Exact Safety Verification of Hybrid Systems Based on Bilinear SOS Representation

In this paper, we address the problem of safety verification of nonlinear hybrid systems. A hybrid symbolic-numeric method is presented to compute exact inequality invariants of hybrid systems efficiently. Some numerical invariants of a hybrid system can be obtained by solving a bilinear SOS programming via PENBMI solver or iterative method, then the modified Newton refinement and rational vector recovery techniques are applied to obtain exact polynomial invariants with rational coefficients, which {\it exactly} satisfy the conditions of invariants. Experiments on some benchmarks are given to illustrate the efficiency of our algorithm.

💡 Research Summary

The paper tackles the safety verification problem for nonlinear hybrid systems by introducing a hybrid symbolic‑numeric framework that produces exact polynomial invariants. The authors first model the continuous dynamics and discrete transitions of a hybrid system as polynomial equalities and inequalities, defining the safe set through a set of polynomial constraints. Traditional sum‑of‑squares (SOS) based verification solves a semidefinite program to find a polynomial invariant, but the resulting coefficients are floating‑point numbers that only approximately satisfy the SOS conditions, making them unsuitable for formal proof systems.

To overcome this limitation, the authors reformulate the invariant search as a bilinear SOS program. In this formulation the decision variables are split into two groups: the coefficients of the candidate invariant polynomial and the SOS multipliers (Lagrange multipliers). The only nonlinearities are bilinear products between these two groups, which allows the use of existing non‑convex solvers such as PENBMI or simple iterative schemes to obtain a numerical solution quickly.

Once a numerical solution is available, the paper applies a modified Newton refinement step. This step treats the residuals of the SOS constraints as a nonlinear system and iteratively updates the coefficients to drive the residuals toward zero, thereby improving the numerical accuracy of the solution while preserving the bilinear structure. After refinement the coefficients are still real numbers, but they are now close enough to exact rational values that a rational reconstruction can be performed reliably.

The rational reconstruction is carried out using an LLL‑based lattice reduction algorithm. By constructing a lattice from the refined real coefficients, the algorithm finds a short lattice vector that corresponds to a set of rational numbers with a common denominator. The resulting rational coefficients exactly satisfy the SOS constraints, yielding a mathematically rigorous invariant.

The authors implement the entire pipeline and evaluate it on several benchmark hybrid systems, including temperature‑control loops, electrical circuits, and robotic arm models. The experimental results show that (i) the bilinear SOS formulation can be solved in comparable or shorter time than standard SOS approaches, (ii) the Newton refinement converges in a few iterations, and (iii) the rational reconstruction produces low‑degree invariants with small integer coefficients. Because the final invariants are expressed with exact rational numbers, they can be directly fed into interactive theorem provers (e.g., Coq, Isabelle) or model‑checking tools, eliminating the need for ad‑hoc numerical tolerance arguments.

In summary, the paper makes three key contributions: (1) a bilinear SOS representation that enables efficient numerical computation of candidate invariants for nonlinear hybrid systems, (2) a modified Newton refinement that bridges the gap between approximate numerical solutions and exact algebraic conditions, and (3) an LLL‑based rational recovery technique that yields invariants with provable correctness. This combination of techniques provides a practical pathway from fast numerical optimization to formally verified safety guarantees, which is especially valuable for safety‑critical cyber‑physical systems where both efficiency and mathematical rigor are essential.