Skip to main content
SpringerLink
Log in
Book cover

Discrete-Time and Discrete-Space Dynamical Systems pp 87–104Cite as

Observability of Boolean Control Networks

  • Chapter
  • First Online:

  • 677 Accesses

Part of the book series: Communications and Control Engineering ((CCE))

Abstract

Given a dynamical system, as the system evolves, a state trajectory is generated. Generally speaking, a quantitative analysis of the system closely depends on states of a trajectory. Particularly for a deterministic system, if the initial state has been determined, then the corresponding trajectory will be naturally determined by using an input sequence. That is, the initial state will help understand the whole information of the corresponding trajectory for deterministic systems.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Parts of Sects. 4.2 and 4.3 were reproduced from Zhang and Zhang (2014) with permission @ 2016 IEEE. Some of the material in Sect. 4.4 were reproduced from Laschov et al. (2013) with permission @ 2013 Elsevier Ltd.

  2. 2.

    Actually a sufficient but not necessary condition for BCNs.

  3. 3.

    Also actually a sufficient but not necessary condition for BCNs.

  4. 4.

    In order to verify whether \(\mathcal {L}(\mathcal {A}_{\{x_0,x_0'\}})=(\Delta _M)^*\) holds, it is a very intuitive way to verify the completeness of \(\mathcal {A}_{\{x_0,x_0'\}}\) by Proposition 4.1.

  5. 5.

    Recall that (see Sect. 2.2) Moore machine as a generalization of a BCN is a deterministic total finite-transition system, where every state can be initial and its output map does not depend on any input.

  6. 6.

    Following the symbol \((HL)_{x_0}^{2^n}(U)\) with U being of length \(2^n\), since there is only one input sequence, we remove (U) for short.

References

  • Cheng D, Li C, He F (2018) Observability of Boolean networks via set controllability approach. Syst Control Lett 115:22–25

    Article  MathSciNet  Google Scholar 

  • Cheng D, Qi H (2009) Controllability and observability of Boolean control networks. Automatica 45(7):1659–1667

    Article  MathSciNet  Google Scholar 

  • Cheng D, Zhao Y (2011) Identification of Boolean control networks. Automatica 47(4):702–710

    Article  MathSciNet  Google Scholar 

  • Cheng D et al (2016) A note on observability of Boolean control networks. Syst Control Lett 87:76–82

    Article  MathSciNet  Google Scholar 

  • Fornasini E, Valcher ME (2013) Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control 58(6):1390–1401

    Article  MathSciNet  Google Scholar 

  • Kalman RE, Falb PL, Arbib MA (1969) Topics in mathematical system theory. International series in pure and applied mathematics. McGraw-Hill

    Google Scholar 

  • Kalman RE (1963) Mathematical description of linear dynamical systems. J Soc Ind Appl Math Ser Control 1(12):152–192

    Article  MathSciNet  Google Scholar 

  • Laschov D, Margaliot M, Even G (2013) Observability of Boolean networks: a graphtheoretic approach. Automatica 49(8):2351–2362

    Article  MathSciNet  Google Scholar 

  • Li R, Yang M, Chu T (2015) Controllability and observability of Boolean networks arising from biology. Chaos: Interdiscip J Nonlinear Sci 25(2), 023104–15

    Article  MathSciNet  Google Scholar 

  • Li R, Yang M, Chu T (2014) Observability conditions of Boolean control networks. Int J Robust Nonlinear Control 24(17):2711–2723

    Article  MathSciNet  Google Scholar 

  • Linz P (2006) An introduction to formal language and automata. Jones and Bartlett Publishers Inc, USA

    MATH  Google Scholar 

  • Moore EF (1956) Gedanken-experiments on sequential machines. Autom Stud, Ann Math Stud 34, 129–153

    Google Scholar 

  • Sontag ED (1979) On the observability of polynomial systems, I: finite-time problems. SIAM J Control Optim 17:139–151

    Article  MathSciNet  Google Scholar 

  • Wonham WM (1985) Linear multivariable control: a geometric approach, 3rd edn. Springer, New York

    Book  Google Scholar 

  • Zhang K, Zhang L (2016) Observability of Boolean control networks: a unified approach based on finite automata. IEEE Trans Autom Control 61(9):2733–2738

    Article  MathSciNet  Google Scholar 

  • Zhang K, Zhang L (2014) Observability of Boolean control networks: a unified approach based on the theories of finite automata and formal languages. In: Proceedings of the 33rd Chinese control conference, pp 6854-6861

    Google Scholar 

  • Zhao Y, Qi H, Cheng D (2010) Input-state incidence matrix of Boolean control networks and its applications. Syst Control Lett 59(12):767–774

    Article  MathSciNet  Google Scholar 

  • Zhu Q et al (2018) Observability of Boolean control networks. Sci China Inf Sci 61(9), 092201

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden

    Kuize Zhang

  2. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, China

    Lijun Zhang

  3. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore

    Lihua Xie

Authors
  1. Kuize Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lijun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lihua Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kuize Zhang .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zhang, K., Zhang, L., Xie, L. (2020). Observability of Boolean Control Networks. In: Discrete-Time and Discrete-Space Dynamical Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-25972-3_4

Download citation

Publish with us

Policies and ethics

Search

Navigation