⚡ electrical engineering

Novel Stability Criteria for Discrete and Hybrid Systems via Ramanujan Inner Products

This paper proposes a novel stability framework for discrete and hybrid systems by introducing a Ramanujan inner product and norm, establishing new $\epsilon$ - $\delta$ conditions that link system stability to number-theoretic properties and demonstrate enhanced robustness compared to traditional Euclidean metrics.

Original authors: Shyam Kamal, Sunidhi Pandey, Thach Ngoc Dinh, Cao Thanh Tinh

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shyam Kamal, Sunidhi Pandey, Thach Ngoc Dinh, Cao Thanh Tinh

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 listen to a specific song in a noisy room.

The Old Way (Euclidean Norm):
Traditional control theory, which has been used for decades, is like wearing noise-canceling headphones that treat all noise the same. If someone is shouting, a baby crying, or a car honking, the headphones just try to lower the volume of everything equally. It works well for general noise, but if the noise has a specific pattern (like a rhythmic drumbeat), the headphones might miss the fact that the rhythm is actually helping you hear the song, or they might fail to filter out a rhythmic hum that keeps coming back.

The New Way (Ramanujan Inner Products):
This paper introduces a new, super-smart pair of headphones designed by mathematicians who love numbers. Instead of treating all noise as a blur, these headphones can "hear" the mathematical rhythm of the noise. They know that some sounds only happen on Tuesdays, some only when the clock strikes a prime number (like 2, 3, 5, 7), and some follow a secret code based on remainders.

Here is a breakdown of the paper's ideas using simple analogies:

1. The Problem: The "Prime Number" Noise

Imagine a robot that is supposed to stay still. But every time the clock hits a prime number (2, 3, 5, 7, 11...), someone gives it a tiny little push.

Traditional View: To the old math, these pushes look like a chaotic, endless stream of energy. Since there are infinitely many prime numbers, the math says, "This robot will never stop shaking; it's unstable!" It sees the total energy as infinite.
The Reality: The pushes are actually very sparse. They happen rarely. The robot is actually quite stable; it just wiggles a bit occasionally. The old math is being too pessimistic because it doesn't understand the pattern of the pushes.

2. The Solution: The "Ramanujan" Lens

The authors, Shyam Kamal and his team, invented a new mathematical tool called the Ramanujan Inner Product. Think of this as a special lens that looks at the world through the eyes of number theory (the study of how numbers relate to each other).

The Analogy: Imagine you are sorting a pile of mixed-up socks.
- Euclidean Method: You just count the total weight of the socks. If the pile is heavy, you say, "This is a mess."
- Ramanujan Method: You sort the socks by their "remainder" when divided by a number (like 5). You realize that the "messy" socks only appear in specific piles (e.g., only socks that leave a remainder of 2 when divided by 5).
- The Magic: Because you know the pattern, you can ignore the "noise" that doesn't fit the pattern. You realize the system is actually very organized, even if it looks messy to the naked eye.

3. How It Works in the Real World

The paper proves that if you use this new "Ramanujan lens" to check if a system (like a drone, a power grid, or a self-driving car) is stable, you get better results:

Robustness: If a drone is hit by wind gusts that happen only at prime-numbered seconds, the old math might say, "The drone will crash!" The new math says, "No, the drone is fine because the wind hits it in a predictable, sparse pattern that the drone can handle."
Hybrid Systems: Many modern systems are "hybrid"—they run continuously (like a car driving) but also jump suddenly (like a traffic light changing or a packet of data dropping). The new math is perfect for these because it understands the "jump" patterns (like prime numbers or specific cycles) better than the old math.

4. The Simulation Results (The "Proof")

The authors ran computer simulations to prove their point:

The Test: They created a system with a "Prime Number Disturbance."
The Result:
- Old Math (Euclidean): The graph of the system's stability looked like a jagged mountain range with spikes everywhere. It never settled down.
- New Math (Ramanujan): The graph was a smooth, calm line going straight down to zero. It showed that the system was actually stabilizing perfectly.

Why This Matters

This is like upgrading from a black-and-white TV to a high-definition 3D TV.

Old Math sees the system as a flat, blurry mess of energy.
New Math sees the hidden structure, the rhythm, and the "arithmetic DNA" of the system.

In a nutshell:
This paper gives engineers a new tool to prove that systems are stable even when they are being disturbed by weird, patterned noise (like prime numbers or regular cycles) that used to confuse traditional math. It allows us to design systems that are more efficient and less "conservative" (less afraid of the unknown) because we finally understand the hidden math behind the chaos.

1. Problem Statement

Traditional stability analysis for control systems, particularly discrete-time and hybrid systems, relies heavily on Euclidean metrics (standard $L_2$ norms and inner products). While effective for geometric convergence, these metrics often fail to capture the arithmetic and combinatorial structures inherent in modern control systems.

The Gap: Systems involving hybrid switching, sampled-data control, or discrete scheduling often exhibit dynamics driven by periodic sequences, residue classes, or number-theoretic patterns (e.g., prime-numbered events).
The Limitation: Euclidean norms treat all time steps equally. Consequently, they may yield overly conservative stability results or fail to prove asymptotic stability when disturbances are "arithmetically sparse" (e.g., occurring only at prime indices) but have infinite $L_2$ energy. The paper argues that a metric sensitive to the arithmetic structure of the signal is required to accurately assess stability in these contexts.

2. Methodology

The authors propose a novel framework replacing the standard Euclidean inner product with a Ramanujan Inner Product, derived from number theory.

A. Ramanujan Inner Product and Norm

Ramanujan Sum ( $c_q(n)$ ): The foundation is the Ramanujan sum, an arithmetic sum of exponential functions defined as $c_q(n) = \sum_{(k,q)=1} e^{2\pi i k n / q}$ . This function captures divisibility and residue-class information.
Inner Product Definition: For two sequences $a$ and $b$ , the Ramanujan inner product is defined as a limit of weighted averages over residue classes:
$\langle a, b \rangle_R = \lim_{Q\to\infty} \frac{1}{Q} \sum_{q=1}^Q \frac{1}{q} \sum_{r=0}^{q-1} c_q(r) \sum_{n \equiv r \pmod q} a_n b_n$
Norm: The associated norm is $\|a\|_R = \sqrt{\langle a, a \rangle_R}$ .
Key Property: Unlike the Euclidean norm, this metric performs a "number-theoretic correlation." It is sensitive to arithmetic periodicities. The paper proves that for sequences admitting a Ramanujan expansion, this defines a valid norm satisfying positivity, homogeneity, and the triangle inequality.

B. Stability Framework

The authors extend classical control concepts to this new metric:

Ramanujan Stability: Defined via $\epsilon-\delta$ conditions using the Ramanujan norm.
Ramanujan Gain Function: A measure of the system's response to initial conditions within the Ramanujan metric.
Ramanujan-Lyapunov Functions: A Lyapunov-like function $V(x)$ is constructed where the decrease conditions ( $\dot{V} \leq -\alpha(\|x\|_R)$ and jump conditions) are evaluated using the Ramanujan norm rather than the Euclidean norm.

3. Key Contributions

Introduction of Ramanujan Metrics to Control Theory: The paper formally introduces Ramanujan inner products and norms as a viable alternative to Euclidean metrics for stability analysis.
Novel Stability Theorems:
- Discrete-Time Systems: Proves a stability criterion where the system state is bounded by a convolution involving Ramanujan sums. If the "Ramanujan gain" $G < 1$ , the system is stable.
- Hybrid Systems: Extends Lyapunov stability theorems to hybrid systems (continuous flows + discrete jumps) using the Ramanujan metric, proving asymptotic stability for sets defined by the Ramanujan distance.
Enhanced Robustness to Structured Disturbances: Demonstrates that systems stable in the Ramanujan metric possess superior robustness against structured arithmetic disturbances (e.g., periodic packet drops, cyclostationary noise, or prime-indexed events) that would destabilize or cause unbounded growth in the Euclidean sense.
Theoretical Proof of Equivalence and Superiority: Shows that while the Ramanujan norm is bounded by the Euclidean norm ( $\|x\|_R \leq \|x\|_2$ ), it can be strictly smaller for specific arithmetic structures, allowing for less conservative stability certification.

4. Results and Validation

The paper validates the theory through numerical examples comparing Ramanujan analysis against traditional Euclidean analysis.

Example 1: Discrete-Time System with Prime-Indexed Disturbance
- Setup: A linear system with a disturbance $w_k$ that is non-zero only when $k$ is a prime number.
- Euclidean Result: The disturbance is not absolutely summable ( $\sum \|w_k\|_2 = \infty$ ). Classical ISS analysis shows the state remains bounded but does not converge to zero (persistent spikes in the Euclidean norm).
- Ramanujan Result: The Ramanujan norm effectively "filters" the prime-indexed sparsity. The state trajectory $\|x_k\|_R$ decays smoothly to zero, proving asymptotic stability where Euclidean analysis could only prove boundedness.
Example 2: Hybrid System with Prime-Triggered Jumps
- Setup: A mass-spring-damper system where energy dissipation (jumps) occurs only at prime time instants.
- Euclidean Result: Due to the logarithmic growth of gaps between primes, the continuous damping is too weak to guarantee convergence in the Euclidean sense; the energy exhibits fluctuations.
- Ramanujan Result: By correlating the jump sequence with the Ramanujan basis, the analysis captures the arithmetic sparsity. The Ramanujan Lyapunov function proves that the system energy converges to zero, establishing asymptotic Ramanujan stability.
Disturbance Filtering Mechanism: The discussion highlights that by tuning the modulus $q$ of the Ramanujan sum, one can selectively filter out disturbances. If the disturbance period shares factors with $q$ such that the Ramanujan sum vanishes (e.g., using the Möbius function properties), the disturbance energy in the Ramanujan norm becomes zero, ensuring stability.

5. Significance

This work represents a paradigm shift in stability analysis by bridging control theory and analytic number theory.

Reduced Conservatism: It provides a mathematical tool to prove stability for systems that appear unstable or only marginally stable under traditional geometric metrics.
Structure-Aware Analysis: It acknowledges that in digital and hybrid systems, time is not just a continuous variable but often carries discrete, arithmetic significance (e.g., scheduling cycles, network packet loss patterns).
Future Applications: The framework opens avenues for analyzing Networked Control Systems (NCS) with periodic dropouts, event-triggered control, and systems subject to cyclostationary noise, offering a more precise characterization of robustness than current $L_2$ or $L_\infty$ methods.

In conclusion, the paper establishes that for systems with inherent arithmetic dynamics, the Ramanujan metric is not just an alternative but a superior tool for certifying asymptotic stability and robustness.