finite-step-invariant-sets-hybrid-systems

---
name: finite-step-invariant-sets-hybrid-systems
description: "Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching b... Activation: safety-critical systems, invariant sets, hybrid systems, saddle-point dynamics, constrained optimization."
---

# Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees

## Overview
Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map captures the evolution of the hybrid system on a guard surface, reducing the stability analysis of a periodic orbit to that of a discrete-time system. While linearization provides local stability information, assessing robustness to disturbances requires identifying invariant sets of the state space under the return dynamics. However, computing such invariant sets is computationally difficult, especially when system dynamics are only available through forward simulation. In this work, we propose an algorithmic framework leveraging sampling-based optimization to compute a finite-step invariant ellipsoid around a nominal periodic orbit using sampled evaluations of the return map. The resulting solution is accompanied by probabilistic guarantees on finite-step invariance satisfying a user-defined accuracy threshold. We demonstrate the approach on two low-dimensional systems and a compass-gait walking model.

## Source Paper
- **Title**: Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees
- **Authors**: Varun Madabushi, Elizabeth Dietrich, Hanna Krasowski, Maegan Tucker
- **arXiv**: 2604.05102v1
- **Published**: 2026-04-06
- **Categories**: eess.SY, cs.RO

## Core Concepts

### Key Contributions
1. Novel methodology for addressing In this work, we propose an algorithmic framework leveraging sampling-based opti...
2. Theoretical analysis with rigorous analysis
3. Practical applicability in real-world systems

### Technical Framework
This research contributes to systems engineering by providing:
- Advanced control methodologies
- Distributed system optimization techniques
- Practical implementation strategies

## Applications

### Primary Use Cases
- Large-scale distributed systems
- Multi-agent coordination
- Safety-critical control systems
- Resource optimization

### Example Scenarios
1. **Industrial Deployment**: Manufacturing and robotics
2. **Cloud Infrastructure**: Kubernetes and container orchestration
3. **Autonomous Systems**: Multi-robot coordination
4. **Network Optimization**: Wireless and communication systems

## Implementation Considerations

### Prerequisites
- Understanding of control theory fundamentals
- Familiarity with distributed systems
- Programming experience in Python or similar

### Key Parameters
| Parameter | Description | Typical Range |
|-----------|-------------|---------------|
| TBD | To be determined from paper | - |

## References

- Varun Madabushi et al. (2026). "Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees." arXiv:2604.05102v1.
- PDF: https://arxiv.org/pdf/2604.05102v1

## Related Skills
- See other systems engineering skills in ai_collection
- Cross-reference with control theory and distributed systems

## Activation Keywords
- safety-critical systems
- invariant sets
- hybrid systems
- saddle-point dynamics
- constrained optimization
- primal-dual

---

*Generated from arXiv research on 2026-04-06*