The paper presents an algorithm for the solution of the consensus problem of a linear multi-agent system composed of identical agents. The control of the agents is delayed, however, these delays are, in general, not equal in all agents. The control algorithm design is based on the $H_{\infty }$-control, the results are formulated by means of linear matrix inequalities. The dimension of the resulting convex optimization problem is proportional to the dimension of one agent only but does not depend on the number...

The paper is divided in two parts. In the first part a deep investigation is made on some system theoretical aspects of periodic systems and control, including the notions of $H_2$ and $H_{\infty }$ norms, the parametrization of stabilizing controllers, and the existence of periodic solutions to Riccati differential equations and/or inequalities. All these aspects are useful in the second part, where some parametrization and control problems in $H_2$ and $H_{\infty }$ are introduced and solved.

This paper deals with some state-feedback $H_2/H_{\infty }$ control problems for continuous time periodic systems. The derivation of the theoretical results underlying such problems has been presented in the first part of the paper. Here, the parametrization and optimization problems in $H_2$, $H_{\infty }$ and mixed $H_2/H_{\infty }$ are introduced and solved.