Skip to main content
SpringerLink
Log in

Preliminaries

  • Chapter
  • First Online:
Book cover Discrete-Time and Discrete-Space Dynamical Systems

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

  • 645 Accesses

Abstract

In this chapter, we introduce basic theoretical tools used throughout the book, that is, graph theory, the semitensor product of matrices, finite automata, and topology. As in the reading flow shown in Fig. 0.1, topology is used to study invertibility (Sect. 3.2) of Boolean control networks and generalized reversibility (Chap. 11) of cellular automata; graph theory is used to characterize nonsingularity (Sect. 3.3) of Boolean control networks, observability (Sect. 4.2) and detectability (Sect. 5.2) of Boolean control networks and large-scale Boolean control networks (Chap. 6); the semitensor product of matrices is used to give an intuitive matrix representation for Boolean control networks (Chap. 3); the theory of finite automata is used to investigate observability (Sect. 4.2) and detectability (Sect. 5.2) of Boolean control networks, observability (Chap. 7) and detectability (Chap. 8) of nondeterministic finite-transition systems, and detectability of finite-state automata (Chap. 9); some existing theoretical results in labeled Petri nets are used to study detectability of labeled Petri nets (Chap. 10).

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

Access this chapter

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

Institutional subscriptions

References

  • Cassandras CG, Lafortune S (2010) Introduction to discrete event systems. 2nd edn. Springer Publishing Company, Incorporated, Berlin

    Google Scholar 

  • Cheng D (2001) Semi-tensor product of matrices and its application to Morgen’s problem. Sci China Ser: Inf Sci 44(3):195–212

    MathSciNet  MATH  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, Qi H, Li Z (2011) Analysis and control of Boolean networks: a semi-tensor product approach. Springer, London

    Book  Google Scholar 

  • Cheng D, Yang G, Xi Z (2007) Nonlinear systems possessing linear symmetry. Int J Robust Nonlinear Control 17(1):51–81

    Article  MathSciNet  Google Scholar 

  • Cheng D, Zhang L (2003) On semi-tensor product of matrices and its applications. Acta Math Appl Sin 19(2):219–228

    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) On decomposed subspaces of finite games. IEEE Trans Autom Control 61(11):3651–3656

    Article  MathSciNet  Google Scholar 

  • Coornaert M (2015) Topological dimension and dynamical systems. Springer International Publishing, Berlin

    Google Scholar 

  • Čech E (1933) Příspěvek k teorii dimense. Časopis pro pěstování matematiky a fysiky 62(8):277–291

    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 

  • Fornasini E, Valcher ME (2014) Optimal control of Boolean control networks. IEEE Trans Autom Control 59(5):1258–1270

    Article  MathSciNet  Google Scholar 

  • Kari J (2016a) A lecture note on automata and formal languages. http://users.utu.fi/jkari/automata/

  • Kari J (2016b) Cellular automata. http://users.utu.fi/jkari/ca2016/

  • Kůrka P (2003) Topological and symbolic dynamics. Société mathématique de France

    Google Scholar 

  • Li H, Wang Y, Xie L (2015) Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica 59:54–59

    Article  MathSciNet  Google Scholar 

  • Li H, Xie L, Wang Y (2016) On robust control invariance of Boolean control networks. Automatica 68:392–396

    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 

  • Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, New York

    Book  Google Scholar 

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

    MATH  Google Scholar 

  • Lu J et al (2016) On pinning controllability of Boolean control networks. IEEE Trans Autom Control 61(6):1658–1663

    Article  MathSciNet  Google Scholar 

  • Qi H et al (2016) Vector space structure of finite evolutionary games and its application to strategy profile convergence. J Syst Sci Complex 29(3):602–628

    Article  MathSciNet  Google Scholar 

  • Ramadge PJ, Wonham WM (1987) Supervisory control of a class of discrete event processes. SIAM J Control Optim 25(1):206–230

    Article  MathSciNet  Google Scholar 

  • Reutenauer C (1990) The mathematics of petri nets. Prentice-Hall Inc, Upper Saddle River

    MATH  Google Scholar 

  • Shu S, Lin F, Ying H (2007) Detectability of discrete event systems. IEEE Trans Autom Control 52(12):2356–2359

    Article  MathSciNet  Google Scholar 

  • Sipser M (1996) Introduction to the theory of computation, 1st edn. International Thomson Publishing, Stamford

    Article  Google Scholar 

  • Stearns RE, Hartmanis J, Lewis PM (1965) Hierarchies of memory limited computations. In: 6th annual symposium on switching circuit theory and logical design (SWCT 1965), pp 179-190

    Google Scholar 

  • Wonham WM, Cai K (2019) Supervisory control of discrete-event systems. Springer International Publishing, Berlin

    Chapter  Google Scholar 

  • Wu Y, Shen T (2017) Policy iteration algorithm for optimal control of stochastic logical dynamical systems. IEEE Trans Neural Netw Learn Syst 99, 1–6 (2017)

    Google Scholar 

  • Xu X, Hong Y (2013a) Observability analysis and observer design for finite automata via matrix approach. IET Control Theory Appl 7(12):1609–1615

    Google Scholar 

  • Xu X, Hong Y (2013b) Solvability and control design for synchronization of Boolean networks. J Syst Sci Complex 26(6):871–885

    Article  MathSciNet  Google Scholar 

  • Zhang K, Zhang L, Su R (2016) A weighted pair graph representation for reconstructibility of Boolean control networks. SIAM J Control Optim 54(6):3040–3060

    Article  MathSciNet  Google Scholar 

  • Zhang K, Zhang L, Xie L (2015) Invertibility and nonsingularity of Boolean control networks. Automatica 60:155–164

    Article  MathSciNet  Google Scholar 

  • Zhang Z, Leifeld T, Zhang P (2017) Identification of Boolean control networks incorporating prior knowledge. In: 2017 IEEE 56th annual conference on decision and control (CDC), pp 5839–5844

    Google Scholar 

  • Zhao Y, Li Z, Cheng D (2011) Optimal control of logical control networks. IEEE Trans Autom Control 56(8):1766–1776

    Article  MathSciNet  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 

  • Zou Y, Zhu J (2017) Graph theory methods for decomposition w.r.t. outputs of Boolean control networks. J Syst Sci Complex 30(3):519–534

    Article  MathSciNet  Google Scholar 

  • Zou Y, Zhu J (2015) Kalman decomposition for Boolean control networks. Automatica 54:65–71

    Article  MathSciNet  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). Preliminaries. In: Discrete-Time and Discrete-Space Dynamical Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-25972-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation