This paper considers the dynamic output feedback control for a class of nonlinear rectangular descriptor semi-Markov jump systems (R-DSMJSs). There are no any constraints for the dimension of differential matrix, and the nonlinearities satisfy incremental quadratic constraints. A class of general rectangular dynamic output feedback controllers with free dimensions are designed, which can guarantee that the corresponding closed-loop systems are square descriptor semi-Markov jump systems (S-DSMJSs). First, sufficient condition is proposed to ensure that the closed-loop systems are regular, impulse-free, have unique solution, and are stochastically stable. Then a novel linear matrix inequality (LMI) sufficient condition is given based on the bounds of the time-varying transition rates and some matrix decoupling techniques, and controllers are obtained. Last, numerical examples are provided to demonstrate the effectiveness and advantages of the obtained methods and results.
DuanG.Analysis and design of descriptor linear systems. Springer, 2010.
3.
XuSLamJ.Robust control and filtering of singular systems. Springer, 2006.
4.
GuopingLDanielWCH. Generalized quadratic stability for continuous time singular systems with nonlinear perturbation. IEEE Trans Automat Contr2006; 51(5): 818–823.
5.
LiuLWenQLiY, et al. State and adaptive disturbance observer co-design for incrementally quadratic nonlinear descriptor systems with nonlinear outputs. Int J Robust Nonlinear Control2023; 33(17): 10678–10697.
6.
HeKSunLYangR.Robust stabilization and adaptive H∞ control of nonlinear singular systems via static output feedback. Proc Inst Mech Eng I J Syst Control Eng2024; 238(1): 87–96.
7.
IshiharaJYTerraMH.Impulse controllability and observability of rectangular descriptor systems. IEEE Trans Automat Contr2001; 45(6): 991–994.
8.
ZhuJMaSChengZ.Singular LQ problem for nonregular descriptor systems. IEEE Trans Automat Contr2002; 47(7): 1128–1133.
9.
LinCChenJChenB, et al. Fuzzy normalization and stabilization for a class of nonlinear rectangular descriptor systems. Neurocomputing2017; 219: 263–268.
10.
MoysisLGuptaMKMishraV, et al. Observer design for rectangular descriptor systems with incremental quadratic constraints and nonlinear outputs—application to secure communications. Int J Robust Nonlinear Control2020; 30(18): 8139–8158.
11.
ZhangXZhaoZ.Robust stabilization for rectangular descriptor fractional order interval systems with order 0 < α < 1. Appl Math Comput2020; 366: 124766.
12.
GengWTLinCChenB, et al. State-equivalent form and minimum-order compensator design for rectangular descriptor systems. IEEE Trans Automat Contr2024; 69(7): 4798–4804.
13.
GengWTLinCWangQG.Admissibility and controller design for rectangular descriptor time-delay systems. Automatica2025; 173: 112082.
14.
BoukasEK.Control of singular systems with random abrupt changes. Springer, 2008.
15.
WangGZhangQXingY.Analysis and design of singular Markovian jump systems. Springer, 2015.
16.
MaSBoukasEChinniahY.Stability and stabilization of discrete-time singular Markov jump systems with time-varying delay. Int J Robust Nonlinear Control2010; 20(5): 531–543.
17.
SongSMaSZhangC.Stability and robust stabilization for a class of non-linear uncertain discrete-time descriptor Markov jump systems. IET Control Theory Appl2012; 6(16): 2518–2527.
18.
KaoYXieJWangC.Stabilization of singular Markovian jump systems with generally uncertain transition rates. IEEE Trans Automat Contr2014; 59(9): 2604–2610.
19.
KwonNKParkISParkP, et al. Dynamic output-feedback control for singular Markovian jump system: LMI approach. IEEE Trans Automat Contr2017; 62(10): 5396–5400.
20.
SunMZhuangGXiaJ, et al. Robust dynamic output feedback stabilization for uncertain singular Markovian jump systems with time-varying delays. Int J Robust Nonlinear Control2022; 32(6): 3890–3908.
21.
RenJFengLFuJ, et al. Admissibility analysis and passive output feedback control for one-sided Lipschitz nonlinear singular Markovian jump systems with uncertainties. Appl Math Comput2021; 409: 126405.
22.
ZhangYMuX.Event-triggered output quantized control of discrete Markovian singular systems. Automatica2022; 135: 109992.
23.
TianJMaS.Existence of non-impulsive unique solution and stability for continuous-time linear rectangular descriptor Markov jump systems. Automatica2020; 117: 108953.
24.
LiYMaS.Indefinite linear quadratic optimal control and stabilization problem for discrete-time rectangular descriptor Markov jump systems with noise. IEEE Trans Syst Man Cybern Syst2024; 54(2): 841–853.
25.
SongXMaS.Robust H∞ dynamic output feedback control for nonlinear time-varying delay rectangular descriptor Markov jump systems. J Franklin Inst2023; 360(16): 11553–11577.
26.
SongXMaS.Finite-time H∞ dynamic output feedback control for one-sided Lipschitz nonlinear rectangular descriptor Markov jump system. Circuits Syst Signal Process2024; 43(5): 2695–2722.
27.
ZhaoZShiHMaS.A general dynamic output feedback control for rectangular nonlinear discrete-time descriptor Markov jump systems. Int J Syst Sci2025; 56(7): 1489–1505.
28.
ZhaoZShiHMaS.Novel indefinite guaranteed cost dynamic output feedback control for nonlinear discrete-time rectangular descriptor Markov jump systems with time-varying delays. Commun Nonlinear Sci Numer Simul2026; 152: 109347.
29.
CostaOLVFragosoMDTodorovMG.Continuous-time Markovian jump linear systems. Springer-Verlag, 2012.
30.
ZhangLBoukasEK.Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica2009; 45(2): 463–468.
31.
CaiBWengRZhangR, et al. Stabilization of a class of fuzzy stochastic jump systems with partial information on jump and sojourn parameters. Sci China Technol Sci2021; 64(2): 353–363.
32.
WangJWuJShenH, et al. Fuzzy H∞ control of discrete-time nonlinear Markov jump systems via a novel hybrid reinforcement Q-learning method. IEEE Trans Cybern2023; 53(11): 7380–7391.
33.
WangSWuZGWuZ. Asynchronous output regulation control for continuous-time Markovian jump systems with colored-noise. J Syst Sci Complex2023; 36(4): 1463–1479.
34.
WangJShenH.Passivity-based fault-tolerant synchronization control of chaotic neural networks against actuator faults using the semi-Markov jump model approach. Neurocomputing2014; 143: 51–56.
35.
JiangBKarimiHR.Further criterion for stochastic stability analysis of semi-Markovian jump linear systems. Int J Robust Nonlinear Control2020; 30(7): 2689–2700.
36.
WangBZhuQLiS.Stability analysis of discrete-time semi-Markov jump linear systems with time delay. IEEE Trans Automat Contr2023; 68(11): 6758–6765.
37.
HuangJShiY.Stochastic stability and robust stabilization of semi-Markov jump linear systems. Int J Robust Nonlinear Control2013; 23(18): 2028–2043.
38.
WuXTangYMaoS, et al. Stability analysis and stabilization of semi-Markov jump linear systems with unavailable sojourn-time information. Sci China Inf Sci2024; 67(7): 1–13.
39.
CaiBZhangLShiY.Observed-mode-dependent state estimation of hidden semi-Markov jump linear systems. IEEE Trans Automat Contr2020; 65(1): 442–449.
40.
QiWLiRZongG, et al. Asynchronous dynamic output feedback control for discrete nonlinear networked semi-Markov jump models with cyber-attacks and applications. IEEE Trans Autom Sci Eng2025; 22: 6316–6326.
41.
WangJMaSZhangC.Stability analysis and stabilization for nonlinear continuous-time descriptor semi-Markov jump systems. Appl Math Comput2016; 279: 90–102.
42.
JiangBKaoYKarimiHR, et al. Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates. IEEE Trans Automat Contr2018; 63(11): 3919–3926.
43.
WangBZhuQ.The stabilization problem for a class of discrete-time semi-Markov jump singular systems. Automatica2025; 171: 111960.
44.
WangJMaSZhangC.Finite-time H∞ control for T–S fuzzy descriptor semi-Markov jump systems via static output feedback. Fuzzy Sets Syst2019; 365: 60–80.
45.
ZhangJMaS.Robust H∞ indefinite guaranteed cost dynamic output feedback control of singular semi-Markov jump systems. J Franklin Inst2023; 360(16): 12436–12462.
46.
ZhangCKaoYHuangH, et al. Observer-based sliding mode control of discrete-time delayed singular semi-Markov jump systems with random false data injection attacks. J Franklin Inst2025; 362(1): 107400.
47.
AçıkmeşeBCorlessM.Observers for systems with nonlinearities satisfying incremental quadratic constraints. Automatica2011; 47(7): 1339–1348.
48.
YangAZhaoZShiH, et al. Composite event-triggered H∞ control for nonlinear switched singular systems with a mode-dependent triggering strategy. Int J Robust Nonlinear Control2023; 33(14): 8290–8314.
49.
FridmanEShakedU.An improved stabilization method for linear time-delay systems. IEEE Trans Automat Contr2002; 47(11): 1931–1937.
