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Semitensor product approach to controllability, reachability, and stabilizability of probabilistic finite automata. (English) Zbl 1435.93157

Summary: This paper proposes a matrix-based approach to investigate the controllability, reachability, and stabilizability of probabilistic finite automata (PFA). Firstly, the state transition probabilistic structure matrix is constructed for PFA, based on which a kind of controllability matrix is defined for PFA. Secondly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of PFA with positive probability by using the controllability matrix. Finally, an illustrate example is given to validate the obtained new results.

MSC:

93E03 Stochastic systems in control theory (general)
68Q45 Formal languages and automata
93B05 Controllability
93D15 Stabilization of systems by feedback
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93C65 Discrete event control/observation systems

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References:

[1] Xu, X.; Hong, Y., Matrix expression and reachability analysis of finite automata, Control Theory and Technology, 10, 2, 210-215 (2012) · doi:10.1007/s11768-012-1178-4
[2] Yan, Y.; Chen, Z.; Liu, Z., Semi-tensor product approach to controllability and stabilizability of finite automata, Journal of Systems Engineering and Electronics, 26, 1, 134-141 (2015) · doi:10.1109/JSEE.2015.00018
[3] Geffert, V.; Mereghetti, C.; Pighizzini, G., Converting two-way nondeterministic unary automata into simpler automata, Theoretical Computer Science, 295, 1-3, 189-203 (2003) · Zbl 1045.68080 · doi:10.1016/S0304-3975(02)00403-6
[4] Keroglou, C.; Hadjicostis, C. N., Verification of detectability in probabilistic finite automata, Automatica, 86, 192-198 (2017) · Zbl 1375.93118 · doi:10.1016/j.automatica.2017.08.027
[5] Wen, Y.; Ray, A., Vector space formulation of probabilistic finite state automata, Journal of Computer and System Sciences, 78, 4, 1127-1141 (2012) · Zbl 1244.68048 · doi:10.1016/j.jcss.2012.02.001
[6] Cassandras, C. G.; Lafortune, S., Introduction to Discrete Event Systems (1999), Dordrecht, Netherlands: Kluwer Academic Publishers, Dordrecht, Netherlands · Zbl 0934.93001 · doi:10.1007/978-1-4757-4070-7
[7] Kumar, R.; Garg, V. K., Modeling and Control of Logical Discrete Event Systems (1995), Dordrecht, Netherlands: Kluwer Academic Publishers, Dordrecht, Netherlands · Zbl 0875.68980
[8] Delvenne, J.-C.; Blondel, V. D., Complexity of control on finite automata, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 51, 6, 977-986 (2006) · Zbl 1368.68231 · doi:10.1109/TAC.2006.876948
[9] Lygeros, J.; Tomlin, C.; Sastry, S., Controllers for reachability specifications for hybrid systems, Automatica, 35, 3, 349-370 (1999) · Zbl 0943.93043 · doi:10.1016/S0005-1098(98)00193-9
[10] Xu, X.; Hong, Y., Matrix approach to model matching of asynchronous sequential machines, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 58, 11, 2974-2979 (2013) · Zbl 1369.93354 · doi:10.1109/TAC.2013.2259957
[11] Jimenez, E.; Julvez, J.; Recalde, L.; Silva, M., On controllability of timed continuous petri net systems: the join free case, Proceedings of the 44th IEEE Conference On Decision and Control · doi:10.1109/CDC.2005.1583396
[12] Júlvez, J.; Jiménez, E.; Recalde, L.; Silva, M., On observability and design of observers in timed continuous petri net systems, IEEE Transactions on Automation Science and Engineering, 5, 3, 532-537 (2008) · doi:10.1109/TASE.2008.917016
[13] Latorre, J.; Jimenez, E.; Julvez, J.; Perez, M., Macro-reachability tree exploration for D.E.S design optimization, Proceedings of the 6th EUROSIM Congress on Modelling and Simulation
[14] Latorre, J.; Jimenez, E.; Julvez, J.; Perez, M., Control of discrete event systems modelled by petri nets, Proceedings of the 7th EUROSIM Congress on Modelling and Simulation
[15] Han, X.; Chen, Z.; Zhang, K.; Liu, Z.; Zhang, Q., Modeling, reachability and controllability of bounded petri nets based on semi-tensor product of matrices, Asian Journal of Control (2019) · doi:10.1002/asjc.1915
[16] Vidal, E.; Casacuberta, F., Probabilistic finite-state machines-Part I, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27, 7, 1013-1025 (2005) · doi:10.1109/TPAMI.2005.147
[17] Vidal, E.; Casacuberta, F., Probabilistic finite-state machines-Part II, IEEE Transactions on Pattern Analysis and Machine Intelligence, 27, 7, 1026-1039 (2005) · doi:10.1109/TPAMI.2005.148
[18] Zhang, Z.; Chen, Z.; Liu, Z., Modelling and reachability of probabilistic finite automata based on the semi-tensor product of matrices, Science China Information Sciences, 61, 12, 129-202 (2018) · doi:10.1007/s11432-018-9507-7
[19] Shu, S.; Lin, F.; Ying, H.; Chen, X., State estimation and detectability of probabilistic discrete event systems, Automatica, 44, 12, 3054-3060 (2008) · Zbl 1153.93512 · doi:10.1016/j.automatica.2008.05.025
[20] Li, Y.; Lin, F.; Lin, Z. H., Supervisory control of probabilistic discrete-event systems with recovery, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 44, 10, 1971-1975 (1999) · Zbl 0956.93038 · doi:10.1109/9.793788
[21] Cheng, D. Z.; Qi, H. S.; Zhao, Y., An Introduction to Semi-Tensor Product of Matrices and Its Applications (2012), Singapore: World Scientific Publishing, Singapore · Zbl 1273.15029 · doi:10.1142/8323
[22] Han, X. G.; Chen, Z. Q.; Liu, Z. X., STP-based judgment method of reversibility and liveness of bounded Petri nets, Journal of Systems Science and Mathematical Sciences, 36, 361-370 (2016) · Zbl 1363.93165
[23] Guo, Y.; Wang, P.; Gui, W.; Yang, C., Set stability and set stabilization of Boolean control networks based on invariant subsets, Automatica, 61, 106-112 (2015) · Zbl 1327.93347 · doi:10.1016/j.automatica.2015.08.006
[24] Li, F.; Li, H.; Xie, L.; Zhou, Q., On stabilization and set stabilization of multivalued logical systems, Automatica, 80, 41-47 (2017) · Zbl 1370.93238 · doi:10.1016/j.automatica.2017.01.032
[25] Zou, Y.; Zhu, J., Kalman decomposition for Boolean control networks, Automatica, 54, 65-71 (2015) · Zbl 1318.93043 · doi:10.1016/j.automatica.2015.01.023
[26] Cheng, D.; Qi, H.; Liu, Z., From STP to game-based control, Science China Information Sciences, 61, 1 (2018) · doi:10.1007/s11432-017-9265-2
[27] Li, H.; Wang, Y., Lyapunov-based stability and construction of Lyapunov functions for Boolean networks, SIAM Journal on Control and Optimization, 55, 6, 3437-3457 (2017) · Zbl 1373.93249 · doi:10.1137/16M1092581
[28] Shen, X.; Wu, Y.; Shen, T., Logical control scheme with real-time statistical learning for residual gas fraction in IC engines, Science China Information Sciences, 61, 1 (2018)
[29] Chen, H.; Li, X.; Sun, J., Stabilization, controllability and optimal control of Boolean networks with impulsive effects and state constraints, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 60, 3, 806-811 (2015) · Zbl 1360.93093 · doi:10.1109/TAC.2014.2330432
[30] Li, H.; Wang, Y., Further results on feedback stabilization control design of Boolean control networks, Automatica, 83, 303-308 (2017) · Zbl 1373.93259 · doi:10.1016/j.automatica.2017.06.043
[31] Li, Y.; Li, H.; Sun, W., Event-triggered control for robust set stabilization of logical control networks, Automatica, 95, 556-560 (2018) · Zbl 1402.93221 · doi:10.1016/j.automatica.2018.06.030
[32] Lu, J.; Li, H.; Liu, Y.; Li, F., Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory & Applications, 11, 13, 2040-2047 (2017) · doi:10.1049/iet-cta.2016.1659
[33] Meng, M.; Liu, L.; Feng, G., Stability and l1 gain analysis of Boolean networks with Markovian jump parameters, Institute of Electrical and Electronics Engineers Transactions on Automatic Control, 62, 8, 4222-4228 (2017) · Zbl 1373.93368 · doi:10.1109/TAC.2017.2679903
[34] Zheng, Y.; Li, H.; Ding, X.; Liu, Y., Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization, Journal of The Franklin Institute, 354, 17, 7812-7827 (2017) · Zbl 1380.93205 · doi:10.1016/j.jfranklin.2017.09.024
[35] Zhong, J.; Lu, J.; Huang, T.; Ho, D. W. C., Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks, IEEE Transactions on Cybernetics, 47, 11, 3482-3493 (2017) · doi:10.1109/TCYB.2016.2560240
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