Optimizing stability certificates improves safety areas for polynomial systems
Optimizing Lyapunov Certificates via Stability-Preserving Quadratization for Polynomial Systems
Symbolic Computation
Summary
Checking how stable a system is—such as predicting if it will stay safe over time—can get very complex for systems described by polynomial equations. The authors studied a way, called quadratization, that simplifies this checking by turning complicated equations into quadratic ones, which are easier to handle. They improved the method by jointly tuning certain parameters, called gains and gauges, that affect the accuracy of verifying stability. Their approach finds better settings that certify larger safe regions compared to previous methods, especially shown in tests on multiple example systems.
What this means in practice
- •For control systems engineers: Improve stability verification for nonlinear control systems modeled by polynomial equations to certify larger safe operating regions.
- •For robotics developers: Increase safety guarantees for robot motion and control algorithms using polynomial dynamic models by better verifying stability regions.
Authors
Yubo Cai, Gioele Zardini
Abstract
Region-of-attraction (ROA) certificates for polynomial systems become expensive as state dimension and degree grow: direct sum-of-squares (SOS) formulations require combinatorially growing monomial bases. Quadratization represents a polynomial vector field exactly on an invariant manifold of a quadratic system, allowing a quadratic Lyapunov function to certify the ROA. For a fixed lift, stabilizer gains shape the off-manifold extension and transverse dynamics, while representation gauges change the matrix representation but not the vector field. Both affect the spectral-norm certificate, yet prior work fixes the gain by a feasibility heuristic before optimizing the gauge. We formulate optimal dissipative quadratization (ODQ), jointly designing gains and gauges for a fixed monomial lift, reference extension, stabilizer factorization, and Lyapunov weight $Q=I$. Gains lie in a prescribed compact Hurwitz box. At each gain, an exact semidefinite program globally minimizes the spectral-norm bound over the gauge. Residual-aware bounds yield a certified closed Lyapunov sublevel set, accounting for the floating-point Lyapunov residual. Under our stated assumptions, every accumulation point of the idealized outer search is box-Clarke stationary. A finite run returns the best independently verified candidate; global optimality of the gain search is not claimed. On a planar quintic, optimizing the gain increases the certified area by a factor of $2.238$ over a matched zero-gain gauge. Across 16 heterogeneous polynomial systems with stabilizer freedom, ODQ improves on both fixed-gain lifted baselines. All 36 ODQ runs on the relay benchmark complete, and all 27 repeat-level comparisons across nine fully paired cases favor ODQ over an SOS baseline with a fixed quadratic Lyapunov function in both the fixed-direction proxy and construction time. Broader comparisons with direct SOS methods remain mixed.