Achievability problem for desired formations of linear multi-agent systems

A fundamental problem in formation control is under what conditions one can design protocols for a multi-agent system to achieve desired formations. By examining the property of a class of formations that are exactly achieved in a linear multi-agent system, an interesting fact is observed that there is a connection between the achievability of desired formations and the consistency of time-varying linear equations. From the asymptotic consistency of time-varying linear equations, we propose the asymptotic consistency concept for desired formations and study its properties using the Moore-Penrose generalized inverse. With this concept, a result on achievability of desired formations is presented for linear multi-agent systems: all asymptotically consistent formations are achievable as long as the agent dynamics and communication network satisfy the controllability and connectivity conditions, respectively. We design state feedback protocols for asymptotically consistent formations using the weighted controllability Gramian and the Moore-Penrose generalized inverse. If further, the agent dynamics are observable, the observer-based protocols can be designed by the separation principle. Finally, a simulation example is given to illustrate the obtained results.