⚡ electrical engineering

Change-Of-Bases Abstractions for Non-Linear Systems

This paper presents abstraction techniques that transform non-linear dynamical systems into linear or polynomial systems via change-of-basis transformations, enabling the use of linear analysis methods to infer invariants for the original non-linear systems across continuous, discrete, and hybrid domains.

Original authors: Sriram Sankaranarayanan

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 3.0

Original authors: Sriram Sankaranarayanan

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ 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 understand a chaotic, swirling storm. The wind speeds, temperatures, and pressures are changing in complex, non-linear ways. Trying to predict exactly what will happen next is incredibly difficult because the rules governing the storm are messy and tangled.

This paper proposes a clever trick to solve that problem: Change the lens through which you look at the storm.

Instead of trying to track the raw, messy variables (like wind speed at a specific point), the authors suggest inventing a new set of "virtual variables" that are combinations of the original ones. If you choose the right combination, the chaotic storm suddenly looks like a calm, straight line or a simple, predictable pattern.

Here is a breakdown of their ideas using everyday analogies:

1. The Core Idea: The "Magic Translator"

Think of a complex non-linear system (like a car engine with many interacting parts) as a foreign language that is very hard to read. The authors present a method to find a "Change-of-Bases" (CoB) transformation.

The Analogy: Imagine you are trying to solve a puzzle where the pieces are all jagged and irregular. It's impossible to see the picture. The authors' method is like finding a special pair of glasses that, when you put them on, magically reshape all those jagged pieces into perfect squares and circles. Suddenly, the puzzle becomes easy to solve.
The Reality: They take the original, messy equations and map them to new variables (the "glasses"). If the mapping is done correctly, the new system behaves like a simple, linear system (like a straight line) or a system of simple polynomials.

2. Why Do This? (The "Safety Net")

Why go through the trouble of translating the system? Because we have powerful tools to analyze simple, linear systems, but we don't have good tools for messy, non-linear ones.

The Analogy: Suppose you want to know if a rollercoaster will ever go off the tracks. Calculating the exact path of a real, twisting rollercoaster is hard. But if you can translate the rollercoaster's path into a straight line on a piece of paper, you can easily draw a box around it to prove it stays safe.
The Reality: Once the authors find this "translation," they use standard tools to find invariants (rules that never break, like "the car never goes above 100mph") for the simple, translated system. Because the translation is a perfect mirror, those rules automatically apply to the original, messy system.

3. How They Find the "Magic Glasses"

The hardest part is figuring out which new variables to create. You can't just guess; you need a systematic way to find the right combination.

The Analogy: Imagine you are a chef trying to find the perfect recipe. You have a list of ingredients (variables like $x$ $x$ , $y$ $y$ , $x^2$ $x^{2}$ , $xy$). You want to mix them together to create a new dish (the new system) that tastes simple (linear).
- The authors' method is like a smart filter. They start with a big bowl of all possible ingredient combinations.
- They test each combination: "If I use this mix, does the cooking process stay simple?"
- If a combination gets messy (the "flavor" gets too complex), they throw it out.
- They repeat this filtering process until only the "perfect" combinations remain. These remaining ingredients form the new, simple system.

4. What Kinds of Systems Can They Fix?

The paper claims this "magic translator" works for three types of systems:

Continuous Systems (ODEs): Like water flowing in a river or a chemical reaction happening over time.
Discrete Systems: Like computer programs that run in loops (e.g., a program counting numbers).
Hybrid Systems: A mix of both, like a self-driving car that drives continuously but makes sudden decisions (braking, turning) at specific moments.

5. Real-World Examples from the Paper

The authors tested their method on several real-world scenarios to prove it works:

The Toda Lattice: A model of particles connected by springs. They found a way to translate the complex bouncing of particles into a simple, linear system where the energy is conserved.
Collision Avoidance: A system modeling two airplanes trying to avoid crashing. They translated the complex trigonometric movements of the planes into a simple linear system to prove they wouldn't crash.
Computer Programs: They took a program that calculates the sum of squares (a math problem) and found a way to translate it into a linear version, allowing them to prove the program works correctly without errors.

Summary

In short, this paper is about simplifying the complex.

The authors built a mathematical "machine" that takes a messy, non-linear problem and searches for a specific translation that turns it into a clean, linear problem. Once the problem is clean, we can easily prove it is safe or correct. Then, we translate those safety rules back to the original messy problem, giving us confidence that the real-world system is safe, even though we couldn't analyze it directly.

They don't just say "it's possible"; they built a prototype tool that actually does this search and successfully simplified several difficult engineering and programming problems.

Technical Summary: Change-Of-Bases Abstractions for Non-Linear Systems

Problem Statement
The paper addresses the challenge of analyzing non-linear dynamical systems, including continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems modeled as transition systems, and hybrid systems combining both. Direct analysis of non-linear systems is often computationally intractable for automated verification tools. While techniques exist for linear or affine systems (e.g., polyhedral abstract interpretation), extending these to non-linear domains remains difficult. The core problem is to determine if a given non-linear system can be transformed into a simpler system (linear or bounded-degree polynomial) via a change-of-bases (CoB) transformation, such that invariants of the simpler system can be mapped back to infer invariants for the original system.

Methodology
The proposed methodology relies on finding a smooth mapping $\alpha: \mathbb{R}^n \to \mathbb{R}^m$ that relates the state variables of the original system to a new set of variables. The transformation is valid if it satisfies specific simulation conditions:

Simulation: The abstract system must simulate the original system. For continuous systems, this requires that the vector field $G$ of the abstract system satisfies $G(\alpha(\vec{x})) = J_\alpha \cdot F(\vec{x})$ , where $J_\alpha$ is the Jacobian of $\alpha$ and $F$ is the vector field of the original system. For discrete systems, the update functions must satisfy $F'(\alpha(\vec{x})) = \alpha(F(\vec{x}))$ .
Algebraic/Linear Abstraction: The goal is to find a map $\alpha$ such that the resulting dynamics in the abstract variables are described by polynomials of a bounded degree $d$ . If $d=1$ , the abstraction is affine (linear).

The authors propose a search technique based on vector space closure:

Basis Selection: An initial basis of functions $B_0$ (e.g., monomials up to a certain degree) is selected.
Iterative Refinement: The algorithm constructs a sequence of nested vector spaces $V_0 \supseteq V_1 \supseteq \dots \supseteq V^*$ $V_{0} \supseteq V_{1} \supseteq \dots \supseteq V^{*}$ .
- For continuous systems, a space $V$ is refined to $V_{new}$ by retaining only functions $g \in V$ whose Lie derivative $L_F(g)$ lies within the space of power products of $V$ up to degree $d$ (denoted $V\langle d \rangle$ ).
- For discrete systems, the refinement retains functions $g$ where the functional pre-image $f_{pre}(g, t)$ under every transition $t$ lies within $V\langle d \rangle$ .
Convergence: The process converges to a subspace $V^*$ that is "d-closed." The basis functions of $V^*$ form the components of the transformation $\alpha$ . If $V^*$ is non-trivial, it yields an abstract system with dynamics of degree at most $d$ .

Key Contributions

Generalized CoB Transformations: The paper defines CoB transformations for continuous, discrete, and hybrid systems, providing formal conditions under which a transformation defines a valid simulation.
Search for Degree-Bounded Abstractions: Unlike previous work that focused primarily on linearizing transformations or specific templates (ansatz), this paper presents a general search technique for finding transformations that result in polynomial systems with a user-defined degree bound $d$ . This includes linear ( $d=1$ ) and higher-degree polynomial abstractions.
Unification of Discrete and Continuous: The theory is extended from purely continuous ODEs to discrete transition systems and hybrid systems. For hybrid systems, the approach merges the closure conditions for Lie derivatives (continuous flow) and functional pre-images (discrete jumps) across multiple locations.
Connection to Conserved Quantities: The paper demonstrates that the technique generalizes the search for strong and constant scale consecution (conserved quantities). It shows that if a system has linearly independent conserved quantities, the iterative search is guaranteed to find a linearizing transformation.
Implementation and Evaluation: A prototype implementation was developed and tested on a variety of benchmarks, including ODEs (e.g., Toda lattice, Fermi-Pasta-Ulam-Tsingou system, biochemical networks) and discrete programs (e.g., Fermat factorization, geometric summation).

Results
The prototype implementation successfully identified non-trivial CoB transformations for several non-linear benchmarks:

Continuous Systems: The tool found linearizing transformations for systems like the Toda lattice with boundary particles and a biochemical reaction network. For the "Roundabout" collision avoidance model, it discovered a 14-dimensional affine abstraction. In cases where no non-trivial linearization existed (e.g., Van der Pol oscillator, Fitzhugh-Nagumo), the algorithm correctly converged to a trivial solution (the zero map).
Discrete Systems: The approach successfully linearized programs with non-linear assignments, such as a program computing the sum of squares and a geometric summation loop, allowing polyhedral analysis to derive non-linear invariants for the original code.
Hybrid Systems: The technique was applied to hybrid models, successfully handling the interplay between continuous flows and discrete jumps.

Significance and Claims
The authors claim that this work provides a systematic framework for discovering abstractions that simplify non-linear dynamics, thereby enabling the use of mature linear analysis techniques (like polyhedral abstract interpretation) on complex non-linear problems.

The paper emphasizes that the search for a CoB transformation is distinct from and potentially more powerful than directly searching for conserved quantities, as it can implicitly handle transformations involving exponentials, sines, and cosines by reasoning purely over polynomial vector spaces.
The authors note that while the technique is powerful, it is not universal; many transformations do not yield valid abstractions, and the search may converge to trivial results for non-integrable systems.
The work extends previous results (specifically from HSCC 2011) by generalizing from linearizing transformations to degree-bounded polynomial transformations and by extending the theory to discrete and hybrid systems.
The paper modestly acknowledges limitations, such as the current reliance on manual translation for guard sets and invariants in the prototype, and the computational cost of quantifier elimination for deriving abstract guards. It identifies the application of Lie symmetries to reduce the search space and the use of abstractions for proving termination as promising future directions.