⚡ electrical engineering

Safe Output Regulation of Coupled Hyperbolic PDE-ODE Systems

This paper proposes a robust safe output regulation control strategy for coupled hyperbolic PDE-ODE systems subject to distributed disturbances, utilizing a nonovershooting backstepping method and observer-based estimation to guarantee exponential tracking convergence, boundedness, and strict adherence to safety constraints for the regulated output.

Original authors: Ji Wang, Miroslav Krstic

Published 2026-03-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ji Wang, Miroslav Krstic

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to guide a very long, wobbly rope that has a heavy weight (a payload) hanging from the end. You are holding the top of the rope, and you want the weight at the bottom to follow a specific path (like a delivery drone dropping a package) without hitting any walls or obstacles along the way.

This is exactly the problem the paper solves, but in the world of advanced mathematics and physics. Here is a simple breakdown of what the authors did, using everyday analogies.

1. The Problem: The "Wobbly Rope" and the "Ghost Wind"

The system they studied is a coupled PDE-ODE system.

The ODE (Ordinary Differential Equation): This is the heavy weight at the end of the rope. It's a simple object with a position and speed.
The PDE (Partial Differential Equation): This is the rope itself. It's not just one point; it's a whole line of points that can wiggle, stretch, and wave.
The Challenge: You can only touch the top of the rope (the control input). You want the bottom weight to follow a target path perfectly. But there are two big problems:
1. The "Ghost Wind": There are invisible gusts of wind blowing on the rope and the weight, pushing them off course. You can't see these winds.
2. The "Safety Zone": You must ensure the weight never hits a wall. If it starts too close to a wall, you need to pull it back safely. If it's already safe, you must keep it safe.

2. The Solution: The "Smart Guide" System

The authors created a control strategy (a set of rules for how to move the top of the rope) that does three things simultaneously:

Tracks the Target: Makes the weight follow the desired path.
Estimates the Invisible: Figures out where the weight is and how hard the "ghost wind" is blowing, even though you can't measure them directly.
Enforces Safety: Acts like a digital "guardian angel" that prevents the weight from crashing, even if it starts in a dangerous spot.

3. How It Works: The Three Magic Tricks

Trick #1: The "Backstepping" Ladder (Untangling the Rope)

The rope and the weight are tangled together mathematically. To fix this, the authors use a method called Backstepping.

Analogy: Imagine trying to untangle a knot in a long string. You don't pull the whole string at once; you work from the end of the knot backward, loosening one loop at a time.
In the paper: They mathematically "untangle" the complex waves of the rope and the movement of the weight into a simpler, predictable shape. This allows them to design a controller that knows exactly how to move the top of the rope to calm the waves at the bottom.

Trick #2: The "Crystal Ball" (The Observer)

Since you can't see the wind or the exact position of every point on the rope, the system needs to guess.

Analogy: Imagine a detective trying to solve a crime. They can't see the criminal, but they see footprints and hear noises. They build a "mental model" of what the criminal is doing based on the clues.
In the paper: They built a State Observer. It's a second, virtual computer running alongside the real system. It watches the few things you can measure (like the top of the rope) and uses math to predict the invisible wind and the hidden movements of the rope. The paper proves this "Crystal Ball" gets more and more accurate over time.

Trick #3: The "Safety Bubble" (Control Barrier Functions)

This is the most exciting part. The system doesn't just try to reach the target; it respects a "Safety Bubble."

Analogy: Imagine driving a car with a very sensitive alarm. If you get too close to a wall, the alarm screams, and the steering wheel automatically turns you away.
In the paper: They use something called Control Barrier Functions (CBFs).
- Scenario A (Safe Start): If the weight starts inside the safe zone, the controller gently guides it to the target while keeping it inside the bubble.
- Scenario B (Unsafe Start): If the weight starts outside the safe zone (e.g., already touching a wall), the controller has a special "rescue mode." It guarantees that within a specific time you set (like "get me back to safety in 2 seconds"), it will pull the weight back into the safe zone. It's like a lifeboat that automatically righting a capsized ship.

4. The Real-World Test: The Drone Delivery

To prove it works, they simulated a UAV (drone) delivery.

The Setup: A drone is flying, holding a package on a long cable. The wind is blowing, and there are obstacles (buildings) nearby.
The Result: The drone successfully moved the package to the drop-off point. Even when the wind pushed the package toward a building, the controller pulled it back. Even if they started the simulation with the package dangerously close to a wall, the system "rescued" it and brought it back to a safe path.

Summary

This paper is about teaching a machine how to control a wobbly, wind-blown rope and weight system. They invented a "Smart Guide" that:

Untangles the physics to understand the system.
Guesses the invisible wind and hidden movements.
Guarantees that the payload never crashes, even if it starts in a dangerous spot, by pulling it back to safety within a set time.

It's a major step forward for autonomous systems (like drones or robots) that need to operate safely in messy, unpredictable real-world environments.

1. Problem Formulation

The paper addresses the safe output regulation problem for a class of systems modeled by coupled $2 \times 2$ hyperbolic Partial Differential Equations (PDEs) interconnected with an Ordinary Differential Equation (ODE) subsystem.

System Structure: The plant consists of a strict-feedback linear ODE subsystem (representing, for example, a payload or vehicle dynamics) coupled with two counter-transporting hyperbolic PDEs (representing wave-like dynamics, such as a cable).
Disturbances: The system is subject to fully distributed disturbances acting throughout the PDE domain and the ODE, as well as external reference signals and disturbances generated by an exogenous model (e.g., sinusoidal or constant signals).
Control Objective:
1. Output Regulation: Drive the system output (specifically the state furthest from the control input) to track a desired reference signal with exponential convergence.
2. Safety: Ensure the regulated output remains within a time-varying safe region defined by a barrier function $h(e(t), t) \geq 0$ .
3. Robustness: Handle unmeasurable states (both PDE and ODE states) and unknown external disturbances.
4. Rescue Capability: If the system starts outside the safe region, the controller must guarantee the state returns to safety within a prescribed time set by the user.

2. Methodology

The authors propose a control framework that integrates Backstepping, Control Barrier Functions (CBFs), and Observer Design.

A. Nominal State-Feedback Design

Coordinate Transformations:
- First Transformation: Converts the ODE subsystem into a chain of integrators form to facilitate tracking error dynamics.
- Second Transformation (CBF Construction): Constructs a sequence of barrier functions ( $h_1, \dots, h_n$ ) based on the tracking error and its derivatives. This transforms the ODE into a high-relative-degree CBF structure. A specific auxiliary function $\sigma(t)$ is introduced to handle initial states that are already unsafe, ensuring a "rescue" to the safe set within a finite time $t_a$ .
- Third Transformation (Backstepping): Applies a backstepping transformation to the PDE subsystem to decouple the in-domain coupling terms and eliminate dependence on the exogenous signal model. This maps the original system to a target system consisting of a stable ODE (the CBF chain) and decoupled transport PDEs with zero boundary conditions.
Control Law: The control input is derived explicitly to stabilize the target system, ensuring the barrier functions remain positive (safety) and the tracking error converges to zero.

B. Output-Feedback Design (Handling Uncertainties)

Since full state measurement is often unavailable, the authors design an extended observer:

Observer Structure: An observer estimates the unmeasurable PDE states ( $z, w$ ), ODE states ( $Y$ ), and the exogenous disturbance/reference signals ( $v$ ).
Disturbance Estimation: The observer utilizes the measurable output $z(1, t)$ and the reference signal to reconstruct the distributed disturbances.
Error Bounds: Explicit bounds on the estimation errors are derived using Lyapunov analysis. These bounds decay exponentially.
Robust Safe Controller: The output-feedback control law replaces the true states with estimates and adds a robustification term (a sign function scaled by the estimated error bound). This term shrinks the safe set slightly to account for estimation errors, ensuring safety is maintained even with imperfect state knowledge.

3. Key Contributions

First Safe Output Regulation for PDEs: This work presents the first result on safe output regulation for PDE systems, extending safety concepts (CBFs) from finite-dimensional ODEs to infinite-dimensional PDE-ODE cascades.
Output-Feedback with Distributed Disturbances: Unlike previous safe control works that assumed full state availability, this paper develops a controller that handles unmeasured PDE states and fully distributed disturbances simultaneously.
General Barrier Functions & Prescribed-Time Rescue:
- The framework supports general barrier functions (not limited to simple positivity).
- It introduces a mechanism to guarantee that if the system starts in an unsafe state, it will be "rescued" to the safe region within a user-defined time $t_a$ , rather than just maintaining safety from a safe initial condition.
Rigorous Stability and Safety Proofs: The paper provides rigorous proofs for:
- Exponential convergence of tracking errors to zero.
- Exponential convergence of observer errors to zero.
- Guaranteed safety (invariance or finite-time rescue) of the regulated output.
- Boundedness of all closed-loop signals.

4. Results and Validation

The proposed method was validated through a UAV delivery scenario with a cable-suspended payload:

Model: The system models the UAV (ODE) and the cable (Wave PDEs) subject to wind disturbances (distributed) and payload dynamics.
Simulation Cases:
- Case 1 (One-sided Barrier): The payload must stay above a certain reference line. The controller successfully tracked the reference and maintained safety. When initialized in an unsafe region, the payload was rescued to the safe zone in 1.85 seconds (within the prescribed limit).
- Case 2 (Two-sided Barrier): The payload must stay within a corridor defined by upper and lower barriers. The controller successfully prevented collisions with both barriers while tracking the reference.
Performance:
- The observer accurately estimated the distributed states and external disturbances.
- The system remained stable even with anti-damping introduced to simulate open-loop instability.
- The control inputs remained bounded, and the tracking error converged exponentially.

5. Significance

This paper bridges a critical gap between safe control (typically ODE-based) and infinite-dimensional systems (PDEs). Its significance lies in:

Autonomous Safety: Providing a theoretical guarantee that autonomous systems governed by wave dynamics (e.g., cranes, UAVs with cables, fluid pipes) can operate safely even in the presence of unknown, distributed environmental disturbances.
Practical Applicability: The ability to handle unmeasured states makes the solution viable for real-world applications where full state sensing of distributed parameters is impossible.
Prescribed-Time Safety: The "rescue" capability offers a new level of safety assurance, allowing systems to recover from transient violations of safety constraints within a guaranteed timeframe, which is crucial for high-stakes autonomous operations.

In conclusion, Wang and Krstic have developed a robust, theoretically sound framework that ensures both precise trajectory tracking and strict safety adherence for complex coupled PDE-ODE systems under uncertainty.