Lin Li; Yanan Li; Wei Feng

Abstract

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 (MAREs), which are special cases of GAREs. The purpose of this paper is to derive new information on vm. Main results are as follows: (i) 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. (ii) New analytical bounds of vm are established. (iii) 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.

Paper Linkage:https://ieeexplore.ieee.org/document/10919099