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Erschienen in:
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2016 | OriginalPaper | Buchkapitel

12. Hybrid System Identification

verfasst von : René Vidal, Yi Ma, S. Shankar Sastry

Erschienen in: Generalized Principal Component Analysis

Verlag: Springer New York

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Abstract

Hybrid systems are mathematical models that are used to describe continuous processes that occasionally exhibit discontinuous behaviors due to sudden changes of dynamics. For instance, the continuous trajectory of a bouncing ball results from alternating between free fall and elastic contact with the ground. However, hybrid systems can also be used to describe a complex process or time series that does not itself exhibit discontinuous behaviors, by approximating the process or series with a simpler class of dynamical models. For example, a nonlinear dynamical system can be approximated by switching among a set of linear systems, each approximating the nonlinear system in a subset of its state space. As another example, a video sequence can be segmented to different scenes by fitting a piecewise linear dynamical model to the entire sequence.

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Vorheriges Kapitel Motion Segmentation
Nächstes Kapitel Final Words
Fußnoten
1
ARX systems are an extremely popular class of dynamical models that are widely used in control, signal processing, communications, and economics. In image/video processing, they can be used to model videos of dynamical scenes.
 
2
Notice that this scheme is impractical, since it requires one to obtain the typically infinitely long output sequence {y t }.
 
3
That is, the polynomials \(z^{\max (n_{c}-n_{a},0)}(z^{n_{a}} - z^{n_{a}-1}a_{1} - z^{n_{a}-2}a_{2} -\cdots - a_{n_{ a}})\) and \(z^{n_{c}-1}c_{1} + z^{n_{c}-2}c_{2} + \cdots + c_{n_{ c}}\) are coprime.
 
4
Only when the initial conditions \(\{y_{t_{0}-1},\ldots,y_{t_{0}-\bar{n}_{a}}\}\) are arbitrary do the data span a hyperplane in \(\mathbb{R}^{D}\) with \(\boldsymbol{b}\) as the only normal vector.
 
5
If n c was known, then we would have \(m =\bar{ n}_{a} + n_{c} + 1\).
 
6
One way to ensure this is to assume that for all \(i\neq j = 1,\ldots,n\), \(\tilde{H}_{i}(z)\) and \(\tilde{H}_{j}(z)\) do not have all their zeros and poles in common. That is, there is no ARX system that can simulate another ARX system with a smaller order. However, this is unnecessary, because two ARX systems can have different configuration spaces even if one system’s zeros and poles are a subset of the other’s.
 
7
This is the case when a particular switching sequence visits only a subset of all the discrete states.
 
8
Rn is the set of homogeneous polynomials of degree n; see Appendix C.
 
9
This is easily verifiable from the fact that the derivatives of the polynomials in \(\mathfrak{a}(Z)\) are exactly the normal vectors of the subspaces.
 
10
\(R_{\leq n}\) is the set of polynomials of degree up to n; see Appendix C.
 
11
That is, the coefficient vector \(\boldsymbol{b} =\boldsymbol{ e}_{1}\) corresponds to the “system” y t  = 0 with \(n_{a} = n_{c} = 0\), which is a trivial ARX system.
 
12
Since here the system is an affine ARX model with a constant input, we need to slightly modify our algorithm by using the homogeneous representation for the regressor \(\boldsymbol{x}_{t}\), i.e., appending an entry of “1.”
 
13
We thank Prof. A. Juloski for providing us with the data sets.
 
14
We take one out of every 80 samples.
 
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Metadaten
Titel
Hybrid System Identification
verfasst von
René Vidal
Yi Ma
S. Shankar Sastry
Copyright-Jahr
2016
Verlag
Springer New York
Buch
Generalized Principal Component Analysis

Print ISBN: 978-0-387-87810-2

Electronic ISBN: 978-0-387-87811-9

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-0-387-87811-9_12

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