Robust \(H_\infty\) control for fractional order singular systems \(0 < \alpha < 1\) with uncertainty. (English) Zbl 1531.93080
Summary: This article studies robust \({H}_{\infty}\) control for fractional order singular systems (FOSS) \(0<\alpha <1\) with uncertainty. First, the condition based on the linear matrix inequality (LMI) is obtained for fractional order systems with \(0<\alpha <1\) in Corollary 1. Compared with existing results, by using two matrices to replace the complex matrix, the condition is easier to solve. Based on Corollary 1, the condition of \({H}_{\infty}\) control based on non-strict LMI for FOSS without uncertainty is proposed. The strict LMI-based conditions of \({H}_{\infty}\) control are improved to overcome the equality constraints. Finally, the LMI-based conditions of robust \({H}_{\infty}\) control are proposed for FOSS. Four examples are shown to illustrate the effectiveness of the method.
{© 2022 John Wiley & Sons Ltd.}
{© 2022 John Wiley & Sons Ltd.}
MSC:
| 93B36 | \(H^\infty\)-control |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 34A08 | Fractional ordinary differential equations |
Keywords:
fractional order singular systems; \({H}_{\infty}\) control; linear matrix inequality; robust \({H}_{\infty}\) controlReferences:
| [1] | XuSY, LamJ. Robust Control and Filtering of Singular Systems. Springer; 2006. · Zbl 1114.93005 |
| [2] | HanYY, ZhouSS. Novel criteria of stochastic stability for discrete‐time Markovian jump singular systems via supermartingale approach. IEEE Trans Automat Contr. 2022;1‐8. |
| [3] | DingK, ZhuQX. Extended dissipative anti‐disturbance control for delayed switched singular semi‐Markovian jump systems with multi‐disturbance via disturbance observer. Automatica. 2021;128:1‐13. · Zbl 1461.93229 |
| [4] | MuXW, HuZH. Stability analysis for semi‐Markovian switched singular stochastic system. Automatica. 2020;118:1‐8. · Zbl 1447.93377 |
| [5] | YuXK, LiJX. Optimal state estimation for singular system with stochastic uncertain parameter. Optim Control Appl Meth. 2020;41:1001‐1015. · Zbl 1469.93126 |
| [6] | Chavez‐FuentesJR, CostaEF, TerraMH, RochaKDT. The linear quadratic optimal control problem for discrete‐time Markov jump linear singular systems. Automatica. 2021;127:1‐8. · Zbl 1461.93540 |
| [7] | LiJR, ZhangHY, FengZG, ZhaoYX, ShiJ. Dissipative filter design for discrete‐time interval type‐2 fuzzy singular systems with mixed delays. Optim Control Appl Meth. 2020;41:1016‐1033. · Zbl 1469.93101 |
| [8] | MukherjeeS, HeB, ChakraborttyA. Reduced‐dimensional reinforcement learning control using singular perturbation approximations. Automatica. 2021;126:1‐11. · Zbl 1461.93330 |
| [9] | ZhangQL, LiL, YanXG, SpurgeonSK. Sliding mode control for singular stochastic Markovian jump systems with uncertainties. Automatica. 2017;79:27‐34. · Zbl 1371.93048 |
| [10] | ZhaoYH, LiuYF, MaYC. Robust finite‐time sliding mode control for discrete‐time singular system with time‐varying delays. J Frank Inst. 2021;358(9):4848‐4863. · Zbl 1465.93037 |
| [11] | ChenJ, YuJP, LamHK. New admissibility and admissibilization criteria for nonlinear discrete‐time singular systems by switched fuzzy models. IEEE Trans Cyber. 2022;52(9):9240‐9250. |
| [12] | LiuXF, XieYF, LiFB, GuiWH. Sliding‐mode‐based admissible consensus tracking of nonlinear singular multiagent systems under jointly connected topologies. IEEE Trans Cyber. 2021;1‐10. |
| [13] | WangX, ZhuangGM, ChenGL, MaQ, LuJW. Asynchronous mixed [( {H}_{\infty } \]\) and passive control for fuzzy singular delayed Markovian jump system via hidden Markovian model mechanism. Appl Math Comput. 2022;425:1‐16. · Zbl 1510.93084 |
| [14] | PodlubnyI. Fractional Differential Equations. Academie Press; 1999. · Zbl 0924.34008 |
| [15] | LakshmikanthamV, VatsalaAS. Basic theory of fractional differential equations. Nonlinear Anal Theory Methods Appl. 2008;69(8):2677‐2682. · Zbl 1161.34001 |
| [16] | KaczorekT.Analysis of fractional linear electrical circuits in transient state. Proceedings of the Conference Logitrans; 2010:1695‐1704; Szczyrk, Poland. |
| [17] | MatignonD. Stabilty properties for generalized fractional differential systems. ESAIM: Proc. 1998;5:145‐158. · Zbl 0920.34010 |
| [18] | SabatierJ, MozeM, FargesC. LMI stability conditions for fractional order systems. Comp Math Appl. 2010;59(5):1594‐1609. · Zbl 1189.34020 |
| [19] | LuJG, ChenYQ. Robust stability and stabilization of fractional‐order interval systems with the fractional order α satisfying 0 < α < 1. IEEE Trans Automat Contr. 2010;55(1):152‐158. · Zbl 1368.93506 |
| [20] | LuJG, ZhuZ, MaYD. Robust stability and stabilization of multi‐order fractional‐order systems with interval uncertainties: an LMI approach. Int J Robust Nonlinear Control. 2021;31:4081‐4099. · Zbl 1526.93190 |
| [21] | ZhangXF, LinC, ChenYQ, BoutatD. A unified framework of stability theorems for LTI fractional order systems with 0 < α < 2. IEEE Trans Circuits Syst II Express Briefs. 2020;67(12):3237‐3241. |
| [22] | ShenJ, LamJ. State feedback [( {H}_{\infty } \]\) control of commensurate factional‐order systems. Int J Syst Sci. 2014;45:363‐372. · Zbl 1307.93146 |
| [23] | WeiX, LiuDY, BoutatD. Nonasymptotic pseudo‐state estimation for a class of fractional order linear systems. IEEE Trans Automat Contr. 2017;62(3):1150‐1164. · Zbl 1366.93085 |
| [24] | N’DoyeI, Laleg‐KiratiTM, DarouachM, VoosH. H∞ adaptive observer for nonlinear factional‐order systems. Int J Adapt Control Signal Process. 2017;31(3):314‐331. · Zbl 1362.93025 |
| [25] | LiangS, WeiYH, PanJW, GaoQ, WangY. Bounded real lemmas for fractional order systems. Int J Autom Comput. 2015;12(2):192‐198. |
| [26] | FargesC, FadigaL, SabatierJ. H∞ analysis and control of commensurate fractional order systems. Mechatronics. 2013;23(7):772‐780. |
| [27] | ChenL, WuR, YuanL, YinL, ChenYQ, XuS. Guaranteed cost control of fractional‐order linear uncertain systems with time‐varying delay. Optim Control Appl Meth. 2021;1‐17. |
| [28] | BoukalY, DarouachM, ZasadzinskiM, RadhyNE. Robust [( {H}_{\infty } \]\) observer‐based control of fractional‐order systems with gain parametrization. IEEE Trans Automat Contr. 2017;62(11):5710‐5723. · Zbl 1390.93632 |
| [29] | LiH, YangGH. Dynamic output feedback [( {H}_{\infty } \]\) control for fractional‐order linear uncertain systems with actuator faults. J Frank Inst. 2019;356:4442‐4466. · Zbl 1412.93031 |
| [30] | DoyeaIN, DarouachbM, ZasadzinskiM. Robust stabilization of uncertain descriptor fractional‐order systems. Automatica. 2013;49:1907‐1913. · Zbl 1360.93620 |
| [31] | WeiYH, PeterW, YaoZ, WangY. The output feedback control synthesis for a class of singular fractional order systems. ISA Trans. 2017;69:1‐9. |
| [32] | WuQ, SongQK, HuBX, ZhaoZJ, LiuYR, AlsaadiFE. Robust stability of uncertain fractional order singular systems with neutral and time‐varying delays. Neurocomputing. 2020;401:145‐152. |
| [33] | BaiZ, QiuT. Existence of positive solution for singular fractional differential equation. Appl Math Comput. 2008;215(7):2761‐2767. · Zbl 1185.34004 |
| [34] | MarirS, ChadliM, BouagadaD. New admissibility conditions for singular linear continuous‐time fractionalorder systems. J Frank Inst. 2017;354:752‐766. · Zbl 1355.93039 |
| [35] | MarirS, ChadliM. Robust admissibility and stabilization of uncertain singular fractional‐order linear time‐invariant systems. IEEE/CAA J Automat Sin. 2019;6(3):685‐692. |
| [36] | ZhangXF, ChenYQ. Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order [( \alpha \]\): the [( 0<\alpha <1 \]\) case. ISA Trans. 2017;82:42‐50. |
| [37] | ZhangQH, LuJG. Bounded real lemmas for singular fractional‐order systems: the [( 1<\alpha <2 \]\) case. IEEE Trans Circuits Syst II Express Briefs. 2021;68(2):732‐736. |
| [38] | LiY, WeiYH, ChenYQ, WangY. H∞ bounded real lemma for singular fractional‐order systems. Int J Syst Sci. 2022;53(6):1125‐1137. · Zbl 1490.93106 |
| [39] | LiY, WeiYH, ChenYQ, WangY. A universal framework of the generalized Kalman‐Yakubovich‐Popov lemma for singular fractional‐order systems. IEEE Trans Syst Man Cybern Syst. 2021;51(8):5209‐5217. |
| [40] | ZhangQH, LuJG. H∞ control for singular fractional‐order interval systems: the [( 0<\alpha <1 \]\) case. ISA Trans. 2021;110:105‐116. |
| [41] | CuiRJ, LuJG. H−/H∞ fault detection observer design for fractional‐order singular systems in finite frequency domains. ISA Trans. 2022. doi:10.1016/j.isatra.2022.02.042 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
