⚡ electrical engineering

Bernstein-based polynomial approach to study the stability of switched systems and formal verification using HOL Light

This paper proposes a preliminary approach to study the stability of switched systems by transforming them into polynomial expressions via Bernstein interpolation to construct Lyapunov functions, which are then formally verified using the HOL Light proof assistant.

Original authors: Loïc Michel

Published 2026-06-04

📖 4 min read☕ Coffee break read

CC BY 3.0

Original authors: Loïc Michel

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 driving a car that has two different engines: a Sport Engine for racing and a Eco Engine for saving fuel. A "switched system" is like a car that instantly jumps between these two engines based on a rule (like "if the road is steep, use Sport; if it's flat, use Eco").

The big question for engineers is: Will this car stay on the road and not crash, no matter how often it switches engines?

This paper proposes a clever new way to answer that question using math, specifically a method called Bernstein Interpolation. Here is how the author breaks it down:

1. The Problem: Too Many Rules

Usually, to check if the car is safe, you have to look at the Sport Engine rules and the Eco Engine rules separately, then try to figure out what happens when you flip the switch back and forth. It's like trying to predict the weather by looking at two different forecast models and guessing how they mix. It gets messy and complicated.

2. The Solution: Blending the Engines into One Smooth Recipe

The author suggests a trick: instead of treating the "switch" as a sudden jump, let's describe it as a smooth, continuous curve using a special type of math called Bernstein polynomials.

Think of it like a dimmer switch for a light, rather than an on/off button.

Instead of saying "Engine A is ON, Engine B is OFF," the math creates a single, smooth formula that blends the two engines together.
If the switch is 30% toward Sport and 70% toward Eco, the formula creates a "hybrid" engine that is exactly 30% Sport and 70% Eco.
This turns a complex, jumping system into one single, smooth polynomial equation (a fancy algebraic recipe).

3. The Safety Check: The "Lyapunov" Test

Once the car is described by this single smooth recipe, the author wants to prove it's safe. In engineering, there's a famous safety test called a Lyapunov function.

The Metaphor: Imagine the car is a ball rolling inside a bowl. If the ball always rolls toward the bottom (the center), the system is stable. If it rolls toward the edge and falls out, it's unstable.
The Lyapunov function is like a mathematical map that proves the ball must roll to the bottom.

4. The "HOL Light" Computer Lawyer

Here is the most unique part of the paper. Usually, humans check these safety maps, and they can make mistakes.

The author uses a computer tool called HOL Light.
Think of HOL Light as a super-strict computer lawyer. It doesn't just guess; it reads the mathematical proof step-by-step and verifies every single logical move with 100% certainty.
If the computer lawyer says, "Yes, this Lyapunov function proves the ball stays in the bowl," then you know for a fact the system is stable. There is no room for human error.

What the Paper Actually Did

The author didn't invent a new car or a new engine. They demonstrated this method with two examples:

A Stable System: They showed a car that naturally stays safe. Their method successfully blended the engines and the computer lawyer confirmed the safety.
An Unstable System: They showed a car that naturally crashes. Their method blended the engines, and the computer lawyer correctly identified that the safety map failed (the ball would roll out of the bowl).

The Bottom Line

This paper is a "preliminary work," meaning it's a proof-of-concept. The author is saying:
"We found a way to turn a complex, jumping system into a single smooth math formula using Bernstein polynomials. Then, we used a computer lawyer (HOL Light) to formally prove whether that formula represents a safe system or a dangerous one."

The goal is to make the process of checking if these switching systems are safe more reliable and easier to verify using computers.

Technical Summary: Bernstein-based Polynomial Approach to Switched Systems Stability and Formal Verification

Problem Statement
Switched systems, composed of multiple subsystems governed by a switching signal, present significant challenges in stability analysis. Traditional methods often struggle to provide a unified mathematical description of the system's full behavior, particularly when the switching signal is complex (e.g., state-dependent or time-dependent). While numerous methods exist to analyze these systems, there is a need for a formulation that can transform the discrete switching logic into a continuous polynomial expression, thereby facilitating the construction of Lyapunov functions and enabling rigorous formal verification.

Methodology
The paper proposes a strategy to reformulate switched systems using Bernstein interpolation. The core approach involves the following steps:

Polynomial Reformulation: The switching signal $\sigma$ , which dictates the activation of subsystems $f_i$ , is approximated using Bernstein polynomials. By expressing the switching signal as a linear combination of Bernstein basis functions, the entire switched system (originally defined by piecewise differential equations) is transformed into a single, unified polynomial expression.
- The system is initially described as a convex combination of subsystems: $\dot{x} = \sum f_i(x)\sigma_i$ , where $\sum \sigma_i = 1$ and $\sigma_i \geq 0$ .
- The switching signal $\sigma_i$ is replaced by its Bernstein series approximation $B_{i,m}^\sigma(\sigma(\nu), \nu)$ , resulting in a "B-switched system" described by a single polynomial differential equation.
Handling Switching Types: The method accommodates different switching behaviors:
- State-dependent switching: Modeled using sign functions (e.g., $\sigma \equiv \text{sign}(\delta x_1 x_2)$ ), approximated by Bernstein polynomials of a specific degree $m$ .
- Time-dependent switching: Modeled using pulse functions to represent periodic activation.
- Flexibility: The degree $m$ of the Bernstein polynomial can be adjusted to refine the approximation, or composition rules can be applied for higher precision, though this increases the complexity of the resulting expressions.
Formal Verification: Once the system is represented as a polynomial, the authors propose using the HOL Light proof assistant to formally verify the stability properties.
- A candidate Lyapunov function $V(x)$ is identified for the polynomial system.
- The verification task is reduced to proving the inequality $\dot{V}(x) < \epsilon$ (where $\epsilon$ is a small positive constant) over a defined domain $D$ .
- This is achieved using interval arithmetic with Taylor approximations, a method previously validated in the Flyspeck project. This allows for the rigorous certification that the Lyapunov derivative is negative (or sufficiently close to zero) within the specified bounds.

Key Results
The paper illustrates the proposed approach through two numerical examples derived from existing literature, comparing the "B-switched" polynomial representation against the "standard" switched system:

Example 1 (Stable System): A linear switched system with two stable subsystems was analyzed. The Bernstein approximation successfully mimicked the stabilizing behavior of the standard system.
Example 2 (Unstable System): A linear switched system with unstable subsystems was analyzed. The method correctly captured the divergent behavior, noting that as the state product $x_1 x_2$ grows beyond the interval $[0, 1]$ , the Bernstein series naturally diverges, reflecting the system's instability.
Formal Verification: A basic linear stable system was used to demonstrate the HOL Light verification process. The authors successfully generated a formal proof certificate confirming that the derivative of a quadratic Lyapunov function remained below a threshold ( $\epsilon = 0.01$ ) within the interval $[0, 1]$ .

Significance and Claims
The authors present this work as a preliminary study with the following specific contributions and limitations:

Unification: The primary contribution is the ability to "unify" multiple switching subsystems into a single polynomial expression via Bernstein interpolation. This simplifies the search for Lyapunov candidate functions.
Formal Certification: The approach bridges the gap between numerical simulation and formal proof, allowing the HOL Light assistant to verify the stability properties of the derived Lyapunov functions.
Modest Scope: The paper explicitly states that it assumes the stability properties of the B-switched system are equivalent to the standard switched system (at least regarding Lyapunov stability) but does not claim to have proven this equivalence rigorously yet.
Future Work: The authors identify several areas for future investigation, including:
- Proving the complete mathematical equivalence between standard and B-switched representations regarding stability.
- Extending the method to nonlinear and non-polynomial systems.
- Utilizing Sum-of-Squares (SOS) tools to calculate "optimal" Lyapunov functions rather than just verifying identified candidates.

In conclusion, the paper proposes a novel framework that leverages Bernstein polynomials to convert switched system analysis into a polynomial problem, solvable via formal verification tools, offering a potential pathway for rigorous stability certification.