In this paper, by using Lyapunov–Krasovskii stability theory and linear matrix inequality approach, the control problem for fractional-order systems under seismic waves with time-varying delays is studied. A novel switching surface is proposed. Based on the linear quadratic regulator and sliding mode control, some sufficient conditions are derived to ensure the fractional system under control. Mittag-Leffler stability theorem is adopted to prove that the switching surface could converge to zero. Finally, a numerical simulation example is provided to validate the advantages and effectiveness of the proposed results.
AbdulwahhabOW (2020) Design of a complex fractional order PID controller for a first order plus time delay system. ISA Transactions99: 154–158.
2.
AbolvafaeiMGanjefarS (2021) Two novel approaches to capture the maximum power from variable speed wind turbines using optimal fractional high-order fast terminal sliding mode control. European Journal of Control60: 78–94.
3.
AgrawalOP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics38(1–4): 323–337.
4.
AhmadSRahmanMUArfanM (2021) On the analysis of semi-analytical solutions of Hepatitis B epidemic model under the Caputo–Fabrizio operator. Chaos, Solitons & Fractals146: 110892.
5.
AmmourASDjennouneSAggouneW, et al. (2015) Stabilization of fractional-order linear systems with state and input delay. Asian Journal of Control17(5): 1946–1954.
6.
AnwarTKumamPAsifa, et al. (2020) Generalized unsteady MHD natural convective flow of Jeffery Model with ramped wall velocity and Newtonian heating; a Caputo–Fabrizio approach. Chinese Journal of Physics68: 849–865.
7.
BahaaGM (2019) Optimal control problem for variable-order fractional differential systems with time delay involving Atangana–Baleanu derivatives. Chaos, Solitons & Fractals122: 129–142.
8.
BongulwarMRPatreBM (2017) Stability regions of closed loop system with one non-integer plus time Ddlay plant by fractional order PID controller. International Journal of Dynamics and Control5(1): 159–167.
9.
CaiG-PHuangJ-ZYangSX (2003) An optimal control method for linear systems with time delay. Computers & Structures81(15): 1539–1546.
10.
ChenPLuoYPengY, et al. (2021) Optimal robust fractional order PIλD controller synthesis for first order plus time delay systems. ISA Transactions114: 136–149.
11.
Coronel-EscamillaALavín-DelgadoJEGómez-AguilarJF, et al. (2020) Fractional dynamics and synchronization of Kuramoto oscillators with nonlocal, nonsingular and strong memory. Alexandria Engineering Journal59(4): 1941–1952.
12.
DalirMBigdeliN (2020) The design of a new hybrid controller for fractional-order uncertain chaotic systems with unknown time-varying delays. Applied Soft Computing Journal87: 106000.
13.
GengW-TLinCChenB (2020) Observer-based stabilizing control for fractional-order systems with input delay. ISA Transactions100: 103–108.
14.
HeSSongJLiuF (2018) Robust finite-time bounded controller design of time-delay conic nonlinear systems using sliding mode control strategy. IEEE Transactions on Systems, Man, and Cybernetics: Systems48(11): 1863–1873.
15.
HeydariMH (2020) Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu–Caputo variable-order fractional derivative. Chaos, Solitons & Fractals130(1): 109401.
16.
HuBSongQZhaoZ (2020) Robust state estimation for fractional-order complex-valued delayed neural networks with interval parameter uncertainties: LMI approach. Applied Mathematics and Computation373: 125033.
17.
HuTHeZZhangX, et al. (2021) New fractional-order integral inequalities: application to fractional-order systems with time-varying delay. Journal of the Franklin Institute358(7): 3847–3867.
18.
JajarmiAHajipourMMohammadzadehE, et al. (2018) A new approach for the nonlinear fractional optimal control problems with external persistent disturbances. Journal of the Franklin Institute355(9): 3938–3967.
19.
LanY-HZhouY (2011) LMI-based robust control of fractional-order uncertain linear systems. Computers & Mathematics with Applications62(3): 1460–1471.
20.
LiYChenYPodlubnyI (2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers & Mathematics with Applications59(5): 1810–1821.
21.
MathiyalaganKSangeethaG (2019) Finite-time stabilization of nonlinear time delay systems using LQR-based sliding mode control. Journal of the Franklin Institute356(7): 3948–3964.
22.
MathiyalaganKSangeethaG (2020) Second-order sliding mode control for nonlinear fractional-order systems. Applied Mathematics and Computation383: 125264.
23.
OrtigueiraMDRodríguez-GermáLTrujilloJJ (2011) Complex Grünwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions. Communications in Nonlinear Science and Numerical Simulation16(11): 4174–4182.
24.
OustaloupALevronFMathieuB, et al. (2000) Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications47(1): 25–39.
25.
PodlubnyI (1999) Fractional Differential Equations. New York: Mathematics in Science and Engineering.
26.
PhatVNiamsupPThuanMV (2020) A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay. European Journal of Control56: 124–131.
27.
RakhshanSAEffatiS (2020) Fractional optimal control problems with time-varying delay: a new delay fractional Euler–Lagrange equations. Journal of the Franklin Institute357(10): 5954–5988.
28.
RenH-PWangXFanJ-T, et al. (2019) Fractional order sliding mode control of a pneumatic position servo system. Journal of the Franklin Institute356: 6160–6174.
29.
DasSPanIDasS (2015) Multi-objective LQR with optimum weight selection to design FOPID controllers for delayed fractional order processes. ISA Transactions58: 35–49.
30.
ŞenolBDemirogluUMatušůR (2020) Fractional order proportional derivative control for time delay plant of the second order: the frequency frame. Journal of the Franklin Institute357(12): 7944–7961.
31.
ShahriESAAlfiAMachadoJAT (2020) Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation. Applied Mathematical Modelling81: 663–672.
32.
ShakibaieSBalochianS (2017) Stability and stabilization of linear fractional order systems with input delay using linear matrix inequalities. National Academy Science Letters40(6): 429–433.
33.
TaoBXiaoMJiangG, et al. (2020) Stability, bifurcation prediction and optimal control of a delayed integer-order small-world network based on the fractional-order PD control policy of variable order. Journal of the Franklin Institute357(15): 10288–10311.
34.
WuQSongQHuB, et al. (2020) Robust stability of uncertain fractional order singular systems with neutral and time-varying delays. Neurocomputing401: 145–152.
ZhouHWYangSZhangSQ (2019) Modeling non-Darcian flow and solute transport in porous media with the Caputo-Fabrizio derivative. Applied Mathematical Modelling68: 603–615.
