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switched systems; robust stabilization; quantization
This paper deals with the robust stabilization of a class of nonlinear switched systems with non-vanishing bounded perturbations. The nonlinearities in the systems satisfy a quasi-Lipschitz condition. An observer-based linear-type switching controller with quantized and sampled output signal is considered. Using a dwell-time approach and an extended version of the invariant ellipsoid method (IEM) sufficient conditions for stability in a practical sense are derived. These conditions are represented as Bilinear Matrix Inequalities (BMI's). Finally, two examples are given to verify the efficiency of the proposed method.
[1] Aihara, K., Suzuki, H.: Theory of hybrid dynamical systems and its applications to biological and medical systems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 368 (2010), 4893-4914. DOI 10.1098/rsta.2010.0237 | Zbl 1211.37099
[2] Azhmyakov, V.: On the geometric aspects of the invariant ellipsoid method: Application to the robust control design. In: Proc. 50th IEEE Conference on Decision and Control and demonstratedntrol Conference, Orlando 2011, pp. 1353-1358. DOI 10.1109/cdc.2011.6161180
[3] Balluchi, A., Benvenuti, L., Benedetto, M. D. Di, Sangiovanni-Vincentelli, A.: The design of dynamical observers for hybrid systems: Theory and application to an automotive control problem. Automatica 49 (2013), 915-925. DOI 10.1016/j.automatica.2013.01.037 | MR 3029107 | Zbl 1284.93155
[4] Yazdi, M. Barkhordari, Jahed-Motlagh, M. R.: Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chemical Engrg. J. 155 (2009), 838-843. DOI 10.1016/j.cej.2009.09.008
[5] Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Birkhauser, Boston 2008. DOI 10.1007/978-0-8176-4606-6_9 | MR 2359816 | Zbl 1140.93001
[6] Branicky, M.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automat. Control 57 (1998), 3038-3050. DOI 10.1109/tac.2012.2199169 | MR 1617575 | Zbl 0904.93036
[7] Donkers, M. C. F., Hemmels, W. P. M. H., Wouw, N. Van den, Hetel, L.: Stability analysis of networked control systems using a switched linear systems approach. IEEE Trans. Automat. Control 56 (2011), 9, 2101-2115. DOI 10.1109/tac.2011.2107631 | MR 2865767
[8] Filipov, A. F.: Differential Equations with Discontinuous Right-hand Side. Kluwer, Dordrecht 1988.
[9] Fridman, E.: Descriptor discretized Lyapunov functional method: Analysis and design. IEEE Trans. Automat. Control 51 (2006), 890-897. DOI 10.1109/tac.2006.872828 | MR 2232620
[10] Fridman, E., Niculescu, S. I.: On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays. Int. J. Robust Nonlinear Control 18 (2008), 3, 364-374. DOI 10.1002/rnc.1230 | MR 2378407 | Zbl 1284.93206
[11] Fridman, E., Dambrine, M.: Control under quantization, saturation and delay: An LMI approach. Automatica 45 (2009), 10, 2258-2264. DOI 10.1016/j.automatica.2009.05.020 | MR 2890785 | Zbl 1179.93089
[12] Fu, M., Xie, L.: The sector bound approach to quantized feedback control. IEEE Trans. Automat. Control 50 (2005), 11, 1698-1711. DOI 10.1109/tac.2005.858689 | MR 2182717
[13] Gao, H., Chen, T.: A new approach to quantized feedback control systems. Automatica 44 (2008), 2, 534-542. DOI 10.1016/j.automatica.2007.06.015 | MR 2530803 | Zbl 1283.93131
[14] Geromel, J. C., Colaneri, P.: Stability and stabilization of continuous-time switched linear systems. SIAM J. Control Optim. 45 (2006), 5, 1915-1930. DOI 10.1137/050646366 | MR 2272172 | Zbl 1130.34030
[15] Glover, J. D., Schweppe, F. C.: Control of linear dynamic systems with set constrained disturbance. IEEE Trans. Automat. Control 16 (1971), 5, 411-423. DOI 10.1109/tac.1971.1099781 | MR 0287947
[16] Gonzalez-Garcia, S., Polyakov, A., Poznyak, A.: Linear feedback spacecraft stabilization using the method of invariant ellipsoids. In: Proc. 41st Southeastern Symposium on System Theory 2009, pp. 195-198. DOI 10.1109/ssst.2009.4806834
[17] Gonzalez-Garcia, S., Polyakov., A., Poznyak, A.: Output linear controller for a class of nonlinear systems using the invariant ellipsoid technique. In: American Control Conference, St. Louis 2009, pp. 1160-1165. DOI 10.1109/acc.2009.5160434
[18] Hartman, P.: Ordinary Differential Equations. Second edition. Society for Industrial and Applied Mathematics, Philadelphia 2002. DOI 10.1137/1.9780898719222 | MR 1929104
[19] Hespanha, J. P., Morse, A. S.: Stability of switched systems with average dwell-time. In: Proc. 38th IEEE Conference on Decision and Control, Phoenix 1999, pp. 2655-2660. DOI 10.1109/cdc.1999.831330
[20] Kruszewski, A., Jiang, W. J., Fridman, E., Richard, J. P., Toguyeni, A.: A switched system approach to exponential stabilization through communication network. IEEE Trans. Control Systems Technol. 20 (2012), 887-900. DOI 10.1109/tcst.2011.2159793
[21] Khurzhanski, A. B., Varaiya, P.: Ellipsoidal techniques for reachability under state constraints. SIAM J. Control Optim. 45 (2006), 1369-1394. DOI 10.1137/s0363012903437605 | MR 2257227
[22] Li, J., Liu, Y., Mei, R., Li, B.: Robust H$_\infty$ output feedback control of discrete time switched systems via a new linear matrix inequality formulation. In: Proc. 8th World Congress on Intelligent Control and Automation 2010, pp. 3377-3382. DOI 10.1109/wcica.2010.5553817
[23] Liberzon, D.: Switching in systems and control. In: Systems & Control. Foundations & Applications, Birkhauser, Boston 2003. DOI 10.1007/978-1-4612-0017-8 | MR 1987806 | Zbl 1036.93001
[24] Liberzon, D.: Stabilizing a switched linear system by sampled-data quantized feedback. In: Proc. 50th IEEE Conference on Decision and Control and European Control Conference, Orlando 2011, pp. 8321-8328. DOI 10.1109/cdc.2011.6160212
[25] Liberzon, D.: Finite data-rate feedback stabilization of switched and hybrid linear systems. Automatica 50 (2014), 2, 409-420. DOI 10.1016/j.automatica.2013.11.037 | MR 3163788
[26] Lin, H., Antsaklis, P. J.: Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Trans. Automat. Control 54 (2009), 308-322. DOI 10.1109/tac.2008.2012009 | MR 2491959
[27] Liu, Y., Niu, Y., Ho, D.: Sliding mode control for linear uncertain switched systems. In: Proc. 31st Chinese Control Conference, Hefei 2012, pp. 3177-3181.
[28] Liu, T., Jiang, Z. P., Hill, D. J.: Small-gain based output-feedback controller design for a class of nonlinear systems with actuator dynamic quantization. IEEE Trans. Automat. Control 57 (2012),5, 1326-1332. DOI 10.1109/tac.2012.2191870 | MR 2923898
[29] Liu, T., Jiang, Z. P., Hill, D. J.: A sector bound approach to feedback control of nonlinear systems with state quantization. Automatica 48 (2012), 1, 145-152. DOI 10.1016/j.automatica.2011.09.041 | MR 2879422 | Zbl 1244.93066
[30] Lozada-Castillo, N. B., Alazki, H., Poznyak, A. S.: Robust control design through the attractive ellipsoid technique for a class of linear stochastic models with multiplicative and additive noises. IMA J. Math. Control Inform. 30 (2013), 1-19. DOI 10.1093/imamci/dns008 | MR 3037694 | Zbl 1273.93167
[31] Nair, G. N., Fagnani, F., Zampieri, S., Evans, R. J.: Feedback control under data rate constraints: an overview. Proc. of the IEEE 95 (2007), 108-137. DOI 10.1109/jproc.2006.887294
[32] Nie, H., Song, Z., Li, P., Zhao, J.: Robust H$_\infty$ dynamic output feedback control for uncertain discrete-time switched systems with time-varying delays. In: Proc. 2008 Chinese Control and Decision Conference, Yantai-Shandong 2008, pp. 4381-4386. DOI 10.1109/ccdc.2008.4598158
[33] Ordaz, P., Alazki, H., Poznyak, A.: A sample-time adjusted feedback for robust bounded output stabilization. Kybernetika 49 (2013), 6, 911-934. MR 3182648 | Zbl 1284.93242
[34] Peng, C., Tian, Y. C.: Networked H$_\infty$ control of linear systems with state quantization. Inform. Sci. 177 (2007), 5763-5774. DOI 10.1016/j.ins.2007.05.025 | MR 2362219 | Zbl 1126.93338
[35] Polyak, B. T., Nazin, S. A., Durieu, C., Walter, E.: Ellipsoidal parameter or state estimation under model uncertainty. Automatica 40 (2004), 1171-1179. DOI 10.1016/j.automatica.2004.02.014 | MR 2148312 | Zbl 1056.93063
[36] Polyak, B. T., Topunov, M. V.: Suppression of bounded exogenous disturbances: Output feedback. Autom. Remote Control 69 (2008), 801-818. DOI 10.1134/s000511790805007x | MR 2437453 | Zbl 1156.93338
[37] Poznyak, A. S.: Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques. Elsevier, Amsterdam 2008. DOI 10.1134/s0005117909110174 | MR 2374025
[38] Poznyak, A. S., Azhmyakov, V., Mera, M.: Practical output feedback stabilization for a class of continuous-time dynamic system under sample-data outputs. Int. J. Control 84 (2011), 1408-1416. DOI 10.1080/00207179.2011.603097 | MR 2830870
[39] Shorten, R., Wirth, F., Manson, O., Wulff, K., King, C.: Stability criteria for switched and hybrid systems. SIAM Rev. 49 (2007), 545-592. DOI 10.1137/05063516x | MR 2375524
[40] Sun, Z., Ge, S. S.: Stability Theory of Switched Dynamical Systems, Communications and Control Engineering. Springer-Verlag, London 2011. DOI 10.1007/978-0-85729-256-8 | MR 3221851
[41] Tatikonda, S., Mitter, S.: Control under communication constraints. IEEE Trans. Automat. Control 49 (2004), 1056-1068. DOI 10.1109/tac.2004.831187 | MR 2071934
[42] Wang, Y., Gupta, V., Antsaklis, P.: On passivity of a class of discrete-time switched nonlinear systems. IEEE Trans. Automat. Control 59 (2014), 692-702. DOI 10.1109/tac.2013.2287074 | MR 3188475
[43] Yanyan, L., Jun, Z., Dimirovski, G.: Passivity, feedback equivalence and stability of switched nonlinear systems using multiple storage functions. In: Proc. 30th Chinese Control Conference, Yantai 2011, pp. 1805-1809.
[44] Zhang, W. A., Yu, L.: Output feedback stabilization of networked control systems with packet dropouts. IEEE Trans. Automat. Control 52 (2007), 1705-1710. DOI 10.1109/tac.2007.904284 | MR 2352449
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