Further results on dynamic-algebraic Boolean control networks
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摘要
Restricted coordinate transformation, controllability, observability and topological structures of dynamic-algebraic Boolean control networks are investigated under an assumption. Specifically, given the input-state at some point, assume that the subsequent state is certain or does not exist. First, the system can be expressed in a new form after numbering the elements in admissible set. Then, restricted coordinate transformation is raised, which allows the dimension of new coordinate frame to be different from that of the original one. The system after restricted coordinate transformation is derived in the proposed form.Afterwards, three types of incidence matrices are constructed and the results of controllability, observability and topological structures are obtained. Finally, two practical examples are shown to demonstrate the theory in this paper.
Restricted coordinate transformation, controllability, observability and topological structures of dynamic-algebraic Boolean control networks are investigated under an assumption. Specifically, given the input-state at some point, assume that the subsequent state is certain or does not exist. First, the system can be expressed in a new form after numbering the elements in admissible set. Then, restricted coordinate transformation is raised, which allows the dimension of new coordinate frame to be different from that of the original one. The system after restricted coordinate transformation is derived in the proposed form.Afterwards, three types of incidence matrices are constructed and the results of controllability, observability and topological structures are obtained. Finally, two practical examples are shown to demonstrate the theory in this paper.
Restricted coordinate transformation, controllability, observability and topological structures of dynamic-algebraic Boolean control networks are investigated under an assumption. Specifically, given the input-state at some point, assume that the subsequent state is certain or does not exist. First, the system can be expressed in a new form after numbering the elements in admissible set. Then, restricted coordinate transformation is raised, which allows the dimension of new coordinate frame to be different from that of the original one. The system after restricted coordinate transformation is derived in the proposed form.Afterwards, three types of incidence matrices are constructed and the results of controllability, observability and topological structures are obtained. Finally, two practical examples are shown to demonstrate the theory in this paper.
引文
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8 Zhao J T,Chen Z Q,Liu Z X.Modeling and analysis of colored petri net based on the semi-tensor product of matrices.Sci China Inf Sci,2018,61:010205
9 Liu G J,Jiang C J.Observable liveness of Petri nets with controllable and observable transitions.Sci China Inf Sci,2017,60:118102
10 Cheng D Z,Qi H S.Controllability and observability of Boolean control networks.Automatica,2009,45:1659-1667
11 Cheng D Z,Li Z Q,Qi H S.Realization of Boolean control networks.Automatica,2010,46:62-69
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16 Meng M,Lam J,Feng J E,et al.l1-gain analysis and model reduction problem for Boolean control networks.Inf Sci,2016,348:68-83
17 Cheng D Z,Qi H S,Liu T,et al.A note on observability of Boolean control networks.Syst Control Lett,2016,87:76-82
18 Zhang K Z,Zhang L J.Observability of Boolean control networks:a unified approach based on the theories of finite automata.IEEE Trans Autom Control,2014,61:6854-6861
19 Zhu Q X,Liu Y,Lu J Q,et al.Observability of Boolean control networks.Sci China Inf Sci,2018,61:092201
20 Cheng D Z.Input-state approach to Boolean networks.IEEE Trans Neural Netw,2009,20:512-521
21 Zhao Y,Qi H S,Cheng D Z.Input-state incidence matrix of Boolean control networks and its applications.Syst Control Lett,2010,59:767-774
22 Liu G B,Xu S Y,Wei Y L,et al.New insight into reachable set estimation for uncertain singular time-delay systems.Appl Math Comput,2018,320:769-780
23 Liu L S,Li H D,Wu Y H,et al.Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions.J Nonlinear Sci Appl,2017,10:243-262
24 Liu L S,Sun F L,Zhang X G,et al.Bifurcation analysis for a singular differential system with two parameters via to topological degree theory.Nonlinear Anal Model Control,2017,22:31-50
25 Zheng Z W,Kong Q K.Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indices.J Math Anal Appl,2018,461:1672-1685
26 Cheng D Z,Zhao Y,Xu X R.Mix-valued logic and its applications.J Shandong Univ(Natl Sci),2011,46:32-44
27 Feng J E,Yao J,Cui P.Singular Boolean networks:semi-tensor product approach.Sci China Inf Sci,2013,56:112203
28 Meng M,Feng J E.Optimal control problem of singular Boolean control networks.Int J Control Autom Syst,2015,13:266-273
29 Liu Y,Li B W,Chen H W,et al.Function perturbations on singular Boolean networks.Automatica,2017,84:36-42
30 Qiao Y P,Qi H S,Cheng D Z.Partition-based solutions of static logical networks with applications.IEEE Trans Neural Netw Learn Syst,2018,29:1252-1262
31 Guo Y X.Nontrivial periodic solutions of nonlinear functional differential systems with feedback control.Turkish JMath,2010,34:35
32 Ma C Q,Li T,Zhang J F.Consensus control for leader-following multi-agent systems with measurement noises.JSyst Sci Complex,2010,23:35-49
33 Qin H Y,Liu J W,Zuo X,et al.Approximate controllability and optimal controls of fractional evolution systems in abstract spaces.Adv Diff Equ,2014,2014:322
34 Sun W W,Peng L H.Observer-based robust adaptive control for uncertain stochastic Hamiltonian systems with state and input delays.Nonlinear Anal Model Control,2014,19:626-645
35 Khatri C G,Rao C R.Solutions to some functional equations and their applications to characterization of probability distributions.Indian J Stat Ser A,1968,30:167-180
36 Heidel J,Maloney J,Farrow C,et al.Finding cycles in synchronous Boolean networks with applications to biochemical systems.Int J Bifurcation Chaos,2003,13:535-552
37 Ashenhurst R L.The decomposition of switching functions.In:Proceedings of an International Symposium on the Theory of Switching,1957.74-116
38 Curtis H A.A New Approach to the Design of Switching Circuits.New York:Van Nostrand Reinhold,1962
39 Sasao T,Butler J T.On Bi-Decompositions of Logic Functions.Technical Report,DTIC Document,1997
40 Sasao T.Application of multiple-valued logic to a serial decomposition of plas.In:Proceedings of the 19th International Symposium on Multiple-Valued Logic,1989.264-271
41 Muroga S.Logic Design and Switching Theory.New York:Wiley,1979
2 Akutsu T,Miyano S,Kuhara S.Inferring qualitative relations in genetic networks and metabolic pathways.Bioinformatics,2000,16:727-734
3 Albert R,Barab′asi A L.Dynamics of complex systems:scaling laws for the period of Boolean networks.Phys Rev Lett,2000,84:5660-5663
4 Zhang S Q,Ching W K,Chen X,et al.Generating probabilistic Boolean networks from a prescribed stationary distribution.Inf Sci,2010,180:2560-2570
5 Zhao Q C.A remark on“scalar equations for synchronous Boolean networks with biological applications”by C.Farrow,J.Heidel,J.Maloney,and J.Rogers.IEEE Trans Neural Netw,2005,16:1715-1716
6 Cheng D Z.Semi-tensor product of matrices and its application to Morgen’s problem.Sci China Ser F-Inf Sci,2001,44:195-212
7 Cheng D Z,Qi H S.A linear representation of dynamics of Boolean networks.IEEE Trans Autom Control,2010,55:2251-2258
8 Zhao J T,Chen Z Q,Liu Z X.Modeling and analysis of colored petri net based on the semi-tensor product of matrices.Sci China Inf Sci,2018,61:010205
9 Liu G J,Jiang C J.Observable liveness of Petri nets with controllable and observable transitions.Sci China Inf Sci,2017,60:118102
10 Cheng D Z,Qi H S.Controllability and observability of Boolean control networks.Automatica,2009,45:1659-1667
11 Cheng D Z,Li Z Q,Qi H S.Realization of Boolean control networks.Automatica,2010,46:62-69
12 Cheng D Z,Zhao Y.Identification of Boolean control networks.Automatica,2011,47:702-710
13 Cheng D Z,Qi H S,Li Z Q,et al.Stability and stabilization of Boolean networks.Int J Robust Nonlinear Control,2011,21:134-156
14 Cheng D Z.Disturbance decoupling of Boolean control networks.IEEE Trans Autom Control,2011,56:2-10
15 Liu Y,Li B W,Lu J Q,et al.Pinning control for the disturbance decoupling problem of Boolean networks.IEEETrans Autom Control,2017,62:6595-6601
16 Meng M,Lam J,Feng J E,et al.l1-gain analysis and model reduction problem for Boolean control networks.Inf Sci,2016,348:68-83
17 Cheng D Z,Qi H S,Liu T,et al.A note on observability of Boolean control networks.Syst Control Lett,2016,87:76-82
18 Zhang K Z,Zhang L J.Observability of Boolean control networks:a unified approach based on the theories of finite automata.IEEE Trans Autom Control,2014,61:6854-6861
19 Zhu Q X,Liu Y,Lu J Q,et al.Observability of Boolean control networks.Sci China Inf Sci,2018,61:092201
20 Cheng D Z.Input-state approach to Boolean networks.IEEE Trans Neural Netw,2009,20:512-521
21 Zhao Y,Qi H S,Cheng D Z.Input-state incidence matrix of Boolean control networks and its applications.Syst Control Lett,2010,59:767-774
22 Liu G B,Xu S Y,Wei Y L,et al.New insight into reachable set estimation for uncertain singular time-delay systems.Appl Math Comput,2018,320:769-780
23 Liu L S,Li H D,Wu Y H,et al.Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions.J Nonlinear Sci Appl,2017,10:243-262
24 Liu L S,Sun F L,Zhang X G,et al.Bifurcation analysis for a singular differential system with two parameters via to topological degree theory.Nonlinear Anal Model Control,2017,22:31-50
25 Zheng Z W,Kong Q K.Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indices.J Math Anal Appl,2018,461:1672-1685
26 Cheng D Z,Zhao Y,Xu X R.Mix-valued logic and its applications.J Shandong Univ(Natl Sci),2011,46:32-44
27 Feng J E,Yao J,Cui P.Singular Boolean networks:semi-tensor product approach.Sci China Inf Sci,2013,56:112203
28 Meng M,Feng J E.Optimal control problem of singular Boolean control networks.Int J Control Autom Syst,2015,13:266-273
29 Liu Y,Li B W,Chen H W,et al.Function perturbations on singular Boolean networks.Automatica,2017,84:36-42
30 Qiao Y P,Qi H S,Cheng D Z.Partition-based solutions of static logical networks with applications.IEEE Trans Neural Netw Learn Syst,2018,29:1252-1262
31 Guo Y X.Nontrivial periodic solutions of nonlinear functional differential systems with feedback control.Turkish JMath,2010,34:35
32 Ma C Q,Li T,Zhang J F.Consensus control for leader-following multi-agent systems with measurement noises.JSyst Sci Complex,2010,23:35-49
33 Qin H Y,Liu J W,Zuo X,et al.Approximate controllability and optimal controls of fractional evolution systems in abstract spaces.Adv Diff Equ,2014,2014:322
34 Sun W W,Peng L H.Observer-based robust adaptive control for uncertain stochastic Hamiltonian systems with state and input delays.Nonlinear Anal Model Control,2014,19:626-645
35 Khatri C G,Rao C R.Solutions to some functional equations and their applications to characterization of probability distributions.Indian J Stat Ser A,1968,30:167-180
36 Heidel J,Maloney J,Farrow C,et al.Finding cycles in synchronous Boolean networks with applications to biochemical systems.Int J Bifurcation Chaos,2003,13:535-552
37 Ashenhurst R L.The decomposition of switching functions.In:Proceedings of an International Symposium on the Theory of Switching,1957.74-116
38 Curtis H A.A New Approach to the Design of Switching Circuits.New York:Van Nostrand Reinhold,1962
39 Sasao T,Butler J T.On Bi-Decompositions of Logic Functions.Technical Report,DTIC Document,1997
40 Sasao T.Application of multiple-valued logic to a serial decomposition of plas.In:Proceedings of the 19th International Symposium on Multiple-Valued Logic,1989.264-271
41 Muroga S.Logic Design and Switching Theory.New York:Wiley,1979