AI 中文总结
本文针对连续-离散投影滤波器预测阶段的数值不稳定性问题，引入平方和（SOS）松弛方法，推导了可保持正性约束的投影常微分方程，并给出高斯情形下的显式演化方程，为非线性状态估计提供了更可靠的求解方案。
AI 中文摘要
连续-离散投影滤波器为近似非线性状态估计问题的解提供了严谨框架，但在预测阶段，当积分误差导致自然参数超出其容许域时，会出现数值不稳定性问题。为解决该问题，我们引入平方和（SOS）松弛以约束自然参数在容许域内演化。通过使用对数-乔列斯基映射对底层SOS矩阵进行参数化，我们推导得到了一个投影常微分方程（ODE），该方程可固有地保持必要的正性约束，无需采用以往半无限规划方法所需的计算成本高昂的在线优化检查。我们对该正性保持传播方案进行了理论推导，并给出了高斯情形下显式的SOS松弛演化方程。
英文摘要
The continuous-discrete projection filter offers a rigorous framework to approximate the solution of the nonlinear state estimation problems. However, it suffers from numerical instability during the prediction phase when integration errors force the natural parameters outside their admissible domain. To address this issue, we introduce the sum-of-squares (SOS) relaxation to constrain the evolution of the natural parameters within the admissible domain. By parameterizing the underlying SOS matrix using the log-Cholesky map, we derive a projected ordinary differential equation (ODE) that inherently preserves the necessary positivity constraints without requiring the computationally expensive online optimization checks associated with previous semi-infinite programming approach. We provide a theoretical derivation of this positivity-preserving propagation scheme and present the explicit SOS-relaxed evolution equations for the Gaussian case.
CommentsThis paper has been presented at ASCC 2026, Bali (https://ascc2026.org/)
