Abstract
This paper deals with the dynamic output feedback stabilization problem of deterministic finite automata (DFA). The static form of this problem is defined and solved in previous studies via a set of equivalent conditions. In this paper, the dynamic output feedback (DOF) stabilization of DFAs is defined in which the controller is supposed to be another DFA. The DFA controller will be designed to stabilize the equilibrium point of the main DFA through a set of proposed equivalent conditions. It has been proven that the design problem of DOF stabilization is more feasible than the static output feedback (SOF) stabilization. Three simulation examples are provided to illustrate the results of this paper in more details. The first example considers an instance DFA and develops SOF and DOF controllers for it. The example explains the concepts of the DOF controller and how it will be implemented in the closed-loop DFA. In the second example, a special DFA is provided in which the DOF stabilization is feasible, whereas the SOF stabilization is not. The final example compares the feasibility performance of the SOF and DOF stabilizations through applying them to one hundred random-generated DFAs. The results reveal the superiority of the DOF stabilization.
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References
Kobetski, A., & Fabian, M. (2009). Time-optimal coordination of flexible manufacturing systems using deterministic finite automata and mixed integer linear programming. Discrete Event Dynamic Systems, 19(3), 287–315.
Passino, K., Michel, A., & Antsaklis, P. (1994). Lyapunov stability of a class of discrete event systems. IEEE Transaction on Automatic Control, 39(2), 269–279.
Daniel, R., & Markus, L. (2014). Automata with modulo counters and nondeterministic counter bounds. Kybernetika, 50(1), 66–94.
Tiwari, S. P., & Srivastava, A. K. (2005). On a decomposition of fuzzy automata. Fuzzy Sets Systems, 151(3), 503–511.
Lu, J., Li, H., Liu, Y., & Li, F. (2017). Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory and Applications, 11(13), 2040–2047.
Kauffman, S. A. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22(3), 437–467.
Cheng, D., He, F., Qi, H., & Xu, T. (2015). Modeling, analysis and control of networked evolutionary games. IEEE Transaction on Automatic Control, 60(9), 2402–2415.
Li, H., & Wang, Y. (2012). Boolean derivative calculation with application to fault detection of combinational circuits via the semi-tensor product method. Automatica, 48(4), 688–693.
Ozveren, F., & Willsky, A. (1991). Output stabilizability of discrete-event dynamic systems. IEEE Transaction on Automatic Control, 36(8), 925–935. https://doi.org/10.1109/9.133186.
Bof, N., Fornasini, E., & Valcher, M. (2015). Output feedback stabilization of Boolean control networks. Automatica, 57, 21–28.
Li, H., & Wang, Y. (2013). Output feedback stabilization control design for Boolean control networks. Automatica, 49(12), 3641–3645.
Yan, Y., Chen, Z., & Liu, Z. (2014). Solving type-2 fuzzy relation equations via semi-tensor product of matrices. Control Theory and Technology, 12(2), 173–186.
Cheng, D. (2011). Disturbance decoupling of Boolean control networks. IEEE Transaction on Automatic Control, 56(1), 2–10.
Zhang, Z., Xia, C., & Chen, Z. (2020). On the stabilization of nondeterministic finite automata via static output feedback. Applied Mathematics and Computation, 365, 124687. https://doi.org/10.1016/j.amc.2019.124687.
Zhang, Z., Chen, Z., Han, X., & Liu, Z. (2019). On the static output feedback stabilisation of discrete event dynamic systems based upon the approach of semi-tensor product of matrices. International Journal of Systems Science, 50(8), 1595–1608.
Syrmos, V., Abdallah, C., & Dorato, P. (1997). Static output feedback—A survey. Automatica, 33(2), 125–137.
Kong, X. S., Wang, S. L., Li, H. T., & Alsaadi, F. E. (2020). New developments in control design techniques of logical control networks. Frontiers of Information Technology and Electronic Engineering, 21, 220–233.
Meng, M., Liu, L., & Feng, G. (2017). Stability and \(l_1\) gain analysis of Boolean networks with Markovian jump parameters. IEEE Transactions on Automatic Control, 62(8), 4222–4228.
Possieri, C., & Teel, A. R. (2017). Asymptotic stability in probability for stochastic Boolean networks. Automatica, 83, 1–9.
Li, H., & Ding, X. (2019). A control Lyapunov function approach to feedback stabilization of logical control networks. SIAM Journal on Control and Optimization, 57(2), 810–831.
Lu, J., Liu, R., Lou, J., & Liu, Y. (2019). Pinning stabilization of Boolean control networks via a minimum number of controllers. IEEE Transactions on Cybernetics,. https://doi.org/10.1109/TCYB.2019.2944659.
Wang, X. F., & Chen, G. (2002). Pinning control of scale-free dynamical networks. Physica A: Statistical Mechanics and its Applications, 310(3–4), 521–531.
Li, F. (2015). Pinning control design for the stabilization of Boolean networks. IEEE Transactions on Neural Networks and Learning Systems, 27(7), 1585–1590.
Zhang, Z., Chen, Z., Han, X., & Liu, Z. (2018). On the static output feedback stabilization of deterministic finite automata based upon the approach of semi-tensor product of matrices. Kybernetika, 54(1), 41–60.
Cheng, D., & Qi, H. (2010). A linear representation of dynamics of Boolean networks. IEEE Transaction on Automatic Control, 55(10), 2251–2258.
Xu, X., & Hong, Y. (2012). Matrix expression and reachability analysis of finite automata. Journal of Control Theory and Applications, 10(2), 210–215.
Han, X., Chen, Z., Liu, Z., & Zhang, Q. (2018). The detection and stabilisation of limit cycle for deterministic finite automata. International Journal of Control, 91(4), 874–886.
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Abolpour, R., Raji, M. & Moradi, P. Dynamic output feedback stabilization of deterministic finite automata via the semi-tensor product of matrices approach. Control Theory Technol. 19, 170–182 (2021). https://doi.org/10.1007/s11768-020-00026-8
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DOI: https://doi.org/10.1007/s11768-020-00026-8