Extension of S-procedure in the Analysis of Multivariable Control Systems

In many problems of investigation of stability, attraction domain estimation, attainability sets estimation, the following question arises. Under what conditions certain quadratic form is positive definite in the region, defined by another 'm' quadratic forms. Commonly used method, called S-procedure, consists in checking of whether there exist such multipliers that a linear combination of quadratic forms is positive definite. Well known is that if m>1 then S-procedure gives only sufficient conditions for positive definiteness of the quadratic form under quadratic constraints. Necessity takes place only for m=1. This property is called "lossness" of the S-procedure for multiple constraints. Using only sufficient conditions leads to too conservative criteria. Several cases of extension of S-procedure necessary and sufficient conditions are presented in the paper.