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On state estimation for nonlinear systems under random access wireless protocols

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Abstract

This article is dedicated to Eduardo D. Sontag on the occasion of his 70th birthday. We build upon fundamental stability concepts developed by Sontag, such as input-to-state stability and its related properties, to study a relevant application in industrial internet of things, namely estimation for wireless networked control systems. Particularly, we study emulation-based state estimation for nonlinear plants that communicate with a remote observer over a shared wireless network subject to packet losses. To reduce bandwidth usage, a stochastic communication protocol is employed to determine which node should be given access to the network. Each node has a different successful transmission probability. We describe the overall closed-loop system as a stochastic hybrid model, which allows us to capture the behaviour both between and at transmission instants, whilst covering network features such as random transmission instants, packet losses and stochastic scheduling. We then provide sufficient conditions on the transmission rate that guarantee an input-to-state stability property (in expectation) for the corresponding estimation error system. We illustrate our results in the design of circle criterion observers.

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Notes

  1. Note that the right-hand side inequality in (12) implies \(\varvec{\lambda }>|A|\), more details in the proof in Sect. 6.1.

  2. It suffices to verify that the holding times \(S_{M+1}-S_M\) are i.i.d. and have positive finite mean. Since \(T_k\)’s and \(\tau _k\)’s are i.i.d., then the holding time \(S_{M+1}-S_M\) is also i.i.d.. Next, we have \(\mathbb {E}\left\{ S_{M+1}-S_M\right\} =\mathbb {E}\left\{ T_M\right\} \mathbb {E}\left\{ \tau _{M,i}\right\} \). Particularly, \(\mathbb {E}\left\{ \tau _{M,i}\right\} =1/\varvec{\lambda }\), and \(\mathbb {E}\left\{ T_M\right\} \) can be found in Lemma 1. Consequently, \(\mathbb {E}\left\{ S_{M+1}-S_M\right\} =\frac{1}{\varvec{\lambda }}\sum _{j=1}^{N}\frac{N}{[N-(j-1)]p_j}\), which is positive and finite since \(p_j\in (0,1]\) for all \(j\in \mathcal {N}\), \(\varvec{\lambda }\in (0,\infty )\), and \(N\ge 1\).

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Funding

This work was supported by the Australian Research Council under the Discovery Grant DP200101303.

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, 3010, VIC, Australia

    Alejandro I. Maass & Dragan Nešić

  2. CRAN, CNRS, Université de Lorraine, 5400, Nancy, France

    Romain Postoyan

  3. Department of Mechanical Engineering, The University of Melbourne, Melbourne, 3010, VIC, Australia

    Ying Tan

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  1. Alejandro I. Maass
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  2. Dragan Nešić
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  4. Ying Tan
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Correspondence to Alejandro I. Maass.

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This paper is dedicated to Eduardo Sontag on the occasion of his 70th birthday.

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Maass, A.I., Nešić, D., Postoyan, R. et al. On state estimation for nonlinear systems under random access wireless protocols. Math. Control Signals Syst. 35, 187–213 (2023). https://doi.org/10.1007/s00498-022-00337-y

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