⚡ electrical engineering

Simultaneous Online System Identification and Control using Composite Adaptive Lyapunov-Based Deep Neural Networks

This paper presents a novel control framework that simultaneously achieves online system identification and trajectory tracking for uncertain nonlinear systems by employing composite adaptive Lyapunov-based update laws for all layers of a deep neural network, thereby guaranteeing uniform ultimate boundedness and exponential convergence of errors while enabling robust performance during intermittent state feedback loss.

Original authors: Omkar Sudhir Patil, Emily J. Griffis, Wanjiku A. Makumi, Warren E. Dixon

Published 2026-05-25

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CC BY 4.0

Original authors: Omkar Sudhir Patil, Emily J. Griffis, Wanjiku A. Makumi, Warren E. Dixon

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 robot to walk a tightrope. The robot has a "brain" (a Deep Neural Network, or DNN) that tries to guess how the wind and gravity will push it around so it can adjust its balance.

The Problem with the Old Way
Traditionally, engineers would teach this robot brain offline. They'd run thousands of simulations, let the robot learn, and then freeze its brain. Once the robot starts walking the real tightrope, its brain can't learn anything new. If the wind changes unexpectedly, the robot might fall because its frozen brain doesn't know how to adapt.

Some newer methods let the robot learn while it walks. However, these methods only care about one thing: staying on the rope. They adjust the brain's internal settings just enough to keep the robot from falling, but they don't actually teach the brain what the wind is doing. The robot might stay upright, but it doesn't truly "understand" the physics of the situation. If the robot loses its balance sensors for a moment (like a sudden blackout), it has no idea how to predict the wind and keep going.

The New Solution: The "Double-Duty" Brain
This paper introduces a new method where the robot learns two things at the same time:

How to stay on the rope (Tracking Control).
How the wind actually works (System Identification).

Think of it like a student taking a test.

Old Method: The student just memorizes the answers to the specific questions on the test. If the teacher asks a slightly different question, the student fails.
This Paper's Method: The student not only answers the test questions correctly but also writes down a detailed study guide explaining why the answers are what they are. If the teacher asks a new question later, the student can use their study guide to figure it out.

How It Works (The Analogy)
The authors created a special "learning rule" (an adaptation law) that acts like a double-check system:

The "Right Now" Check (Tracking Error): The robot looks at where it is versus where it should be. If it's drifting, it adjusts its brain to correct the path. This is the standard way.
The "What If" Check (Prediction Error): This is the magic part. The robot has a special "observer" (like a second set of eyes) that tries to guess what the wind force should be based on the robot's movement. It compares this guess to what the robot's brain predicted the wind would be.
- If the brain's guess is wrong, the robot doesn't just fix the path; it rewrites the brain's internal rules to make the guess more accurate for the future.

By combining these two checks, the robot's brain becomes a true expert. It learns the actual physics of the system, not just how to cheat its way to staying upright.

The "Flashlight" Requirement (Persistence of Excitation)
The paper mentions a condition called "Persistence of Excitation" (PE). Imagine trying to learn the shape of a dark room. If you only walk in a straight line, you only learn about the floor in front of you. To learn the whole room, you need to move around, touch the walls, and look in different directions.

PE means the robot needs to move around enough (not just stay still or move in a boring loop) so the brain gets enough "data" to learn the full picture of the wind and gravity. If the robot moves enough, the learning becomes incredibly fast and accurate.

The Superpower: Surviving Blackouts
The biggest win of this method is what happens when the robot loses its sensors (the "tightrope" sensors go dark).

Old Methods: When sensors fail, the robot panics. It doesn't know the wind, so it falls.
This Method: Because the robot's brain has already learned the "study guide" (the system identification), it can predict what the wind will do even without sensors. It uses its learned model to keep walking through the blackout until the sensors come back online.

The Results
The authors tested this on two things: a robotic arm (like a factory robot) and an underwater vehicle.

Accuracy: The new method kept the robot on its path much better than the old methods.
Understanding: The robot's brain learned the physics of the system much more accurately.
Resilience: When the sensors were turned off (simulating a blackout), the new method kept the robot stable, while the old methods struggled or failed.

In Summary
This paper gives robots a brain that doesn't just react to mistakes; it learns the rules of the game while playing. This allows the robot to perform better, understand its environment deeply, and keep working even when it temporarily loses its senses.

Technical Summary: Simultaneous Online System Identification and Control using Composite Adaptive Lyapunov-Based Deep Neural Networks

Problem Formulation
Deep neural network (DNN)-based controllers are widely used for uncertain nonlinear dynamic systems, yet traditional approaches typically rely on offline pre-training. In these scenarios, DNN weights are fixed during execution, limiting adaptability and raising concerns regarding model reliability when operating outside the training dataset. While recent Lyapunov-based (Lb) adaptive control methods allow for online weight updates to ensure trajectory tracking, they rely solely on tracking error feedback. Consequently, these methods guarantee tracking error convergence but fail to provide guarantees on system identification (parameter estimation). Without accurate identification, the DNN cannot generalize to unexplored trajectories, and the controller cannot effectively compensate for uncertainties during intermittent loss of state feedback. Furthermore, extending composite adaptation laws—which combine tracking and prediction errors—to DNNs is technically challenging due to the nonlinearity-in-parameters (NIP) structure of DNNs, where inner-layer weights are embedded within nested nonlinear activation functions.

Methodology
The paper proposes a composite adaptive Lb-DNN controller that performs simultaneous online system identification and trajectory tracking for second-order nonlinear systems. The core methodology involves the following components:

Dynamic State-Derivative Observer: To address the difficulty of obtaining the prediction error (the discrepancy between actual dynamics and the DNN estimate) without direct access to the unknown state derivative, a dynamic state-derivative observer is designed. This observer estimates the drift function $f(x, \dot{x})$ using available state measurements and the control input, effectively eliminating the need for the unknown state derivative in the adaptation law.
Composite Adaptation Law: A new prediction error formulation is constructed using the observer's estimate of the drift. This prediction error is combined with the tracking error to form a least-squares-based composite adaptation law.
Handling NIP Structure: To overcome the challenges posed by the nested NIP structure of DNNs, the adaptation law utilizes the Jacobian of the DNN with respect to its weights. A first-order Taylor series expansion is applied to the DNN output to express the prediction error in terms of the parameter estimation error, allowing the adaptation law to be formulated despite the nonlinearity.
Stability Analysis: A combined Lyapunov-based stability analysis is performed. The analysis incorporates the tracking error, state-derivative estimation error, and DNN weight estimation errors. It utilizes a projection operator to ensure parameter estimates remain within a bounded set and employs a time-varying least-squares gain matrix with a forgetting factor.

Key Contributions
The paper claims the following primary contributions:

First Result on Simultaneous Identification and Control: This work presents the first result achieving simultaneous online system identification and trajectory tracking for nonlinear systems using online updates for all layers of a DNN.
Novel Prediction Error Formulation: The development of a prediction error based on a dynamic state-derivative observer, which is integrated with the tracking error to construct a composite adaptation law suitable for NIP uncertainties.
Stability Guarantees: The analysis proves that the tracking error, state-derivative estimation error, and DNN weight estimation errors are uniformly ultimately bounded (UUB).
Exponential Convergence under PE: Under the persistence of excitation (PE) condition, the tracking and weight estimation errors are shown to exponentially converge to a neighborhood of the origin. The size of this neighborhood and the convergence rate depend on control gains and the PE factor.
Generalization for Intermittent Feedback: By achieving accurate system identification, the identified DNN model can be propagated forward in time to predict and compensate for uncertain dynamics during intermittent loss of state feedback, provided sufficient dwell-time conditions are met.

Results
Comparative simulations were conducted on two systems: a two-link robot manipulator and an unmanned underwater vehicle (UUV).

Robot Manipulator: The developed composite adaptive Lb-DNN controller demonstrated significant performance improvements over a tracking-error-based adaptive Lb-DNN, an observer-based disturbance rejection controller, a nonlinear PD controller, and nonlinear model predictive control (MPC). Specifically, the composite method reduced the root mean square (RMS) tracking error by approximately 50% compared to the tracking-error-based baseline and achieved a 72% improvement in the mean function approximation error on a test dataset, indicating superior generalization. The control effort was comparable to or lower than the baselines.
Unmanned Underwater Vehicle (UUV): In simulations involving intermittent loss of state feedback, the composite adaptive Lb-DNN maintained superior tracking performance compared to all baselines. While the tracking-error-based method's predictions diverged rapidly during feedback loss due to a lack of parameter estimation guarantees, the composite method successfully predicted and compensated for drift dynamics. The observer-based approach performed comparably when feedback was available but degraded significantly during feedback loss and required higher control effort.

Significance
The paper asserts that this work provides a foundational step toward robust adaptive control where the controller not only tracks a trajectory but also learns the underlying system dynamics. By ensuring parameter estimation, the identified model gains the ability to generalize beyond the explored trajectory. This capability is critical for safety-critical applications involving intermittent sensor failures, where the system must rely on an internal model to maintain stability. The results demonstrate that integrating system identification directly into the control loop via composite adaptation yields tangible performance benefits in both tracking accuracy and dynamic compensation compared to methods that focus solely on tracking error minimization.

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