New extreme point results on robust strict positive realness

This paper considers the robust strict positive real (SPR) problem for a family of plants of the form G(s,q)=N(s,q_n)D^-1 (s,q_d)-α, where N(s,q_n) and D(s,q_d) are multiaffine in uncertain parameters q_n and q_d, respectively, and α>0. In the discrete-time setting, this problem plays an important role in digital quantization. Several results are presented. First, we prove that this plant family is robustly SPR if and only if all “corner plants” in the family are SPR. We then show that, if this plant family is robustly SPR, it admits a multiaffine Lyapunov matrix for the Kalman-Yakubovic-Popov (KYP) inequality, i.e., the Lyapunov matrix is multiaffine in the uncertain parameters. This result is useful in robustness analysis of time-varying systems. Finally, we relate the robust SPRness of this plant family to the robust strict bounded realness (SBRness) of a plant family involving the inverse of N(s,q_n)D^-1 (s,q _d). We show that multiaffine Lyapunov matrix for the KYP inequality of the first plant family yields a multiaffine Lyapunov matrix for the bounded real inequality of the second plant family. Finally, the robust SPR problem is considered for a more general plant family with applications in circuits and communication systems