⚡ electrical engineering

Data-Driven Safe Output Regulation of Strict-Feedback Linear Systems with Input Delay

This paper proposes a data-driven safe control framework for strict-feedback linear systems with unknown parameters, disturbances, and input delays by combining Koopman operator theory and batch least-squares identification with control barrier functions and backstepping to ensure both state estimation and rigorous safety constraint satisfaction.

Original authors: Zhenxu Zhao, Ji Wang, Weiyao Lan

Published 2026-02-10

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Zhenxu Zhao, Ji Wang, Weiyao Lan

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 teaching a self-driving car how to follow a lead vehicle in a high-speed convoy (a "platoon"). This is much harder than it sounds because of three "invisible enemies":

The Fog (Uncertainty): You don't know the exact weight of the car, the friction of the road, or how much the wind is pushing it.
The Lag (Input Delay): When you press the gas pedal, there is a tiny, unpredictable delay before the engine actually responds.
The Ghost (Disturbances): Random gusts of wind or bumps in the road act like invisible hands pushing the car off course.

This paper presents a "smart brain" (a control framework) that allows a vehicle to navigate these three enemies safely, even if it starts with almost zero knowledge about its own mechanics.

Here is how the "brain" works, broken down into three simple steps:

1. The "Detective" Phase (Data-Driven Identification)

Most robots are programmed with a manual: "If you weigh 1,000kg and the wind is 5mph, do X." But this paper assumes the robot has no manual.

Instead, the robot acts like a detective. For the first few seconds, it doesn't even try to drive; it just "listens" to how it moves when it's idling. It uses a mathematical tool called Krylov DMD (think of this as a high-speed pattern recognition engine). By watching how the car wobbles or reacts to tiny movements, the detective reconstructs the "manual" from scratch. It figures out the car's weight, the wind's strength, and even exactly how long the "lag" is.

2. The "Safety Bubble" (Control Barrier Functions)

Now that the robot is learning, it could easily make a mistake and crash. To prevent this, the researchers use something called Control Barrier Functions (CBFs).

Imagine the car is surrounded by an invisible, glowing "safety bubble." As long as the car is inside the bubble, it can drive however it wants. But as the car approaches the edge of the bubble (the danger zone), the CBF acts like a powerful, invisible spring. The closer you get to the edge, the harder the spring pushes you back toward the center. This ensures that even while the "detective" is still learning, the car is mathematically forbidden from crossing the line into a collision.

3. The "Blindfolded Driver" (Output Feedback)

In the most advanced part of the paper, the researchers imagine the car is "blindfolded." It can't see its own internal engine temperature or its exact speed; it can only see its position on the road (the "output").

To solve this, they build a "Digital Twin" (an Observer). The car looks at its position, compares it to where it expected to be, and says, "Aha! I'm slightly further left than I thought; there must be a gust of wind pushing me." It uses this logic to "hallucinate" its internal states with incredible accuracy, allowing it to drive smoothly even though it's technically flying blind.

The Big Picture

In short, this paper provides a way for a machine to:

Learn its own physics by watching itself move.
Predict how much lag it has to deal with.
Guarantee it will never hit anything by using a mathematical "safety bubble."

The Result: A vehicle that can join a convoy, figure out its own quirks on the fly, and stay perfectly in line—all while staying safely away from the car in front of it.

Technical Summary: Data-Driven Safe Output Regulation of Strict-Feedback Linear Systems with Input Delay

1. Problem Statement

The paper addresses the safe output regulation problem for linear systems characterized by a strict-feedback structure. The core challenge lies in the high degree of uncertainty: the plant parameters, external disturbances, and the magnitude of the input delay are all unknown.

Specifically, the authors consider a system where:

Input Delay ( $D$ ): The control signal is delayed, which can be modeled as a transport Partial Differential Equation (PDE).
Strict-Feedback Structure: The system is a chain of integrators with unknown coefficients, making it highly sensitive to modeling errors.
Output-Feedback Constraints: In practical scenarios, only the state furthest from the control input (the output) is measurable, necessitating the estimation of unmeasured internal states and disturbance dynamics.
Safety Requirements: The system must track a reference trajectory while rigorously ensuring that the error states remain within a predefined "safe set" (defined by Control Barrier Functions), even during the initial transient phase or when starting from an unsafe state.

2. Methodology

The authors propose a multi-stage framework that combines data-driven identification with robust safe control.

A. Data-Driven Identification (The "Learning" Phase):

Koopman Operator & Krylov DMD: To extract system dynamics without a model, the authors leverage Koopman operator theory. They use Krylov Dynamic Mode Decomposition (DMD) to process snapshots of data. By utilizing the initial "uncontrolled" period (the interval $[0, D]$ where the input delay prevents control from affecting the plant), they extract the system matrix $A$ , the disturbance matrix $S_d$ , and the disturbance gain $\bar{G}$ using only output data.
Batch Least-Squares Identification (BaLSI): Once the structural dynamics are identified via DMD, the remaining unknown parameters—specifically the input delay $D$ and the input gain $b$ —are identified using a batch least-squares method.

B. Control Design (The "Regulation & Safety" Phase):

Backstepping & Delay Compensation: To handle the strict-feedback structure and the input delay, the authors employ a three-step backstepping transformation. This converts the complex delayed system into a stable target system.
Control Barrier Functions (CBFs): To guarantee safety, the authors design high-relative-degree CBFs. They introduce a "rescue" function ( $\varsigma(t)$ ) that allows the system to return to the safe set within a user-defined finite time if it starts in an unsafe state.
Observer Design: For the output-feedback case, an extended state observer is constructed to estimate the unmeasured plant states and the exogenous disturbance signals, using the identified parameters to ensure convergence.

3. Key Contributions

Comprehensive Uncertainty Handling: Unlike previous works that assume a small number of unknown parameters, this framework handles a "nearly entire" unknown plant, including disturbances and delays.
Integration of DMD and CBF: This is, to the authors' knowledge, the first work to combine Koopman-based Krylov DMD for data-driven identification with CBF-based safe control for continuous-time systems with input delays.
Output-Feedback Capability: The methodology provides a way to achieve safe regulation using only measurable output signals and actuation data, rather than requiring full-state knowledge.
Robustness to Initial Unsafety: The design includes a mechanism to handle "initial unsafe" conditions, ensuring the system recovers to the safe set in a prescribed time.

4. Results and Significance

Results:

Theoretical Guarantees: The paper proves that the unknown parameters are identified exactly in finite time ( $t_f$ ) and that the tracking error converges exponentially to zero while maintaining safety.
Simulation Validation: The method was tested on a vehicle platooning application. The simulation demonstrated that even with significant mass, damping, and delay uncertainties, the vehicles could maintain safe inter-vehicle distances and track the leader's trajectory, successfully recovering from unsafe initial distances.

Significance:
This research bridges the gap between data-driven control (which excels when models are unknown) and safety-critical control (which is essential for autonomous systems). It provides a mathematically rigorous way to deploy autonomous agents (like self-driving cars or drones) in complex, uncertain environments where traditional model-based controllers would fail due to delays or unmodeled disturbances.