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2017 | OriginalPaper | Buchkapitel

5. Coordination and Consensus of Linear Systems

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Abstract

In this chapter, we will see how the theory of asymptotic tracking can be fruitfully extended to address problems in which a large set of systems is controlled in such a way that certain variables of interest asymptotically coincide. The specific challenge addressed in this chapter resides in the fact that there is a limited exchange of information between individual systems, each one of which has access only to measurements of the outputs of limited number of neighbors.

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Fußnoten
1
Note that all such systems have the same number of input and output components.
 
2
Further motivations for the interest of an information pattern such as the one described by (5.5) can be found in [16] and [20].
 
3
Background material on graphs can be found, e.g., in [7, 8]. All objects defined below are assumed to be independent of time. In this case the communication graph is said to be time-invariant. Extensions of definitions, properties and results to the case of time-varying graphs are not covered in this book and the reader is referred, e.g. to [9, 10].
 
4
Note that in a connected graph there may be more than one node with such property. See the example in Fig. 5.1.
 
5
In fact, if the graph \(\mathscr {G}\) possesses a root from which information can propagate to all other nodes along paths, the same is true for the graph \(\hat{\mathscr {G}}\), because the nonzero entries of \(\hat{A}\) coincide with the nonzero entries of A.
 
6
From now on, we drop the assumption—considered in the previous section—that \(p=m=1\).
 
7
For a couple of matrices \(A\in \mathbb R^{m\times n}\) and \(B\in \mathbb R^{p\times q}\), their Kroeneker product—denoted by \(A\otimes B\)—is the \(mp\times nq\) matrix defined as
$$ A\otimes B = \left( \begin{matrix}a_{11}B &{} a_{12}B &{} \cdots &{} a_{1n}B \\ a_{21}B &{} a_{22}B &{} \cdots &{} a_{2n}B \\ \cdot &{} \cdot &{} \cdots &{} \cdot \\ a_{m1}B &{} a_{m2}B &{} \cdots &{} a_{mn}B \\ \end{matrix}\right) .$$
 
8
We use in what follows the property \((A\otimes B)(C\otimes D)=(AC\otimes BD)\), which—in particular—implies \((T^{-1}\otimes I_n)^{-1}=(T\otimes I_n)\).
 
9
See, e.g., [11].
 
10
Note that in this case the elements of R may be complex numbers.
 
11
The existence of such \(P>0\) is guaranteed by the assumption that the pair (AC) is observable. See, e.g., [12].
 
12
See proof of Theorem A.2 in Appendix A.
 
13
See, e.g., [13]. See also [14, 15] and [2123] for further reading.
 
14
The approach described hereafter is based on the work [16].
 
15
See, e.g., [17, p. 22].
 
16
That is, using \(x^*X^*Yy + y^*Y^*Xx \le d x^*X^*Xx + {1\over d} y^*Y^*Yy\), which holds for any \(d>0\).
 
17
The approach described in this section is motivated by the works of [18] and [19]. In particular, the work [18] shows that the approach outline above is in some sense necessary for the solution of the problem in question.
 
18
Note that the latter guarantees the existence a (unique) the solution pair of (5.47).
 
Literatur
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Metadaten
Titel
Coordination and Consensus of Linear Systems
verfasst von
Alberto Isidori
Copyright-Jahr
2017
DOI
https://doi.org/10.1007/978-3-319-42031-8_5

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