AI 中文总结
本文针对含 i.i.d. 乘性不确定性和加性噪声的线性离散时间系统，建立协方差递推、矩阵谱性质与均方稳定性的联系，提出易处理的 LMI 状态反馈条件，验证了协方差表征的实用性。
AI 中文摘要
本文研究受独立同分布（i.i.d.）乘性不确定性和加性噪声影响的线性离散时间系统，建立了协方差递推、相关克罗内克矩阵的谱性质与均方稳定性之间的主要联系，并利用这些联系推导了控制器综合的易处理条件。首先，在基于 tube 的随机模型预测控制（SMPC）框架内，使用基于克罗内克积的矩阵增广推导了确定性协方差递推。对于仅含乘性不确定性而无加性噪声的线性随机系统，证明了协方差递推产生的全空间矩阵表示与其对称空间对应矩阵具有相同的谱半径，结合现有的对称空间表征，确定全空间增广矩阵的舒尔稳定性等价于均方稳定性。对于状态反馈设计，提出了新的充分线性矩阵不等式（LMI）条件，与传统的充要条件相比，其规模更小，数值上更易处理。数值测试表明，该协方差表征可在不依赖基于采样的方法的情况下递推估计协方差，还评估了所提 LMI 条件的计算负担及其相对于充要条件的保守性。
英文摘要
This paper studies linear discrete-time systems affected by independent and identically distributed (i.i.d.) multiplicative uncertainties and additive noise. It establishes the main links between covariance recursions, the spectral properties of associated Kronecker-based matrices, and mean-square stability, and exploits these links to derive tractable conditions for controller synthesis. We first derive a deterministic covariance recursion within the tube-based Stochastic Model Predictive Control (SMPC) framework using a Kronecker product based matrix augmentation. For linear stochastic systems with multiplicative uncertainty and without additive noise, we show that the full-space matrix representation arising from the covariance recursion has the same spectral radius as its symmetric-space counterpart. Combined with the existing symmetric-space characterization, this establishes that Schur stability of the full-space augmented matrix is equivalent to mean-square stability. For state-feedback design, we propose new sufficient Linear Matrix Inequality (LMI) conditions that are numerically more tractable owing to their reduced size compared with the conventional necessary and sufficient conditions. Numerical tests illustrate the usefulness of the covariance characterization for recursively estimating the covariance without relying on sampling-based methods. We also assess the computational burden of the proposed LMI conditions and their conservatism relative to the necessary and sufficient ones.
