A unified eigenvalue theory for time-varying linear circuits and systems

Linear time-varying circuits and systems of the vector form dx /dt=A(t)x+bu and the scalar form y⁽ⁿ⁾+α_n(t)y ^(n-1)+ . . . +α₂(t)dy/dt+α₁ (t)y=u can be studied as operators over a differential ring K of almost everywhere C^∞ functions. A unified (time-varying) eigenvalue theory has been recently developed for such operators, relative to a class of equivalence transformations on K^n×n. This unified eigenvalue theory leads to the natural time-varying counterparts of eigenvalues, eigenvectors, characteristics, equations, modal matrices, stability criteria, etc., as traditionally used for time-invariant linear circuits and systems. The main results of the theory are summarized. The time-varying counterparts of transfer functions, series/parallel realizations, and pole-assignment control technique for the scalar form are presented