Sequential Quadratic Sum-of-squares Programming for Nonlinear Control Systems
Jan Olucak, Torbjørn Cunis
TL;DR
The paper addresses solving nonconvex sum-of-squares (SOS) problems arising in nonlinear control analysis and design, where local properties like region-of-attraction or constrained reachability are certified by polynomial inequalities. It introduces a filter line-search algorithm that solves a sequence of quadratic SOS subproblems, providing local convergence guarantees and improved scalability over existing nonconvex SOS methods. The authors develop SOS-specific design elements, including efficient constraint-violation checks and a feasibility restoration phase, and demonstrate practical performance with an open-source CaΣoS implementation and benchmarks across ROA estimation, control synthesis, and reachability. The results indicate substantial reductions in iterations and computation time, enabling more practical SOS-based control tooling in local regions.
Abstract
Many problems in nonlinear systems analysis and control design, such as local region-of-attraction estimation, inner-approximations of reachable sets or control design under state and control constraints can be formulated as nonconvex sum-of-squares programs. Yet tractable and efficient solution methods are still lacking, limiting their application in control engineering. To address this gap, we propose a filter line-search algorithm that solves a sequence of quadratic subproblems. Numerical benchmarks demonstrate that the algorithm can significantly reduce the number of iterations, resulting in a substantial decrease in computation time compared to established methods for nonconvex sum-of-squares programs. An open-source implementation of the algorithm along with the numerical benchmarks is provided