Abstract
This paper is concerned with the design of robust state-feedback controllers for uncertain positive Takagi–Sugeno systems with time-delay. The \(\alpha \)-stabilization by static state-feedback controllers with memory is investigated, taking into account the positivity and the interval uncertainties of the delayed closed-loop system into account. For this, new sufficient \(\alpha \)-stabilization conditions are established in terms of LMIs. An example illustrates of the proposed approach.
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A. Benzaouia, A. El Hajjaji, Delay-dependent stabilization conditions of controlled positive T–S fuzzy systems with time-varying delay. Int. J. Innov. Comput. Inf. Control 7(4), 1533–1547 (2011)
A. Benzaouia, R. Oubah, A. El Hajjaji, F. Tadeo, Stability and stabilization of positive T–S fuzzy continuous systems with delays, in Proceedings of the 50th IEEE Conference on Decision and Control Conference (CDC-ECC), Orlando, USA (2011), pp. 8279–8284
A. Benzaouia, R. Oubah, A. El Hajjaji, Stabilization of positive T–S fuzzy discrete-time systems with multiple delays and bounded controls. J. Frankl. Inst. 351(7), 3719–3733 (2014)
A. Benzaouia, R. Oubah, Stability and stabilization by output-feedback control of positive T–S fuzzy discrete-time systems with multiple delays. Nonlinear Anal.: Hybrid Syst. 11, 154–170 (2014)
M. Bolajraf, Robust Control and Estimation of Positive Systems. PhD dissertation (School of Industrial Engineering, Valladolid University, Spain, 2012)
Y. Cao, P.M. Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi–Sugeno fuzzy models. Fuzzy Sets Syst. 124(2), 213–229 (2001)
H. Du, N. Zhang, L. Wang, Switched control of vehicle suspension based on motion-mode detection. Vehicle Syst. Dyn.: Int. J. Veh. Mech. Mobil. 52(1), 142–165 (2013)
L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)
A. Hmamed, A. Benzaouia, M.A. Rami, F. Tadeo, Positive stabilization of discrete-time systems with unknown delay and bounded controls, in Proceedings of the European Control Conference (ECC), Kos, Greece (2007), pp. 5616–5622
A. Hmamed, A. Benzaouia, F. Tadeo, M.A. Rami, Memoryless control of delayed continuous-time systems within the nonnegative orthant, in Proceedings of the IFAC World Congress, Seoul, Korea (2008), pp. 3934–3939
D. Ichalal, B. Marx, J. Ragot, D. Maquin, State estimation of Takagi–Sugeno systems with unmeasurable premise variables. IET Control Theory Appl. 4(5), 897–908 (2010)
Z. Kardous, N.B. Braiek, A. Elkamel, On the residue formulation in multimodel control, in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Tunisia, vol. 6 (2002)
H. Li, X. Jing, H.K. Lam, P. Shi, Fuzzy sampled-data control for uncertain vehicle suspension systems. IEEE Trans. Cybern. 44(7), 1111–1126 (2013)
H. Li, X. Jing, H.R. Karimi, Output-feedback-based \(\text{ H }_\infty \) control for vehicle suspension systems with control delay. IEEE Trans. Ind. Electron. 61(1), 436–446 (2013)
H. Li, H. Gao, P. Shi, X. Zhao, Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach. Automatica 50(7), 1825–1834 (2014)
D.G. Luenberger, Introduction to Dynamic Systems: Theory, Models and Applications (Wiley, New York, 1985)
B. Marx, D. Koenig, J. Ragot, Design of observers for Takagi–Sugeno descriptor systems with unknown inputs and application to fault diagnosis. IET Control Theory Appl. 1(5), 1487–1495 (2007)
M.A. Rami, F. Tadeo, Linear programming approach to impose positiveness in closed-loop and estimated states, in Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan (2006)
M.A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II 54(2), 151–155 (2007)
K. Tanaka, H. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (Wiley, USA, 2001)
T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. Soc. 15(1), 116–132 (1985)
H.O. Wang, K. Tanaka, M. Griffin, Parallel distributed compensation of nonlinear systems by Takagi–Sugeno fuzzy model, in International Joint Conference on Fuzzy systems and the Second International Fuzzy Engineering Symposium, Yokohama, Japan, vol. 2 (1995), pp. 531–538
I. Zaidi, F. Tadeo, M. Chaabane, Positive observer design for continuous-time Takagi–Sugeno systems, in Proceedings of the 52nd Conference on Decision and Control (CDC), Firenze, Italy (2013), pp. 5030–5035
I. Zaidi, M. Chaabane, F. Tadeo, A. Benzaouia, Static state-feedback controller and observer design for interval systems with time-delay. IEEE Trans. Circuits Syst. II 62(5), 506–510 (2014)
P. Zhang, S.X. Ding, Fault tolerant measurement and monitoring of a nonlinear two-tank system, in Proceedings of the 13th Mediterranean Conference on Control and Automation, Limassol, Cyprus (2005), pp. 1355–1360
X.M. Zhang, Q.L. Han, On designing network-based \(H_\infty \) controllers for stochastic systems. Int. J. Robust Nonlinear Control 25(1), 52–71 (2015)
X. Zhao, L. Zhang, P. Shi, H.R. Karimi, Novel stability criteria for T–S fuzzy systems. IEEE Trans. Fuzzy Syst. 22(2), 313–323 (2014)
X. Zhao, L. Zhang, P. Shi, M. Liu, Stability of switched positive linear systems with average dwell switching. Automatica 48(6), 1132–1137 (2012)
X. Zhao, X. Liu, S. Yin, H. Li, Improved results on stability of continuous-time switched positive linear systems. Automatica 50(2), 614–621 (2014)
Acknowledgments
This work is funded by MiCInn DPI2010-21589-C05-05. We would like to thank Dr. Abdellah Benzaouia, Professor at LAEPT (Laboratory of Automatic Control, Environment and Transfer Processes) and CNRST (National Center of Scientific and Technical Research) of the University of Cadi Ayyad (Marrakech, Morocco) for collaborating with us.
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Zaidi, I., Abid, H., Tadeo, F. et al. Robust \(\alpha \)-Stabilization of Uncertain Positive Takagi–Sugeno Systems with Time-Delay via Memory State-Feedback. Circuits Syst Signal Process 35, 619–633 (2016). https://doi.org/10.1007/s00034-015-0075-8
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DOI: https://doi.org/10.1007/s00034-015-0075-8