Large-scale dynamical systems often consist of a number of subsystems that are interconnected according to a hierarchical multi-scale network. This paper introduces a hierarchical control scheme as an efficient strategy to handle such systems and proposes a method for designing a hierarchical linear quadratic optimal regulator. The proposed framework employs an algebraic approach. We first characterize a hierarchy of systems as an algebra based on semigroups, the Kronecker product, and the linear combination. This allows us to prove that the stabilizing solution of the Riccati equation inherits a hierarchy if system matrices and weights in the cost function belong to the corresponding common algebra that characterizes the hierarchy. A couple of classes of systems that can be treated by our algebraic framework are also provided in the paper. We will see that the derived result gives a unified insight into several related previous works.
