Abstract
In this chapter, we consider mainly the consensus for multiple generic linear system dynamics with switching network topology.
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Notes
- 1.
The very special and trivial case that \(\gamma <0\) is not considered here, since all the agents, including the leader, finally approach zeros even in the absence of interactions among agents due to the asymptotical stability of each individual agent [25].
- 2.
\(\mathrm {Ran}(\mathbf {U}) \perp \mathrm {Ran}(\mathbf {H})\) means that for any \(\mathbf {x}\in \mathrm {Ran}(\mathbf {U})\) and \(\mathbf {y}\in \mathrm {Ran}(\mathbf {H})\), \(\mathbf {x}\perp \mathbf {y}\)
- 3.
\(\omega _i\) represents an N-dimensional vector with all 0’s except for the i-th entry being 1.
- 4.
\(\mathbf {v}^j(t)\otimes \psi ^j(t)\) denotes the projection of \(\mathbf {x}(t)\) onto the space spanned by \(\{\mathbf {v}^j(t)\otimes \omega _k,k=1,\ldots ,n \}\).
- 5.
Indeed, there are infinite such \(t_*\)’s.
- 6.
Although associated with any \(t\ge t_0\) there might exist a T(t) such that the union of \(G_{\sigma (t)}\) over the interval \([t,t+T(t))\) contains a \(\delta \)-graph, T(t) grows unbounded as t increases.
- 7.
see Lemma 3.31 for the definition of \(\mathcal {S}_1\) and \(\mathcal {S}_2\).
- 8.
\(\hat{\mathbf {L}}^{1/2}\) represents the unique positive semi-definite matrix that satisfies \(\hat{\mathbf {L}}^{1/2}\hat{\mathbf {L}}^{1/2}=\hat{\mathbf {L}}\) for positive semi-definite matrix \(\hat{\mathbf {L}}\) [65].
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Qin, J., Ma, Q., Gao, H., Zheng, W.X., Kang, Y. (2022). Generic Linear Systems over Undirected Network: Controllability and Connectivity. In: Consensus Over Switching Network Topology: Characterizing System Parameters and Joint Connectivity. Studies in Systems, Decision and Control, vol 393. Springer, Cham. https://doi.org/10.1007/978-3-030-85657-1_3
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DOI: https://doi.org/10.1007/978-3-030-85657-1_3
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