In this paper, we introduce a new concept of stochastic finite-time stability for a class of nonlinear Markovian switching systems with impulsive effects. Based on the linear matrix inequality approach, sufficient conditions for the system to be stochastic finite-time stable are derived. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed conditions.
References
1.
Boukas
, E.
, 2006, Stochastic Switching Systems: Analysis and Design
, Birkhäuser
, Boston
.2.
Feng
, J.
, Lam
, J.
, and Shu
, Z.
, 2010, “Stabilization of Markovian Systems via Probability Rate Synthesis and Output Feedback
,” IEEE Trans. Autom. Control
, 55
(3
), pp. 773
–777
. 3.
Wang
, Z.
, Liu
, Y.
, and Liu
, X.
, 2010, “Exponential Stabilization of a Class of Stochastic System With Markovian Jump Parameters and Mode-Dependent Mixed Time-Delays
,” IEEE Trans. Autom. Control
, 55
(7
), pp. 1656
–1662
. 4.
Nguang
, S.
, and Shi
, P.
, 2006, “Robust h∞ Output Feedback Control Design for Takagi-Sugeno Systems With Markovian Jumps: A Linear Matrix Inequality Approach
,” ASME J. Dyn. Syst., Meas., Control
, 128
(3
), pp. 617
–625
. 5.
Zhang
, H.
, Feng
, G.
, and Dang
, C.
, 2009, “Stability Analysis and h∞ Control for Uncertain Stochastic Piecewise-Linear Systems
,” IET Control Theory Appl.
, 3
(8
), pp. 1059
–1069
. 6.
Huang
, R.
, Lin
, Y.
, and Lin
, Z.
, 2010, “Robust Fuzzy Tracking Control Design for a Class of Nonlinear Stochastic Markovian Jump Systems
,” ASME J. Dyn. Syst., Meas., Control
, 132
(5
), 510051
. 7.
Lakshmikantham
, V.
, Bainov
, D.
, and Simeonov
, P.
, 1989, Theory of Impulsive Differential Equations
, World Scientific
, Singapore
.8.
Yang
, T.
, 2001, Impulsive Systems and Control: Theory and Applications
, Springer
, Berlin
.9.
Liu
, B.
, 2008, “Stability of Solutions for Stochastic Impulsive Systems via Comparison Approach
,” IEEE Trans. Autom. Control
, 53
(9
), pp. 2128
–2133
. 10.
Chen
, W.
, and Zheng
, W.
, 2009, “Input-to-State Stability and Integral Input-to-State Stability of Nonlinear Impulsive Systems With Delays
,” Automatica
, 45
(6
), pp. 1481
–1488
. 11.
Sakthivel
, R.
, Mahmudov
, N.
, and Lee
, S.
, 2009, “Controllability of Non-Linear Impulsive Stochastic Systems
,” Int. J. Control
, 82
(5
), pp. 801
–807
. 12.
Lu
, J.
, Ho
, D.
, and Cao
, J.
, 2010, “A Unified Synchronization Criterion for Impulsive Dynamical Networks
,” Automatica
, 46
(7
), pp. 1215
–1221
. 13.
Wu
, H.
, and Sun
, J.
, 2006, “p-Moment Stability of Stochastic Differential Equations With Impulsive Jump and Markovian Switching
,” Automatica
, 42
(10
), pp. 1753
–1759
. 14.
Pan
, S.
, Sun
, J.
, and Zhao
, S.
, 2008, “Stabilization of Discrete-Time Markovian Jump Linear Systems via Time-Delayed and Impulsive Controllers
,” Automatica
, 44
(11
), pp. 2954
–2958
. 15.
Dong
, Y.
, and Sun
, J.
, 2008, “On Hybrid Control of a Class of Stochastic Non-Linear Markovian Switching Systems
,” Automatica
, 44
(4
), pp. 990
–995
. 16.
Raouf
, J.
, and Boukas
, E.
, 2009, “Stabilisation of Singular Markovian Jump Systems With Discontinuities and Saturating Inputs
,” IET Control Theory Appl.
, 3
(7
), pp. 971
–982
. 17.
Kamenkov
, G.
, 1953, “On Stability of Motion Over a Finite Interval of Time
,” J. Appl. Math. Mech.
, 17
, pp. 529
–540
.18.
Lebedev
, A.
, 1954, “The Problem of Stability in a Finite Interval of Time
,” J. Appl. Math. Mech.
, 18
, pp. 75
–94
.19.
Lebedev
, A.
, 1954, “On Stability of Motion During a Given Interval of Time
,” J. Appl. Math. Mech.
, 18
, pp. 139
–148
.20.
Dorato
, P.
, 1961, “Short-Time Stability in Linear Time-Varying Systems
,” Proceedings of the IRE International Convention Record
, Part 4, pp. 83
–87
.21.
Weiss
, L.
, and Infante
, E.
, 1967, “Finite Time Stability Under Perturbing Forces and on Product Spaces
,” IEEE Trans. Autom. Control
, 12
(1
), pp. 54
–59
. 22.
Amato
, F.
, Ariola
, M.
, and Dorato
, P.
, 2001, “Finite-Time Control of Linear Systems Subject to Parametric Uncertainties and Disturbances
,” Automatica
, 37
(9
), pp. 1459
–1463
. 23.
Liu
, L.
, and Sun
, J.
, 2008, “Finite-Time Stabilization of Linear Systems via Impulsive Control
,” Int. J. Control
, 81
(6
), pp. 905
–909
. 24.
Amato
, F.
, Ambrosino
, R.
, Ariola
, M.
, and Cosentino
, C.
, 2009, “Finite-Time Stability of Linear Time-Varying Systems With Jumps
,” Automatica
, 45
(5
), pp. 1354
–1358
. 25.
Zhao
, S.
, Sun
, J.
, and Liu
, L.
, 2008. “Finite-Time Stability of Linear Time-Varying Singular Systems With Impulsive Effects
,” Int. J. Control
, 81
(11
), pp. 1824
–1829
. 26.
Xu
, J.
, and Sun
, J.
, 2010, “Finite-Time Stability of Linear Time-Varying Singular Impulsive Systems
,” IET Control Theory Appl.
, 4
(10
), pp. 2239
–2244
. 27.
Amato
, F.
, Ariola
, M.
, and Cosentino
, C.
, 2010, “Finite-Time Control of Discrete-Time Linear Systems: Analysis and Design Conditions
,” Automatica
, 46
(5
), pp. 919
–924
. 28.
Amato
, F.
, Cosentino
, C.
, and Merola
, A.
, 2010, “Sufficient Conditions for Finite-Time Stability and Stabilization of Nonlinear Quadratic Systems
,” IEEE Trans. Autom. Control
, 55
(2
), pp. 430
–434
. 29.
He
, S.
, and Liu
, F.
, 2010, “Stochastic Finite-Time Stabilization for Uncertain Jump Systems via State Feedback
,” ASME J. Dyn. Syst., Meas., Control
, 132
(3
), 345041
. 30.
Petersen
, I.
, 1987, “A Stabilization Algorithm for a Class of Uncertain Linear Systems
,” Syst. Control Lett.
, 8
(4
), pp. 351
–357
. 31.
Skorokhod
, A.
, 1989, Asymptotic Methods in the Theory of Stochastic Differential Equations
, American Mathematical Society
, Providence
.32.
Gahinet
, P.
, Nemirovskii
, A.
, Laub
, A.
, and Chilali
, M.
, 1995, The LMI Control Toolbox
, The Mathworks Inc.
, Natick
.Copyright © 2012
by American Society of Mechanical Engineers
You do not currently have access to this content.
