Abstract
We study the ergodicity of backward product of stochastic and doubly stochastic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of convergence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochastic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1477-1488 |
| Number of pages | 12 |
| Journal | Automatica |
| Volume | 48 |
| Issue number | 8 |
| DOIs | |
| State | Published - Aug 2012 |
| Externally published | Yes |
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Keywords
- Absolute infinite flow property
- Averaging control
- Discrete inclusion systems
- Distributed control
- Doubly stochastic matrices
- Ergodicity
- Product of stochastic matrices
- Switching control
ASJC Scopus subject areas
- Control and Systems Engineering
- Electrical and Electronic Engineering
Cite this
On backward product of stochastic matrices. / Touri, Behrouz; Nedich, Angelia.
In: Automatica, Vol. 48, No. 8, 08.2012, p. 1477-1488.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On backward product of stochastic matrices
AU - Touri, Behrouz
AU - Nedich, Angelia
PY - 2012/8
Y1 - 2012/8
N2 - We study the ergodicity of backward product of stochastic and doubly stochastic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of convergence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochastic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.
AB - We study the ergodicity of backward product of stochastic and doubly stochastic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of convergence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochastic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.
KW - Absolute infinite flow property
KW - Averaging control
KW - Discrete inclusion systems
KW - Distributed control
KW - Doubly stochastic matrices
KW - Ergodicity
KW - Product of stochastic matrices
KW - Switching control
UR - http://www.scopus.com/inward/record.url?scp=84864471562&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84864471562&partnerID=8YFLogxK
U2 - 10.1016/j.automatica.2012.05.025
DO - 10.1016/j.automatica.2012.05.025
M3 - Article
AN - SCOPUS:84864471562
VL - 48
SP - 1477
EP - 1488
JO - Automatica
JF - Automatica
SN - 0005-1098
IS - 8
ER -