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Topological conditions for the unique solvability of linear time‐invariant and time‐varying networks
Author(s) -
Poletti Mario,
Terreni Pierangelo
Publication year - 1987
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.4490150402
Subject(s) - mathematics , bounded function , independence (probability theory) , generality , topology (electrical circuits) , invariant (physics) , integer (computer science) , discrete mathematics , computer science , combinatorics , mathematical analysis , statistics , psychology , mathematical physics , psychotherapist , programming language
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.