Robust stabilisation of fractional-order interval systems via dynamic output feedback: an LMI approach. (English) Zbl 1483.93510
Summary: This paper addresses the problem of robust dynamic output stabilisation of FO-LTI interval systems with the fractional order \(0<\alpha <2\), in terms of linear matrix inequalities (LMIs). Our purpose is to design a robust dynamic output feedback controller that asymptotically stabilises interval fractional-order linear time-invariant (FO-LTI) systems. Sufficient conditions are obtained for designing a stabilising controller with a predetermined order, which can be chosen to be as low as possible. The LMI-based procedures of designing robust stabilising controllers are preserved in spite of the complexity of assuming the most complete model of linear controller, with direct feedthrough parameter. Finally, some numerical examples with simulations are presented to demonstrate the effectiveness and correctness of the theoretical results.
MSC:
| 93D21 | Adaptive or robust stabilization |
| 93D15 | Stabilization of systems by feedback |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional-order system; interval uncertainty; linear matrix inequality (LMI); robust stabilisation; dynamic output feedbackReferences:
| [1] | Ahn, H. S.; Chen, Y., Necessary and sufficient stability condition of fractional-order interval linear systems, Automatica, 44, 11, 2985-2988 (2008) · Zbl 1152.93455 · doi:10.1016/j.automatica.2008.07.003 |
| [2] | Ahn, H. S.; Chen, Y.; Podlubny, I., Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, Applied Mathematics and Computation, 187, 1, 27-34 (2007) · Zbl 1123.93074 · doi:10.1016/j.amc.2006.08.099 |
| [3] | Alagoz, B. B., A note on robust stability analysis of fractional order interval systems by minimum argument vertex and edge polynomials, IEEE/CAA Journal of Automatica Sinica, 3, 4, 411-421 (2016) · doi:10.1109/JAS.2016.7510088 |
| [4] | Badri, P.; Amini, A.; Sojoodi, M., Robust fixed-order dynamic output feedback controller design for nonlinear uncertain suspension system, Mechanical Systems and Signal Processing, 80, 137-151 (2016) · doi:10.1016/j.ymssp.2016.04.020 |
| [5] | Badri, P.; Sojoodi, M., Robust fixed-order dynamic output feedback controller design for fractional-order systems, IET Control Theory & Applications, 12, 9, 1236-1243 (2018) · doi:10.1049/iet-cta.2017.0608 |
| [6] | Badri, P.; Sojoodi, M., Stability and stabilization of fractional-order systems with different derivative orders: An LMI approach Wiley online library, Asian Journal of Control, 21, 1-10 (2019) · Zbl 1422.00024 · doi:10.1002/asjc.2054 |
| [7] | Badri, V.; Tavazoei, M. S., Fractional order control of thermal systems: Achievability of frequency-domain requirements, Nonlinear Dynamics, 80, 4, 1773-1783 (2015) · doi:10.1007/s11071-014-1394-1 |
| [8] | Badri, V.; Tavazoei, M. S., Simultaneous compensation of the gain, phase, and phase-slope, Journal of Dynamic Systems, Measurement, and Control, 138, 12, 121002 (2016) · doi:10.1115/1.4034073 |
| [9] | Badri, V.; Tavazoei, M. S., Some analytical results on tuning fractional-order [proportional-integral] controllers for fractional-order systems, IEEE Transactions on Control Systems Technology, 24, 3, 1059-1066 (2016) · doi:10.1109/TCST.2015.2462739 |
| [10] | Badri, V.; Tavazoei, M. S., On time-constant robust tuning of fractional order [proportional derivative] controllers, IEEE/CAA Journal of Automatica Sinica (2017) |
| [11] | Chen, W.; Dai, H.; Song, Y.; Zhang, Z., Convex Lyapunov functions for stability analysis of fractional order systems, IET Control Theory & Applications, 11, 7, 1070-1074 (2017) · doi:10.1049/iet-cta.2016.0950 |
| [12] | Farges, C., Moze, M., & Sabatier, J. (2009). Pseudo state feedback stabilization of commensurate fractional order systems. Control Conference (ECC), 2009 European. · Zbl 1204.93094 |
| [13] | Gabano, J. D.; Poinot, T.; Kanoun, H., Identification of a thermal system using continuous linear parameter-varying fractional modelling, IET Control Theory & Applications, 5, 7, 889-899 (2011) · doi:10.1049/iet-cta.2010.0222 |
| [14] | Gao, Z., Robust stabilization criterion of fractional-order controllers for interval fractional-order plants, Automatica, 61, 9-17 (2015) · Zbl 1327.93335 · doi:10.1016/j.automatica.2015.07.021 |
| [15] | Gherfi, K.; Charef, A.; Abbassi, H. A., Proportional integral-fractional filter controller design for first order lag plus time delay system, International Journal of Systems Science, 50, 5, 885-904 (2018) · Zbl 1483.93192 · doi:10.1080/00207721.2018.1431815 |
| [16] | Hamidian, H.; Beheshti, M. T., A robust fractional-order PID controller design based on active queue management for TCP network, International Journal of Systems Science, 49, 1, 211-216 (2018) · Zbl 1385.93022 · doi:10.1080/00207721.2017.1397801 |
| [17] | Henrion, D., Sebek, M., & Kucera, V. (2001). LMIs for robust stabilization of systems with ellipsoidal uncertainty. Process Control Conference. |
| [18] | Higham, D.; Higham, N., MATLAB guide. (2005), Siam: Cambridge University Press, Siam · Zbl 1067.68180 |
| [19] | Hua, C.; Zhang, T.; Li, Y.; Guan, X., Robust output feedback control for fractional order nonlinear systems with time-varying delays, IEEE/CAA Journal of Automatica Sinica, 3, 4, 477-482 (2016) · doi:10.1109/JAS.2016.7510106 |
| [20] | Ibrir, S.; Bettayeb, M., New sufficient conditions for observer-based control of fractional-order uncertain systems, Automatica, 59, 216-223 (2015) · Zbl 1326.93104 · doi:10.1016/j.automatica.2015.06.002 |
| [21] | Jin, Y.; Chen, Y. Q.; Xue, D., Time-constant robust analysis of a fractional order [proportional derivative] controller, IET Control Theory & Applications, 5, 1, 164-172 (2011) · doi:10.1049/iet-cta.2009.0543 |
| [22] | Lan, Y. H.; Huang, H. X.; Zhou, Y., Observer-based robust control of a (. ) fractional-order uncertain systems: A linear matrix inequality approach, IET Control Theory & Applications, 6, 2, 229-234 (2012) · doi:10.1049/iet-cta.2010.0484 |
| [23] | Lan, Y. H.; Zhou, Y., Non-fragile observer-based robust control for a class of fractional-order nonlinear systems, Systems & Control Letters, 62, 12, 1143-1150 (2013) · Zbl 1282.93095 · doi:10.1016/j.sysconle.2013.09.007 |
| [24] | Li, C.; Wang, J., Robust stability and stabilization of fractional order interval systems with coupling relationships: The. case, Journal of the Franklin Institute, 349, 7, 2406-2419 (2012) · Zbl 1287.93063 · doi:10.1016/j.jfranklin.2012.05.006 |
| [25] | Liang, T.; Chen, J.; Zhao, H., Robust stability region of fractional order. controller for fractional order interval plant, International Journal of Systems Science, 44, 9, 1762-1773 (2013) · Zbl 1277.93063 · doi:10.1080/00207721.2012.670291 |
| [26] | Lofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. Computer Aided Control Systems Design, 2004 IEEE International Symposium on. |
| [27] | Lu, J. G.; Chen, G., Robust stability and stabilization of fractional-order interval systems: An LMI approach, IEEE Transactions on Automatic Control, 54, 6, 1294-1299 (2009) · Zbl 1367.93472 · doi:10.1109/TAC.2009.2013056 |
| [28] | Lu, J. G.; Chen, Y. Q., Robust stability and stabilization of fractional-order interval systems with the fractional order. : The. case, IEEE Transactions on Automatic Control, 55, 1, 152-158 (2010) · Zbl 1368.93506 · doi:10.1109/TAC.2009.2033738 |
| [29] | Ma, Y.; Lu, J.; Chen, W., Robust stability and stabilization of fractional order linear systems with positive real uncertainty, ISA Transactions, 53, 2, 199-209 (2014) · doi:10.1016/j.isatra.2013.11.013 |
| [30] | Park, J. H., Synchronization of cellular neural networks of neutral type via dynamic feedback controller, Chaos, Solitons & Fractals, 42, 3, 1299-1304 (2009) · Zbl 1198.93182 · doi:10.1016/j.chaos.2009.03.024 |
| [31] | Podlubny, I., Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1998), San Diego, California, US: Elsevier, San Diego, California, US · Zbl 0922.45001 |
| [32] | Sabatier, J.; Moze, M.; Farges, C., LMI stability conditions for fractional order systems, Computers & Mathematics with Applications, 59, 5, 1594-1609 (2010) · Zbl 1189.34020 · doi:10.1016/j.camwa.2009.08.003 |
| [33] | Shen, J.; Lam, J., State feedback H∞ control of commensurate fractional-order systems, International Journal of Systems Science, 45, 3, 363-372 (2014) · Zbl 1307.93146 · doi:10.1080/00207721.2012.723055 |
| [34] | Sontag, E. D., Mathematical control theory: Deterministic finite dimensional systems (2013), New York, New York, US: Springer Science & Business Media, New York, New York, US |
| [35] | Sturm, J. F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11, 1-4, 625-653 (1999) · Zbl 0973.90526 · doi:10.1080/10556789908805766 |
| [36] | Tavakoli-Kakhki, M., Implementation of fractional-order transfer functions in the viewpoint of the required fractional-order capacitors, International Journal of Systems Science, 48, 1, 63-73 (2017) · Zbl 1358.93092 · doi:10.1080/00207721.2016.1152519 |
| [37] | Xing, S. Y.; Lu, J. G., Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach, Chaos, Solitons & Fractals, 42, 2, 1163-1169 (2009) · Zbl 1198.93169 · doi:10.1016/j.chaos.2009.03.017 |
| [38] | Yusof, N. M., Ishak, N., Adnan, R., Tajjudin, M., & Rahiman, M. H. F. (2015). Identification of electro-hydraulic actuator system using commensurate fractional-order. 2015 IEEE Conference on Systems, Process and Control (ICSPC). |
| [39] | Zheng, S., Robust stability of fractional order system with general interval uncertainties, Systems & Control Letters, 99, 1-8 (2017) · Zbl 1353.93085 · doi:10.1016/j.sysconle.2016.11.001 |
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