Abstract

This paper studies the feedback stabilization problem of $k$-valued logical control networks (KVLCNs), and proposes a control Lyapunov function (CLF) approach for this problem. First, the CLF is defined for KVLCNs, and it is proved that the existence of state feedback stabilizers is equivalent to the existence of CLF. Second, two necessary and sufficient conditions are presented for the existence of CLF, based on which, all possible state feedback stabilizers are characterized by finding all admissible sets of control Lyapunov inequalities. Third, the concept of convergence index vector is defined for KVLCNs, and it is shown that for a given admissible set of control Lyapunov inequalities, the CLF is unique in the sense of convergence index vector. Finally, the obtained new results are applied to regulation of the lactose operon in Escherichia coli, stabilization of switched KVLCNs, and strategy consensus of networked evolutionary games, respectively.

Keywords

  1. logical control network
  2. feedback stabilization
  3. control Lyapunov function
  4. convergence index vector
  5. semi-tensor product of matrices

MSC codes

  1. 06E30
  2. 93B05
  3. 93B52
  4. 93C55

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
F. Ay, F. Xu, and T. Kahveci, Scalable steady state analysis of Boolean biological regulatory networks, PLoS ONE, 4 (2009), e7992.
2.
A. Bemporad and M. Morari, Control of systems integrating logic, dynamics, and constraints, Automatica J. IFAC, 35 (1999), pp. 407--427.
3.
N. Bof, E. Fornasini, and M. Valcher, Output feedback stabilization of Boolean control networks, Automatica J. IFAC, 57 (2015), pp. 21--28.
4.
H. Chen, X. Li, and J. Sun, Stabilization, controllability and optimal control of Boolean networks with impulsive effects and state constraints, IEEE Trans. Automat. Control, 60 (2015), pp. 806--811.
5.
H. Chen and J. Sun, Output controllability and optimal output control of state-dependent switched Boolean control networks, Automatica J. IFAC, 50 (2014), pp. 1929--1934.
6.
D. Cheng, H. Qi, and Z. Li, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach, Springer-Verlag, London, 2011.
7.
D. Cheng, H. Qi, Z. Li, and J. Liu, Stability and stabilization of Boolean networks, Internat. J. Robust Nonlinear Control, 21 (2011), pp. 134--156.
8.
D. Cheng, F. He, H. Qi, and T. Xu, Modeling, analysis and control of networked evolutionary games, IEEE Trans. Automat. Control, 60 (2015), pp. 2402--2415.
9.
D. Cheng, C. Li, and F. He, Observability of Boolean networks via set controllability approach, Systems Control Lett., 115 (2018), pp. 22--25.
10.
E. Fornasini and M. Valcher, On the periodic trajectories of Boolean control networks, Automatica J. IFAC, 49 (2013), pp. 1506--1509.
11.
E. Fornasini and M. Valcher, Recent developments in Boolean networks control, J. Control Decis., 3 (2016), pp. 1--18.
12.
R. Goebel, C. Prieur, and A. Teel, Smooth patchy control Lyapunov functions, Automatica J. IFAC, 45 (2009), pp. 675--683.
13.
P. Guo, Y. Wang, and H. Li, Algebraic formulation and strategy optimization for a class of evolutionary networked games, Automatica J. IFAC, 49 (2013), pp. 3384--3389.
14.
Y. Guo, P. Wang, W. Gui, and C. Yang, Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica J. IFAC, 61 (2015), pp. 106--112.
15.
I. Karafyllis and Z. Jiang, Global stabilization of nonlinear systems based on vector control Lyapunov functions, IEEE Trans. Automat. Control, 58 (2013), pp. 2550--2562.
16.
S. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theoret. Biol., 22 (1969), pp. 437--467.
17.
D. Laschov and M. Margaliot, Controllability of Boolean control networks via the Perron-Frobenius theory, Automatica J. IFAC, 48 (2012), pp. 1218--1223.
18.
F. Li and Y. Tang, Set stabilization for switched Boolean control networks, Automatica J. IFAC, 78 (2017), pp. 223--230.
19.
H. Li and Y. Wang, Output feedback stabilization control design for Boolean control networks, Automatica J. IFAC, 49 (2013), pp. 3641--3645.
20.
H. Li and Y. Wang, Controllability analysis and control design for switched Boolean networks with state and input constraints, SIAM J. Control Optim., 53 (2015), pp. 2955--2979, https://doi.org/10.1137/120902331.
21.
H. Li, L. Xie, and Y. Wang, On robust control invariance of Boolean control networks, Automatica J. IFAC, 68 (2016), pp. 392--396.
22.
H. Li and Y. Wang, Further results on feedback stabilization control design of Boolean control networks, Automatica J. IFAC, 83 (2017), pp. 303--308.
23.
H. Li and Y. Wang, Lyapunov-based stability and construction of Lyapunov functions for Boolean networks, SIAM J. Control Optim., 55 (2017), pp. 3437--3457, https://doi.org/10.1137/16M1092581.
24.
M. Li, J. Lu, J. Lou, Y. Liu, and F. E. Alssadi, The equivalence issue of two kinds of controllers in Boolean control networks, Appl. Math. Comput., 321 (2018), pp. 633--640.
25.
R. Li, M. Yang, and T. Chu, State feedback stabilization for Boolean control networks, IEEE Trans. Automat. Control, 58 (2013), pp. 1853--1857.
26.
Y. Li, H. Li, X. Ding, and G. Zhao, Leader-follower consensus of multiagent systems with time delays over finite fields, IEEE Trans. Cybern., to appear, https://doi.org/10.1109/TCYB.2018.2839892.
27.
J. Liang, H. Chen, and Y. Liu, On algorithms for state feedback stabilization of Boolean control networks, Automatica J. IFAC, 84 (2017), pp. 10--16.
28.
Y. Liu, H. Chen, J. Lu, and B. Wu, Controllability of probabilistic Boolean control networks based on transition probability matrices, Automatica J. IFAC, 52 (2015), pp. 340--345.
29.
Y. Liu, B. Li, J. Lu, and J. Cao, Pinning control for the disturbance decoupling problem of Boolean networks, IEEE Trans. Automat. Control, 62 (2017), pp. 6595--6601.
30.
J. Lu, M. Li, Y. Liu, D. W. C. Ho, and J. Kurths, Nonsingularity of grain-like cascade FSRs via semi-tensor product, Sci. China Inf. Sci., 61 (2018), 010204.
31.
J. Lu, H. Li, Y. Liu, and F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory Appl., 11 (2017), pp. 2040--2047.
32.
Y. Lu and W. Zhang, A piecewise smooth control-Lyapunov function framework for switching stabilization, Automatica J. IFAC, 76 (2017), pp. 258--265.
33.
M. Meng, L. Liu, and G. Feng, Stability and $l_1$ gain analysis of Boolean networks with Markovian jump parameters, IEEE Trans. Automat. Control, 62 (2017), pp. 4222--4228.
34.
C. Possieri and A. Teel, Asymptotic stability in probability for stochastic Boolean networks, Automatica J. IFAC, 83 (2017), pp. 1--9.
35.
R. Robeva and T. Hodge, Mathematical Concepts and Methods in Modern Biology: Using Modern Discrete Models, Academic Press, San Diego, 2013.
36.
R. Sanfelice, On the existence of control Lyapunov functions and state-feedback laws for hybrid systems, IEEE Trans. Automat. Control, 58 (2013), pp. 3242--3248.
37.
G. Scutari, S. Barbarossa, and D. P. Palomar, Potential games: A framework for vector power control problems with coupled constraints, in Proceedings of ICASSP, IEEE, 2006, pp. 241--244.
38.
J. Suo and J. Sun, Asymptotic stability of differential systems with impulsive effects suffered by logic choice, Automatica J. IFAC, 51 (2015), pp. 302--307.
39.
H. Tian, H. Zhang, Z. Wang, and Y. Hou, Stabilization of $k$-valued logical control networks by open-loop control via the reverse-transfer method, Automatica J. IFAC, 83 (2017), pp. 387--390.
40.
G. Vahedi, B. Faryabi, J. Chamberland, A. Datta, and E. Dougherty, Optimal intervention strategies for cyclic therapeutic methods, IEEE Trans. Biomed. Eng., 56 (2009), pp. 281--291.
41.
U. Vaidya, P. Mehta, and U. Shanbhag, Nonlinear stabilization via control Lyapunov measure, IEEE Trans. Automat. Control, 55 (2010), pp. 1314--1328.
42.
Y. Wang and D. Cheng, Stability and stabilization of a class of finite evolutionary games, J. Franklin Inst., 354 (2017), pp. 1603--1617.
43.
Y. Wu and T. Shen, An algebraic expression of finite horizon optimal control algorithm for stochastic logical dynamical systems, Systems Control Lett., 82 (2015), pp. 108--114.
44.
X. Xu and Y. Hong, Matrix approach to model matching of asynchronous sequential machines, IEEE Trans. Automat. Control, 58 (2013), pp. 2974--2979.
45.
K. Zhang, L. Zhang, and L. Xie, Invertibility and nonsingularity of Boolean control networks, Automatica J. IFAC, 60 (2015), pp. 155--164.
46.
Z. Zhang, T. Leifeld, and P. Zhang, Finite horizon tracking control of Boolean control networks, IEEE Trans. Automat. Control, 63 (2018), pp. 1798--1805.
47.
J. Zhong, J. Lu, Y. Liu, and J. Cao, Synchronization in an array of output-coupled Boolean networks with time delay, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), pp. 2288--2294.
48.
J. Zhong, J. Lu, T. Huang, and D. Ho, Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks, IEEE Trans. Cybern., 47 (2017), pp. 3482--3493.
49.
B. Zhu, X. Xia, and Z. Wu, Evolutionary game theoretic demand-side management and control for a class of networked smart grid, Automatica J. IFAC, 70 (2016), pp. 94--100.
50.
Q. Zhu, Y. Liu, J. Lu, and J. Cao, Observability of Boolean control networks, Sci. China Inf. Sci., 2018 (61), 092201.
51.
Y. Zou and J. Zhu, System decomposition with respect to inputs for Boolean control networks, Automatica J. IFAC, 50 (2014), pp. 1304--1309.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 810 - 831
ISSN (online): 1095-7138

History

Submitted: 12 February 2018
Accepted: 11 December 2018
Published online: 7 March 2019

Keywords

  1. logical control network
  2. feedback stabilization
  3. control Lyapunov function
  4. convergence index vector
  5. semi-tensor product of matrices

MSC codes

  1. 06E30
  2. 93B05
  3. 93B52
  4. 93C55

Authors

Affiliations

Funding Information

National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 61873150, 61503225
Natural Science Foundation of Shandong Province https://doi.org/10.13039/501100007129 : ZR2015FQ003, JQ201613
Shandong University https://doi.org/10.13039/100009108 : SCX201805

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media