Controllability and Observability in Linear Time-Variable Networks With Arbitrary Symmetry Groups

In the past two decades, engineers and applied mathematicians have devoted a great deal of attention to diverse aspects of linear timevariable networks and systems. However, one problem that has not been treated in depth is that of analyzing time-varying networks displaying arbitrary geometrical symmetries. A symmetric network may be regarded as a set of identical subnetworks connected in a symmetric pattern. Such a circuit may be more easily implemented in an integrated form than is a nonsymmetric network, especially when the circuit is time-variable and the construction and synchronization of the variable elements are major technical problems. Since the trend in integrated circuit technology is towards large-scale integration, it may soon become 507 508 T H E BELL S Y S T E M T E C H N I C A L J O U R N A L , FEBRUARY 1972 practically important to consider large networks displaying arbitrary geometrical symmetries. The present research was undertaken partly as a possible first step toward developing a modular approach to linear network design. While it has long been known that network symmetries can be used to facilitate analysis, previous work on symmetric networks dealt mainly with bisection techniques for networks with mirror-plane symmetry and has not incorporated general types of symmetries into an analysis scheme. The present work treats arbitrary symmetries by utilizing the mathematics of group theory, a natural tool for studying symmetry. Network controllability and observability are important concepts in analysis and synthesis, and group theory may be employed in determining symmetry-constrained noncontrollability and nonobservability of the network.