椭球状态定界的数值稳定算法

Numerically Stable Ellipsoidal State Bounding Algorithm

针对带有未知但有界噪声的线性离散时间系统,提出了一种数值稳定的集员状态估计递推算法.算法采用椭球集合来描述状态的不确定性和噪声的界限.椭球形状矩阵的计算采用奇异值分解技术,以提高算法的数值稳定性.同时,给出了包含时间更新椭球和在状态空间中与量测量和量测噪声相一致的椭球交集的次最小容积椭球的计算方法,以避免受病态矩阵求逆的影响.蒙特卡洛仿真结果表明,数值稳定算法所得到的均方误差和椭球容积与最优算法得到的十分接近.此外,当存在舍入误差时,数值稳定算法可以保证形状矩阵的正定性,而最优算法有时难以保证,说明该算法比最优算法具有更好的数值稳定性.

A numerically stable recursive set membership state estimation algorithm for linear discrete-time systems with unknown but bounded noises is proposed, where ellipsoidal sets are adopted to describe the state uncertainties and to bound the process and observation noises. With the purpose of getting high numerical stability, singular value decomposition is used in the propagation of the shape-defining matrix of the ellipsoid. Besides, a subminimal-volume ellipsoid con- taining the intersection of the time-updated ellipsoid and the ellipsoidal set of state values consistent with the current observation and noise bounds is computed to circumvent inverse of ill-conditioned matrix. Monte Carlo simulations are performed on a digital computer for different models to demonstrate the effectiveness of the proposed algorithm. The simulation results show that the proposed algorithm not only matches the performance of the optimal algorithm closely in terms of mean ellipsoid volumes and mean square errors, but also keeps shape-defining matrix positive definite consistently.