Poles and Eigenvalues in LTI Systems

Abstract

This text is strictly didactic and does not intend to present any scientific novelty. Its purpose is to serve as supporting material for Control Theory classes, organizing classical definitions and results regarding the relationship between the eigenvalues of a state-space realization and the poles and zeros of the transfer matrix. In particular, it is demonstrated that the finite poles of $\mathbf{G}(s)$ (defined invariantly for the MIMO case via the Smith-McMillan form) always satisfy the inclusion $\{\text{finite poles of } \mathbf{G}\} \subseteq \text{spec}(\mathbf{A})$.

Resumo em Português: Este texto é estritamente didático e organiza definições e resultados clássicos sobre a relação entre autovalores de uma realização em espaço de estados e polos/zeros da matriz de transferência em sistemas lineares invariantes no tempo (SLIT).