Peano-Baker Series Visualizer

An interactive Streamlit web application that visualizes the Peano-Baker series, a powerful mathematical tool for approximating the state transition matrix $\Phi(t, t_0)$ of Linear Time-Varying (LTV) systems.

This tool aims to bridge the gap between abstract control theory and intuitive understanding by providing real-time visualizations, error bounds, and practical control case studies.

🚀 Key Features

1. Interactive Simulation

• Custom LTV Systems: Define your own time-varying matrix $A(t)$ using Python syntax (e.g., [[0, 1], [-2 - 0.5*cos(t), -1]]) or choose from preset physical systems like the Damped Oscillator or Mathieu Equation.
• Real-Time Phase Portraits: Visualize 2D state trajectories ($x_1$ vs $x_2$) and time-series evolution.
• Visual Convergence: Adjust the number of series terms ($N$) and watch the approximation converge to the true solution in real-time.

2. Rigorous Error Analysis

• ** Theoretical Upper Bounds**: Compares the actual numerical error against the theoretical analytical upper bound derived from the work of Baake & Schlägel.
• Validity Horizon: Automatically calculates the time horizon for which the approximation is guaranteed to stay within a specified error threshold.

3. Sensitivity & Stability

• Parameter Sensitivity: Explore how changes in initial conditions ($x_0$) and system dynamics magnitude ($|A|$) affect convergence.
• Gradient Visualization: Compute and visualize 3D surfaces showing the sensitivity of the solution to system parameters (e.g., stiffness vs. perturbation).

4. Advanced Analytics

• Convergence Heatmap: A "Wavefront" visualization showing the error magnitude across Time vs. Number of Terms.
• Term Magnitude: Analyze the Frobenius norm of individual series terms to verify decay and convergence.
• Computation Benchmarks: Analyze the $O(N \cdot M)$ complexity of the cumulative integration scheme.

5. Case Study: Robot Walking Control

• Variable-Height LIPM: A demonstration of LTV control applied to a walking humanoid robot that changes height (squatting).
• LTI vs LTV: Visually compares a standard "Naive" controller (assuming constant height) vs. the "Peano" controller (accounting for time-varying height), showing why LTV theory is critical for dynamic robots.

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💻 Installation & Usage

Prerequisites

• Python 3.8+
• pip

1. Install Dependencies

Clone the repository, create a virtual environment, and install the required Python packages:

git clone https://github.com/yourusername/Peano-Baker-Series-Visualizer.git
cd Peano-Baker-Series-Visualizer

# Create a virtual environment
python3 -m venv .venv
source .venv/bin/activate  # On Windows: .venv\Scripts\activate

# Install dependencies
pip install -r peano_baker_viz/requirements.txt

2. Run Locally

Start the Streamlit development server:

streamlit run peano_baker_viz/app.py

The app will automatically open in your default browser at http://localhost:8501.

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🌐 Deployment to GitHub Pages

This project is deployed to GitHub Pages using stlite (Serverless Streamlit).

How it works

Unlike traditional Streamlit apps that require a backend Python server, this project uses stlite to run the entire application directly in the user's browser using WebAssembly (Pyodide).

1. Static Files: The docs/ directory contains a static index.html file and copies of the Python source code (app.py, utils.py, case_studies.py).
2. WebAssembly Runtime: When a user visits the page, index.html loads the stlite runtime, which downloads the Pyodide Python interpreter.
3. Browser Execution: The Python code is executed locally within the browser tab. This eliminates the need for a dedicated backend server, allowing free hosting on GitHub Pages.

GitHub Pages is configured to serve the /docs folder from the main branch.

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📂 Project Structure

• peano_baker_viz/: Source code for local development.
  • app.py: Main application entry point.
  • utils.py: Core mathematical engine.
  • case_studies.py: Robot simulation logic.
• docs/: Static site assets for GitHub Pages deployment (contains index.html and mirrors the python files).

The pages webiste hosts a preview of the streamlit app. It is not exactly the same as what is seen if running the streamlit app itself.

You can see the app in action at Peano-Baker-Series-Visualizer.