We investigate the fractional-order systems that are perturbed by stochastic input to achieve stabilization via sliding mode control (SMC) approach. It is assumed that the system states are unknown and there is uncertainty and time-delay in the system. We utilize the diffusive representation of the stochastic fractional-order dynamics to transform the system into an integer-order system perturbed by Brownian motion. Provided that some linear matrix inequalities (LMIs) are feasible, it is proven that the estimation error system is stochastically stabilized and the overall closed-loop system is stable in probability. A numerical simulation shows the effectiveness of the results.
AngelLViolaJ (2018) Fractional order PID for tracking control of a parallel robotic manipulator type delta. ISA Transactions79: 172–188.
2.
BettayebMDjennouneS (2016) Design of sliding mode controllers for nonlinear fractional-order systems via diffusive representation. Nonlinear Dynamics84(2): 593–605.
3.
BoydSEl GhaouiLFeronEBalakrishnanV (1994) Linear Matrix Inequalities in System and Control Theory (Vol. 15). Philadelphia, PA: Siam.
4.
HuangCLiuHChenX, et al. (2020) Extended feedback and simulation strategies for a delayed fractional-order control system. Physica A: Statistical Mechanics and its Applications545: 123127.
5.
KhandaniK (2019) A sliding mode observer design for uncertain fractional Ito stochastic systems with state delay. International Journal of General Systems48(1): 48–65.
6.
KhandaniKMajdVJTahmasebiM (2016) Robust stabilization of uncertain time-delay systems with fractional stochastic noise using the novel fractional stochastic sliding approach and its application to stream water quality regulation. IEEE Transactions on Automatic Control62(4): 1742–1751.
7.
KhandaniKParvizianMEfeMÖ (2021) A non-fragile robust observer design for uncertain time-delay fractional Itô stochastic systems with input nonlinearity: An SMC approach. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering. DOI:10.1177/09596518211040008.
8.
KheirizadIJalaliAAKhandaniK (2012) Stabilization of fractional-order unstable delay systems by fractional-order controllers. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering226(9): 1166–1173.
9.
KheirizadIJalaliAAKhandaniK (2013) Stabilization of all-pole unstable delay systems by fractional-order [PI] and [PD] controllers. Transactions of the Institute of Measurement and Control35(3): 257–266.
10.
MonjeCAChenYVinagreBM, et al. (2010) Fractional-order Systems and Controls: Fundamentals and Applications. London: Springer-Verlag.
11.
MontsenyG (1998) Diffusive representation of pseudo-differential time-operators. ESAIM: Proceedings5: 159–175.
12.
MoornaniKAHaeriM (2010) Robustness in fractional proportional–integral–derivative-based closed-loop systems. IET Control Theory & Applications4(10): 1933–1944.
13.
ParvizianMKhandaniK (2021a) Mean square exponential stabilization of uncertain time-delay stochastic systems with fractional Brownian motion. International Journal of Robust and Nonlinear Control. https://doi.org/10.1002/rnc.5764.
14.
ParvizianMKhandaniK (2021b) A diffusive representation approach toward sliding mode control design for fractional-order Markovian jump systems. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering235(7): 1154–1163.
15.
PodlubnyI (1999) Fractional differential equations. In: Mathematics in Science and Engineering (Vol. 198). London: Academic Press.
16.
SayyafNTavazoeiMS (2017) Robust fractional-order compensation in the presence of uncertainty in a pole/zero of the plant. IEEE Transactions on Control Systems Technology26(3): 797–812.
17.
SadeghianHSalariehHAlastyAMeghdariA (2013) On the general Kalman filter for discrete time stochastic fractional systems. Mechatronics23(7): 764–771.
18.
SadeghianHSalariehHAlastyAMeghdariA (2014a) On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment. Applied Mathematical Modelling38(3): 961–973.
19.
SadeghianHSalariehHAlastyAMeghdariA (2014b) On the linear-quadratic regulator problem in one-dimensional linear fractional stochastic systems. Automatica50(1): 282–286.
20.
SakthivelRRevathiPRenY (2013) Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Analysis: Theory, Methods & Applications81: 70–86.
21.
SierociukDDzielinskiA (2006) Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation. International Journal of Applied Mathematics and Computer Science16(1): 129–140.
22.
SierociukDTejadoIVinagreBM (2011) Improved fractional Kalman filter and its application to estimation over lossy networks. Signal Processing91: 542–552.
23.
SonDTHuongPTKloedenPETuanHT (2018) Asymptotic separation between solutions of Caputo fractional stochastic differential equations. Stochastic Analysis and Applications36(4); 654–664.
24.
SunHZhangYBaleanuD, et al. (2018) A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation64: 213–231.
25.
TrigeassouJCMaamriNSabatierJOustaloupA (2011) A Lyapunov approach to the stability of fractional differential equations. Signal Processing91(3): 437–445.
26.
TsaiJHSChienTHGuoSM, et al. (2007) State-space self-tuning control for stochastic fractional-order chaotic systems. IEEE Transactions on Circuits and Systems54(3): 632–642.
27.
XuJZhangZCaraballoT (2019) Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay. Communications in Nonlinear Science and Numerical Simulation75: 121–139.
