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Shen, Shen, Tedrake, Russ.  2020.  Sampling Quotient-Ring Sum-of-Squares Programs for Scalable Verification of Nonlinear Systems. 2020 59th IEEE Conference on Decision and Control (CDC). :2535–2542.
This paper presents a novel method, combining new formulations and sampling, to improve the scalability of sum-of-squares (SOS) programming-based system verification. Region-of-attraction approximation problems are considered for polynomial, polynomial with generalized Lur'e uncertainty, and rational trigonometric multi-rigid-body systems. Our method starts by identifying that Lagrange multipliers, traditionally heavily used for S-procedures, are a major culprit of creating bloated SOS programs. In light of this, we exploit inherent system properties-continuity, convexity, and implicit algebraic structure-and reformulate the problems as quotient-ring SOS programs, thereby eliminating all the multipliers. These new programs are smaller, sparser, less constrained, yet less conservative. Their computation is further improved by leveraging a recent result on sampling algebraic varieties. Remarkably, solution correctness is guaranteed with just a finite (in practice, very small) number of samples. Altogether, the proposed method can verify systems well beyond the reach of existing SOS-based approaches (32 states); on smaller problems where a baseline is available, it computes tighter solution 2-3 orders of magnitude faster.