Theory and application of linear periodic differential equations which contain a small parameter

Abstract

New results in the theory and application of linear periodic differential equations containing a small parameter are presented. First, results on the convergence of an asymptotic solution algorithm for these equations are given. Second, a theorem on multiplier invariance is presented. Third, a method is given for the construction of discs which contain the spectrum of an equation. Finally, the theory is applied to the phase locked loop and synchronous linear machine. An asymptotic solution algorithm for a class of differential equations with periodic coefficients is given. A necessary condition, due to Yakubovich, for the divergence of this algorithm is presented. Two additional hypotheses are added to this result to produce a sufficient condition for the divergence of the algorithm. A theorem is proved which states sufficient conditions for a periodic differential equation containing a parameter to have at least one multiplier which is independent of the parameter. Application of the theorem produces the invariant multiplier (or multipliers) of the equation. A method is given for the construction of discs containing the spectrum of an equation. These discs will lie in the complex plane and have radii related to a norm of the system perturbation. The procedure can be augmented with a known asymptotic reduction technique to produce discs of arbitrarily small radii. A new phase locked loop (PPL) model is presented as the first application of the theory. The model's innovative feature is the inclusion of the multiplier sum term. The steady state phase error in the model satisfies a nonhomogeneous periodic differential equation when the loop is locked to a constant frequency reference. This periodic phase error is shown to cause spurious frequency components in the VCO output. ...