A necessary and sufficient condition for designing formation of discrete-time multi-agent systems with delay

This paper establishes a necessary and sufficient condition for designing formation of discrete-time second-order multi-agent systems with one sample period delay, which are networked by digraphs. We assume that the control input of each agent is constructed based on weighted difference between its states and those of its neighbor agents. Using the formation information in a decentralized control input with design parameters, we first transform the formation problem into an asymptotically stabilization problem, which is equivalent to designing a Schur polynomial with complex coefficients. Then, we obtain a necessary and sufficient condition by performing a bilinear transformation between Schur polynomials and Hurwitz polynomials with complex coefficients. We also discuss how to solve the condition efficiently, and provide numerical examples to demonstrate effectiveness of the theoretical results.