Generalized S-procedure for inequality conditions on one-vector-lossless sets and linear system analysis

The generalized S-procedure, introduced by Iwasaki et al., has proved to be very useful for robustness analysis and synthesis of control systems. This procedure provides a nonconservative way to convert inequality conditions on lossless sets into numerically verifiable conditions represented by linear matrix inequalities (LMIs). In this paper, we introduce a new notion, one-vector-lossless sets, and propose a generalized S-procedure to reduce inequality conditions on one-vector-lossless sets into LMIs without any conservatism. By means of the proposed generalized S-procedure, we can examine various properties of matrix-valued functions over some regions on the complex plane. To illustrate the usefulness, we show that full rank property analysis problems of polynomial matrices over some specific regions on the complex plane can be reduced into LMI feasibility problems. It turns out that many existing results such as Lyapunov's inequalities for stability analysis of linear systems and LMIs for state-feedback controller synthesis can be viewed as particular cases of this result.