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

Abstract

In this paper, two new spectral canonical relizations are developed for the class of time-varying Scalar Linear Dynamical Systems (SLDS) of the form: y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n)</sup> +α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (t)y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n-1)</sup> +...+α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (t)y+ α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (t)y=ß(t)u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n)</sup> +ß <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> (t)u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n-1)</sup> +...+ ß <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (t)u+ß <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (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.