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Realization theory for poset-causal systems: controllability, observability and duality

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Abstract

Poset-causal systems form a class of decentralized systems introduced by Shah and Parrilo (47th IEEE conference on decision and control, IEEE, 2008) and studied mainly in the context of optimal decentralized control. In this paper, we develop part of the realization theory for poset-causal systems. More specifically, we investigate several notions of controllability and observability, and their relation under duality. These new notions extend concepts of controllability and observability in the context of coordinated linear systems (Kempker et al. in Linear Algebra Appl 437:121–167, 2012). While for coordinated linear systems there is a clear hierarchical structure with a single (main) coordinator, for poset-causal systems there need not be a single coordinator, and the communication structure between the decentralized systems allows for more intricate structures, governed by partial orders. On the other hand, we show that the class of poset-causal systems is closed under duality, which is not the case for coordinated linear systems, and that duality relations between the various notions of observability and controllability exist.

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Acknowledgements

This work is based on research supported in part by the National Research Foundation of South Africa (NRF) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material are that of the authors and the NRF and CoE-MaSS do not accept any liability in this regard.

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Authors and Affiliations

  1. Department of Mathematics, Research Focus Area: Pure and Applied Analytics, North-West University, Potchefstroom, 2531, South Africa

    S. ter Horst & J. Zeelie

  2. DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    S. ter Horst

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  1. S. ter Horst
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  2. J. Zeelie
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Correspondence to S. ter Horst.

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This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 118513 and 127364).

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ter Horst, S., Zeelie, J. Realization theory for poset-causal systems: controllability, observability and duality. Math. Control Signals Syst. 33, 197–236 (2021). https://doi.org/10.1007/s00498-021-00284-0

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