Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees

A team of researchers from Caltech and the University of California has developed a new algorithmic framework for analyzing the stability of complex hybrid systems, such as walking robots and power electronics. Published on arXiv under the title 'Finite-Step Invariant Sets for Hybrid Systems with Probabilistic Guarantees,' the work addresses a fundamental challenge in control theory: proving that a system will remain stable within a safe region of operation despite disturbances. The authors—Varun Madabushi, Elizabeth Dietrich, Hanna Krasowski, and Maegan Tucker—focus on systems with switching behavior, which are notoriously difficult to analyze with traditional linear methods.

The core innovation is a sampling-based optimization method that computes a 'finite-step invariant ellipsoid'—a mathematically defined safe zone—around a system's periodic orbit. Instead of requiring a perfect analytical model, the algorithm works by taking sampled evaluations from forward simulations of the system's Poincaré return map. This makes it applicable to complex, real-world systems where only simulation data is available. The result is accompanied by a user-defined probabilistic guarantee, meaning engineers can state with a specific confidence level (e.g., 95%) that the system will stay within the safe set for a finite number of steps.

The team validated their framework on two low-dimensional academic systems and a more practical compass-gait walking model, a classic benchmark in bipedal robotics. This demonstrates a path toward certifying the safety and robustness of autonomous systems that operate in uncertain environments. By providing a computationally tractable method with statistical assurances, this research could accelerate the deployment of reliable legged robots, advanced power converters, and other cyber-physical systems where failure is not an option.