Preliminaries

Abstract

This chapter introduces some basic definitions and results on graph theory, consensus decomposition of linear space theory, matrix theory, linear system theory, and singular system theory, which will be used in the following chapters. First, the definitions of directed graph, spanning tree, and Laplacian matrix, etc. are given, and properties of Laplacian matrix are addressed. Then the concepts of consensus subspace, complement consensus subspace, and state/output space decomposition are defined. Third, the properties of Kronecker product and Schur complement lemma are introduced. Moreover, the definitions and criteria for controllability, observability, and stability of linear time-invariant systems are summarized, and some results on partial stability of linear systems are also given. Finally, the definitions and results on the regularity, equivalent form, admissibility, and controllability of singular systems are introduced.