Well-defined series and parallel D-spectra for linear time-varying systems

Abstract

An nth-order, scalar, linear time-varying (LTV) systems y(n)+/spl Sigma//sub k=1//sup n//spl alpha//sub k/(t)/sup (k-1)/=0 can be conveniently represented as /spl Dscr//sub /spl alpha//{y}=0 using the scalar polynomial. Differential operator (SPDO) /spl Dscr//sub /spl alpha//=/spl delta//sup n/+/spl Sigma//sub k=1//sup n//spl alpha//sub k/(t)/spl delta//sup k-1/, where /spl delta/=d/dt is the derivative operator. Based on a classical result of Floquet (1879) on the factorization of SPDO /spl Dscr//sub /spl alpha//=(/spl delta/-/spl lambda//sub n/(t))...(/spl delta/-/spl lambda//sub 1/(t)), a unified eigenvalue theory has recently been developed. In that theory the collection {/spl lambda//sub k/(t)}/sub k=1//sup n/ are called a series D-spectrum (SD-spectrum) for /spl Dscr//sub /spl alpha// and an n-parameter family {/spl rho//sub k/(t)=/spl lambda//sub 1,k/(t)}/sub k=1//sup n/ are called a parallel D-spectrum (PD-spectrum) for /spl Dscr//sub /spl alpha//, where /spl lambda//sub 1,k/(t) are particular solutions for /spl lambda//sub 1/(t) satisfying some nonlinear independence constraints. Although more than a century old, the important Floquet factorization has apparently not been fully harnessed in LTV system theory and control, due mainly to the well-known fact that even for a 2nd-order time-invariant SPDO, the PD-eigenvalue /spl rho/(t)=/spl lambda//sub 1/(t) satisfying the scalar Riccati equation /spl lambda//spl dot//sub 1/+/spl lambda//sub 1//sup 2/+/spl alpha//sub 2//spl lambda//sub 1/+/spl alpha//sub 1//spl lambda//sub 1/=0 may suffer from finite-time singularities known as finite-escapes. In this paper, necessary and sufficient conditions for the existence of well-defined (free of finite-time singularities) SD- and PD-spectra for SPDOs with complex- and real-valued coefficients are presented. The new results will have significant impact on applications of the unified eigenvalue theory to the analysis and control of LTV systems, and its further development.