This paper studies the fixed-time stability problem for positive nonlinear systems defined by cooperative vector fields. A criterion is derived to ensure fixed-time stabilization of the positive nonlinear systems using Lyapunov method. In addition, a sufficient condition is also presented for fixed-time stability of switched positive nonlinear systems under arbitrary switching. Simulation examples are provided to demonstrate the obtained results.
AndrieuVPralyLAstolfiA (2008) Homogeneous approximation, recursive observer design, and output feedback. SIAM Journal on Control and Optimization47(4): 1814–1850.
2.
BhatSPBernsteinDS (2000) Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization38(3): 751–766.
3.
BokharaieVSMasonOWirthF (2011) Stability and positivity of equilibria for subhomogeneous cooperative systems. Nonlinear Analysis: Theory, Methods & Applications74(17): 6416–6426.
4.
ChangXHLiuQWangYM, et al. (2018) Fuzzy peak-to-peak filtering for networked nonlinear systems with multipath data packet dropouts. IEEE Transactions on Fuzzy Systems27(3): 436–446.
5.
ChangXHYangCXiongJ (2019) Quantized fuzzy output feedback H1 control for nonlinear systems with adjustment of dynamic parameters. IEEE Transactions on Systems, Man, and Cybernetics: Systems49(10): 2005–2015.
6.
DongJG (2015) On the decay rates of homogeneous positive systems of any degree with time-varying delays. IEEE Transactions on Automatic Control60(11): 2983–2988.
7.
DongJG (2016) Stability of switched positive nonlinear systems. International Journal of Robust and Nonlinear Control26(14): 3118–3129.
8.
FarinaLRinaldiS (2011) Positive Linear Systems: Theory and Applications. New York: John Wiley & Sons.
9.
FeyzmahdavianHRCharalambousTJohanssonM (2014a) Asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays. SIAM Journal on Control and Optimization52(4): 2623–2650.
10.
FeyzmahdavianHRCharalambousTJohanssonM (2014b) Exponential stability of homogeneous positive systems of degree one with time-varying delays. IEEE Transactions on Automatic Control59(6): 1594–1599.
11.
GaoFWuYYuanF (2015) Semi-global finite-time stabilization of uncertain nonholonomic systems via output feedback. Transactions of the Institute of Measurement and Control37(1): 122–130.
12.
LiYSunYMengF, et al. (2018) Exponential stabilization of switched time-varying systems with delays and disturbances. Applied Mathematics and Computation324: 131–140.
13.
LiuLXingHCaoX, et al. (2019) Asynchronously input-output finite-time control of discrete-time nonlinear impulsive positive switched systems. Transactions of the Institute of Measurement and Control41(14): 4157–4166.
14.
LiuX (2015) Stability analysis of a class of nonlinear positive switched systems with delays. Nonlinear Analysis: Hybrid Systems16: 1–12.
15.
LiuXYuWWangL (2010) Stability analysis for continuous-time positive systems with time-varying delays. IEEE Transactions on Automatic Control55(4): 1024–1028.
16.
LiuXZhaoQZhongS (2018) Stability analysis of a class of switched nonlinear systems with delays: A trajectory-based comparison method. Automatica91: 36–42.
17.
MaJYeMZhengY, et al. (2019) Consensus analysis of hybrid multiagent systems: A game-theoretic approach. International Journal of Robust and Nonlinear Control29(6): 1840–1853.
18.
MasonOVerwoerdM (2009) Observations on the stability properties of cooperative systems. Systems & Control Letters58(6): 461–467.
19.
PolyakovA (2012) Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Transactions on Automatic Control57(8): 2106–2110.
20.
PolyakovAEfimovDPerruquettiW (2015) Finite-time and fixed-time stabilization: Implicit Lyapunov function approach. Automatica51: 332–340.
21.
QiWGaoX (2016) Finite-time L1 control for positive Markovian jump systems with constant time delay and partly known transition rates. Transactions of the Institute of Measurement and Control38(3): 348–355.
22.
RenWBeardRWAtkinsE M (2007) Information consensus in multivehicle cooperative control. IEEE Control Systems Magazine, 27(2): 71–82.
23.
ShenJWangW (2017) Finite-time stability and boundedness for positive switched systems with time-varying delay under state-dependent switching. Transactions of the Institute of Measurement and Control39(1): 43–51.
24.
SmithHL (2008) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Providence: American Mathematical Society.
25.
SunY (2016) Stability analysis of positive switched systems via joint linear copositive Lyapunov functions. Nonlinear Analysis: Hybrid Systems19: 146–152.
26.
SunYTianYXieXJ (2017) Stabilization of positive switched linear systems and its application in consensus of multiagent systems. IEEE Transactions on Automatic Control62(12): 6608–6613.
27.
ValcherMEMisraP (2014) On the stabilizability and consensus of positive homogeneous multi-agent dynamical systems. IEEE Transactions on Automatic Control59(7): 1936–1941.
28.
XiaJGaoHLiuM, et al. (2018a) Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems. Journal of the Franklin Institute355(6): 3031–3049.
29.
XiaJZhangJFengJ, et al. (2019) Command filter-based adaptive fuzzy control for nonlinear systems with unknown control directions. IEEE Transactions on Systems, Man, and Cybernetics: Systems. DOI: 10.1109/TSMC.2019.2911115.
30.
XiaJZhangJSunW, et al. (2018b) Finite-time adaptive fuzzy control for nonlinear systems with full state constraints. IEEE Transactions on Systems, Man, and Cybernetics: Systems49(7): 1541–1548.
31.
XieDShiLJiangF (2019) Second-order group consensus for linear multi-agent systems with average dwell time switching. Transactions of the Institute of Measurement and Control41(2): 484–493.
32.
YuJYuSLiJ, et al. (2019) Fixed-time stability theorem of stochastic nonlinear systems. International Journal of Control92(9): 2194–2200.
33.
ZhengYWangL (2012) Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Systems & Control Letters61(8): 871–878.
34.
ZhengYMaJWangL (2018) Consensus of hybrid multi-agent systems. IEEE Transactions on Neural Networks and Learning Systems29(4): 1359–1365.
35.
ZhengYZhuYWangL (2014) Finite-time consensus of multiple second-order dynamic agents without velocity measurements. International Journal of Systems Science45(3): 579–588.
36.
ZhuYLiSMaJ, et al. (2018) Bipartite consensus in networks of agents with antagonistic interactions and quantization. IEEE Transactions on Circuits and Systems II: Express Briefs65(12): 2012–2016.