Phase-variable canonical forms for linear, multi-input, multi-output systems†

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Abstract
Three types of system equivalent relations are introduced to unify existing definitions of system equivalence. Called α-, β- and γ-equivalence, their characteristics are examined mathematically and physically. Based on those definitions, two phase-variable canonical forma are developed for linear, multi-input, multi-output systems, resulting in a canonical form for control and a canonical form for observation. It is shown that a characteristic quantity called a stage sequence can be defined for a given system to uniquely determine its equivalent system in the canonical forms. The compact structures of these canonical forms are convenient representations of the mathematical and physical compositions of systems and for subsequent system analyses.

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