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.
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References
Ren W, Beard RW (2005) Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control 50(5):655–661
Xi JX, Cai N, Zhong YS (2010) Consensus problems for high-order linear time-invariant swarm systems. Phys A 389(24):5619–5627
Godsil C, Royle G (2001) Algebraic graph theory. Springer, New York
Horn RA, Johnson CR (1989) Topics in matrix analysis. Cambridge University Press, Cambridge
Boyd S, Ghaoui LE, Feron E et al (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia
Williams RL, Lawrence DA (2007) Linear state-space control systems. Wiley, Hoboken
Chen CT (1999) Linear system theory and design. Oxford University Press, New York
Xi JX, Shi ZY, Zhong YS (2012) Output consensus analysis and design for high-order linear swarm systems: partial stability method. Automatica 48(9):2335–2343
Zhang QL, Liu C, Zhang X (2012) Complexity, analysis and control of singular biological systems. Springer, New York
Dai L (1989) Singular control systems. Springer, Berlin
Duan GR (2010) Analysis and design of descriptor linear systems. Springer, New York
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Dong, X. (2016). Preliminaries. In: Formation and Containment Control for High-order Linear Swarm Systems. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47836-3_2
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DOI: https://doi.org/10.1007/978-3-662-47836-3_2
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Online ISBN: 978-3-662-47836-3
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