The Variance Margin for the Stabilization of Stochastic Systems and Applications to the Solvability of GAREs

• 成果类型: Article
• 署名作者: Li, Lin; Li, Yanan; Feng, Wei
• 署名单位: Shandong Agricultural University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3550077
• 发表日期: 2025
• 页码: 5608-5615
• 关键词: Eigenvalues and eigenfunctions; Riccati equations; Symmetric matrices; Stochastic systems; Robustness; POLYNOMIALS; optimization; vectors; training; optimal control; discrete-time systems; generalized algebraic Riccati equations (GAREs); multiplicative noise; robust stabilization; variance margin

摘要

A variance margin problem for linear systems with multiplicative noise is considered. The variance margin vm is the maximal variance tolerance such that the systems can be robustly stabilized by a state-feedback control. vm also provides a fundamental limit for the solvability of a generalized algebraic Riccati equation (GARE). Fundamental limits in literature are only valid for modified algebraic Riccati equations, which are special cases of GAREs. The purpose of this article is to derive new information on vm. Main results are as follows: 1) The variance margin of a control is established in direct and analytical ways and a new characterization of vm is proposed via the solution to a spectral optimization problem. 2) New analytical bounds of vm are established. 3) vm is completely determined by the optimal cost of a standard homogeneous linear quadratic regulation problem with a specified initial state if the systems have no state dependent noise and have a single input. If additionally, certain eigenvalue and controllability conditions are satisfied, an analytical expression of vm is available.