This paper considers a linear quadratic optimal hierarchical control problem for large-scale dynamical systems modeled by an interconnected system under multi-scale information exchange networks. We first propose an algebraic characterization of hierarchies by using semigroups the Kronecker product. The multiplication rule of the Kronecker product quite fits to the property of semigroups. As a result, a condition under which the stabilizing solution of the Riccati equation inherits the hierarchy is obtained with the aid of the previous result. Furthermore, the proposed framework makes it possible to understand several previous results on decentralized optimal control from a unified viewpoint.

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.

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