New Spectral Canonical Realizations for Time-Varying Linear Systems using a Unified Eigenvalue Concept

In this paper, two new spectra¿ canonical relizations are developed for the class of time-varying Scalar Linear Dynamical Systems (SLDS) of the form: y⁽ⁿ⁾+¿n(t)y^(n-1)+...+¿2(t)¿+ ¿1(t)y=ß(t)u⁽ⁿ⁾+ßn(t)u^(n-1)+...+ ß2(t)u¿+ß1(t)u, and the class of completely controllable time-varying Vector Linear Dynamical Systems (VLDS) of the form: x¿=A(t)x+b(t)u, y=C(t)x+d(t)u, where A(t), b(t), C(t) and d(t) are, respectively, n×n, n×1, m×n and m×1 matrices; y, x, are vectors of dimension m, n, respectively; and u is a scalar. These new spectral canonical realizations are based on a recently established unified eigenvalue concept, spectral canonical forms and canonical coordinate transformations for matrices over a differential ring. Therefore, they are natural extensions and unifications of the well-known canonical realizations traditionally used for time-invariant SLDS and VLDS using the conventional (time-invariant) eigenvalues. The new results presented here have important applications in the areas of analysis, synthesis, controller design, simulations and implementation for time-varying linear systems.