Pattern Formation Via Eigenstructure Assignment: General Theory and Multi-Agent Application

Complex dynamical systems often exhibit formation of a pattern in observed variables in the steady state. An important special case is when the system consists of multiple subsystems (or “agents”) subjected to local interactions to reach consensus or an arbitrary pattern specified by their relative positioning in the state space. This paper formulates a general pattern formation problem as the design of a feedback controller such that selected outputs of a linear plant exponentially converge to $Re^{\Lambda t}\rho _o$ for some vector $\rho _o$ , with prescribed matrices $R$ and $\Lambda$ . We show that the problem reduces equivalently to an eigenstructure assignment problem, and provide a necessary and sufficient condition for existence of a feasible controller as well as a parameterization of all such controllers. This general theory is further specialized to give a complete solution to a heterogeneous multi-agent synchronization problem. Two numerical examples are provided to demonstrate the efficacy of the proposed design method: one illustrates the importance of adaptive pattern formation through sensory feedback and another suggests an extension for achieving stable limit cycles by additional nonlinearities.