⚡ electrical engineering

Data-driven nonlinear output regulation via data-enforced incremental passivity

This paper proposes a data-driven nonlinear output regulation framework that achieves asymptotic tracking and disturbance rejection by designing a feedback controller to ensure incremental passivity via data-dependent linear matrix inequalities, thereby decoupling the controller design from the internal model and enabling efficient solutions for both regulation and non-zero equilibrium stabilization problems.

Original authors: Yixuan Liu, Meichen Guo

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yixuan Liu, Meichen Guo

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 teach a very stubborn, wiggly robot dog to walk in a perfect circle, even though the floor is shaking under it and you don't have the robot's blueprints. You can't see the gears or the math inside its brain; you only have a video camera recording its movements and the commands you sent it. This is the challenge the authors, Yixuan Liu and Meichen Guo, tackled in their new work.

Their main finding is a clever "data-driven" recipe that lets you build a controller for these tricky, nonlinear systems using only recorded experiments, without ever needing to know the exact equations of how the robot moves. They proved that by using a specific mathematical trick called "incremental passivity," they can force the system to follow a desired path perfectly, eventually making the error (the distance between where it is and where it should be) shrink to exactly zero.

The "Black Box" Problem

Usually, to control a complex machine, engineers need a perfect mathematical model—a set of rules describing exactly how the machine reacts to every push and pull. But in the real world, things like ecosystems, brain networks, or even a simple pendulum can be too messy to model perfectly. The authors argue against the idea that you must have a perfect model to get a perfect result. Instead, they show you can skip the modeling step entirely and go straight from "data" to "control."

However, they are very clear about what they don't do. They don't use "black box" guessing games like neural networks that need massive amounts of data and might fail without a guarantee. They also don't rely on finding a "stabilizing" controller first to get the ball rolling, which is a common requirement in other advanced methods. Their method is direct: here is the data, here is the math, here is the controller.

The "Magic Mirror" and the "Ghost"

To explain how they do it, let's use a metaphor. Imagine the robot dog is a chaotic dancer. You want it to dance to a specific song (the reference), but someone is constantly bumping into it (the disturbance).

Step 1: The "Passivity" Makeover
First, the authors design a controller that acts like a "magic mirror." They take the chaotic dancer and, using only past video footage of the dance, they tweak the dancer's moves so that they become "incrementally passive." In plain English, this means they make the dancer so well-behaved that if you nudge them, they don't go wild; instead, they gently absorb the energy and settle back into rhythm. They proved that you can find this "nudge" (the feedback controller) by solving a specific set of math puzzles called Linear Matrix Inequalities (LMIs) using the data you collected.

Step 2: The "Ghost" Partner
Once the dancer is well-behaved, they introduce a "ghost" partner. This ghost is a simple, predictable machine (an internal model) that knows exactly what the song and the bumps look like. The ghost doesn't need to know the dancer's secret moves; it just needs to know the rhythm of the music and the bumps. The authors connect the well-behaved dancer to this ghost. Because both are "passive" (well-behaved), they lock into a perfect harmony. The ghost leads, the dancer follows, and the bumps are ignored.

The "Decoupled" Advantage

Here is the really cool part: The authors designed the "magic mirror" (the controller) and the "ghost" (the internal model) separately. They are independent.

If the song changes (the reference changes), you don't have to re-teach the dancer how to move. You just swap out the ghost.
If the dancer's style changes slightly, you don't have to re-design the ghost.
This "decoupled" design is a huge win for efficiency. In older methods, if the song changed, you often had to re-simulate the entire system from scratch. Here, you just collect a little more data and update the ghost.

What They Actually Did (and Didn't Do)

The authors didn't just suggest this might work; they mathematically proved it works for a specific class of nonlinear systems. They showed that if you have enough data (specifically, a data set length $T$ that is at least as large as the number of functions you use to describe the system plus the input and disturbance dimensions), you can guarantee the error goes to zero.

They tested this with two simulations:

A Simple Pendulum: They simulated a swinging pendulum with a noisy wind blowing on it. Using data from just 20 samples (collected every 0.5 seconds), they designed a controller that made the pendulum swing exactly to the desired rhythm, no matter where they started it. The error dropped to zero in all cases.
A Non-Zero Equilibrium: They tackled a harder problem: stabilizing a system at a spot that isn't "zero." Imagine balancing a broom on your hand, but you want it to stay balanced at a weird angle, and you don't know exactly how much force is needed to hold it there. Their method found a way to stabilize it without ever needing to calculate that mysterious "holding force."

The Limits and the Future

The paper is very honest about its boundaries. Their "magic mirror" works best if the messy parts of the system (the nonlinearities) are known and can be cancelled out perfectly. They admit that if the noise in the data is random and unpredictable (not following a pattern like a sine wave), their current method might struggle. They suggest that future work will need to add "robustness" to handle that kind of messy noise.

Also, they note that their method currently assumes you can measure the system's speed (the derivative of the state) directly. If you have to guess the speed from noisy position data, that introduces uncertainty they haven't fully solved yet.

In short, Liu and Guo have built a bridge from raw data to perfect control for a specific type of wiggly, noisy system. They didn't just say "it's possible"; they built the bridge, showed the math that holds it up, and walked across it in a simulation to prove it doesn't collapse. It's a step toward controlling the messy, real world without needing a perfect map.

Technical Summary: Data-driven Nonlinear Output Regulation via Data-enforced Incremental Passivity

Problem Formulation
This work addresses the output regulation problem for a class of nonlinear systems subject to external disturbances, where the system dynamics are unknown. The plant is modeled as $\dot{x} = AZ(x) + Bu + Ew$ and $e = CZ(x) + Fw$, where $Z(x)$ is a known vector of continuous functions (including the state and nonlinear terms), but the matrices $A, B, C, E,$ and $F$ are unknown. The reference trajectory and disturbances are generated by a known linear exosystem $\dot{w} = Sw$ . The objective is to design a feedback control input $u$ using only a finite set of offline input-state-output data ( $D_S$ ) such that the regulation error $e$ asymptotically converges to zero, while ensuring all closed-loop solutions remain bounded.

Methodology
The proposed approach relies on enforcing incremental passivity on the closed-loop system via data-driven feedback, followed by interconnection with an internal model. The methodology proceeds in two main stages:

Data-Driven Incremental Passivation:
The first step designs a static state-feedback controller of the form $u = KZ(x) + v$, where $v$ is a virtual input. The goal is to render the closed-loop system incrementally passive with respect to the regulation error $e$ and the virtual input $v$ .
- Model-Based Foundation: The authors first establish that if there exist a gain $K$ and a positive definite matrix $P$ satisfying specific Linear Matrix Inequalities (LMIs), the system becomes incrementally passive.
- Data-Driven Extension: Leveraging the "fundamental lemma" of data-driven control, the authors derive conditions where the system matrices are replaced by data-dependent matrices constructed from $D_S$ . Specifically, they formulate a set of data-dependent LMIs (Theorem 1) involving matrices $Y$ and $G_2$ . Solving these LMIs yields the feedback gain $K = U_0 Y P^{-1}$ without explicitly identifying the system matrices $A, B, C, E, F$ . This design assumes the nonlinearities in $Z(x)$ are known and perfectly cancellable by the control input.
Regulator Synthesis via Internal Model:
Once the closed-loop system is rendered incrementally passive, an incrementally passive internal model is designed based on the known exosystem dynamics. The internal model is given by $\dot{\eta} = S\eta - \alpha \Xi e$ and $\tilde{v} = \Xi^\top \eta$ .
- Interconnection: The closed-loop system and the internal model are interconnected. By Lemma 1, the feedback interconnection of two incrementally passive systems remains incrementally passive.
- Stability Proof: The overall regulator is formed by closing the loop with a stabilizing gain on the error. Using a composite storage function and Barbalat's lemma, the authors prove that the regulation error $e(t)$ converges asymptotically to zero, provided the exosystem satisfies standard assumptions (simple roots on the imaginary axis) and the feasibility condition (existence of a linear steady-state solution) holds.

Key Contributions
The paper highlights four primary contributions:

Global Asymptotic Regulation from Data: The work solves the output regulation problem for nonlinear plants directly from offline data, guaranteeing asymptotic convergence of the regulation error to zero. This contrasts with recent data-driven results that either focus on linear systems or only guarantee bounded errors for nonlinear systems.
Decoupled Design: A significant feature is the decoupling of the feedback passivation controller design from the internal model design. The passivation controller is independent of the internal model, and the regulator design does not require data collected from the internal model. This offers high data efficiency and flexibility, as changes in the exosystem do not necessitate a complete redesign of the passivation controller.
Characterization of Passivation Controllers: The authors characterize feedback incremental passivation controllers for nonlinear systems and derive data-based LMIs for their design. This extends direct data-driven control beyond verification or analysis of dissipativity properties to the actual synthesis of passivity-enforcing controllers.
Non-Zero Equilibrium Stabilization: The framework is extended to solve the stabilization problem for non-zero equilibria where the equilibrium input is neither known nor estimated. By defining a virtual error based on the distance to the equilibrium, the output regulation framework is applied to achieve asymptotic stability without coordinate transformation.

Results and Validation
The theoretical results are validated through two numerical examples:

Simple Pendulum: A nonlinear pendulum subject to sinusoidal and constant disturbances is controlled. The data-driven regulator successfully drives the tracking error to zero from various initial conditions.
Non-Zero Equilibrium Stabilization: A nonlinear system is stabilized at a specific non-zero equilibrium point ( $x_e = [-2, -1]^\top$ ) despite an unknown equilibrium input and time-varying disturbances. The state trajectories converge to the target equilibrium.

Significance and Claims
The paper claims that its significance lies in providing a rigorous, data-driven solution to nonlinear output regulation that ensures global asymptotic stability without requiring explicit system identification. The decoupled architecture is highlighted as a major advantage over traditional augmented system approaches, particularly in data-driven settings where collecting data from the augmented system (including the internal model) is complex.

The authors are modest regarding the scope of their assumptions. They explicitly note that their approach assumes:

The nonlinearities in $Z(x)$ are known and perfectly cancellable (matched channels).
The experimental data is free of measurement noise (or that noise can be modeled by the exosystem).
The existence of a linear steady-state solution (Assumption 2).

Future work directions identified by the authors include handling general classes of noise via robust control techniques and extending the approach to output-feedback designs using only input-output data.

The "Black Box" Problem

The "Magic Mirror" and the "Ghost"

The "Decoupled" Advantage

What They Actually Did (and Didn't Do)

The Limits and the Future

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