Finite-time stability and stabilization of discrete-time hybrid systems. (English) Zbl 07885795
Summary: This paper is concerned with the finite-time stability and stabilization problems for a class of hybrid systems, which consists of discrete-time continuous-valued and Boolean dynamics. First, we introduce the so-called systems briefly and give the algebraic form of the systems via Khatri-Rao product and semi-tensor product (STP, i.e., Cheng product). Next, we originally propose the concept of finite-time stability (FTS) for the hybrid systems. Furthermore, some criteria of FTS for the hybrid systems are provided. Via a lemma we present, the computational complexity of the condition on FTS for the logical part could be reduced to \(O (1)\). Based on these obtained results, two classes of controllers, state feedback controllers and logical controllers, are designed. Finally, two numerical examples are given to demonstrate the effectiveness of the theoretical results.
MSC:
| 93D40 | Finite-time stability |
| 93C55 | Discrete-time control/observation systems |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 93B52 | Feedback control |
| 93C29 | Boolean control/observation systems |
References:
| [1] | Antsaklis, P. J., Special issue on hybrid systems: Theory and applications a brief introduction to the theory and applications of hybrid systems, Proc. IEEE, 88, 7, 879-887, 2000 |
| [2] | Bemporad, A.; Giorgetti, N., A SAT-based hybrid solver for optimal control of hybrid systems, (Hybrid Systems: Computation and Control, 2004, Springer: Springer Berlin, Heidelberg), 126-141 · Zbl 1135.93348 |
| [3] | Bemporad, A.; Morari, M., Control of systems integrating logic, dynamics, and constraints, Automatica, 35, 3, 407-427, 1999 · Zbl 1049.93514 |
| [4] | Suo, J.; Sun, J., Asymptotic stability of differential systems with impulsive effects suffered by logic choice, Automatica, 51, 302-307, 2015 · Zbl 1309.93132 |
| [5] | Xue, Y.; Sun, J.; Zhang, Y., Existence of open-loop equilibria in differential games with impulsive effects suffered by logic choice, Systems Control Lett., 185, Article 105729 pp., 2024 · Zbl 1537.91040 |
| [6] | Zhang, J.; Sun, J., Exponential synchronization of complex networks with continuous dynamics and Boolean mechanism, Neurocomputing, 307, 146-152, 2018 |
| [7] | Sun, T.; Zhao, X.; Sun, X.-M., Switched dynamic systems with logic switching and its stability analysis, SIAM J. Control Optim., 59, 2, 1188-1217, 2021 · Zbl 1465.93102 |
| [8] | Wang, Q.; Sun, J., Finite-time stability of hybrid systems with continuous and Boolean dynamics, Nonlinear Anal. Hybrid Syst., 49, Article 101375 pp., 2023 · Zbl 1520.93480 |
| [9] | Yao, Y.; Sun, J.; Zhang, Y., Multitask synthesis of hybrid systems via temporal logic, IEEE Trans. Autom. Control, 68, 11, 6883-6890, 2023 · Zbl 1535.93009 |
| [10] | Braga, M. F.; Morais, C. F.; Tognetti, E. S.; Oliveira, R. C.; Peres, P. L., Discretization and event triggered digital output feedback control of LPV systems, Systems Control Lett., 86, 54-65, 2015 · Zbl 1325.93031 |
| [11] | Raman, R.; Grossmann, I., Relation between MILP modelling and logical inference for chemical process synthesis, Comput. Chem. Eng., 15, 2, 73-84, 1991 |
| [12] | Liu, D.; Yu, W.; Baldi, S.; Cao, J.; Huang, W., A switching-based adaptive dynamic programming method to optimal traffic signaling, IEEE Trans. Syst. Man Cybern. Syst., 50, 11, 4160-4170, 2020 |
| [13] | Guo, Y.; Wu, Y.; Gui, W., Stability of discrete-time systems under restricted switching via logic dynamical generator and STP-based mergence of hybrid states, IEEE Trans. Autom. Control, 67, 7, 3472-3483, 2022 · Zbl 1537.93578 |
| [14] | Kamenkov, G., On stability of motion over a finite interval of time, J. Appl. Math. Mech., 17, 529-540, 1953 · Zbl 0055.32101 |
| [15] | P. Dorato, Short time stability in linear time-varying systems, in: Proceedings of the IRE International Convention Record Part 4, 1961, pp. 83-87. |
| [16] | Zhang, J.; Wang, Q.-G.; Sun, J., On finite-time stability of nonautonomous nonlinear systems, Internat. J. Control, 93, 4, 783-787, 2020 · Zbl 1436.93125 |
| [17] | Mele, A.; Pironti, A., Assessing the finite-time stability of nonlinear systems by means of physics-informed neural networks, Systems Control Lett., 178, Article 105580 pp., 2023 · Zbl 1520.93472 |
| [18] | Amato, F.; Ariola, M.; Cosentino, C., Finite-time stability of linear time-varying systems: Analysis and controller design, IEEE Trans. Autom. Control, 55, 4, 1003-1008, 2010 · Zbl 1368.93457 |
| [19] | Yang, Y.; Li, J.; Chen, G., Finite-time stability and stabilization of nonlinear stochastic hybrid systems, J. Math. Anal. Appl., 356, 1, 338-345, 2009 · Zbl 1163.93033 |
| [20] | Amato, F.; Ambrosino, R.; Ariola, M.; Cosentino, C.; De Tommasi, G., Finite-time stability and control, in: Lecture notes in control and information sciences, 2014, Springer: Springer London · Zbl 1297.93001 |
| [21] | Guan, K.; Luo, R., Finite-time stability of impulsive pantograph systems with applications, Systems Control Lett., 157, Article 105054 pp., 2021 · Zbl 1480.93365 |
| [22] | Hu, M.-J.; Park, J. H., Finite-time \(\mathcal{L}_1\) control via hybrid state feedback for uncertain positive systems with impulses, Internat. J. Robust Nonlinear Control, 31, 5, 1600-1620, 2021 · Zbl 1526.93191 |
| [23] | Xi, Q.; Liu, X., Finite-time stability and controller design for a class of hybrid dynamical systems with deviating argument, Nonlinear Anal. Hybrid Syst., 39, Article 100952 pp., 2021 · Zbl 1478.93609 |
| [24] | Friedkin, N. E.; Proskurnikov, A. V.; Tempo, R.; Parsegov, S. E., Network science on belief system dynamics under logic constraints, Science, 354, 6310, 321-326, 2016 · Zbl 1355.91060 |
| [25] | Glass, L.; Edwards, R., Hybrid models of genetic networks: Mathematical challenges and biological relevance, J. Theoret. Biol., 458, 111-118, 2018 · Zbl 1406.92241 |
| [26] | Cheng, D.; Qi, H., A linear representation of dynamics of Boolean networks, IEEE Trans. Autom. Control, 55, 10, 2251-2258, 2010 · Zbl 1368.37025 |
| [27] | Chen, H.; Sun, J., A new approach for global controllability of higher order Boolean control network, Neural Netw., 39, 12-17, 2013 · Zbl 1327.93073 |
| [28] | Zhao, Y.; Qi, H.; Cheng, D., Input-state incidence matrix of Boolean control networks and its applications, Systems Control Lett., 59, 12, 767-774, 2010 · Zbl 1217.93026 |
| [29] | Li, F.; Sun, J., Controllability of probabilistic Boolean control networks, Automatica, 47, 12, 2765-2771, 2011 · Zbl 1235.93044 |
| [30] | Meng, M.; Lam, J.; Feng, J.-E.; Li, X., \( l_1\)-gain analysis and model reduction problem for Boolean control networks, Inform. Sci., 348, 68-83, 2016 · Zbl 1398.93151 |
| [31] | Li, F.; Tang, Y., Set stabilization for switched Boolean control networks, Automatica, 78, 223-230, 2017 · Zbl 1357.93079 |
| [32] | Zhu, S.; Lu, J.; Liu, Y., Asymptotical stability of probabilistic Boolean networks with state delays, IEEE Trans. Autom. Control, 65, 4, 1779-1784, 2020 · Zbl 1533.93626 |
| [33] | Yao, Y.; Sun, J., Optimal control of multi-task Boolean control networks via temporal logic, Systems Control Lett., 156, Article 105007 pp., 2021 · Zbl 1478.49019 |
| [34] | Liu, Y.; Wang, L.; Yang, Y.; Wu, Z.-G., Minimal observability of Boolean control networks, Systems Control Lett., 163, Article 105204 pp., 2022 · Zbl 1497.93022 |
| [35] | Li, W.; Li, H.; Yang, X., Edge removal towards asymptotical stabilizability of Boolean networks under asynchronous stochastic update, Systems Control Lett., 181, Article 105639 pp., 2023 · Zbl 1530.93405 |
| [36] | Wang, Q.; Sun, J., On asymptotic stability of discrete-time hybrid systems, IEEE Trans. Circuits Syst. II, 70, 6, 2047-2051, 2023 |
| [37] | Cheng, D.; Qi, H.; Li, Z., Analysis and Control of Boolean Networks: A Semi-tensor Product Approach, 2011, Springer: Springer London · Zbl 1209.93001 |
| [38] | Liu, S.; Trenkler, G., Hadamard, Khatri-Rao, Kronecker and other matrix products, Int. J. Inf. Syst. Sci., 4, 1, 160-177, 2008 · Zbl 1159.15008 |
| [39] | Xu, J.; Sun, J., Finite-time filtering for discrete-time linear impulsive systems, Signal Process., 92, 11, 2718-2722, 2012 |
| [40] | 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 |
| [41] | Bhat, S. P.; Bernstein, D. S., Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38, 3, 751-766, 2000 · Zbl 0945.34039 |
| [42] | Haddad, W. M.; Lee, J., Finite-time stability of discrete autonomous systems, Automatica, 122, Article 109282 pp., 2020 · Zbl 1453.93212 |
| [43] | F. Amato, M. Carbone, M. Ariola, C. Cosentino, Finite-time stability of discrete-time systems, in: Proceedings of the 2004 American Control Conference, vol. 2, 2004, pp. 1440-1444. |
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