On Design of Stabilizing Controllers Using Compression Operators for Linear Systems with Time-Varying and Time-Invariant Stochastic Parameters

Abstract

This work involves the design of controllers to stabilize a linear stochastic system. The system contains time-invariant and time-varying stochastic parameters, which makes it difficult to ensure stability. To overcome this challenge, the system is expanded into a linear system with only the time-invariant parameter. A compression operator is used to reduce the dimension of the expanded system. Guarantee of stochastic stability of the original system reduces to that of robust stability of the expanded system. A condition for the robust stability is characterized using matrix inequalities because the expanded system can be treated as a well-known linear polytopic system. The inequalities are just cubic matrix inequalities (CMIs) of a state feedback gain for the original system and decision variables. This paper presents a technique to transform the CMIs into quadratic matrix inequalities (QMIs). Solving the QMIs derives a stabilizing controller with the feedback gain.