computeSOSFunnels

A MATLAB Framework for Invariant Funnel Synthesis using Sum-of-Squares Optimization.

This repository implements a modular pipeline to compute invariant funnels around nominal trajectories for nonlinear dynamical systems. It integrates trajectory optimization, time-varying LQR (TVLQR) stabilization, and Sum-of-Squares (SOS) programming to generate rigorous Lyapunov certificates of stability.

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🚀 Key Features

• Trajectory Optimization: Direct collocation (using IPOPT) to compute nominal trajectories and feedforward inputs.
• Feedback Control: Computation of TVLQR gains and solution to the Riccati differential equation (useful in initializing the SOS program).
• Taylor Approximation: Automated symbolic expansion of nonlinear dynamics into polynomial form (required for SOS verification).
• Funnel Synthesis: Bilinear alternation scheme to compute invariant funnels via SOS programming.
• Multi-System Support: Ready-to-use implementations for:
  • Unicycle (Default)
  • Cart-Pole (Branch: cartPole)
  • Quadrotor (Branch: quadrotor)

📄 Documentation

A comprehensive technical report and user tutorial is available: Download the Tutorial PDF

🛠️ Prerequisites

• MATLAB (R2021b or later recommended)
• YALMIP (Optimization interface)
• IPOPT (Nonlinear solver)
• SOSTOOLS (SOS parsing)
• SDP Solver: MOSEK (Recommended) or SeDuMi.

📦 Installation

git clone https://github.com/khalid2696/computeSOSFunnels.git
cd computeSOSFunnels

Open MATLAB and run main.m. The script handles path setup automatically.

🏃 Quick Start

1. Open main.m in MATLAB.
2. Run the script to execute the pipeline:
   • Step 1: Compute Nominal Trajectory
   • Step 2: Synthesize TVLQR Controller
   • Step 3: Polynomialize Deviation Dynamics
   • Step 4: Compute SOS Funnels
3. Visualize results using utils/plottingScript.m.

🤖 Supported Dynamical Systems

This framework is designed to be modular, versatile, and system-agnostic. It is currently implemented and tested on three distinct nonlinear systems, each maintained on a separate branch to serve as a reference template.

System	Branch Name	Complexity	Description
Unicycle	main	3 States, 2 Inputs	A non-holonomic mobile robot model (Default).
Cart-Pole	cartPole	4 States, 1 Input	A classic underactuated benchmark system.
Quadrotor	quadrotor	12 States, 4 Inputs	A high-dimensional system with complex nonlinear dynamics.

Switching Systems

To use a specific system, simply checkout the corresponding branch. For example, to use the quadrotor system:

git fetch origin
git checkout quadrotor

📧 Contact & Support

If you have questions about the codebase, installation issues, or encounter bugs, please open a GitHub Issue — this would allow the community to benefit from the discussions.

For research inquiries, collaboration proposals, or theoretical questions regarding the Funnel Synthesis framework, feel free to reach out:

Mohamed Khalid M Jaffar
🌐 khalid2696.github.io

🖊️ Citation

If you find this code useful in your research, please cite the software:

BibTeX:

@software{jaffar_2025_computeSOSFunnels,
  author       = {M Jaffar, Mohamed Khalid},
  title        = {computeSOSFunnels: A MATLAB Framework for Invariant Funnel Synthesis using Sum-of-Squares Optimization},
  version      = {1.0.0},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18047554},
  url          = {https://doi.org/10.5281/zenodo.18047554},
  year         = {2025}
}

📜 License

This project is licensed under the MIT License - see the LICENSE file for details.