Geometric Control of Linear Patterned Systems

An interesting type of distributed system is a collection of identical subsystems that interact in a distinct pattern. A notable example is a ring, more commonly referred to as a circulant system. It is well known that control problems for circulant systems can be simplified by exploiting their common connection with the shift operator. Based on an examination of the algebraic properties underlying this connection, we identify a broader class of systems that share common base transformations. We call it the class of linear patterned systems. Members with meaningful physical interpretations include symmetric circulant systems, triangular Toeplitz systems and certain hierarchical systems. A geometric approach is employed to study the basic control properties of patterned systems, including controllability, observability and decomposition. Controller synthesis for several stabilization problems is then considered, and we show that a patterned solution to the problems exists if a general solution exists.MAS