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Volume 3 Issue 3
Jul.  2016

IEEE/CAA Journal of Automatica Sinica

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Article Contents
Kai Chen, Junguo Lu and Chuang Li, "The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form," IEEE/CAA J. of Autom. Sinica, vol. 3, no. 3, pp. 311-319, 2016.
 Citation: Kai Chen, Junguo Lu and Chuang Li, "The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form," IEEE/CAA J. of Autom. Sinica, vol. 3, no. 3, pp. 311-319, 2016.

# The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form

Funds:

This work was supported by Natural Science Foundation of Hainan Province (20156218) and National Natural Science Foundation of China (61374030).

• The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.

•  [1] Sabatier J, Agrawal O P, Machado J A T. Advances in Fractional Calculus. Netherlands: Springer, 2007. [2] Mainardi F. Fractional calculus. Fractals and Fractional Calculus in Continuum Mechanics. Vienna: Springer, 1997. 291-348 [3] Meral F C, Royston T J, Magin R. Fractional calculus in viscoelasticity: an experimental study. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(4): 939-945 [4] Baleanu D, Golmankhaneh A K, Golmankhaneh A K, Baleanu M C. Fractional electromagnetic equations using fractional forms. International Journal of Theoretical Physics, 2009, 48(11): 3114-3123 [5] Luo Y, Chen Y Q. Fractional order [proportional derivative] controller for a class of fractional order systems. Automatica, 2009, 45(10): 2446-2450 [6] Luo Y, Chen Y Q. Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica, 2012, 48(9): 2159-2167 [7] Kheirizad I, Jalali A A, Khandani K. Stabilization of all-pole unstable delay systems by fractional-order [PI] and [PD] controllers. Transactions of the Institute of Measurement and Control, 2013, 35(3): 257-266 [8] Hu T S, Lin Z L, Chen B M. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 2002, 38(2): 351-359 [9] Zhang L X, Boukas E K. Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica, 2009, 45(2): 463-468 [10] Ma S, Zhang C, Zhu S. Robust stability for discrete-time uncertain singular Markov jump systems with actuator saturation. IET Control Theory & Applications, 2011, 5(2): 255-262 [11] Chen Y Q, Moore K L. On Dα-type iterative learning control. In: Proceedings of the 40th IEEE Conference on Decision and Control. Orlando, FL: IEEE, 2001. 4451-4456 [12] Li Y, Chen Y, Ahn H S. Fractional order iterative learning control. In: Proceedings of the ICCAS-SICE 2009. Fukuoka, Japan: IEEE, 2009. 3106-3110 [13] Li Y, Chen Y Q, Ahn H S. Fractional-order iterative learning control for fractional-order linear systems. Asian Journal of Control, 2011, 13(1): 54-63 [14] Lu J G, Chen Y Q. Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fractional Calculus and Applied Analysis, 2013, 16(1): 142-157 [15] Li C, Wang J C. Robust stability and stabilization of fractional order interval systems with coupling relationships: the 0 < α < 1 case. Journal of the Franklin Institute, 2012, 349(7): 2406-2419 [16] Lu J G, Ma Y D, Chen W D. Maximal perturbation bounds for robust stabilizability of fractional-order systems with norm bounded perturbations. Journal of the Franklin Institute, 2013, 350(10): 3365-3383 [17] Liao Z, Peng C, Li W, Wang Y. Robust stability analysis for a class of fractional order systems with uncertain parameters. Journal of the Franklin Institute, 2011, 348(6): 1101-1113 [18] Jiao Z, Zhong Y S. Robust stability for fractional-order systems with structured and unstructured uncertainties. Computers & Mathematics with Applications, 2012, 64(10): 3258-3266 [19] Li Y, Chen Y Q, Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45(8): 1965-1969 [20] Yu J M, Hu H, Zhou S B, Lin X R. Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica, 2013, 49(6): 1798-1803 [21] Li C, Wang J, Lu J. Observer-based robust stabilisation of a class of nonlinear fractional-order uncertain systems: an linear matrix inequalitie approach. IET Control Theory & Applications, 2012, 6(18): 2757-2764 [22] Lim Y H, Oh K K, Ahn H S. Stability and stabilization of fractionalorder linear systems subject to input saturation. IEEE Transactions on Automatic Control, 2013, 58(4): 1062-1067 [23] Hu T S, Lin Z L. Composite quadratic Lyapunov functions for constrained control systems. IEEE Transactions on Automatic Control, 2003, 48(3): 440-450 [24] Duarte-Mermoud M A, Aguila-Camacho N, Gallegos J A, Castro-Linares R. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 2015, 22(1-3): 650-659 [25] Hu T S, Lin Z L. Control Systems with Actuator Saturation: Analysis and Design. Basel: BirkhÄauser, 2001. [26] Li C, Lu J G. On the ellipsoidal invariant set of fractional order systems subject to actuator saturation. In: Proceedings of the 34th Chinese Control Conference. Hangzhou, China: IEEE, 2015. 800-805 [27] Li C, Wang J C, Lu J G, Ge Y. Observer-based stabilisation of a class of fractional order non-linear systems for 0 < α < 2 case. IET Control Theory & Applications, 2014, 8(13): 1238-1246

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