Analysis of large scale parameter-varying systems by using scaled diagonal dominance

It has been recognized recently that semidefinite optimization problems (SDP) can be cast into second order cone programs (SOCP) by replacing the positive definiteness constraints with stronger, scaled diagonal dominance (SDD) conditions. Since the scalability of SOCP solvers is much better than that of the SDPs, the new formulation allows solving large dimensional problems more efficiently. However, scaled diagonal dominant matrices form only a subset of the positive definite matrices. Hence, the new problem formulation results in more conservative solutions. This paper analyses the efficiency and conservativeness of the SDD formulation on two particular problems: the stability analysis and induced ℒ₂ gain computation for linear parameter-varying systems. In the paper some important features of the SDD formulation are revealed and numerical examples are provided to demonstrate the efficiency of the approach.