Topological conditions for the unique solvability of linear time-invariant and time-varying networks

Abstract

The problem of knowing whether the non-unique solvability depends on the particular values of the components or on their topological interconnections is studied for linear networks with arbitrary, time-invariant as well as time-varying n-ports. Within every network, the topological notions of its sockets and of their independence are introduced. Networks with independent sockets are shown-at least when there are no relations among the non-zero coefficients, nor repetitions of the same coefficient are allowed, i.e. under suitable generality assumptions-to be uniquely solvable. Networks with dependent sockets are shown to be never uniquely solvable. Polynomially bounded algorithms, requiring only integer arithmetic, to test independence are available. When independence fails, a topological configuration of components which shows fewer topologically independent variables than equations, is proved to exist.