tau, t) = -\partial_\tau \eta_{n+1}(\tau, t) - a_n y(t),$$

with explicit boundary conditions and the measured output given by the outflow at the far boundary.

1. Invertible Coordinate Transformations: Transformations from original state coordinates to HOCF observer coordinates are mediated by Volterra-type integral transformations, constructed via the intermediate observability coordinates.

This construction guarantees that the full system state can be reconstructed from boundary measurements, under the assumption of exact observability and structural controllability conditions on the system.

Explicit Construction for PDE–ODE Cascade Systems

The methodology is instantiated for a relevant class of systems: coupled heterodirectional two-component hyperbolic PDEs with a finite-dimensional ODE at the boundary. The spatial domain is $z \in [0, 1]$, and the PDE system takes the form

$$\partial_z x(z, t) + \Sigma(z) \partial_t x(z, t) = A(z) x(z, t)$$

with boundary coupling to an observable ODE, and output measured at the opposite boundary. The paper details:

• Characteristic-Based Solution Representation: Solutions to the PDE are decomposed along characteristic directions, isolating the influence of spatial propagation and explicitly expressing the distributed state in terms of boundary traces and convolution operators with computable kernels.
• Kernel Operator Formulation: The spatial coupling arising from the PDE coefficients is encoded via a convolution-type integral operator, capturing in-domain coupling effects within the state mapping.
• Functional Differential Equation (FDE) Encoding: The derivation of the system's FDE, encoding the output's evolution in terms of both its current and delayed values, establishes the necessary structure for defining observer coordinates.
• State Transformation Implementation: Procedures for transforming between original coordinates and observer coordinates—including solving Volterra-type integral equations and handling time-shifted trajectories—are specified, ensuring the approach is constructive and implementable.

String–Mass–Spring Example

To illustrate the construction, the paper works through a classical example: a taut string with a mass-spring boundary condition at one end and collocated boundary actuation and measurement at the other. The wave equation governs the in-domain dynamics, while a second-order ODE models the mass-spring subsystem coupled at the boundary. Applying the general HOCF construction:

• Boundary Parameterization: The system state at boundaries is expressed recursively in terms of the measured output and its delayed values.
• Explicit FDE: The system's output satisfies a neutral FDE involving both advanced and retarded arguments, capturing the closed-loop dynamic influence of the boundary ODE.
• Observer Coordinates and Transformations: The observer canonical form is explicitly derived with all its components—transport equation with injection, integrator chain, and corresponding output.
• Invertibility and Regularity: The paper establishes that the requisite transformations are well-posed on suitable Sobolev spaces, with invertibility extended to the full state space via continuity arguments.

Numerical and Theoretical Properties

The proposed HOCF framework:

• Enables explicit state reconstruction from boundary measurements under exact observability conditions.
• Handles spatial in-domain coupling and nontrivial boundary ODE dynamics in a unified fashion.
• Provides a foundation for systematic observer and estimator design by reducing the infinite-dimensional system to an explicit canonical form amenable to further analysis.

The approach gives direct access to the output-based parameterization, avoiding the need for ad hoc observer structure, and supports the development of robust output feedback for distributed-parameter systems.

Implications and Future Directions

The construction of observer canonical forms for hyperbolic PDE-ODE systems with boundary dynamics has significant practical and theoretical implications:

• Observer and Output Feedback Design: The explicit observer forms derived facilitate the design of high-gain or robust estimation algorithms, as well as the synthesis of output feedback controllers leveraging observer state reconstructions.
• Extension to Nonlinear Boundary Dynamics: The methodology provides a blueprint for extending observer canonical form constructions to nonlinear systems, though nonlinear functional differential equations may present substantial analysis challenges.
• Integration with Backstepping and Flatness-Based Control: While previous works have focused on backstepping and flatness approaches, the observer canonical form complements these by providing alternative structural representations for estimation and control.
• Mathematical Analysis of Well-Posedness and Invertibility: The framework's reliance on integral transformations and Volterra equations opens avenues for further analysis regarding regularity, robustness to noise, and implementation in numerical observer algorithms.

Potential future developments may include generalizations to fully nonlinear systems, systems with dynamic boundary actuation, distributed sensing, and incorporation of uncertainty or measurement noise within the observer design procedure.

Conclusion

This paper delivers a rigorous and constructive approach for formulating observer canonical forms for hyperbolic PDEs with boundary dynamics, utilizing the connection to functional differential equations and observability coordinates. The explicit transformations ensure systematic reduction of complex distributed-parameter systems to observer forms suitable for state estimation and output feedback. The practical procedures and theoretical guarantees established lay the groundwork for advanced observer and control synthesis in infinite-dimensional systems with boundary coupling (2604.03009).