⚡ electrical engineering

Boosting the transient performance of reference tracking controllers with neural networks

This paper extends the Performance Boosting framework to reference tracking by characterizing a set of nonlinear controllers that preserve stability while utilizing expressive neural networks to optimize transient performance and ensure robustness against dynamic uncertainties.

Original authors: Nicolas Kirsch, Leonardo Massai, Giancarlo Ferrari-Trecate

Published 2026-02-05

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CC BY 4.0

Original authors: Nicolas Kirsch, Leonardo Massai, Giancarlo Ferrari-Trecate

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 from point A to point B. You give it a basic instruction: "Keep your eyes on the target and walk straight toward it." This is like a standard controller. It works well enough to get the robot to the destination eventually, but it might stumble, bump into furniture, or take a very long, winding path to get there. It's good at the finish line, but not great at the journey.

This paper introduces a new method called rPB (reference Performance Boosting) to fix that journey. Here is how it works, using simple analogies:

1. The Problem: The "Stubborn" Base Controller

Think of the robot's basic controller as a loyal but stubborn GPS.

The Good: It guarantees the robot will eventually reach the destination, no matter what.
The Bad: It doesn't care how it gets there. It might drive through a wall, spin in circles, or waste energy just to get to the point. It lacks "style" or efficiency during the trip.

2. The Solution: The "Creative Co-Pilot" (rPB)

The authors created a "Co-Pilot" (the neural network) that sits next to the GPS.

How it works: Instead of telling the robot where to go, the Co-Pilot tells the GPS to temporarily shift the target.
The Analogy: Imagine you are walking toward a door, but there is a giant vase in the way. The GPS says, "Walk straight to the door." The Co-Pilot whispers to the GPS, "Okay, but for the next few seconds, pretend the door is actually 2 feet to the left."
- The robot walks toward that "fake" door, smoothly gliding around the vase.
- Once the vase is behind them, the Co-Pilot whispers, "Okay, the door is back where it was."
- The robot seamlessly returns to the real target without ever stopping or crashing.

3. The Magic Trick: "Safety Guarantees"

Usually, when you add a smart computer (like a Neural Network) to a robot, it becomes unpredictable. You might train it to avoid a vase, but then it might accidentally drive off a cliff because it learned something weird.

The paper's big breakthrough is that they figured out a way to build this Co-Pilot so that it is mathematically impossible for it to break the robot's safety rules.

They proved that as long as the Co-Pilot follows a specific mathematical structure (like a "contract"), it can be as creative and flexible as it wants to optimize the path, but it will never lose the guarantee that the robot will eventually reach the target.
It's like giving a race car driver a completely free rein to drive however fast or fancy they want, with a magical rule that says, "You can never leave the track."

4. The "One-Size-Fits-All" Training

In the past, if you wanted a robot to handle different targets (e.g., "Go to the kitchen," then "Go to the garage"), you might have to train a new brain for each specific destination.

With rPB, the robot learns one single brain that understands the concept of shifting targets.

The Analogy: Instead of memorizing a map for every single street in the city, the robot learns the rules of traffic and how to navigate around obstacles.
The paper shows that after training on a few scenarios, this single brain could handle many different targets it had never seen before, still avoiding obstacles and taking the shortest path.

5. Real-World Test: The Robot Dance

The authors tested this on two small robots moving through a room full of obstacles (like a mountain range of furniture).

Without the Co-Pilot: The robots followed the basic GPS, crashed into obstacles, or took very long, inefficient paths.
With rPB: The robots danced around the obstacles, taking the shortest, smoothest paths possible, and never crashed. They did this even when the starting points and destinations changed randomly.

Summary

The paper presents a way to upgrade a robot's "brain" so it can be smarter and faster during the trip, without risking the safety guarantee that it will actually arrive at the destination. It turns a rigid, "get there eventually" controller into a flexible, "get there efficiently and safely" controller, all while using a single training session that works for many different goals.

Technical Summary: Boosting the Transient Performance of Reference Tracking Controllers with Neural Networks

Problem Formulation

Reference tracking is a fundamental objective in control systems across domains such as power systems, robotics, and aerospace. While traditional controllers (e.g., PID) ensure steady-state accuracy, they often fail to explicitly optimize transient performance metrics, such as minimizing energy consumption, reducing overshoot, or avoiding obstacles in the presence of nonlinearities and disturbances.

Existing approaches to enhance nonlinear tracking include Model Predictive Control (MPC), which offers flexibility but suffers from high computational costs in real-time; adaptive control, which often provides only local guarantees; and neural network (NN) based methods, which lack formal closed-loop stability and performance guarantees. The authors aim to bridge this gap by developing a framework that leverages the flexibility of neural networks to optimize arbitrary transient cost functions while rigorously preserving $L_p$ -stability and reference tracking properties.

Methodology

The paper introduces rPB (reference Performance Boosting), an extension of the existing Performance Boosting (PB) framework tailored for reference tracking tasks. The methodology relies on an Internal Model Control (IMC) architecture and the following key theoretical steps:

Complete Parametrization of Tracking Controllers:
The authors characterize the complete set of nonlinear controllers that preserve the tracking properties of a given baseline controller. They demonstrate that any such controller can be parametrized by a free causal operator $M$ within an IMC structure. In this structure, the control input $u$ is computed as $u = M(\hat{w}, x_{ref})$ , where $\hat{w}$ is the internal estimate of the disturbance derived from the difference between the actual system state and the nominal model prediction.
Stability and Tracking Guarantees:
- Exact Model Case: If the operator $M$ generates $\ell_p$ signals for $\ell_p$ inputs, the closed-loop system preserves the $L_p$ -stability of the baseline system and ensures asymptotic reference tracking for any reference signal in the set $X_{ref}$ .
- Robustness to Model Mismatch: The framework is extended to handle imperfect model knowledge. By modeling the mismatch as an incrementally finite-gain $\ell_p$ -stable (i.f.g $\ell_p$ -stable) operator, the authors prove that reference tracking is preserved provided the incremental gain of $M$ is sufficiently small relative to the gain of the mismatch and the system dynamics.
Neural Network Implementation:
To make the optimization tractable, the infinite-dimensional operator $M$ is decomposed into two components: $M = M_1 \circ M_2$ .
- $M_1$ is chosen from a class of parametrized $L_p$ -stable operators (specifically, contractive Recurrent Equilibrium Networks or RENs) to ensure stability.
- $M_2$ is a Multi-Layer Perceptron (MLP) with bounded outputs (e.g., using sigmoid activations) to handle the reference signal.
  This decomposition allows the problem to be reformulated as an unconstrained optimization over the network parameters $\theta$ . The authors utilize backpropagation-through-time to minimize a finite-horizon loss function that penalizes tracking errors, collisions, and other transient costs.

Key Contributions

rPB Framework: The introduction of a new neural network control method specifically for reference tracking that decouples performance optimization from stability constraints.
Generalization: Unlike standard PB which may require retraining for specific references, the rPB characterization is independent of the specific signal to be tracked. This allows a single training process to generalize across a wide range of reference signals.
Robustness Guarantees: Theoretical proof that reference tracking guarantees are maintained under model mismatch, provided the mismatch satisfies specific incremental stability conditions.
Unconstrained Optimization: A formulation that transforms the constrained nonlinear optimal control problem into an unconstrained problem over neural network parameters, enabling the use of efficient gradient-based optimizers.

Experimental Results

The authors validated the rPB framework using a simulated cooperative robotics scenario involving two point-mass robots subject to nonlinear drag forces and an integral baseline controller.

Benchmark against Standard PB (bPB):
In a "tight corridor" task, the standard PB controller (trained on a single specific target pair) failed to generalize to new target configurations, resulting in collisions with obstacles. In contrast, rPB, trained on a distribution of reference pairs, successfully generated collision-free, near-optimal trajectories for unseen targets, demonstrating superior generalization and transient performance.
Complex Environment (Mountain Range):
In a more complex environment with an array of obstacles, rPB was trained on 2,000 rollouts. The results showed that the controller could navigate various gaps between obstacles to minimize path length. Across 500 test scenarios with varying initial conditions and targets, the rPB controller achieved zero collisions, whereas the baseline controller alone resulted in collisions. The system successfully optimized transient behavior (avoiding obstacles) while maintaining the steady-state tracking capability of the baseline.

Significance and Claims

The paper claims that rPB offers a principled way to optimize generic transient costs in nonlinear reference tracking problems without sacrificing formal stability guarantees. The primary significance lies in the ability to:

Optimize Transients: Explicitly minimize complex, non-convex cost functions (e.g., obstacle avoidance) that traditional controllers cannot handle.
Ensure Safety: Maintain $L_p$ -stability and reference tracking guarantees even when using expressive neural network models.
Generalize Efficiently: Achieve robust performance across a wide range of reference signals with a single training phase, addressing a key limitation of standard PB and other learning-based control methods.

The authors conclude that this approach represents a significant improvement over standard PB frameworks and holds potential for complex real-world applications where both safety and performance optimization are critical. Future work is suggested to extend rPB to distributed control architectures and systems with input-to-output representations.